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, :COVER

Lower Limit of Detection: Definition andi Elaboration of a Proposed Position for Radiological EHluent and Environmental Measurements

.. . . . . .

Prepared by L. A. Currie

National Bureau of Standards

Prepared for U.S. Nuclear Regulatory Commission

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~ .,*'-r --.- PORTIONS OF THlS REPORT -..-- A R E RELEGISLE. _I__-

I L . + * . k h > ' :* . 1 I .

_. . It has been reproduced frorn ?Re bzst . N U R EG / C R -4007

L .' available copy to permit iS'e bt'clzdest

5 <- possible avaiiabitity.

Lower Limit of Detection: NUREG/CR--4007

TI85 9G0256

1 of a Proposed Position for - . Radiological Effluent and

Environmental Measu rements

Manuscript Completed: August 1984 Date Published: September 1984

Prepared by L. A. Currie

National Bureau of Standards Washington, DC 20234

Prepared for Division of Systems Integration Office of Nuclear Reactor Regulation U.S. Nuclear Regulatory Commission Washington, D.C. 20555 NRC FIN 68615

This document is PUBLICLY RELEASABLE

Authorizing Of3dd Da9E b%h - 6 %

. . . . _. . . . , . . , . , -..,* ./:. ..... , . . - . . .

FOREWORD

The concept of Lower Limit of Detection (LLD) is used routinely in the NRC

Radiological Effluent Technical Specifications (RETS) for measurement of radio-

logical effluent concentrations within a nuclear power plant and of radiological

environmental samples outside of the plant. The definition of LLD is subject

to different interpretations by various groups. Consequently, difficulties arose

when the NRC attempted to apply uniformly requirements on licensees. At

present, NRC relies on documentation on LLDs that has been developed by other

agencies for their own purposes. The material is for the most part difficult

to obtain, and is only partially relatable to Technical Specifications require-

ments.

There was clearly a need to evaluate the various concepts and interpretations

of LLD presented in the literature and to determine the current use and applica-

tion of these concepts in practice in Technical Specifications for operating

nuclear plants. This would then lead to a NUREG/CR document that could assist

the NRC Nuclear Reactor Regulation staff in defining and elaborating its position

relative to LLDs, as well as providing a technically sound basic document on

detection capability for effluent and environmental monitoring.

Dr. Lloyd A. Currie of the National Bureau of Standards, a nationally

recognized expert in statistics, was asked to undertake this task. At the start

Dr. Currie performed an extensive literature search in the area of detection

limits. He discussed concepts and problems of LLD with a number of individuals

from licensed nuclear power plants, from contracting measurement laboratories,

iii

and from NRC Headquarters and Regional Offices. He then integrated these nuclear-

power oriented questions and concepts into his extensive experience in low-level

measurement to develop a comprehensive document covering the problems of LLD in

radiological effluent and environmental measurements.

It should be emphasized that this document represents Dr. Currie's inter-

pretation of the situations he encountered and his recommendations to the NRC

staff relative to these problems. It cannot of itself represent NRC policy. It

will, however, be used by NRC staff in development of potential modifications in

the definitions and bases sections of the model RETS relative to LLD. And of

most immediate importance, it will provide a sound basis to licensees and NRC

staff alike for use in clarifying thoughts and writings in the area of detection

capability of radiological measurement systems.

Frank J. Congel, Chief Radiological Assessment Branch

Charles A . Willis, Leader Effluent Treatment Section

NRC Division of Systems Integration

iv

.-

ABSTRACT

A manual is provided to define and illustrate a proposed use of the Lower

Limit of Detection (LLD) for Radiological Effluent and Environmental Measure-

ments. The manual contains a review of information regarding LLD practices

gained from site visits; a review of the literature and a summary of basic

principles underlying the concept of detection in Nuclear and Analytical

Chemistry; a detailed presentation of the application of LLD principles to

a range of problem categories (simple counting to multinuclide spectroscopy),

including derivations, equations, and numerical examples; and a brief exami-

nation of related issues such as reference samples, numerical quality control,

and instrumental limitations. An appendix contains a summary of notation

and terminology, a bibliography, and worked-out examples.

EXECUTIVE SUMMARY

This document defines and illustrates a proposed use of the concept of

Lower Limit of Detection (LLD) for Radiological Effluent and Environmental

Measurements. It contains a review of information regarding LLD practices

gained from nuclear plant site visits, a review of the literature and a

summary of basic principles underlying the concept of detection in Nuclear

and Analytical Chemistry, and a detailed presentation of the application of

LLD principles to a range of problem categories (simple counting to multi-

nuclide spectroscopy), including derivations, equations, and numerical

examples. It also contains a brief examination of related issues such as

reference samples, numerical quality control, and instrumental limitations.

An appendix contains a summary of notation and terminology, a bibliography,

and worked-out examples.

The detection capability of any measurement process ( M P ) is one of

its most important performance characteristics. When one is concerned with

pressing an MP to its lower limit'or with designing an ME' t o meet an extreme

measurement requirement, an objective measure of this capability is just as

important for characterizing the MP as is the more commonly understood

characteristics "precision1' and ''accuracy.''

the detection capability cannot be specified quantitatively unless the MP is

I

A s with these other characteristics,

rigorously defined and in a state of control. In the monitoring environment,

for'*low.levels of ef-fluent and environmental radioactivity associated with

the operation of nuclear power reactors, MPs must be capable of detecting the

relevant radionuclides at levels well below those of concern to the public

health and safety.

vii

t

Much confusion surrounds the nomenclature, formulation, and assumptions

associated with this important measurement process characteristic. For the

purposes of this document the term "Lower Limit of Detection" (LLD) is used

to describe the MP characteristic, and the same terminology, with appropriate

adjustments for scale and dimensions is applied to amounts of radioactivity,

concentrations, release rates, etc. In short, the same notation, LLD, is used

as a universal descriptor for all of the MPs in question. The assumptions

and mathematical and numerical formulations underlying LLDs are treated

explicitly, and the practical usage (and limitations thereof) is illustrated

with appropriate numerical examples. In particular, the special opportunities

and pitfalls associated with "Poisson counting statistics" are duly noted.

Section I of the report provides an introduction that sets the stage for

the technical sections that follow. Considerations that enter into an NRC

Technical Position on LLD are recorded, including theoretical background,

technical issues, policy issues, and implementation and documentation. High-

lights from site visits are next presented, providing perspective on the

problems and actual practices regarding LLD from the viewpoints o f : the NRC

(regional offices and inspectors), a trade assoc.iation, nuclear utility labo-

ratories, the EPA cross-check laboratory, and contracting laboratories.

' I The primary hktorical and theoretical background on detection decisions

and detection limits is presented in Section 11. The lack of and need for

uniform practice,< which was ascertained during the site visits, is underlined

in the historical review of the literature. The basis for the approach to

viii

LLD adopted here, hypothesis testing, is outlined in some detail. This is

followed by an examination of several crucial issues of general concern such

as the role of detection decisions, the meaning of a priori in the case of

interference, the treatment of systematic error, and the calibration func-

tion. The basic concepts are next applied to radioactivity, and to specific

issues related to the blank, counting technique, measurement process design

(to meet the requisite LLD), quality in communication and monitoring (control),

and the increase required in LLD to meet the demands of multiple detection

decisions.

Section I11 builds on the theory developed in Section 11. Basic and

simplified formulations are presented in "stand-alone" form, with sufficient

notes, that they might be adapted for use in Radiological Efluent Technical

Specifications (RETS). The heart of Section I11 comprises detailed algebraic

reductions of the general equations for a variety of radioactivity measure-

ment situations, ranging from "simple counting" to multicomponent spectroscopy.

The treatment of extreme low-level counting is illustrated, as well as ordinary

Poisson error treatment and systematic error treatment in relation to the LLD.

The Appendix includes a condensed summary of notation, an index to the

tutorial notes in Section 111, a more extended literature survey and biblio-

graphy, and worked-out numerical examples.

i I , L

7 1

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TABLE OF CONTENTS Pages

I . INTRODUCTION .................................................... 1

A . Introdwtory Remark ......................................... B . Plan for the Report ......................................... C . Considerations for an NRC Technical Posltion ................ D . Highlights from Site Visits .................................

I1 . BASIC CONCEPTS .................................................. A . Overview. Historical Perspective ............................ B . Signal Detection (Principles) ...............................

1 . Alternative Approaches .................................. 2 . Simple Hypothesis Testing ............................... Limitations ................................................. 1 . Detection Decisions vs Detection Limits ................. 2 . A Priori vs A Posteriori (Interference) ................. 3 . Continxi trof Hypotheses; Unprovability ................. 4 . The Calibration Function and LLD ........................ 5 . Bounds for Systematic Error .............................

D . Special Topics Concerning the LLD and Radioactivity ......... 1 . The Blank. BEA. and Regions of Validity ................. 2 . Deduction of Signal Detection Limits for Specific

Counting Techniques ..................................... 3 . Design and Optimization ................................. 5 . Multiple Detection Decisions ............................

C . General Formglation of LLD - Major Assgmptions and

4 . Quality ................................................. I11 . PROPOSED APPLICATION TO RADIOLOGICAL EFFLUENT TECHNICAL

SPECIFICATIONS .................................................. A . Basic Formulation ...........................................

1 . Definition .............................................. 2 . Tutorial Extension and Notes [ A ] ........................

B . Simplified Formglation ...................................... 1 . Definition .............................................. 2 . Tutorial Extension and Notes [B] ........................

'C . Development and Use of Equations for Specific Counting Applications ................................................ 1 . Extreme Low-Level Counting .............................. 2 . Reduction of the General Equations ...................... 3 . Derivation of Expressions for a (standard deviation of

the net signal under the null hypothesis) [Simple

resolution] ............................................. ' counting; mutual interference; least squares

IV . APPENDIX ........................................................ A . Summary of Notation and Terminology ......................... B . Guide to Tutorial Extensions and Notes ...................... C . Survey of the Literature ....................................

1 . Bibliography ............................................ D . Worked-out Numerical Examples ...............................

15

16 19 19 21

25 27 27 31 32 38 40 40

45 52 57 64

67

67 67 70 78 78 80

84 84 89

90

115

115 116 119 122 133

xi

1.

2.

4.

5.

6.

7.

8.

9 .

10.

LIST OF FIGURES

i I , .

Pages I

Hypothesis Testing - Critical Level and Detection Limit .......... 23

Systematic and Excess Random Error: of Degrees of Freedom .......................................%.... 26 Detection Limits - vs Nambers

A Priori vs A Posteriori: Sequential Relationship and Design of the Measurement Process .......................................... 30

-

Reporting of Non-Detected Results: The Problem of Bias .......... 59

IAEA Detection Limit Intercomparison Y-ray Spectrum .............. 62

Summary of Resalts for IAEA Simdated Ge(Li) Spectrum Intercomparison .................................................. 63

Redaced Activity Plot for Extreme Low-Level Counting ............. 87

Simple Counting: Detection Limit for a Spectrum Peak ............ 98

Simple Counting: Detection Limit for a Decay Curve .............. 101 Baseline Bias in Spectram Peak Fitting ........................... 102

x i i

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LIST OF TABLES

Pages

1. Historical Perspective of Detection Limit Terminology ............. 17

2. Approaches for the Formxlation of Signal Detection Limits ......... 18

3. Approaches and Difficxlties in the Form’Jlation of Concentration Detection Limits .................................................. 19

4. The Blank ......................................................... 42

5 . LLD Variations with Comting Time and Nxnber of Blank (Backgromd, Baseline) Comts ........... ....................................... 55

6. LLD Estimation by Replication: Student’s-t and ( o / s ) - Bomds - vs Number of Observations ............................................ 80

7. Extreme Low-Level Counting: Critical Levels and Detection Limits ............................................................ 89

x i i i

I . I N T R O D U C T I O N

A . Introductorv Remark'

I

The detect ion capabi l i ty of any measurement process (MP) i s one of its

most important performance c h a r a c t e r i s t i c s . When one is concerned w i t h

pressing an MP t o i ts lower l i m i t or w i t h designing an MP t o meet an extreme

measurement requirement, an object ive measure of t h i s capabi l i ty is j u s t a s

important for character iz ing the MP as is the more commonly understood

c h a r a c t e r i s t i c s "precision" and lfaccuracy .If A s w i t h these other character is-

t i c s , the detect ion capabi l i ty cannot be specif ied quant i ta t ive ly unless the

MP is rigorously defined and i n a s t a t e of control . (Thus , a secondary issue

\ \

of major importance is the qual i ty control of the measurement procedure.) I n

the monitoring environment -- i n the present case, f o r low l e v e l s of e f f luent

and environmental rad ioac t iv i ty associated w i t h the operation of nuclear

power r e a c t o r s -- MPs must be capable of detect ing the relevant

radionuclides a t l e v e l s well below those of concern t o the public heal th and

safe ty . ( T h i s need may be contrasted w i t h o thers where, f o r example,

adequate detect ion capabi l i ty may be required t o monitor biological condi-

t i o n s , na tura l hazards, i n d u s t r i a l processes and mater ia ls proper t ies ,

in te rna t iona l agreements, e t c . )

Much confusion surrounds the nomenclature, formulation, and assumptions

associated w i t h t h i s important measurement process c h a r a c t e r i s t i c . For the

purposes of t h i s document, we s h a l l somewhat a r b i t r a r i l y s e l e c t the term

"Lower L i m i t of Detection" ( L L D ) t o describe the MP c h a r a c t e r i s t i c , and we

s h a l l apply the same terminology, w i t h appropriate adjustments f o r sca le and

dimensions, t o amounts of r a d i o a c t i v i t y , concentrations, re lease r a t e s , e t c .

-- i n s h o r t , w e s h a l l use the same nota t ion , LLD, a s a universal descr iptor

I n t h i s report reference numbers a r e placed i n parentheses and special numbered notes (preceded by s e r i e s l e t t e r A o r B), i n brackets.

i I ,

7

f o r a l l of t h e MPs i n q u e s t i o n . The a s s u m p t i o n s and mathematical and

n u m e r i c a l f o r m u l a t i o n s u n d e r l y i n g L L D ' s w i l l be t rea ted e x p l i c i t l y , and t h e

p r a c t i c a l >sage (and l i m i t a t i o n s t h e r e o f ) w i l l be i l l u s t r a t e d w i t h

a p p r o p r i a t e n u m e r i c a l examples . I n p a r t i c u l a r , t h e s p e c i a l o p p o r t u n i t i e s and

p i t f a l l s a s s o c i a t e d w i t h I fPo i s son c o u n t i n g s t a t i s t i c s t f w i l l be d u l y n o t e d .

0. P l a n f o r t h e Repor t

The o b j e c t i v e and background f o r a n NRC T e c h n i c a l p o s i t i o n ( f o l l o w i n g

s e c t i o n ) sets the s t a g e f o r t h i s r epor t -manua l o n LLD. Nex t , p e r s p e c t i v e is

g i v e n on t h e p rob lems and a c t u a l p r a c t i c e s from t h e v i e w p o i n t s o f : t he N R C

( r e g i o n a l o f f i c e s and i n s p e c t o r s ) , a trade a s s o c i a t i o n , n u c l e a r u t i l i t y

l a b o r a t o r i e s , t h e EPA cross-check l a b o r a t o r y , and c o n t r a c t i n g l a b o r a t o r i e s .

The p r i m a r y h i s t o r i c a l and t h e o r e t i c a l background on d e t e c t i o n d e c i s i o n s

and d e t e c t i o n l i m i t s i s p r e s e n t e d i n s e c t i o n 11. The l a c k o f and need f o r

un i fo rm p r a c t i c e , which was a s c e r t a i n e d d u r i n g t h e s i t e v i s i t s , is u n d e r l i n e d

i n t h e h i s t o r i c a l r e v i e w of the l i t e r a t u r e . The bas i s f o r t h e a p p r o a c h t o

LLD a d o p t e d here , h y p o t h e s i s t e s t i n g , is o u t l i n e d i n some d e t a i l . T h i s is

f o l l o w e d by a n e x a m i n a t i o n of s e v e r a l c r u c i a l i s s u e s of g e n e r a l c o n c e r n s u c h

as t h e r o l e of d e t e c t i o n d e c i s i o n s , t h e meaning o f a p r i o r i i n t h e case o f

i n t e r f e r e n c e , t h e t r e a t m e n t of s y s t e m a t i c e r r o r , and t h e c a l i b r a t i o n f u n c -

t i o n . The bas i c c o n c e p t s are n e x t a p p l i e d t o r a d i o a c t i v i t y , and t o s p e c i f i c

i s s u e s r e l a t e d t o t h e b l a n k , c o u n t i n g t e c h n i q u e , measurement p r o c e s s d e s i g n

( t o meet the r e q u i s i t e L L D ) , q u a l i t y i n communication and m o n i t o r i n g

( c o n t r o l ) , and t h e i n c r e a s e r e q u i r e d i n LLD t o meet t h e demands of m u l t i p l e

d e t e c t i o n d e c i s i o n s .

2

S e c t i o n I11 b u i l d s on t h e t h e o r y d e v e l o p e d i n s e c t i o n 11. Basic and

s i m p l i f i e d f o r m u l a t i o n s are p r e s e n t e d i n " s t and-a lone r f fo rm, w i t h s u f f i c i e n t

n o t e s , t h a t t h e y might be a d a p t e d f o r u s e i n R a d i o l o g i c a l E f f l t l e n t T e c h n i c a l

S p e c i f i c a t i o n s ( R E T S ) . ( T h i s l e d t o some n e c e s s a r y r edundancy w i t h ideas

p r e s e n t e d i n s e c t i o n 11.) The hea r t of s e c t i o n I11 c o m p r i s e s d e t a i l e d

a l g e b r a i c r e d u c t i o n s of t h e g e n e r a l e q u a t i o n s f o r a v a r i e t y of r a d i o a c t i v i t y

measurement s i t u a t i o n s , r a n g i n g from " s i m p l e c o u n t i n g f f t o mul t icomponent

s p e c t r o s c o p y . The t r e a t m e n t of ex t r eme low- leve l c o u n t i n g is i l l u s t r a t e d , as

well as o r d i n a r y P o i s s o n e r r o r t r e a t m e n t and s y s t e m a t i c e r ror t r e a t i n e n t i n

r e l a t i o n t o t he LLD.

The Appendix i n c l Q d e s a condensed sammary of n o t a t i o n , a n i n d e x t o t h e

t u t o r i a l n o t e s i n s e c t i o n 111, a more e x t e n d e d l i t e ra t t l re s u r v e y and

b i b l i o g r a p h y , and worked-out n u m e r i c a l examples .

C . C o n s i d e r a t i o n s €or a n NRC T e c h n i c a l P o s i t i o n

1 . O b j e c t i v e of t h e NRC P o s i t i o n

Adequate measwement c a p a b i l i t i e s f o r e f f l a e n t and e n v i r o n m e n t a l

r a d i o a c t i v i t y are r e q u i r e d t o assure t h e s a f e t y of t h e p u b l i c , as p u t f o r t h

i n 10 CFR Parts 20 a n d 50 w h i c h mandate a p p r o p r i a t e r a d i o l o g i c a l e f f l u e n t and

e n v i r o n m e n t a l m o n i t o r i n g programs. I n order t o ass's-e a d e q u a t e d e t e c t i o n

c a p a b i l i t y f o r r a d i o n u c l i d e s t o meet these r e q u i r e m e n t s , t h e N R C has

e s t a b l i s h e d n u m e r i c a l l e v e l s f o r Lower L i m i t s of D e t e c t i o n ( L L D ) which are

c o n s i s t e n t w i t h a s u f f i c i e n t c a p a c i t y f o r d e t e c t i n g e f f l u e n t and e n v i r o n -

men ta l r a d i o n u c l i d e s w e l l below l e v e l s of c o n c e r n f o r t h e p u b l i c h e a l t h and

s a f e t y . For such ,LLDs t o be mean ingfu l and u s e f u l , t h e y must ( a ) be s o u n d l y

based i n terms of measurement s c i e n c e , and ( b ) t h e y must be a c c e p t e d ,

u n d e r s t o o d , and a p p l i e d i n a uni form manner by t h e community r e s p o n s i b l e f o r

3

8 ,

p e r fo rming and e v a l u a t i n g t h e r e s p e c t i v e measurements. These l i m i t i n g v a l u e s

a s - L L D s become p a r t of t h e O p e r a t i n g L i c e n s e of a Nuc lea r Power P l a n t t h r o u g h

t h e R a d i o l o g i c a l E f f l u e n t T e c h n i c a l S p e c i f i c a t i o n s (RETS) o f the o p e r a t i n g

l i c e n s e .

2. T h e o r e t i c a l Background

A f i r m basis f o r e v a l u a t i n g LLDs is g i v e n by t h e s t a t i s t i c a l t h e o r y of

h y p o t h e s i s t e s t i n g , which r e c o g n i z e s t h a t t h e i s s u e of d e t e c t i o n i n v o l v e s a

d e c i s i o n ("detected," "no t detected" made on t h e basis of an e x p e r i m e n t a l

o b s e r v a t i o n and a n a p p r o p r i a t e t e s t s t a t i s t i c . Once t h e d e c i s i o n a l g o r i t h m

has Been d e f i n e d , o n e can e v a l u a t e the u n d e r l y i n g d e t e c t i o n c a p a b i l i t y ( L L D )

of t h e measurement process under c o n s i d e r a t i o n . A r b i t r a r y r u l e s f o r d e f i n i n g

L L D ' s which do n o t have a sound base ( such as h y p o t h e s i s t e s t i n g ) y i e l d L L D ' s

w i t h l i t t l e meaning and n e e d l e s s i n c o m p a r a b i l i t y among l a b o r a t o r i e s . The

s y s t e m for computing and e v a l u a t i n g L L D s t o ' b e recommended f o r e f f l u e n t and

e n v i r o n m e n t a l r a d i o a c t i v i t y measurement p r o c e s s e s , is based on e x a c t l y t h e

same p r i n c i p l e s which u n d e r l i e more commonly used and u n d e r s t o o d c o n f i d e n c e

i n t e r v a l s . Key q u a n t i t i e s which a r i se i n t he approach t o L L D s are t h e

p r o b a b i l i t i e s o f f a l se p o s i t i v e s (a) and f a l se n e g a t i v e s ( B ) - b o t h g e n e r a l l y

t a k e n t o be 5%.

3. T e c h n i c a l I s s u e s

o The adop ted t e r m i n o l o g y ( n o t a t i o n ) t o r e f l ec t t h e measurement

( d e t e c t i o n ) c a p a b i l i t y sha l l be lILLD,ff and i t sha l l refer t o t h e i n t r i n s i c

d e t e c t i o n c a p a b i l i t y o f t h e e n t i r e measurement p r o c e s s - sampl ing t h r o u g h

data r e d u c t i o n and r e p o r t i n g .

4

>

.. .- An LLD for simply one stage of the measurement process, such as Y-ray

spectroscopy or @-counting, may in some instances be far smaller than the

overall LLD; as a result, the presumed capability to detect important levels

of (e.g.1 environmental contamination may be much too optimistic.

EJ The LLD shall be defined according to the statistical hypothesis

testing theory, using 5% for both frrisksll (errors of the first and second

kind), taking into consideration possible bounds for systematic error. This

means that the detection decision (based on an experimental outcome) and its

comparison with a critical or decision level must be clearly and consciously

distinguished from the detection limit, which is an inherent performance

characteristic of the measurement process. (Note that physical non-

negativity implies the use of 1-sided significance tests.)

o Both the critical level and the LLD depend upon the precision of the

measurement process (MP) which must be evaluated with some care at and below

the LLD in order for the critical level and LLD to be reliable quantities.

Information concerning the nature and variability of the blank is crucial in

this regard. (For a=@, and symmetric distribution functions, LLD = twice the

critical level, numerically.)

o Given the above statistical (random error) bases it is clear that

the overall random error ( 0 1 of the MP must be evaluated -- via propagation, replication, or "scientific judgment" -- to compute a meaningful LLD. flMeaningful,ft as used here, refers to an LLD which in fact reflects the

desired a, f3 error rates or risks.

-

o A great many assumptions must be recognized and satisfied for the

LLD to be meaningful (or valid). These include: knowledge of the error

distribution function(s) (they may not simply be Poisson or Normal); consid-

5

T

* , . e r a t i o n o f a l l s o u r c e s of random e r r o r ; r e l i ab le e s t i m a t i o n o f random e r r o r s

and a p p r o p r i a t e use o f S t u d e n t ' s - t and c a r e f u l a t t e n t i o n t o s o u r c e s of.

systematic e r r o r .

o S y s t e m a t i c e r r o r d e r i v e s from non- repea ted c a l i b r a t i o n , i n c o r r e c t

mode-1s o r parameters (as i n Y-ray s p e c t r o s c o p y ) , i n c o r r e c t y i e l d s , e f f i c i e n -

c i e s , s a m p l i n g , and f f b l u n d e r s . f ' Bounds f o r systematic e r r o r s h o u l d a lways

be estimated and made small compared t o the i m p r e c i s i o n ( 0 1 , i f p o s s i b l e .

S y s t e m a t i c c a l i b r a t i o n and e s t i m a t i o n e r r o r may become a v e r y s e r i o u s problem

f o r measurements of t f g r o s s r t (a,B) a c t i v i t y where t h e r e s p o n s e depends on t h e

r e l a t i v e mix of h a l f - l i v e s and p a r t i c l e e n e r g i e s .

o C o n t r o l o f the MP a l s o is e s s e n t i a l , and s h o u l d t h e r e f o r e be

g u a r a n t e e d by both i n t e r n a l and e x t e r n a l t t c ros s -checkfT programs. E x t e r n a l

c r o s s - c h e c k s s h o u l d r e p r e s e n t t he same t y p e ( sample m a t r i x , n u c l i d e m i x t u r e )

and l e v e l of a c t i v i t y as t h e "real" e f f l u e n t and e n v i r o n m e n t a l s amples

i n c l u d i n g b l a n k s f o r t h e " p r i n c i p a l r a d i o n u c l i d e s " , and the c r o s s - c h e c k s

s h o u l d be a v a i l a b l e " b l i n d " t o t he measu r ing l a b o r a t o r y . Note that w i t h o u t

a d e q u a t e c o n t r o l o r w i t h o u t n e g l i g i b l e systematic e r r o r , LLD loses meaning

i n t h e p u r e l y p r o b a b i l i s t i c s e n s e . The issues of s e t t i n g bounds f o r r e s i d u a l

systematic e r r o r and bounds f o r p o s s i b l y u n d e t e c t e d a c t i v i t y unde r these

c i r c u m s t a n c e s b o t h d e s e r v e c a r e f u l c o n s i d e r a t i o n , however.

o R a d i o n u c l i d e i n t e r f e r e n c e (and i n c r e a s e d Compton b a s e l i n e )

n e c e s s a r i l y i n f l a t e s t h e LLD, and must be t a k e n i n t o c o n s i d e r a t i o n q u a n t i t a -

t i v e l y . The u s e o f Ita p r i o r i " and p o s t e r i o r i " t o refer t o t h i s i s s u e is

s t r o n g l y d i s c o u r a g e d , because of n e e d l e s s c o n f u s i o n t h e r e b y i n t r o d u c e d

i n v o l v i n g a n o t h e r usage o f these terms ( re la ted t o d e t e c t i o n d e c i s i o n s a n d .

LLD) .

6

o Reporting practices are crucial to the communication and

understanding of data (as well as the validity of the respectiye LLD).

is a special problem for levels at or below the LLD, where sometimes even

negative experimental estimates obtain. Full data reporting is recommended,

from a technical point of view, to alleviate information-loss and the

possibility of introducing bias when periodic averages are required.

policy on uncertainty estimates and significant figures is in order.)

This

( A l s o ,

4. Related Policy Issues

o Once defined and agreed upon, a uniform approach to LLD, statement

of uncertainty, QA assessment (external), and data reporting should be

es tab1 ished . o Issues involving interference (and LLD relaxation) and reliance only

on Poisson counting statistics (vs - adequate replication and full error propa-

gation) must be settled. Other factors such as branching ratios/Y-abundance

should be considered in setting practically-achievable nuclide LLDs.

o Significant distortions which could arise from: a) t*gross7t ( a , @ )

activity measurements, b) sampling systematic errors, and c) concealed

software and bad nuclear parameters must be highlighted and controlled.

(Institution of an external data ttcross-checktt QA program, as the I A E A Y-ray

intercomparison spectra, may be one fruitful approach to the last problem.)

o Difficulties between scientific - vs public (political) perceptions

connected with "detected" - vs ttnon-detectedtt radionuclides especially in

reporting contexts need to be addressed.

i . j :

o Means for dealing with situations where the purely statistical

assumptions underlying LLD may not be satisfied must -be defined. (That is one

purpose of the present report. See section 11' for a catalog of assumption

difficulties.)

ImDlementation and Documentation

A potential basis for the NRC position for effluent and environmental

radioactivity measurement process LLD's is developed and illustrated in this

technical manual (NUREWCR document). This document is designed to provide

explicit information on: a) the history and principles of LLD's; b) practices

actually encountered in the field at the time of this study; c) simple, clear

yet accurate exposition and numerical illustrations of detection decisions

and LLD use, as applied to effluent and environmental radioactivity measure-

ments; and d) special technical issues, data, and bibliographic material (in

the Appendix).

D. Highlights from Site Visits

The highlights developed from a series of site visits are presented as a

synthesis of information gained rather than as a report concerning individual

discussions or specific organizations. The information represents my under-

standing from numerous discussions; the more critical issues may need to be

appropriately verified. Also, it should be understood that the contents in

this section constitute a record of my observations, not necessarily an .

indication that all parts are directly applicable to the Radiological

Effluent Technical Specifications (RETS). (e.g., parts 12 and 1 3 ) .

a

v I . I

.. .a-

O r g a n i z a t i o n s and I n d i v i d u a l s V i s i t ed (besides NRC-Headquarters)

4 November 1982

1 9 November 1982

5 J u l y 1983

6 J u l y 1983

7 J u l y

1 1 J u l y

1 2 J u l y

983

983

983

9 August 1983

21 November 1983

983

984

22 November

9 F e b r u a r y

Dave Harward, Atomic I n d u s t r i a l Forum, Bethesda, MD

Dave McCurdy, Yankee Atomic Elec t r ic Company, Framingham, MA, (Env i ronmen ta l Lab)

J e r ry Hamada ( I n s p e c t o r ) , NRC Region V Office, Walnut Creek, CA

Roger Miller, Rancho Seco Power P l a n t , CA (accompanied by J. Hamada)

Rod Melga rd , EAL, I n c . ( C o n t r a c t i n g L a b . ) , Richmond, CA

Art J a r v i s and Gene E a s t e r l y , EPA - Las Vegas (cross-check program)

J i m Johnson , Colorado S t a t e U n i v e r s i t y , F t . C o l l i n s (measurements f o r F t . S t . V r a i n p l a n t )

Mary B i r c h ' a n d Bob Sorber, Duke Power Co., C h a r l o t t e , NC ( H a , and L a b - a t Oconee s i t e )

Carl Paperiello, (Marty Schumacher, S t e v e Rozak, A 1 J a n u s k a ) NRC Region I11 Office, Glen E l l y n , I L

Leonid Huebner, Te ledyne I s o t o p e s Midwest Lab (formerly H a z e l t o n ) , Nor thb rook , I L

. .

Tom J e n t z , J o h n Campisi , J o a n Grove r , Charlie M a r c i n k i e w i c z , NUS ( C o n t r a c t o r Lab . ) , G a i t h e r s b u r g , MD

1 . Nee-1 and approach fo r the p lanned LLD manual. With o n e e x c e p t i o n , I - came away f rom t h e s e v e r a l m e e t i n g s wi th s t r o n g s u p p o r t f o r the aim o f

p roduc ing a manual. Most o f - t h o s e I v i s i t e d ( e s p e c i a l l y i n t he West) were

q u i t e a n x i o u s t o r e c e i v e a copy of t h e manual as soon as poss ib le . Valuable

s u g g e s t i o n s i n c l u d e d r e q u e s t s t o t rea t t h e basic c o n c e p t s i n a u n i f i e d and

c o m p l e t e , y e t e a s y - t o - g r a s p manner (e.g., h y p o t h e s i s t e s t i n g ) . One approach

wou ld .be t-o i n c l u d e mathematics and appropr ia te r e p r i n t s i n a n a p p e n d i x , b u t

worked-through examples i n t h e t e x t .

2 . D i v e r s Z t y o f t r a i n i n g and e x p e r i e n c e . T h i s was e v i d e n t i n s p e a k i n g

t o p e r s o n n e l r a n g i n g from l a b t e c h n i c i a n s t o l a b managers t o company o f f i -

c i a l s . T h i s d i v e r s i t y u n d e r l i n e s the approach called for i n item 1 . ( I t was

no tewor thy t h a t some of the younger and leas t p r o f e s s i o n a l l y t r a i n e d person-

n e l raised some o f the most p e n e t r a t i n g q u e s t i o n s a b o u t a s s u m p t i o n s ,

a l t e r n a t i v e a p p r o a c h e s t o data p r e s e n t a t i o n and e v a l u a t i o n , e tc . )

3. D i v e r s i t y of t e r m i n o l o g y , u s a g e , etc. D e s p i t e the d e f i n i t i o n and

r e f e r e n c e s p rov ided by t h e NRC f o r LLD (e .g . , t h r o u g h o u t NUREC-0472), there

e x i s t a number o f p o p u l a r terms (LLD, MDA, MDC, . . . I and f o r m u l a t i o n s ( 2 0 ,

S/N, h y p o t h e s i s t e s t i n g r i s k s , . . . I t o the d e t e c t i o n l i m i t , and a n e v e n wider

d i v e r s i t y o f a s s u m p t i o n s r e c o g n i z e d ( o r i g n o r e d ! ) i n p r a c t i c e . Some o f the

more p e r t i n e n t practices ( re : - a s s u m p t i o n s ) w i l l be n o t e d below.

4. P o l i c y I s s u e s . I found many o p p o r t u n i t i e s t o become enmeshed i n

p o l i c y . D e s p i t e my advance l e t te r (and copy o f t h e !!manual!! - work s ta te-

ment), c e r t a i n of my h o s t s seemed t o b e l i e v e I c o u l d s p e a k t o p o l i c y -- i . e . ,

what n u m e r i c a l v a l u e s s h o u l d be established f o r L L D ' s t o be met. I e x p l a i n e d

tha t t h i s was n o t my c h a r g e , though i n c e r t a i n s p e c i a l cases -- e . g . , t h e

effects of s e v e r e r a d i o n u c l i d e i n t e r f e r e n c e on d e t e c t i o n capab i l i t i e s -- it

might be u s e f u l t o c o n s i d e r t he impact of p o l i c y on p r a c t i c a l o p e r a t i o n s (see

be low) .

I n c e r t a i n cases, I was a d v i s e d t ha t the ! ' p rocess environment '! mandated

s p e c i a l a p p r o a c h e s t o t h e e v a l u a t i o n and r e p o r t i n g of data, b e c a u s e o f large

sample l o a d s and t h e need f o r r a p i d d e c i s i o n s . Under some c i r c u m s t a n c e s t h i s

could imply ( s t a t i s t i c a l l y ) c o n s e r v a t i v e l y b i a s e d r e p o r t i n g +of data, and

n o n - s p e c i f i c r a d i o n u c l i d e measurements ( e . g . , 6- c o u n t i n g o f s e p a r a t e d iodine

10

. '

.I'

i s o t o p e s , and t r e a t i n g t h e r e su l t as though i t were a l l 1 -131) . The issue I

p e r c e i v e is whether i t is a p p r o p r i a t e t o recommend d i f f e r e n t LLD a n d / o r

r e p o r t i n g schemes depending on how busy a l a b o r a t o r y is.

5. D e t e c t i o n d e c i s i o n s . I found t h e f u l l r a n g e o f c r i te r ia : from

d e c i s i o n s based on t h e c r i t i c a l l e v e l ( s u c h t h a t Q and 8 risks each e q u a l 5%)

t o t h o s e based on LLD ( s u c h t h a t "false p o s i t i v e s ! ? are i n f i n i t e s i m a l , b u t

" fa l se n e g a t i v e s " are 50%! 1. I have the i m p r e s s i o n t h a t t h e dec i s ion -mak ing

a s p e c t o f d e t e c t i o n -- i . e . , t h e ac tua l t e s t i n g o f t he n u l l h y p o t h e s i s -- is

n o t f u l l y a p p r e c i a t e d by a l l worke r s .

6 . R e p o r t i n g (when " n o t detected!?). Such r e s u l t s are e q u a t e d t o z e r o ,

some uppe r l i m i t , LLD, L L D / 2 , e tc . All of t h o s e I spoke t o r e c o g n i z e d t h a t

a v e r a g i n g ( e .g . , o v e r a q u a r t e r ) o f such r e p o r t e d resul ts is e i ther imposs-

i b l e , o r p o s i t i v e l y o r n e g a t i v e l y biased. I s e n s e d some r e s i s t a n c e t o

r e p o r t i n g t h e o b s e r v e d v a l u e ( e s p e c i a l l y when i t is n e g a t i v e ) , though o n e

g r o u p p r e s e r v e s s u c h i n f o r m a t i o n f o r u n b i a s e d a v e r a g i n g ; b u t t h e n r e p o r t s t h e

same data i n two d i f f e r e n t ( b i a s e d ) ways a c c o r d i n g t o t h e p o l i c i e s mandated

by d i f f e r e n t u s e r s of t h e data! Also, d u r i n g one v i s i t , I l e a r n e d t h a t

company ( ? ) p o l i c y l e a d s t o d i f f e r e n t ways o f r e p o r t i n g "non-de tec t ed"

r e s u l t s between e n v i r o n m e n t a l and e f f l u e n t measurements .

7. R a d i o n u c l i d e i n t e r f e r e n c e . A s i g n i f i c a n t issue. I t is ( u n i v e r s a l l y )

r e c o g n i z e d t h a t i n t e r f e r e n c e i n c r e a s e s d e t e c t i o n limits ( a l l else b e i n g

e q u a l ) . The same example (Ce-144 w i t h v e r y large amounts of Co-58, - 6 0 ) was

raised d u r i n g two v i s i t s , b u t w i t h somewhat d i f f e r e n t ( p o l i c y ) p e r s p e c t i v e s .

I n t h e o n e , .it was s u g g e s t e d t h a t p r e s c r i b e d L L D ' s be r e l a x e d ( o r p o s s i b l y

reinain " p u r e s o l u t i o n f t o r i n t e r f e r e n c e - f r e e L L D ' s ) when e x c e s s i v e

11

interference is present because the relative contribution of Ce-144 (here) is

trivial by comparison. In the other, caution was suggested, because even a

small amount of Ze-144.could be an important indicator for transuranics.

8. Blank, background, baseline. Some ambiguity was noted in the current

proposed NRC definition for LLD. Also, the question of real background

variability and number of degrees of freedom (and Student's-t) were raised.

One laboratory always assumes Poisson-background variability, or, if this

seems exceeded, it shuts down until a problem is identified or expected

behavior resumes.

9. Non-counting errors. Almost universally it was recognized that

actual probabilities of detection (and LLD) depend upon - all sources of error,

yet nearly all workers are using Poisson statistics only (for the blank and

sample, and ignoring errors for efficiency or chemical yield estimates) to

calculate LLD. Since the Relative Standard Deviation ~ 3 0 % at the detection

limit (a = B = 0 . 0 5 ) , this approximation is partly justified. Severe errors,

however, in blank estimates, detection efficiency (e.g., for cartridge

filters and for gross-a deposits), and sampling2 can seriously invalidate

this (Poisson) approximation. Several of the groups are working very hard to

estimate (and minimize) non-counting error, but there is little movement

toward considering its (necessary) effects on the LLD.

One interesting suggestion (mutually developed) was to distribute blind

cross-check samples having radionuclide concentrations slightly (e.g., 50%)

higher than the intended (NRC) LLD's to assess the actual significance of

non-Poisson error on detection capabilities. (This might also include blanks

of "principal radionuclides'' to test a-risk performance.)

_--_-_--___---- 2Sampling Errors -- e.g., involving soil particles, coolant containing sediment, single ion exchange beads, -- were in some cases shown to be overwhelming, reducing all other errors to insignificance.

12

10. Modeling ra ther t h a n d i rec t measurement. Knowing ( a t l eas t

a p p r o x i m a t e l y ) re la t ive d i l u t i o n f a c t o r s ( l a b o r a t o r y , a t m o s p h e r e , c o o l a n t

systems) i n many cases a l l o w s more a c c u r a t e i n f e r e n c e s t o be drawn f rom

r e l a t i v e l y h i g h l e v e l measurements f o l l o w e d by c a l c u l a t i o n -- as opposed t o

d i r ec t measurements of t he d i l u t e d ( d i s p e r s e d ) material. ( T h i s i s f o l l o w e d ,

f o r example , i n p r e p a r a t i o n of t h e EPA cross-check samples.)

1 1 . QA and c r o s s - c h e c k samples. I found some e x c e l l e n t i n t r a l a b Q A , b u t

a t t h e same time I found e x t r e m e l y s t r o n g s u p p o r t f o r e x t e r n a l c r o s s - c h e c k

programs -- e s p e c i a l l y b e c a u s e of t h e wide r a n g e o f ( e . g . ) c o n t r a c t o r o r

t e c h n i c i a n c a p a b i l i t i e s . The EPA sample program is v a l u a b l e ( e s s e n t i a l ,

s i n c e there is no o t h e r ) f o r t h i s p u r p o s e , b u t s e v e r a l u s e f u l e x t e n s i o n s were

s u g g e s t e d : i n c r e a s e d f r e q u e n c y ( p e r h a p s s u i t e d t o QA p e r f o r m a n c e ) , t r u l y

" b l i n d t f s amples ( E P A ' s are c l e a r l y r e c o g n i z a b l e , and o f t e n g i v e n spec ia l

a t t e n t i o n ) , and s a m p l e s which are c l o s e r i n compos i t ion and l e v e l t o t h o s e

e n c o u n t e r e d i n t he v a r i o u s programs ( e n v i r o n m e n t a l , e f f l u e n t , waste).

( S p l i t s , e s p e c i a l l y w i t h m o b i l e laborator ies s e r v e e f f l u e n t QA w e l l , b u t

a v a i l a b i l i t y o f flknownfl samples would be v a l u a b l e . )

12. "De minimisf1 r e p o r t i n g . Media o t h e r t h a n a i r and water are i n many

cases n o t cove red by s p e c i f i e d LLD's (e.g. , o i l , c h a r c o a l , ... ) , so t h a t any

detected a c t i v , i t y must be r e p o r t e d . A p p a r e n t l y , t he s i t u a t i o n is a n a l o g o u s

t o t h a t a r i s i n g from one i n t e r p r e t a t i o n of t h e Delaney Amendment, where

n o n - d e t e c t i o n is t a k e n e q u i v a l e n t t o a b s e n c e ; so t h a t r e p o r t i n g r e q u i r e m e n t s

(and p u b l i c p e r c e p t i o n s ) a re s t r o n g l y affected as m e a s u r e m e n t . t e c h n i q u e s

improve.

13

13. U n c e r t a i n t i e s , r e p o r t i n g l e v e l s , l i t i g a t i o n . I n view o f measurement

u n c e r t a i n t y , o n e o f t e n meets t h e q u e s t i o n of whether a n e .xper imenta1 o b s e r v a -

t i o n i m p l i e s t h a t t he t r u e v a l u e e x c e e d s o r is less t h a n , a specif ied r e g u l a -

t o r y l i m i t . The i s s u e is p e r h a p s compounded when one c o n s i d e r s a summation,

) 2 1 concen t ra t i o n

r e p o r t i n g l e v e l i

as on page 5 o f the NRC R a d i o l o g i c a l Assessment Branch T e c h n i c a l P o s i t i o n

(November 1979) . Both the magn i tude of t he t o t a l e r r o r s and t h e number of

terms ( n ) impact t h i s matter. A c t i o n s and l e g a l d e f e n s e can be rather complex

as a r e s u l t ; so c a u t i o u s a t t e n t i o n must be g i v e n t o matters of r e l a t i v e

"costs11, e x p e r i e n c e d judgment on the p a r t of i n s p e c t o r s , bu rden of p r o o f ,

e t c .

1 4 . Con t inuous and c o n t i n u a l m o n i t o r i n g ; a v e r a g i n g . A d i f f i c u l t area:

v a r i e d equipment age o r q u a l i t y c a n make c o n t i n u o u s m o n i t o r s d i f f i c u l t t o

i n t e g r a t e r e l i a b l y , and e r r o r s i n estimated time c o n s t a n t s and f l o w rates can

be s u b s t a n t i a l . C o n t i n u a l mon i to r ing . ( f o r p e r i o d a v e r a g i n g ) , o n the o t h e r

h a n d , must be done w i t h care t o a v o i d m i s s i n g non-monotonic b e h a v i o r

( e x c u r s i o n s , ...). Random v a r i a t i o n s may be a p p r o x i m a t e l y normal ( g a u s s i a n )

c l o s e t o t h e e m i s s i o n s i t e , b u t log-normal when mixed i n t h e e n v i r o n m e n t a l

sys t em. Averaging p r o c e d u r e s (ari thmetic - v s . g e o m e t r i c mean) may d i f f e r

a c c o r d i n g l y . (Weighted a v e r a g i n g is y e t a n o t h e r t o p i c . )

15. M u l t i p l e d e t e c t i o n d e c i s i o n s . Bas ing a l l d e c i s i o n s on a = 5%

( s i n g l e o b s e r v a t i o n f a l s e p o s i t i v e r i s k ) means t h a t o n the a v e r a g e 1 i n 20

b l a n k s w i l l be r e p o r t e d as detected. Adjus tment so t h a t , e.g. i n a m u l t i -

component Y-ray s p e c t r u m , there is o n l y a 5% change o f - any f a l se p o s i t i v e ,

was a s e e m i n g l y esoteric matter n o t e d by v e r y few of t h o s e I v i s i t e d .

1 4

A l s o , n o t w ide ly a p p r e c i a t e d was t h e t o o l i b e r a l n a t u r e of an o u t l i e r

r u l e ( C h a u v e n e t ' s c r i t e r i o n ) b e i n g sometimes employed.

I

t

I

16. Hidden a l g o r i t h m s , bad p a r a m e t e r s . A w i d e s p r e a d , b u t n o t too w i d e l y

a p p r e c i a t e d problem is t h e n a t u r e and lack of access t o computer programs

used f o r Y-ray spec t rum e v a l u a t i o n . A number of p a r a m e t e r s ( e . g . , b r a n c h i n g

I

r a t i o s ) b o t h i n c e r t a i n n u c l e a r data c o m p i l a t i o n s and i n some "canned"

software r o u t i n e s a re wrong. The a b s e n c e of a d e q u a t e software documen ta t ion

and t h e i n a c c e s s i b i l i t y of s o u r c e code h a s c a u s e d modera t e d i f f i c u l t i e s i n

s e v e r a l l a b o r a t o r i e s -- problems which may be e x a c e r b a t e d f o r small a c t i v i -

t i e s ( 7 L L D ) , f o r h i g h l e v e l s of i n t e r f e r e n c e ( b a s e - l i n e s h a p e ,

p i l e - u p , ... ) , and f o r m u l t i p l e t s . One i n t e r e s t i n g t es t t h a t was d e s c r i b e d ,

r e v e a l e d s o f t w a r e a r t i f a c t s (a lgor i thm s w i t c h i n g ) when computer o u t p u t was

examined f o r a series of s e q u e n t i a l (known) d i l u t i o n s of a g i v e n r a d i o n u c l i d e

sample . (Note t h e s i m i l a r i t y t o the c l a s s i c , S t a n d a r d A d d i t i o n Method t o

r e v e a l or compensa te chemical i n t e r f e r e n c e . )

11. BASIC C O N C E P T S ~

I n o r d e r t o meet t h e u n d e r l y i n g o b j e c t i v e of d e f i n i n g LLD fo r u s e i n

Radio logica l E f f l u e n t T e c h n i c a l Spec i f i ca t ions ( R E T S ) i t i s n e c e s s a r y f i r s t

t o adop t a un i fo rm and r e a s o n a b l e c o n c e p t u a l approach t o t h e s p e c i f i c a t i o n of

d e t e c t i o n c a p a b i l i t y f o r a n MP, and i t is t h e n n e c e s s a r y t o se t f o r t h a

c a r e f u l l y - c o n s t r u c t e d and c o n s i s t e n t scheme of nomenc la tu re and mathematical

s t a t i s t i c a l r e l a t i o n s f o r s p e c i f i c a p p l i c a t i o n t o t h e r a n g e of p rob lems

e n c o u n t e r e d i n measurements of e f f l u e n t and e n v i r o n m e n t a l r a d i o a c t i v i t y . Our

g o a l i n t h i s s e c t i o n is t o o u t l i n e the p r e f e r r e d c o n c e p t u a l a p p r o a c h t o g e t h e r

w i t h a r e a s o n a b l y comple t e c a t a l o g u e of a s s u m p t i o n s and means f o r p u t t i n g i t

--------------- I S e e Appendix A f o r se lec ted n o m e n c l a t u r e and t e r m i n o l o g y .

15

I , 1

i n t o p r a c t i c e . Detai led r e d u c t i o n o f t h e basic f o r m u l a s p r e s e n t e d i n t h i s

s e c t i o n w i l l take place i n t he n e x t s e c t i o n , f o r t h e s e v e r a l common cate-

gories o f n u c l e a r and r a d i o c h e m i c a l measurement; and e x p l i c i t numer i ca l

examples w i l l be g i v e n i n t h e Appendix. Let u s b e g i n w i t h a g l a n c e a t t h e

p a s t .

A . Overview and H i s t o r i c a l P e r s D e c t i v e

Some a p p r e c i a t i o n f o r t he e v o l u t i o n o f methods f o r e x p r e s s i n g d e t e c t i o n

c a p a b i l i t y may be g a i n e d f rom T a b l e 1 . I n t h i s t a b l e , which r e f e r s o n l y t o

d e t e c t i o n c a p a b i l i t y ( n o t d e t e c t i o n d e c i s i o n l e v e l s ) , we o b s e r v e t h a t t he

development of d e t e c t i o n t e r m i n o l o g y and f o r m u l a t i o n s f o r Nuc lea r and

A n a l y t i c a l C h e m i s t r y covers an ex tended p e r i o d of time a n d t h a t i t has been

c h a r a c t e r i z e d by d ive r se and n o n - c o n s i s t e n t a p p r o a c h e s . (Besides a l t e r n a t i v e

terms f o r t h e same c o n c e p t , o n e o c c a s i o n a l l y f i n d s the same term a p p l i e d t o

d i f f e r e n t c o n c e p t s -- v i z . , Kaiser's fTNachweisgrenzelf , which re fers t o the

t e s t or d e t e c t i o n d e c i s i o n l e v e l , is commonly t r a n s l a t e d " d e t e c t i o n l i m i t " ;

-

y e t , i n e n g l i s h " d e t e c t i o n l i m i t 1 ' g e n e r a l l y r e l a t e s t o t h e i n h e r e n t d e t e c t i o n

c a p a b i l i t y of t h e Chemical Measurement P r o c e s s (CMP).) For i n f o r m a t i o n

c o n c e r n i n g the de t a i l ed a s s u m p t i o n s and f o r m u l a t i o n s a s s o c i a t e d w i t h t h e

terms p r e s e n t e d i n T a b l e 1 t h e reader is referred t o t he o r i g i n a l l i t e r a -

ture . The p r i n c i p a l a p p r o a c h e s , however , a re r e p r e s e n t e d by: ( a ) F e i g l

-- s e l e c t i n g a more o r less a r b i t r a r y c o n c e n t r a t i o n (or amoun t ) , based on

e x p e r t judgment of t h e c u r r e n t s t a t e o f the a r t ; ( b ) Kaiser and A l t s h u l e r

-- ground ing d e t e c t i o n t h e o r y on t h e p r i n c i p l e s of h y p o t h e s i s t e s t i n g ; ( c ) S t .

John -- u s i n g s i g n a l / n o i s e (assumed " w h i t e " ) and c o n s i d e r i n g o n l y t h e e r r o r

b \ >

o f the f irst k i n d ; ( d ) N i c h o l s o n -- c o n s i d e r i n g d e t e c t i o n from t h e

~ p e r s p e c t i v e of a s p e c i f i c assumed p r o b a b i l i t y d i s t r i b u t i o n ( P o i s s o n ) ; ( e )

L i t e a n u -- t r e a t i n g d e t e c t i o n i n terms of t h e d i r e c t l y o b s e r v e d f r e q u e n c y

d i s t r i b u t i o n , and ( f ) G r i n z a i d -- a p p l y i n g t h e weaker, b u t more r o b u s t

a p p r o a c h e s of non-paramet r ic s t a t i s t i c s t o t h e problem. The w i d e s p r e a d

p r a c t i c e of i g n o r i n g t h e error of t h e second k i n d is e p i t o m i z e d by I n g l e in

h i s i n f e r e n c e t h a t i t is t o o complex f o r o r d i n a r y chemists t o u s e and

comprehend! T r e a t m e n t of d e t e c t i o n i n t h e p r e s e n c e of p o s s i b l e s y s t e m a t i c

a n d / o r model e r r o r is c o n s i d e r e d b r i e f l y i n Ref. C331.

T a b l e 1 . H i s t o r i c a l P e r s p e c t i v e -- D e t e c t i o n L i m i t Terminology

F e i g l ( ' 2 3 ) A l t s h u l e r ( ' 6 3 ) Kai-ser ( ' 65- '68) S t . John ( ' 6 7 ) C u r r i e ('68) N i c h o l s o n ( ' 6 8 )

IUPAC ( ' 7 2 )

I n g l e ( ' 7 k ) Lochamy ("76) G r i n z a i d ( ' 7 7 ) L i t e a n u ( ' 8 0 )

- L i m i t of I d e n t i f i c a t i o n [Ref . 11 - Minimum Detectable T r u e A c t i v i t y [Ref. 4 1 - L i m j t of G u a r a n t e e f o r P u r i t y [Ref. 21 - L i m i t i n g Detectable C o n c e n t r a t i o n (S/Nrms) [Ref. 31 - D e t e c t i o n L i m i t [Ref. 51 - D e t e c t a b i l i t y [Ref. 361 - S e n s i t i v i t y ; L i m i t of D e t e c t i o n ... [Ref. 22, 231 - ( " [ t o o ] complex. . . n o t common") [Ref. 51 3 - Minimum Detectable A c t i v i t y [Ref. 7 1 - Nonparametr ic . . . D e t e c t i o n L i m i t [Ref. 4 4 1 - F r e q u e n t o m e t r i c D e t e c t i o n [Ref. 311

A condensed summary of t h e p r i n c i p a l a p p r o a c h e s t o s i g n a l d e t e c t i o n is

p r e s e n t e d i n Table 2. The h y p o t h e s i s t e s t i n g a p p r o a c h , which t h i s a u t h o r

f a v o r s , s e r v e s a l so as t h e bas i s f o r t h e more familiar c o n s t r u c t i o n of

c o n f i d e n c e i n t e r v a l s for s i g n a l s which a re de tec ted [831. For more informa-

t i o n on t h e r e l a t i o n s h i p between t h e power of a n h y p o t h e s i s tes t and t h e

significance levels and number of replicates (for normally-distributed data)'

the reader may refer to OC (0perating.Characteristic) carves as compiled by

Natrella C841. There it i s seen, for example, that 5 replicates are neces-

sary if one wishes to establish a detection limit which is no greater than

2 0 , taking [a] and [B] risks at 5% each. (Note the inequality statement;

this arises because of the discrete nature of replication.) Once we leave

the domain of simple detection of signals, and face the question of analyte

or radioactivity concentration detection, we encounter namerous added

Table 2. Detection Limits: Approaches, Difficalties

I -- Signal/Noise (S/N) [Ref's 3,29,30,861

Detection Limit E 2Np-p, ZNrms, 3s (n=16-20)

DC: white noise assumed, 6-error ignored - AC: must consider noise power spectram, non-stationarity, - -

digitization noise

Simple Hypothesis Testing [Ref's 2,5,26,56,83]

S = y - B h

J&: significance test (a-error) - 1-sided confidence interval a: power of test (B-error) - Operating Characteristic Curve Determination of SD requires accurate knowledge of the distribution

fanction for S

If 5 - N(S, a * ) , and a, B = O . O 5 , then SD = 2Sc = 3.29 a

Other Approaches [Ref ' s 28,85,87,881 Decision Analysis (uniformly best, Bayes, minimax), Information and Fuzzy

set theories.

18

problems or difficulties with assumption validity. That is, assumptions

concerning the calibration function or functions -- i.e., the full analytic

model -- and the llpropagation'l of errors (and distributional characteristics)

become crucial. A catalog of some of these issues is given in Table 3;

further discussion will be found in the following subsection. Finally, for

more detailed summary of the relevant literature, the reader is referred to

the review and bibliography in Appendix C.

Table 3. Concentration iletection Limits - Some Problems

o 2 only estimated; Ho-test ok (ts/h), but XD is uncertain

Calibration function estimated, so normality not exactly preserved:

f; = (y-i)/i # linear Fen (observations)

B-distribution (or even magnitude) may not be directly observed

Effects of non-linear regression; effects of "errors in x-

and y" (calibration)

Systematic error, blunders -- e.g., in the shape, parameters of A

[S + A , without continual re-calibration]

Uncertain number of components (and identity)

[Lack of fit tests lose power under multicollinearity]

Multiple detection decisions: (l-a)-+( l-a)"

B. Signal Detection (principles)

1. Alternative Approaches

A necessary, first step in treating signal detection is to consider what

magnitude observed (a posteriori) response (gross signal) constitutes a

statistically significant deviation (increment, or net signal) from the

zero-level (blank or background or baseline in radioactivity measurement).

This increment, which really represents a critical or decision level (SC)

19

1

' ,

w i t h which t h e o b s e r v e d s i g n a l is compared, is d e r i v e d f rom the d i s t r i b u t i o n

f u n c t i o n f o r t h e n o i s e . If t h e n o i s e c a n be c o n s i d e r e d normal ( "Gauss i an f1 )

w i t h parameter -o ( s t a n d a r d d e v i a t i o n ) , SC is g i v e n by a f i x e d m u l t i p l i e r

times a , and t h e d e t e c t i o n p r o c e s s becomes s i m p l y a s i g n i f i c a n c e t e s t based ,

on compar ison o f t h e o b s e r v e d w i t h t h e c r i t i c a l s i g n a l t o n o i s e r a t i o .

C e r t a i n n o n - t r i v i a l p roblems a r i s e i f t h e n o i s e power s p e c t r u m is n o t r rwhitelf

- , ( G a u s s i a n ) and when the s i g n a l is c o n t i n u o u s ( i n time) b u t is sampled

p e r i o d i c a l l y . These i s s u e s are treated i n some d e p t h i n R e f e r e n c e s i n d i c a t e d

i n T a b l e 2.

The t e s t , however , is i n c o m p l e t e ( though w i d e l y p r a c t i c e d ! ) f o r ou r

I t s p e a k s o n l y t o t he q u e s t i o n of s i g n a l d e t e c t i o n ( a p u r p o s e s .

p o s t e r i o r i ) -- i . e . , t h e d e t e c t i o n d e c i s i o n g i v e n t h e n o i s e p r o b a b i l i t y

d e n s i t y f u n c t i o n ( p d f ) and a n o b s e r v e d s i g n a l . I t i s i m p o r t a n t t o u s i n t h a t

t h e s i g n i f i c a n c e l e v e l of t h e t e s t - c1 is e q u i v a l e n t t o t h e fa l se p o s i t i v e

p r o b a b i l i t y o r " e r r o r o f the f i r s t k ind ." (Tha t is , ci e q u a l s t he p r o b a b i l i t y

t h a t one would , by c h a n c e , f a l s e l y c o n c l u d e t h a t a b l ank c o n t a i n e d e x c e s s

r a d i o a c t i v i t y . ) T h i s is i n s u f f i c i e n t , p e r s e , f o r u s t o s p e c i f y the detec-

t i o n c a p a b i l i t y o r L L D , which is an a p r i o r i per formance charac te r i s t ic o f

t h e Measurement P r o c e s s (MP).

A s o l u t i o n i s found i n t he t h e o r y o f H y p o t h e s i s T e s t i n g , where in we use

a n e x p e r i m e n t a l outcome 2 n o t s i m p l y t o t e s t f o r t h e p r e s e n c e o f a s i g n a l b u t

a c t u a l l y t o d i s c r i m i n a t e between two p o s s i b l e s t a t e s o f t h e sys t em: Ho and

H D .

hypo thes i s ! ' and t h e c r i t i c a l l e v e l Sc is set i n such a way t h a t an o p t i m a l

d e c i s i o n ( i n t h e l o n g r u n ) is made between t h e two h y p o t h e s e s . A s t h e

Ho and HD a r e , r e s p e c t i v e l y , t h e " n u l l h y p o t h e s i s " and t h e r r a l t e r n a t i v e

I

20

s u b s c r i p t s imp ly , Ho refers t o s a m p l e s c o n t a i n i n g no n e t r a d i o a c t i v i t y , and

H D , t o s a m p l e s c o n t a i n i n g r a d i o a c t i v i t y a t t h e LLD. I n terms of t he n e t

s i g n a l , Eo: S=O and HJ: S=SD (S b e i n g t h e t r u e , b u t unknown n e t s i g n a l . )

Two of t h e basic fo rms o f H y p o t h e s i s t e s t i n g r e q u i r e i n f o r m a t i o n o r

a s s u m p t i o n s t h a t are n o t g e n e r a l l y a v a i l a b l e f o r s i m p l e chemical o r p h y s i c a l

measurements . The f i rs t i n v o l v e s t h e u s e o f t h e "Bayes C r i t e r i o n " which

r e q u i r e s p r i o r p r o b a b i l i t i e s f o r HO and H D , as well as t h e a s s i g n m e n t of

I

c o s t s f o r making i n c o r r e c t d e c i s i o n s . I n t h i s case SC would be se t t o

minimize t h e a v e r a g e ( l o n g - r u n ) cost . The second a p p r o a c h , which is related

t o game t h e o r y , d o e s n o t r e q u i r e p r i o r p r o b a b i l i t i e s . Rather, i t is d e s i g n e d

t o minimize t h e maximum cost o v e r t h e e n t i r e se t o f p o s s i b l e p r i o r p r o b a b i l i -

t i e s . A p p r o p r i a t e l y , t h i s is termed t h e "Minimax" d e c i s i o n s t r a t e g y .

Lack ing e i ther costs o r p r io r p r o b a b i l i t i e s , w e prefer t o d e f i n e de t ec t ion

c a p a b i l i t y ( L L D ) on t h e basis of s i m p l e h y p o t h e s i s t e s t i n g (vlNeyman-Pearson

c r i t e r i o n 1 ? ) which c o n s i d e r s Ho, HD and SC s i m p l y i n terms of t h e p r o b a b i l i -

t i e s of d rawing fa l se c o n c l u s i o n s when .? is compared t o SC.

t i o n s o f a l l three d e c i s i o n s t r a t e g i e s are g i v e n i n Ref's 28, 29 and 79. A

more c o m p l e t e deve lopment of s i m p l e h y p o t h e s i s t e s t i n g f o r d i rec t a p p l i c a t i o n

t o LLD f o l l o w s .

Lucid e x p o s i -

2. S imple H y p o t h e s i s T e s t i n g and t h e LLD

[ a d a p t e d from Ref. 381

The basic i s s u e we wish t o address is 'whether one p r imary h y p o t h e s i s

[ t h e " n u l l h y p o t h e s i s " , H o l describes t h e s t a t e o f t he sys t em a t t h e p o i n t

( o r time) o f s ampl ing o r whether t he " a l t e r n a t i v e h y p o t h e s i s 1 ? [ H D ] describes

it. The a c t u a l tes t is o n e of c o n s i s t e n c y - i .e. , g i v e n t he e x p e r i m e n t a l

s a m p l e , a r e t h e data c o n s i s t e n t w i t h H o , a t t h e s p e c i f i e d l e v e l o f s i g n i f i -

a

cance, a? That is the first question, and if we draw (unknowingly) the wrong

conclusion, it is called an error of the first kind. This is equivalent to a

false positive in the case of trace analysis - i.e., although the (unknown)

true analyte signal S equals zero (state Ho), the analyst reports,

If detec tedff . The second question relates to discrimination. That is, given a

decision- (or critical-) level S c used for deciding upon consistency of the'

experimental sample with Ho, what true signal level SD can be distinguished

from SC at a level of significance f 3 ?

to HD (S=SD) and we falsely conclude that it is in state Ho, that is called

an error of the second kind, 'and it corresponds in trace analysis to a false

negative. The probabilities of making correct decisions are therefore 1-a

(given Ho) and 1 - 6 (given HD); 1-6 is also known as the ffpowerff of the test,

and it is fixed by 1-a (or S c ) and SD. One major objective in selecting a

particular MP is thus to achieve adequate detection power ( 1 - 6 ) at

the signal level of interest (SD), while minimizing the risk ( a ) of false

positives. Given a and B (commonly taken to be 5% each), there are clearly

two derived quantities of interest; SC for making the detection decision, and

SD the detection limit.

net signal rather than radioactivity concentration, LLD would be taken

equal to SD.) Figure 1 illustrates the interrelation of a , f i , SC and

the detection limit.

If the state of the system corresponds

(If, for RETS, our concern were strictly with the

An assumption underyying the above test procedure is that the estimated

net signal is an independent random variable having a known distribution.

(This is identical to the prerequisite for specifying confidence intervals.)

Thus, knowing (01" having a statistical estimate for) the standard deviation

of the estimated net signal S, one can calculate S c and SD, given the form of A

22

. >

F i g . 1 . H y p o t h e s i s t e s t i n g f e r r o r s of t h e f i r s t and second k i n d s

23

t h e d i s t r i b u t i o n and a and B . If the d i s t r i b u t i o n is Normal w i t h c o n s t a n t a ,

and a = B = 0.05, SD = 3.29 os and Sc = sD/2. Thus, t h e r e l a t i v e s t a n d a r d

d e v i a t i o n of t h e estimated n e t s i g n a l e q u a l s 30% a t the d e t e c t i o n l i m i t ( 5 ) .

I n c i d e n t a l l y , t h e t h e o r y of d i f f e r e n t i a l d e t e c t i o n fol lows e x a c t l y t h a t of

d e t e c t i o n , e x c e p t t h a t ASJND ( t h e " j u s t n o t i c e a b l e d i f f e r e n c e " ) takes the

p l a c e of SD, and for HO r e f e r e n c e is made t o the base l e v e l So of t h e a n a l y t e

ra ther t h a n the zero l e v e l ( b l a n k ) . A small f r a c t i o n a l change (AS/S)D t h u s

r e q u i r e s e v e n smaller i m p r e c i s i o n .

O b v i o u s l y , t h e smallest d e t e c t i o n l i m i t s o b t a i n f o r i n t e r f e r e n c e - f r e e

measurements and i n the a b s e n c e of s y s t e m a t i c error. Allowance f o r these

f ac to r s n o t o n l y i n c r e a s e s SD, b u t ( a t least i n t he case of s y s t e m a t i c error)

d i s t o r t s t h e p r o b a b i l i s t i c s e t t i n g , j u s t as i t does w i t h c o n f i d e n c e i n t e r -

v a l s . Special t r e a t m e n t s f o r these q u e s t i o n s and for non-normal d i s t r i b u -

t i o n s w i l l be g i v e n as appropr i a t e . Not so o b v i o u s pe rhaps is t h e f ac t t h a t

SD d e p e n d s on the s p e c i f i c a lgor i thm selected f o r data r e d u c t i o n .

i n t e r f e r e n c e e f f ec t s on SD, t h i s dependence comes a b o u t b e c a u s e of the e f fec t

on a s , t h e s t a n d a r d d e v i a t i o n of t h e estimated n e t s i g n a l . More e x p l i c i t

c o v e r a g e of these matters w i l l be g i v e n below and de t a i l ed d e r i v a t i o n s and

n u m e r i c a l examples w i l l be found i n s e c t i o n I11 and t h e Appendix of t h i s

r e p o r t , r e s p e c t i v e l y , (see a l s o Ref. 33.) .

A s w i t h

H y p o t h e s i s t e s t i n g is e x t c e m e l y i m p o r t a n t f o r other p h a s e s of chemical

and radiochemical a n a l y s i s , i n a d d i t i o n t o the q u e s t i o n of a n a l y t e d e t e c t i o n

l i m i t s . Through the use of a p p r o p r i a t e test s t a t i s t i c s , o n e may tes t data

sets for b i a s , for unexpec ted random error components , f o r o u t l i e r s , and e v e n

f o r e r r o n e o u s e v a l u a t i o n (data r e d u c t i o n ) models (33 ) . Because of s t a t i s t i -

cal l i m i t a t i o n s of s u c h tests, e s p e c i a l l y when there are r e l a t i v e l y few

degrees of freedom, t h e y are somewhat i n s e n s i t i v e (lack power) e x c e p t f o r

24

quite large effects. For this reason it is worth considerable effort on the

part of the analyst to construct his MP so that it is as free from or

resistant to bias, blunders, and imperfect models as possible.

Figure 2 gives an illustration of the difficulties of detecting both

i systematic error and excess random error. There we see that just to detect

systematic error when it is comparable to the random error ( 0 ) requires about

15 observations; and to detect an extra random error component having a

comparable o requires 47 observations ( 8 9 ) . In a simple case involving model

error it has been shown that analyte components omitted from a least-squares

I multicomponent spectrum fitting exercise must be significantly above their

detection limits (given the correct model) before misfit statistics signal

the error ( 3 3 ) . This limitation in "statistical power". to prevent

significant model error bias, especialy in the fitting of multicomponent

spectra, is one of the most important reasons for developing multidimensional

chemical or instrumental procedures and improved detectors of high

specificity or resolution.

C. General Formulation of LLD - Major Assumptions and Limitations The foregoing discussion provides the basis for deriving specific

expressions for the LLD for signals, given a and 6, and os as a function of

concentration. Before treating concentration detection limits generally, and

radioactivity concentration detection limits specifically, however, it is

necessary to examine a number of basic assumptions connected with the concept

and with the MP.

25

100.0 I I I I I - - - - -

t z J 2

0 0

o x

n

- A

i== - -

- - 0.1 I I I I I .

1 10 100

NUMBER OF OBSERVATIONS

F i g . , 2 . D e t e c t i o n limits v s . number of o b s e r v a t i o n s for e x t r a n e o u s random error ( a x , dashed c u r v e ) and systematic error ( A , s o l i d c u r v e ) :

26

1 ) D e t e c t i o n D e c i s i o n s v s D e t e c t i o n L i m i t s

The s i g n a l d e t e c t i o n l i m i t SD is u n d e f i n e d u n l e s s a or Sc is d e f i n e d - and

a p p l i e d . That i s , d e t e c t i o n d e c i s i o n s are mandatory i f d e t e c t i o n limits ( i n

the h y p o t h e s i s t e s t i n g s e n s e ) are t o be m e a n i n g f u l . The r e l a t i v e l y common

p r a c t i c e of e q u a t i n g these two l e v e l s ( S ~ = S D ) is e q u i v a l e n t t o s e t t i n g t he

fa l se n e g a t i v e r i s k a t 50%. T h a t i s , a d e t e c t i o n l i m i t so d e f i n e d w i l l i n

f a c t be missed h a l f the time! The recommended p r a c t i c e t h e r e f o r e is t o take

a=B=0.05, i n which case,

Sc = = 1.645 o0 ( 1 1

( 2 ) SD = sc + Z i - ~ o o = 2sc = 3.2900

p r o v i d e d t h e s t a n d a r d d e v i a t i o n of t h e n e t s i g n a l OS is known and c o n s t a n t

( a t l e a s t up t o t h e d e t e c t i o n l i m i t ) and i t is n o r m a l l y - d i s t r i b u t e d (z refers

t o t h e i n d i c a t e d p e r c e n t i l e o f t h e s t a n d a r d normal v a r i a t e . ) I n E q ' s ( 1 ) and

( Z ) , oo = OS ( a t S = O ) ; t h i s i n t u r n e q u a l s OB i f t h e a v e r a g e v a l u e of t h e

b l a n k is well-known (Ref. 5 ) .

u s e d f o r t e s t i n g whether an o b s e r v e d s i g n a l 5 is ( s t a t i s t i c a l l y ) d i s t i n g u i s h -

able from t h e b l a n k -- i .e . "detected"; SD r e p r e s e n t s the c o r r e s p o n d i n g MP

per formance cha rac t e r i s t i c , i . e . , t h e d e t e c t i o n l i m i t . Al though SD/SC = 2

g e n e r a l l y , t h i s is n o t u n i v e r s a l l y t r u e . A.number of e x c e p t i o n a l cases which

do o c c u r , e s p e c i a l l y i n e x t r e m e l o w - l e v e l c o u n t i n g and i n n u c l e a r

s p e c t r o s c o p y , are treated i n s e c b i o n I11 of t h i s manual .

(For " p a i r e d o b s e r v a t i o n s " , oo = o ~ c . 1 SC is

-

2 ) A P r i o r i v s A P o s t e r i o r i ; Changes i n t h e MP ( I n t e r f e r e n c e , . . . I

Some c o n f u s i o n e x i s t s i n the u s a g e of these terms which mean "before t h e

fac t" and "af ter the fac t . " The "factfT referred t o is the e x p e r i m e n t a l

outcome -- i . e . , t h e o b s e r v a t i o n of a (random) s i g n a l 5 , associated w i t h t h e

measurement of a p a r t i c u l a r sample . The MP, which n e c e s s a r i l y i n c l u d e s t h e

27

i n f l u e n c e o f t he sample on the character is t ics of the measurement s y s t e m is

- n o t t h e r l factf ' , from the p e r s p e c t i v e of h y p o t h e s i s t e s t i n g .

i n t e l l i g e n t d e c i s i o n s r e g a r d i n g S we need t h e r e f o r e i n f o r m a t i o n c o n c e r n i n g

the MP charac te r i s t ics , n o t a b l y OS a t S='O and t h e v a r i a t i o n o f OS w i t h

c o n c e n t r a t i o n . T h i s i n t u r n is i n f l u e n c e d by t h e l e v e l and n a t u r e of any

i n t e r f e r i n g s p e c i e s i n t he sample i n q u e s t i o n . Also, as soon as we c o n s i d e r

t h e rea l q u a n t i t y o f i n t e r e s t , t h e c o n c e n t r a t i o n d e t e c t i o n l i m i t ( x D ) , we

r e q u i r e i n f o r m a t i o n c o n c e r n i n g t h e o v e r a l l c a l i b r a t i o n f a c t o r f o r t he

p a r t i c u l a r s ample ; t h i s i n c l u d e s the ( r a d i o ) c h e m i c a l y i e l d o r r e c o v e r y ,

d e t e c t i o n e f f i c i e n c y (as p e r t u r b e d by sample m a t r i x e f fec ts : a b s o r p t i o n and

s c a t t e r i n g ) , volume o r mass o f t h e sample , e t c .

I n o r d e r t o make

T h u s p r i o r knowledge c o n c e r n i n g t h e sample i n q u e s t i o n - is r e q u i r e d i n

order t o compute SC which o n e n e e d s fo r t h e a p o s t e r i o r i tes t of S ; i t is

needed a l s o t o compute t he s i g n a l and c o n c e n t r a t i o n d e t e c t i o n l i m i t s (SD,

X D ) f o r t h a t sample. Such p r i o r i n f o r m a t i o n may be o b t a i n e d i n a p r e l i m i n a r y

o r s c r e e n i n g e x p e r i m e n t ; i t may be estimated from data r e s u l t i n g f rom - t h e

e x p e r i m e n t , i t s e l f ; o r i t may be assumed ( n o t recommended) i n d e p e n d e n t of t h e

e x p e r i m e n t . The l a s t a p p r o a c h might be t a k e n i f one were i n t e r e s t e d i n " p u r e

s o l u t i o n " o r ideal sample d e t e c t i o n l i m i t s , where there is no i n t e r f e r e n c e ,

no m a t r i x e f fec ts and p e r f e c t or u n v a r y i n g r e c o v e r i e s . A s l i g h t l y less

d i s a s t r o u s a l t e r n a t i v e , t o assume a v e r a g e v a l u e s f o r such q u a n t i t i e s o r

e f f ec t s , r e s u l t s i n n e e d l e s s i n f o r m a t i o n loss. To c a r i c a t u r e t h e s i t u a t i o n ,

i t ' s e q u i v a l e n t t o p e r m i t t i n g the c o u n t i n g time t o v a r y i n a haphaza rd

f a s h i o n from sample t o sample and g u e s s i n g a n a v e r a g e time for c a l c u l a t i n g

i n d i v i d u a l c o u n t i n g ra tes . The p o i n t is: the c r i t i c a l ( d e c i s i o n ) l e v e l and

d e t e c t i o n l i m i t r e a l l y do v a r y w i t h t h e n a t u r e of t h e sample . So p r o p e r

A

28

a s s e s s m e n t of these q u a n t i t i e s demands r e l e v a n t i n f o r m a t i o n o n each s a m p l e ,

u n l e s s t he v a r i a t i o n s among samples ( e . g . , i n t e r f e r e n c e l e v e l s ) are q u i t e

t r i v i a l .

Some p e r s p e c t i v e and a s u g g e s t e d a p p r o a c h t o t h i s matter a re g i v e n i n '

F i g . 3. Here, we c o n s i d e r three p o s s i b l e outcomes f o r a n e x p e r i m e n t

( " e x p e r i m e n t - a " ) which i s d e s i g n e d ( s a m p l e s i z e , e x p e c t e d i n t e r f e r e n c e l e v e l

o r background a c t i v i t i e s , c o u n t i n g time, e t c . ) a c c o r d i n g t o our p r i o r

knowledge of t he MP. T h i s p r i o r l lknowledgelT, which here i n c l u d e s t he

a s s u m p t i o n of z e r o i n t e r f e r e n c e (I=O), we d e s i g n a t e l l p r i o r ( a ) l l ; i t leads t o a

c o n c e n t r a t i o n d e t e c t i o n l i m i t XD based on a background e q u i v a l e n t a c t i v i t y

Bo. We c o n s i d e r t h e e x p e r i m e n t a d e q u a t e l y d e s i g n e d i f t h i s estimated

d e t e c t i o n l i m i t XD ( a c t u a l LLD) does n o t exceed t h e s p e c i f i e d maximum l e v e l

X R ( p r e s c r i b e d L L D ) .

0

A s s o o n as t h e ( f i r s t ) e x p e r i m e n t is p e r f o r m e d , we g a i n two k i n d s of

i n f o r m a t i o n : new d a t a o n t h e MP-characteristics f o r the sample a t hand , and

a n e x p e r i m e n t a l r e s u l t Ga . d e p i c t e d i n F i g . 3 show p r o g r e s s i v e l y g r e a t e r background- (or- b a s e l i n e - )

e q u i v a l e n t a c t i v i t i e s (B3>B2>B1) and therefore s i m i l a r l y i n c r e a s i n g d e t e c t i o n

l i m i t s ( x D ~ s ) . For outcome-1 , t h e posterior MP charac te r i s t ics

a r e e q u i v a l e n t t o o u r assumed p r i o r MP-characteristics [ l l p r i o r ( a ) r r ] , -- i . e ,

B1 and

t h e e x p e r i m e n t i s a d e q u a t e ( X D 6 XR).

charac te r i s t ics d i f f e r from t h e p r i o r ; there - is i n t e r f e r e n c e (B2 a n d B 3 >

B o ) , so t h e d e t e c t i o n l i m i t is greater.

d e t e c t i o n l i m i t (XD

s u f f i c i e n t l y c o n s e r v a t i v e ( X D < XR) t h a t some i n t e r f e r e n c e c o u l d be

tolerated.

The three p o s s i b l e outcomes (MP charac te r i s t ics )

= BO -- so of course t h e d e t e c t i o n l i m i t is as c a l c u l a t e d ( X D = XD 1 0 For outcomes-2 and -3 t h e p o s t e r i o r

Outcome-2 s t i l l shows a n a d e q u a t e

S X R ) , so o u r t a s k is c o m p l e t e - - the i n i t i a l d e s i g n was 2

0

29

EXPT-a x=o (3 possible outcomes)

+ 0

r

0

EXPT-b

I 0

I I I I I I

I I

I I I

B2 C XD2

MP-CHARACTERISTICS

POST# PRIOR

(Interference) [Ba = B,>B,>B1]

R E-DESIG N I I I I I I

POST(b) = PRIOR(b) [Bb = B31

I

B3 C I xD; 1 A I 'b

Fig . 3. A Pr io r i v s A Pos te r io r i , Sequential Experiments and the Effect of I n t e r f F e n c e

30

The t h i r d se t of MP-characteristics (outcome-3) c o r r e s p o n d t o a sample

h a v i n g so h i g h a l e v e l of i n t e r f e r e n c e t h a t t h e i n i t i a l d e s i g n was i n a d e q u a t e

( X D ~ > X R ) . We therefore must u s e t h i s p o s t e r i o r i n f o r m a t i o n as

o u r - new p r i o r i n f o r m a t l o n ( l r p r i o r ( b ) l t ) t o r e - d e s i g n t h e MP t o y i e l d a d e q u a t e

charac te r i s t ics ( x i S xRj , i n p r e p a r a t i o n f o r a second ( f i n a l ) e x p e r i m e n t . 3

( T h i s i s s t i l l p r o p e r l y c o n s i d e r e d Ira p r i o r i " i n t h e t e c h n i c a l s e n s e of

h y p o t h e s i s t e s t i n g u n t i l the second e x p e r i m e n t a l r e s u l t ];b [ ' I fact" or

o b s e r v a t i o n ] has been o b t a i n e d . ) Such r e - d e s i g n can be based on a n y of t h e

MP-var iab les under o u r c o n t r o l , s u c h as sample s i z e , radiochemical s e p a r a t i o n

or c o n c e n t r a t i o n , or c o u n t i n g time. ( I n F ig . 3 w e i n d i c a t e r e - d e s i g n s i m p l y

as a n e x t e n s i o n of c o u n t i n g time f o r r e l a t i v e l y l o n g - l i v e d r a d i o a c t i v i t y . ) A

1 - l i n e summary of these comments r e g a r d i n g s e q u e n t i a l e x p e r i m e n t s would be

s i m p l y t o s t a t e t h a t o n e ' s p o s t e r i o r becomes a n o t h e r ' s p r i o r .

3. C o n t i n u i t y of Hypotheses ; U n p r o v a b i l i t y

H y p o t h e s i s - t e s t i n g as o u t l i n e d above was dichotomous -- t h a t i s , we

referred t o the n u l l h y p o t h e s i s ( H o : S = O ) and t h e d e t e c t i o n l i m i t h y p o t h e s i s

( H D : S=SD) o n l y . I n f a c t , S is a c o n t i n u o u s q u a n t i t y which may take on a n y

v a l u e from z e r o and some large, r e a s o n a b l e upper l i m i t . '

when we compare .? w i t h SC and make t h e d e t e c t i o n d e c i s i o n is t o c o n c l u d e t h a t

o n e o r t he other of o u r two h y p o t h e s e s (Ho, H D ) is q u i t e u n l i k e l y , o r more

c o r r e c t l y t h a t s u c h a r e s u l t 5 is q u i t e u n l i k e l y (here , 55% c h a n c e of

What t akes p l a c e

\

--------------- I A l o g i c i a n might ob jec t t o t h i s s t a t e m e n t on the bas i s t h a t atoms are discrete; and s u c h a n argument might e v e n seem r e l e v a n t i f we h a d , s a y , 100 atoms of a s h o r t - l i v e d r a d i o n u c l i d e , a n d a p e r f e c t (100% e f f i c i e n t ) d e t e c t o r . We c o u l d c o u n t them a l l . Even here, however , t h e rtStt t h a t as s c i e n t i s t s we're interested i n is n o t t h e number of atoms i n t h a t p a r t i c u l a r s a m p l e , b u t i ts e x p e c t e d v a l u e -- s u c h as t h e l o n g - r u n a v e r a g e t h a t would a r i s e from r e p e a t e d , i d e n t i c a l a c t i v a t i o n a n a l y s e s . The u n d e r l y i n g i s s u e r e l a t e s t o compound p r o b a b i l i t y d i s t r i b u t i o n s ; a t r e a t m e n t f o r t h e case of r a d i o a c t i v i t y is g i v e n i n Ref. 63.

o c c u r r i n g ) g i v e n HO o r H D . The o t h e r h y p o t h e s i s ( H o i f S S S c , HD i f S > Sc)'

is s a i d t o be c o n s i s t e n t w i t h the o b s e r v a t i o n , b u t i t is by no means proved .

An i n f i n i t e number of i n t e r m e d i a t e v a l u e s of S are a l s o c o n s i s t e n t ! (The

most l i k e l y is S = 3 . ) T h i s b i t of l o g i c may seem t r i v i a l and o b v i o u s t o

some, and s u b t l e and i r r e l e v a n t t o o t h e r s , b u t there i s o n e c u r i o u s and

i m p o r t a n t consequence . The h a b i t of " a c c e p t i n g T 1 t h e h y p o t h e s i s t h a t i s n o t

re jec ted , sometimes leads t o b i a s e d r e p o r t i n g of data . For example , i f 2 5

Sc, t h e v a l u e r e p o r t e d may be zero; t h e o t h e r ex t r eme is r e p o r t i n g i t as

b e i n g a t t h e d e t e c t i o n l i m i t , if S > S c . A f a r t h e r comment on t h i s matter is

g i v e n i n t h e s u b s e c t i o n on R e p o r t i n g of R e s u l t s ( s e c t i o n I I . D . 4 ) . (See a l s o

n o t e A13.)

4. The C a l i b r a t i o n F u n c t i o n and LLD.

S i n c e ou r c o n c e r n i s w i t h t he d e t e c t i o n l i m i t f o r r a d i o a c t i v i t y concen-

t r a t i o n -- i . e . , t h e " lower l i m i t of d e t e c t i o n " ( L L D ) -- w e must go beyond

the above e x p o s i t i o n on s i g n a l d e t e c t i o n . I f t h e c a l i b r a t i o n f u n c t i o n ,

r e l a t i n g r e s p o n s e - y t o c o n c e n t r a t i o n - x is l i n e a r ,

y = B + Ax + ey ( 3 )

where B r e p r e s e n t s t h e b l a n k ; A , t h e c a l i b r a t i o n c o n s t a n t o r f a c t o r ; and e y ,

the e r r o r i n t he o b s e r v a t i o n y .

32

The estimated n e t s i g n a l is

; = y - ; A

B b e i n g a n i n d e p e n d e n t estimate f o r B ; and the estimated c o n c e n t r a t i o n

;; = ( y - ;)hi ( 5 )

b e i n g a n i n d e p e n d e n t estimate f o r A . (Here, " i n d e p e n d e n t f f means

i n d e p e n d e n t of the o b s e r v a t i o n y . a l w a y s r e s u l t s , of c o u r s e , when t h e y are b o t h estimated from t h e f i t t i n g of a

s i n g l e s e t of c a l i b r a t i o n da t a . )

I n t e r d e p e n d e n c e [ c o r r e l a t i o n ] of 6 and 5

I d e a l l y we would n e x t d e t e r m i n e a x a s a f u n c t i o n of x e i ther v i a - r e p l i c a t i o n , o r by e r r o r - p r o p a g a t i o n . Complete r e p l i c a t i o n of t h e e n t i r e

c a l i b r a t i o n and sample measurement p r o c e s s f o r t h e f u l l r a n g e of s a m p l e

m a t r i x e s and i n t e r f e r i n g a c t i v i t i e s t o y i e l d and a d e q u a t e number ( n ) of

rep l ica tes : x i f o r I = 1 t o n s p a n n i n g t h e f u l l c o n c e n t r a t i o n r a n g e of

c o n c e r n (from z e r o t o - LLD) would be a v e r y large t a s k . (For t h e estimated

s t a n d a r d d e v i a t i o n t o have a r e l a t i v e u n c e r t a i n t y (95% C I ) of + I O % f o r

example would r e q u i r e a b o u t n = 200 r e p l i c a t e s a t each c o n c e n t r a t i o n ! )

f a v o r therefore error p r o p a g a t i o n , r e s e r v i n g o c c a s i o n a l f u l l r e p l i c a t i o n f o r

c o n t r o l of q u a l i t y and b l u n d e r i d e n t i f i c a t i o n .

A

We

E r r o r - p r o p a g a t i o n i s s t r a i g h t f o r w a r d f o r l i n e a r f u n c t i o n s of normal ly-

d i s t r i b u t e d random v a r i a b l e s . Thus ,

VS = vy VB = o2 (6) S

where V r e p r e s e n t s the v a r i a n c e of t he s u b s c r i p t e d q u a n t i t y . S i n c e E ( y ) ( t h e

e x p e c t e d v a l u e of y ) e q u a l s S + B ,

vo = vS(s=o) = VB + v i ( 7 )

33

, - e.

A

so, if the observations leading to B and y are equivalent, Vo = 2vB or

a. = o g f i as noted earlier.

(assuming still Normality).

Calculation of Sc and SD follow immediately

With the introduction of a random variable in the denominator of E q . 5,

complications set in because we now have a non-linear function (ratio) of

random variables.

small, the distribution of is only slightly skew; however, the appropriate

error propagation formula (not shown), which itself is an approximation,

contains the unknown quantity A. The consequence is that both xc and XD are

themselves uncertain.

testing errors a and B are uncertain.) Full treatment of this matter is

beyond the scope of this document, but further details may be found in

Ref. 76.

If @A (relative standard deviation or R S D of A) is quite

(Or, if we choose values for xc and xD, the hypothesis

The approach adopted for LLD purposes, which we label "S-basedTf is

simpler in concept and straightforward in application. That is, we treat the

detection decision strictly in the signal domain, using 5 and Sc. corresponding signal detection limit SD is then transformed into the fltrueff

concentration detection limit XD using the true calibration factor A, which

we do not know.

The

XD = SD/A = ( ~ i - ~ o ~ + zl-BaD)/~ ( 8 ) A

Using bounds for A; A ? Z I - ~ / ~ ~ A , we can then calculate a confidence interval

for XD. Taking a conservative viewpoint, we go one step further; namely

A, = A-zl-y/2a~ is inserted in the denominator of E q . (8).

quantity is an upper 1irnit.for XD for B = 0.05. (A dual interpretation,

which will not be discussed here, defines XD in conjunction with an upper

limit for B . a , of course, remains at 0.05; and neither S c nor SD suffer

from the A-uncertainty, because they are strictly signal-based. When A is

A

The resulting

34

. n o t randomly sampled , t h e u n c e r t a i n t y i n XD no l o n g e r r e p r e s e n t s a " c o n f i -

dence" i n t e r v a l . I t must be viewed as a s y s t e m a t i c error i n t e r v a l . F i n a l l y ,

i f t h i s c o n s e r v a t i v e estimate (uppe r l i m i t ) f o r XD is less t 9 a n t h e

p r e s c r i b e d r e g u l a t o r y l i m i t ( x R ) , t h e o b j e c t i v e of RETS w i l l have been met.

Recogn iz ing t h e d i s t i n c t i o n between XR -- t h e maximum p e r m i s s i b l e LLD, o r

l l r e g u l a t o r y l i m i t " , and XD -- t h e a c t u a l LLD or " c o n c e n t r a t i o n d e t e c t i o n

l i m i t f 1 f o r a p a r t i c u l a r sample and measurement t e c h n i q u e , and t h e RETS

re qu i remen t :

X D 5 X R (9)

, i t becomes i n t e r e s t i n g t o c o n s i d e r i n e q u a l i t y a p p r o a c h e s . One s u c h

i n e q u a l i t y , f o r c e d on u s because of t h e n o n - l i n e a r r e l a t i o n Eq. 5 , has

a l r e a d y been u s e f u l i n c o n j u n c i o n w i t h Eq. 9. The c r u c i a l p o i n t i s t h a t

Eq. 9 removes t h e n e c e s s i t y t h a t XD be known e x a c t l y o r w i t h a f i x e d small

r e l a t i v e u n c e r t a i n t y . A s l o n g as a r e a s o n a b l y chosen uppe r l i m i t f o r XD

s a t i s f i e s t h i s r e l a t i o n t h e problem i s s o l v e d .

A second t y p e o f i n e q u a l i t y i n v o l v i n g X D , of g r e a t p r a c t i c a l

i m p o r t a n c e , d e r i v e s from upper bounds which c a n be d e r i v e d immedia t e ly from

t h e e x p e r i m e n t a l r e s u l t ( x , o x ) which is n e c e s s a r i l y produced f o r e v e r y

a n a l y s i s . The r e s u l t i n g uppe r bound for x, i f x > XC, c a n be shown always t o

exceed XD. Therefore, i f f o r a g i v e n sample t h a t bound s a t i s f i e s Eq. 9 ,

there is no need t o r e - d e t e r m i n e t h e a c t u a l d e t e c t i o n l i m i t or t o r e - d e s i g n

t h e e x p e r i m e n t . (See t h e comments on s e q u e n t i a l e x p e r i m e n t s , accompanying

F ig . 3 [ s e c t i o n 11, C.21, and t h e n o t e [B4] i n S e c t i o n I11 f o r a s l i g h t l y

ex tended d i s c u s s i o n of t h e u s e o f - i n e q u a l i t i e s f o r r a p i d e s t i m a t i o n of bounds

f o r t h e d e t e c t i o n l i m i t . )

A

A

35

. ' J .

A p u r p o s e l y c o n t r o v e r s i a l , l lnon-de tec ted l l r e s u l t ( < a ) has been shown i n ,

F i g . 3, so t h a t w e may address t h e matter of a n i n a d e q u a t e MP ( X D > X R ) f o r

which a seemingly a d e q u a t e r e s u l t (xa-upper l i m i t < X R ) h a s been o b t a i n e d .

We a d v i s e c a u t i o n . That is , i f XD > X R , t h e u n c e r t a i n t y associated w i t h a n y

g i v e n measurement i s a p t t o y i e l d rather g r a d u a l l y c h a n g i n g s i g n i f i c a n c e

l e v e l s (and f a l s e n e g a t i v e e r rors , 6 ) . I t is a d v i s a b l e i n cases s u c h as

t h i s t o esti-mate d i r e c t l y t h e p r o b a b i l i t y B which would o b t a i n t a k i n g f f ~ as

the upper l i m i t . . That is, assuming n o r m a l i t y

I f t h e 90% C I u p p e r 1 i m j . t (xu = + 1.645 a,) i s smaller t h a n X R , t h e n B is

n e c e s s a r i l y l e s s t h a n 5%. However, as i-s o b v i o u s from Eq 1 0 , t h e s t a t i s t i c a l

s i g n i f i c a n c e of a g i v e n d i f f e r e n c e ( x R - x ~ ) d e c r e a s e s w i t h i n c r e a s i n g ox,

which is t o s a y i t decreases w i t h i n c r e a s i n g LLD ( x D ) . Taking t h e r e s u l t i n

F i g . 3, x0 = xa + 1.645 oxu = 0.9 XR (where XD3 = 1.5 X R ) , we f i n d t h a t

(xR-xU)/ox = 0.219 [assumes a O = ox = c o n s t . ] , s o z1-6' = 1.864 o r

B ' = 0.031. T h i s is n o t s o much smaller t h a n t h e base v a l u e 6 = 0.05 o r , p u t

d i f f e r e n t l y , t he upper l i m i t from a 95% C I would exceed XR. C o n t r a s t t h i s

w i t h outcome-1 i n F i g . 3 , where X D 1 (and t h e r e f o r e a,) is smaller by a f a c t o r

of 3. There, if a n xu were 0.9 X R , z1-6' would be 1.645 + 3 ( 0 . 2 1 9 ) = 2.302,

so B ' = 0.01, and a 98% CI would be r e q u i r e d f o r t h e upper l i m i t t o reach X R s 3

A f i n a l se t of p r e c a u t i o n a r y n o t e s r e g a r d i n g the c a l i b r a t i o n f u n c t i o n are

i n order :

o The presumed s t r a i g h t - l i n e model (Eq. 3 ) is g e n e r a l l y a d e q u a t e o v e r

a small c o n c e n t r a t i o n r a n g e ( " l o c a l l y l i n e a r " ) , s u c h as between

--__----------- 3The numerology i n t h i s p a r a g r a p h takes a n added impact when o n e faces

t h e i s s u e of m u l t i p l e d e t e c t i o n d e c i s i o n s , where s t i l l more s t r i n g e n t r e q u i r e m e n t s ' a re p l a c e d o n a and B . (See s e c t i o n I I . D . 4 . )

36

x = 0 and x = X D . If there is any d o u b t , however , s u c h a p resumpt ion

s h o u l d be checked; a n d , above a l l , t he s l o p e o r t f c a l i b r a t i o n

c o n s t a n t f t A i n t he r e g i o n o f t h e d e t e c t i o n l i m i t s h o u l d n o t be

d e r i v e d f rom remote data ( x > > x ~ ) where t h e c u r v e may e x h i b i t

n o n - l i n e a r i t y (Ref. 7 6 ) .

-

o Imposed ( i n s t r u m e n t a l , software) t h r e s h o l d s , i n p l a c e o f S C , w i l l

n o t o n l y a l t e r ci b u t may change t h e r e l e v e n t t f l o c a l f f s l o p e -- u n l e s s

t h e c a l i b r a t i o n c u r v e is p e r f e c t l y s t r a i g h t (Ref. 7 6 ) .

o The c a l i b r a t i o n f a c t o r A , and any o f t h e f a c t o r s t h a t compr i se i t --

Y ( y i e l d ) , E ( e f f i c i e n c y ) , V ( s ample mass o r vo lume) , T ( c o u n t i n g

time f u n c t i o n ) -- may show i n t e r a c t i o n s w i t h B ( b a c k g r o u n d ,

b a s e l i n e , b l a n k , i n t e r f e r e n c e ) . Such f u r t h e r d i s t o r t i o n s ( o f Eq. 3 )

are d i s c u s s e d b r i e f l y i n s e c t i o n 111.

o If n o n - l i n e a r e s t i m a t i o n t e c h n i q u e - s , s u c h as n o n - l i n e a r l ea s t

s q u a r e s , are employed for n u c l i d e i d e n t i f i c a t i o n o r f o r e s t i m a t i o n

of c a l i b r a t i o n c u r v e p a r a m e t e r s , v a l u e s o f CY and 6 and t h e

d i s t r i b u t i o n o f x c a n be p e r t u r b e d . (Ref. 9 0 ) .

o Obvious , b u t worth s t a t i n g , is t h e f a c t t h a t $A ( R S D o f A ) f o r use

i n c o n n e c t i o n w i t h E q . ( 8 ) is

2 2 2 2 1 /2 @A = 14, + @E + $v + @T 3 ( 1 1 )

p r o v i d e d t ha t a l l t h e c o n s t i t u e n t @ I s are small. (Sampl ing e r r o r s , which

c o u l d be m a n i f e s t i n t he f a c t o r s Y , E , o r V may n o t a l w a y s s a t i s f y t h i s

r e q u i r e m e n t . @T, on t h e o t h e r hand , is e f f e c t i v e l y z e r o i n most c o u n t i n g

s i t u a t i o n s -- though u n c e r t a i n ( t e m p o r a l ) s a m p l i n g i n p u t f u n c t i o n s , o r

u n c e r t a i n h a l f - l i v e s o r r a d i o n u c l i d e mixes c o u l d a f f e c t even t h i s q u a n t i t y . ) . I

37

5. Bounds for Systematic Error

It would be marvelous if all our errors were random and of known

distribution (with known parameters), and even more so if we could rely on

their being Poisson. Such is never the case, so it is inappropriate to apply

the foregoing random-error based hypothesis testing framework for XD-

calculation, except as an asymptotic component. With carefully controlled

experimental work, however, that asymptotic component fortunately can be the

principal component.

A basis for the treatment of detection decisions and detection limits

in the presence of possible (uncorrected) systematic error is given in Ref.

33 for the case of signal detection. We extend that here to include the case

of "S-based" concentration detection, through the introduction of a second

systematic error bound parameter. Building on Eq. (8) for the random-error-

based concentration detection limit, we get

SC = A + Z1-a0o (12)

XD = f(2A + Z1-a~o + Z I - B O D ) / A (13)

where the quantity in the numerator in parentheses in Eq. (13) is SD

(incorporating blank systematic error bounds), and - f is a proportionate

amplification factor to provide a conservative bound for possible systematic

error in A. Thus, if A = YEVT (ignoring the 2.22 pci conversion factor) were

based on a one-time calibration such that random calibration errors became

systematic,

f = 1 + Zl-Y/2 @A (14)

where $A is given by Eq. (11).

or interference systematic error.

where 4~ denotes the relative systematic error bound in the blank (or interference) and B denotes the magnitude of this quantity. (See Eq. 4.)

A represents the a bound for possible blank

It can be further decomposed into ~ B B

38

If we re-cast Eq. ( 1 3 ) in terms of radioactivity, assuming oo = OD and

taking = z1-6 = 1.645

Here, the numerator is in units of counts, and XD, in units of pCi per unit

mass or volume.

Following our A-notation for the relative systematic error bound we

obtain from Eq. (14)

f = 1 + h A (16)

Clearly, the best experimental practice would include exhaustive theoretical

and/or experimental studies to obtain reliable values for AB and 4 A - That empirical evaluation of such quantities is not trivial is shown in

Fig. 2, where we see that just to detect a systematic error equal in magni-

tude to the random error of the MP requires more than ten observations (for

standard error reduction).

In lieu of this, and for the sake of providing explicit, reasonable

limits for the A's, we suggest the following [See notes All and B 3 ] :

A,, = 0.05, A, = 0.01, A, = 0.10 r

where 'IBkTT refers to both the blank and background and rcIrT refers to baseline

or interfering activity effects on B. Systematic error of still another

type, systematic model error is beyond the scope of our discussion though it

is treated briefly in section 111. C and in some detail in Ref. 72.

Equations (12) and (15) thus reduce to

SC = (0.05)B + 1.645 oo [counts] (17)

XD = (0.11)BEA + (0.50) ( [pCi/g or L 1 (18)

39

fo r the case of Blank ( B k ) predominance. If I >> Bk, t h e n t h e c o e f f i c i e n t s

o f - t h e f i r s t terms i n E q ' s ( 1 7 and 18) become 0.01 and 0.022. B , i n Eq. ( 1 7 )

r e p r e s e n t s t he Blank c o u n t s ; and BEA, i n Eq. (18) is t h e Blank E q u i v a l e n t

A c t i v i t y . A s we s h a l l see i n s u b s e q u e n t d i s c u s s i o n s , t h i s is a v e r y impor-

t a n t q u a n t i t y b o t h f o r the c a l c u l a t i o n o f t h e s y s t e m a t i c error bound (term-I,

Eq. ( 1 8 ) ) - and f o r d e r i v a t i o n of the random error-based term-2 ( t h r o u g h o o ) .

oo is t h e s t a n d a r d d e v i a t i o n of the estimated n e t s i g n a l ( c o u n t s ) when i ts

t r u e v a l u e is z e r o . Its magni tude depends on the s p e c i f i c c o u n t i n g (measure-

m e n t ) p r o c e s s , and i t is t h e s u b j e c t of the second f o l l o w i n g s u b s e c t i o n .

E q u a t i o n (18) is the e x p r e s s i o n f o r t h e LLD ( a c t u a l [ X D ] , n o t -

p r e s c r i b e d [ X R ] ) . I t is v a l i d o n l y when u s e d i n c o n j u n c t i o n w i t h Eq. ( 1 7 ) .

Also, i t car r ies the a s s u m p t i o n of n o r m a l i t y , and i t s h o u l d therefore be used

o n l y when t h e " b l a n k e x p e r i m e n t " y i e l d s B j 70 c o u n t s .

t h e t r e a t m e n t of v e r y l o w - l e v e l c o u n t i n g and other s p e c i a l s i t u a t i o n s . )

(See s e c t i o n I11 f o r

D . S p e c i a l T o p i c s Concern ing t h e LLD and R a d i o a c t i v i t y

1 . The B l a n k , Blank E q u i v a l e n t A c t i v i t y ( B E A ) , and Regions of V a l i d i t y

The u l t i m a t e l i m i t of d e t e c t i o n f o r any n u c l e a r o r chemical measurement

p r o c e s s is governed by t h e s y s t e m a t i c and random u n c e r t a i n t y i n B.

read: background, b l a n k , i n t e r f e r e n c e , model error b i a s , e t c . ) For t h i s

( F o r B,

r e a s o n BEA s h o u l d be r e c o g n i z e d as an i m p o r t a n t benchmark i n c o n s i d e r a t i o n s of

d e t e c t i o n c a p a b i l i t i e s . Some u s e f u l p e r s p e c t i v e o n t h e n a t u r e and i m p o r t a n c e

of B - v a r i a t i o n s is of fe red i n t h e f o l l o w i n g three p a r a g r a p h s ( a d a p t e d from

Ref. 38.)

f f U n f o r t u n a t e l y , there is no a l t e r n a t i v e t o e x t r e m e v i g i l e n c e when

t r e a t i n g the l i m i t a t i o n s imposed by the b l a n k . I n the bes t of c i r c u m s t a n c e s

t h e mean v a l u e of t h e b l a n k might be e x p e c t e d t o be c o n s t a n t and i ts

40

f l u c t u a t i o n s ( f l n o i s e f l ) n o r m a l l y d i s t r i b u t e d . Given a n a d e q u a t e number of

o b s e r v a t i o n s , o n e c o u l d estimate t h e s t a n d a r d d e v i a t i o n of t h i s n o i s e and

therefore se t d e t e c t i o n limits and p r e c i s i o n s for trace s i g n a l s . I n s i t u a -

t i o n s where t h e chemical ( a n a l y t e ) b l a n k r e m a i n s small compared t o t h e

i n s t r u m e n t a l n o i s e b l a n k t h i s p r o c e d u r e may be v a l i d , a s i n many l o w - l e v e l 1

I

I

c o u n t i n g e x p e r i m e n t s . Even here, however , t o assume t h a t t h e n o i s e is nor-

m a l l y or P o i s s o n d i s t r i b u t e d , o r t o estimate t h e background from o n e or two

o b s e r v a t i o n s is t o i n v i t e d e c e p t i o n . A s i n d i c a t e d i n T a b l e 4 , there is a

s i g n i f i c a n t c h a n c e (5% f o r n o r m a l l y - d i s t r i b u t e d b l a n k s ) t h a t t he e x p e c t e d

v a l u e of t h e n o i s e ( b l a n k s t a n d a r d d e v i a t i o n ) w i l l exceed t he o b s e r v e d d i f -

f e r e n c e between two b l a n k s by a factor of 16! S u b t l e p e r t u r b a t i o n s ar ise

e v e n i n t h e i n s t r u m e n t a l b l a n k s i t u a t i o n . For example , i f t h e a n a l y t e detec-

t i o n e f f i c i e n c y c h a n g e s d i s c r e t e l y or even f l u c t u a t e s , i t is q u i t e p o s s i b l e

t h a t t h e i n s t r u m e n t a l b l a n k w i l l s u f f e r a d i s p r o p o r t i o n a t e change ( 7 7 ) .

C e r t a i n special cases o c c u r where the b l a n k c a n be r e l i a b l y estimated,

and therefore a d j u s t e d , i n d i r e c t l y . T h i s is t h e s i t u a t i o n : f o r f f o n - l i n e f f

c o i n c i d e n c e c a n c e l l a t i o n of t he cosmic- ray mu-meson component of the back- I

ground i n l o w - l e v e l r a d i o a c t i v i t y measurement (where there is n o t e v e n a

s tochast ic r e s i d u e from t h e a d j u s t m e n t process); f o r t h e a d j u s t m e n t of the

b a s e l i n e (due g e n e r a l l y t o m u l t i p l e i n t e r f e r i n g p r o c e s s e s ) i n t h e f i t t i n g of

spectra o r chromatograms; and f o r c o r r e c t i o n f o r i s o n u c l i d i c c o n t a m i n a t i o n

( d u e t o i n t e r f e r i n g n u c l e a r r e a c t i o n s ) i n h i g h s e n s i t i v i t y n u c l e a r a c t i v a t i o n

a n a l y s i s .

When the b l a n k is due t o c o n t a m i n a t i o n (as opposed t o i n t e r f e r e n c e s or

i n s t r u m e n t a l b a c k g r o u n d ) , h igh q u a l i t y trace a n a l y s i s is a t i ts greatest

r i s k . Assumptions of c o n s t a n c y , n o r m a l i t y or even randomness are n o t t o be

t r u s t e d . An a p p a r e n t a n a l y t e s i g n a l may be almost e n t i r e l y d u e t o

41

Contamina t ion ( 7 8 ) ; and b l a n k c o r r e c t i o n m u s t take i n t o a c c o u n t i ts p o i n t ( s )

of i n t r o d u c t i o n and s u b s e q u e n t a n a l y t e r e c o v e r i e s . The randomness a s s u m p t i o n

may be i n a p p r o p r i a t e because t h e b l a n k may depend upon the s p e c i f i c h i s t o r y

of t h e s a m p l e , c o n t a i n e r o r r e a g e n t s (35) . Also when p r o c e d u r e s are a p p l i e d

t o rea l sample matrices as opposed t o p u r e s o l u t i o n s b l a n k problems abound,

as was o b s e r v e d , f o r example , i n t h e a n a l y s i s o f Pb ( a t a c o n c e n t r a t i o n of 30

n g / g ) i n p o r c i n e b lood i n c o n t r a s t t o aqueous s o l u t i o n s ( 9 3 ) . ( R e f e r e n c e 93

is also commended t o t h e reader f o r a more comple t e t r e a t m e n t o f t h e b l a n k i n

t race a n a l y s i s . ) The most s e v e r e t e s t of t h i s s o r t comes when f l b l i n d f f b l a n k s

t o g e t h e r wi th s a m p l e s a t o r n e a r t h e d e t e c t i o n l i m i t , a l l i n a c t u a l s ample

matrices, are s u b m i t t e d f o r a n a l y s i s . H o r w i t z , f o r example , r e f e r r i n g t o

c o l l a b o r a t i v e tests o f llunknownsll f o r 2378-TCDD i n p u r e s o l u t i o n s , beef f a t ,

and human m i l k , n o t e d t h a t s i g n i f i c a n t numbers of f a l se n e g a t i v e s began t o

appear when c o n c e n t r a t i o n s were l e s s t h a n 9 x ( p g / g ) , and t h a t f a l s e

p o s i t i v e s i n c r e a s e d f rom 19% f o r b l a n k f l s t a n d a r d s l f t o o v e r 90% f o r human m i l k

s a m p l e s (94 1 ! If

~~ ~~

Table 4. The Blank

Direct O b s e r v a t i o n - C r u c i a l f o r D e t e c t i o n L i m i t

Adequate No. of Measurements Needed: With b u t two, OB may be 16 times t h e d i f f e r e n c e

E f f i c i e n c y C o r r e c t i o n May Differ Between Blank and A n a l y t e (Scales , 1963) [Ref. 771

Y i e l d C o r r e c t i o n s Must Recognize P o i n t ( s ) o f I n t r o d u c t i o n o f Blank ( P a t t e r s o n , 1976) [Ref. 781

M u l t i s o u r c e B lanks G e n e r a t e S t r a n g e P r o b a b i l i t y D i s t r i b u t i o n s - Shape and Parameters I m p o r t a n t ( K i n g s t o n , 1983) [Ref. 951

P o i s s o n H y p o t h e s i s Must be Tes ted f o r Coun t ing Background and Blank

42

I n r e l a t i v e l y c o n t r o l l e d e n v i r o n m e n t s , e s p e c i a l l y i f I3 is n o t a n

e x c e s s i v e n m b e r o f c o u n t s , t h e P o i s s o n a s sumpt ion (02 = B ) may b e r e a s o n -

a b l y v a l i d . The p o s s i b i l i t y o f a d d i t i o n a l s y s t e m a t i c and random e r r o r

Components s h o ? i l d n e v e r be d i s m i s s e d , however ; and i t is recommended t h a t

b o t h t y p e s of non-Poisson B-e r ro r be mon i to red v i a i n t e r n a l as wel l as

e x t e r n a l q u a l i t y c o n t r o l p r o c e d u r e s . I t h a s a l r e a d y been shown t h a t s u c h

a

-

I

c o n t r o l is n o t e a s y -- i . e . , i n F i g . 2 (and Ref. 38) i t was shown t h a t more

t h a n 10 and n e a r l y 50 o b s e r v a t i o n s are r e q u i r e d j u s t t o d e t e c t s y s t e m a t i c o r

a d d i t i o n a l random e r r o r , r e s p e c t i v e l y e q u a l i n magni tude t o t h e P o i s s o n

component . The a l t e r n a t i v e of s u b s t i t u t i n g s2 f o r t h e P o i s s o n estimate f o r

t h e a s s e s s m e n t SCand X D h a s some merit; b u t , f o r a n*mber o f r e a s o n s w e

recommend r is ing i t (s2) ra ther as a meas'ire of c o n t r o l . [See n o t e s A I and

A2.1 What h a s been recommended ( p r e c e d i n g s e c t i o n ) t o c o v e r t h e p o s s i b i l i t y

o f non-Poisson e r ror is p r o v i s i o n of a r e l a t i v e s y s t e m a t i c - e r r o r bound 4 ~ .

B

a

I n l e s s - c o n t r o l l e d e n v i r o n m e n t s , r a t h e r s e v e r e e x c u r s i o n s i n B and i n i t s

v a r i a b i l i t y may take p l a c e . I f B comes from c o n t a m i n a t i o n i n sampl ing and

a n a l y s i s ( r e a g e n t ) , i t s d i s t r i b 2 t i o n f u n c t i o n -- which is c r u c i a l f o r

e s t i m a t i n g d e t e c t i o n l i m i t s -- may be d e r i v e d from both normal ( o r

a p p r o x i m a t e l y c o n s t a n t f f o f f s e t T 7 ) and log-normal components ( R e f . 9 5 ) , i n

which case a l a r g e s u i t e of g e n u i n e b l a n k s is a p r e r e q u j s i t e t o xD estima-

t i o n . I n t h e w o r s t of c i r c u m s t a n c e s B f l u c t u a t i o n s may be w i l d and non-

random. I n t h i s case there is no s u b s t i t u t e f o r e x p e r i e n c e d , " e x p e r t

judgmentIT a s t o maximum n o n - s i g n i f i c a n t e x c u r s i o n s . (Modern s t a t i s t i c a l

t o o l s , s u c h as E x p l o r a t o r y Data A n a l y s i s (Ref. 9 6 ) would make s u p e r b

p a r t n e r s f o r " e x p e r t judgment" i n these cases.) F o r m a l l y , t h i s co>ild

c o r r e s p o n d t o s u b s t i t u t i o n of a s i t e - s p e c i f i c , r e a l i s t i c v a l u e of h ~ , i n

43

p l a c e of o s r s s g g e s t e d d e f a x l t va1:;e ( 0 . 0 5 ) .

r e l a t i v e l y s e v e r e f1:;ctuat i o n s migh t be e x p e c t e d wo:ild be c o n t i n s v d s

m o n i t o r s ( co : in t r a t e m e t e r s - a n a l o g o r d i g i t a l ) f o r e f f l s e n t n o b l e g a s e s .

One s i t : i a t i o n i n which s s c h

Model e r r o r , ssch as d e v i a t i o n s o f b a s e l i n e s from s i n g l e f m c t i o n a l

s h a p e s ( l i n e a r , q s a d r a t i c , ...I o r i n c o r r e c t components o r peak s h a p e s when

f i t t i n g complex m s l t i p l e t s o r s ? e c t r a , c o n s t i t s t e s anot i ;er s o u r c e of B - e r r o r .

H e r e , t h e rf3rf i n v o l v e d a c t s a l l y is i n t e r f e r e n c e , and t h e problem is t h a t h i g h

l e v e l s of i n t e r f e r - i n g a c t i v i t i e s can c a a s e ser ims d e v i a t i o n s from o a r

asssmed B ( e . g . , b a s e l i n e ) m c e r t a i n t i e s a n d , h e n c e , e s t i m a t e d d e t e c t i o n

l imits .

Some d i s c u s s i o n and i l l u s t r a t i o n of t h i s p o t e n t i a l l y complex issse is g i v e n

i n s e c t i o n 111 and Ref . 7 2 .

Our d e f a s l t val:;e 41 = 0.01 i s i n t e n d e d t o p r o v i d e some p r o t e c t i o n .

S e f o r e l e a v i n g t h e t o p i c o f t h e B lank , l e t :is c o n s i d e r some r e g i o n s of

v a l i d i t y i n r e l a t i o n t o 3 t y p e s o f e f f e c t s on t h e d e t e c t i o n l i m i t .

t h e s e have been n o t e d a l r e a d y :

d i s t r i b s t e d random e r r o r ( v i a - o o ) . (See E q . 1 5 . ) The t h i r d , of major

c o n c e r n i n ex t r eme l o w - l e v e l co , in t ing i s Po i s son e f f e c t , - v i z . P o i s s o n

d e v i a t i o n s from N o r m a l i t y . For " s i m p l e c o u n t i n g " ( g r o s s s i g n a l minus

backgrosnd) t h i s ( P o i s s o n e f f e c t ) a d d s a term 22 = 2.71 t o t h e p a r e n t h e t i c a l

q d a n t i t y i n t h e numera to r of E q . 15 .

Two of

s y s t e m a t i c e r r o r ( v i a - 4 ~ ) and norma l ly -

( F o r t h e l o w e s t l e v e l c o u n t i n g , where 3

= 0 , S a . 15 mxst be r e p l a c e d w i t h an e x a c t Po i s son t r e a t m e n t .

I I I .C . l . )

4~ = 3 .05 , we can d i r e c t l y compare t h e t h r e e t e r m s which d e l i m i t t h e

d e t e c t i o n of n e t s i g n a l s ( u n i t s : c o X n t s ) :

(See s e c t i o n -

Taking a. e q u a l t o 0 3 = JB f o r t h e lfwell-knowntf b l a n k c a s e , and

[ s y s t e m a t i c ] term-1: 2bBa = 0.10 a ( c o i m t s )

[ c o n v e n t i o n a l ] term-2: 3.29 a. = 3.29 Js ( c o u n t s )

[ P o i s s o n ] term-3: 2 2 = 2.71 ( eo un t s )

44

Two types of question interest us: ( 1 ) the cross-over points where each term

I

becomes predominant, and (2) the points (3-magnitudes) by which the

unconventional" terms-1 and -3 are negligible. For question ( l ) , we set

adjacent terms equai and solve for 3; f o r question (2) we define negligible

as 10% relative. The results:

term-1 < term-2 for 3 < 1082 counts

terv3 < term-2 for 3 > 0.68 counts'

Thus, the conventional, aparoxiaately Normal Poisson expression (term-2)

predoninates for roughly 1 to 1000 background counts observed. (For

interference, substituting 41 = 0.01 for 43, the u ? w limit is

increased to about 27,000 counts.

Terms-1 and -3 are not so easily ignored, however. The systematic error

term-1 exceeds 105 of term-2, for 3 > 10.8 counts; and the extra Poisson

term-3 exceeds 10% of term-2 for 3 < 67.6 counts. Thus, Eq's (15) and (18)

were recommended for use when 3 5 70 counts.

apply strictly to the very common simple-counting, well-known blank case.

Somewhat altered values come about when x is estimated from single or

multicomponent least squares deconvolution.) (See also note B9 for a

discussi-on of the a?proximation 03 = 6 . 1

(The above regicns of validity

2 . Deduction of Signal Detection Limits for Specific Comting Techniques

The concentration detection limit XD or LLD can be expressed as (see Eq's

( 1 3 ) and (15)

XD = const. SEA + const' So/(YEVT) D

( 1 9 )

--------------- 'It is interesting to consider the exact ?oisson treatment in this case. Using Table T in section III.C.l we calculate a detection limit (SD) of 5.63 counts, whereas the sum of terms-2 and -3 gives 5.42 counts.

45

where t h e f i r s t term r e l a t e s p u r e l y t o s y s t e m a t i c u n c e r t a i n t y ( e r r o r bounds )

and b o t h c o n s t a n t s i n c l u d e t h e c a l i b r a t i o n s y s t e m a t i c e r r o r f a c t o r f . So D

i s t h e s i g n a l d e t e c t i o n l i m i t t a k i n g i n t o a c c o u n t random e r r o r o n l y . Apar t

f rom BE.4, t h e LLD i s c o n t r o l l e d by t h e n a t u r e of t h e c o u n t i n g p r o c e s s

( i n c l u d i n g t h e d a t a r e d u c t i o n a l g o r i t h m ) a s r e f l e c t e d i n t h e random e r r o r -

c o n t r o l l e d q u a n t i t y So and t h e c a l i b r a t i o n f a c t o r s Y,E,V,T. I n t h i s

s u b s e c t i o n we s h a l l c o n s i d e r t h e dependence of t h e a l l - i m p o r t a n t q u a n t i t y D

So on t h e n a t u r e of t h e c o u n t i n g p r o c e s s . The c a l i b r a t i o n f a c t o r s w i l l be

d i s c u s s e d i n t h e f o l l o w i n g s u b s e c t i o n on d e s i g n . D

S i g n a l d e c i s i o n ( c r i t i c a l ) l e v e l s and d e t e c t i o n l imits were g i v e n i n E q ' s

So = Z I - ~ oo = 1 .645 o0 ( 1 1 C

( A p r ime has been p l a c e d on E q . ( 2 ) because we do n o t wish t o r e s t r i c t

o u r s e l v e s t o t h e a s sumpt ion t h a t o o = OD a t t h i s p o i n t . ) The c r u c i a l

q u a n t i t i e s g o v e r n i n g t h e s i g n a l d e t e c t i o n l i m i t a r e t h u s o o and OD -- t h e

s t a n d a r d d e v i a t i o n s of t h e es t imated n e t s i g n a l ( 5 ) when i t s t r u e v a l u e i s

S = 0 and S = So. These a re what we s h a l l r e l a t e t o t h e c o u n t i n g s y s t e m .

What f o l l o w s i s s imply a c o n c i s e summary f o r d i f f e r e n t s y s t e m s o f i m p o r t a n c e .

D e r i v a t i o n s and d e t a i l e d e x p o s i t i o n s a re t o be found i n s e c t i o n 1II.C. (No te

D

t h a t i n t h e r ema inde r of t h i s s e c t i o n , s i n c e we s h a l l refer s t r i c t l y t o t h e

random e r r o r component , we s h a l l omi t t h e s u p e r s c r i p t - z e r o on SC and SD

-- f o r e a s e o f p r e s e n t a t i o n . A l s o , o = B = 0.05, so z = 1.645.)

a ) Meaning of oo and OD. These q u a n t i t i e s a re c e n t r a l t o t h e e n t i r e

d i s c u s s i o n . Let u s t h e r e f o r e c o n s i d e r t h e i r d e f i n i t i o n s i n terms of t h e

o b s e r v a t i o n s ( g r o s s c o u n t s ) y1 and y2 , f o r " s i m p l e c o u n t i n g . f f

46

y l = S + 3 + e1 ( ~ r o s s signal) (20)

Y2 = j g =- e 2 (blank) (21 )

(In Eq 21, one can envisage y2 as the s ~ l m of b - masxements of the jlank, so y/ j eq;lals tihe avenage observed blank.)

The estimtec! net signal is

(The coefficients ci a re i3trod;lced for late? genenzlization.) Following the

ndles of error propagstion, and >s ing v = - ’ V -

(Equation (24) defines the coefficient q. )

when S = SD, V1 -

general case (e.g., - non-counting system, or systems where non-?oisson

variations dominate). Th~ls, for va?iance uhicl is relatively independent

3 which may 3 r may not differ from V3 in the most - % +

of signal anplitude, V1 = const = Vg, so VD = Vo. It follows, in this case

that

Sc, = 1.645 a. = 1.545 03 6 (25)

(Thus far we have said nothing about Poisson counting statistics. That will

follow shortly.)

47

First, an important generalization: If we consider a rather more

complicated measurement scheme (e.g., decay curve and/or Y-spectrum fitting

by linear least squares),

yi = C aijSj + Bi + ei (27)

the solution to Eq. (27) is of the form (see section III.C.3),

Sj = C cji yi (22')

or, denoting the component of interest as S i (or simply S) and the respective

coefficients as cli (or simply ci) we write

just like Eq. (22). Therefore,

2 VS = Z Ci Vi ( 2 3 ' )

just like Eq. ( 2 3 ) . Knowing the least squares coefficients (ci) and the

variances (Vi) of the observations (yi), we can proceed according to exactly

the sample principles developed for "simple counting." (Admittedly, non-

trivial issues aust be dealt with concerning Poisson statistics, identity and

amplitudes of interfering components (Sj for j f I ) , and possible serni-

empirical shape functions for fitting the baseline bi. Such complications

will be treated in part below and in part in section 1II.C.)

In any case, Eq's (25) and (26) are the most important results of this

introductory section. The signal detection limit is seen to be directly

p-oportional to the standard deviation of the blank, where the constant of

proportionality (for simple counting) is 3.29 fi. The dimensionless quantity

depends on the relative amount of effort (replicate measurements, counting

time) involved in estimating the mean value of the blank. The bounds for rl

are clearly 1 and 2 (taking b > l ) . -

48

1

b) Use of replication (s2) and Student's-t. We have an enormous

advantage but a subtle trap as a result of Poisson counting statistics. O B

and OD can be estimated directly from the respective number of gross counts.

The trap is that other sources of random error may be operating [Ref. 201.

One solution to this problem is to substitute t, SB for 1.645 OB in Eq.

(23), where t, is Student's-t (also ai the 1-a = 0.95 significance level) for

v-degrees of freedom. (v=b-l according to the convention of Eq. 21) s3 is I

and V2 at S = 0 and S = SD give a valid solutions:

(28)

the square root of the estimated blank variance, i.e.,

2 n (Bi - B12 S a = 1

1 n- 1

where, for our example, n = b.

We strongly recommend the routine calculation of SB -- as a control for the

anticipated Poisson value, fi. If non-Poisson Normal, random error

predominates and is wsll understood and in control, then it is appropriate to

adopt t,sg in place of 1.645 JB. Unless this is assured, blithe application

of t,sB could be foolhardy, for Eq. (28) will give a numerical value even if

the blank is non-Normal or not in control. Further, information which can be

deduced using Poisson statistics (e.g., from Eq's (22') and (23')) is

generally far more general and more precise than what can be derived from a

reasonable number of replicates. [For more on this topic, including the

analogue of Eq. (26) under replication, see notes Al, A2, and B2.l

c) Simple Counting -- Poisson Statistics. If there are at least several

blank counts expected ( B 3 51, substitution of the Poisson variances for V I

Sc = 1.645 JBfi'= 1.645 A (y) (25')

SD = Z* + 2 Sc = 2.71 + 2 Sc (26')

49

The constant 22 in Eq. (24') comes directly from ?oisson statistics and the '

fact that OD > a. [Ref. 53. Thus, it is evident that the detection limit

remains finite even with a zero blank.

d) Multicomponent Counting. When there are two or more mutually

interfering species, a. and O D are not so easily expressed. More detail on

these topics will be found in section III.C, but two of the results will be

highlighted here.

For two mutually intefering components, where a solution is given by

simultaneous equations or linear least squares, it can be shown that

Sc = 1.645 [n>13 (25")

SD Z 2 -k 2 SC [)J>1] (26")

where, now, B, n , and u depend on the specific set of equations defining the

observations in relation to the net signal of interest. ( l l S 1 t and r lB1r remain

useful and even meaningful labels for the components when there are only

two.) These more general relations show that a universal consequence of

Poisson statistics is the inequality: SD/SC > 2. Equality is approached,

however, for simple counting when B > 70 counts. -

For multiple interference, a closed (analytic) solution for SD cannot be

given. One must return to the original defintions, Eq's ( 1 ) and ( 2 ' ) , and

tentatively estimate the corresponding a's from the appropriate diagonal

elements of the inverse least-squares (variance-covariance) matrix. (Non-

linear fitting introduces some rather peculiar problems. See section 111.)

Fortunately, a limiting calculation for XD, which derives from non-

negatively (S>O), - can be made for any specific result (G, ax) of multi- component analysis.

upper bounds for the critical level and detection limit can be immediately

derived. (See note B4.)

Through the use of Inequality Relations (ox 2 o o , etc.)

50

.4 very s i g n i f i c a n t p o i n t w i t h r e s p e c t t o t h e s e more c o m p l i c a t e d ,

mult icomponent c a s e s is a l g o r i t h m dependence . ( S e e s e c t i o n 111.) That i s

t h e p a r t i c u l a r d a t a r e d u c t i o n a l g o r i t h m (model and c h a n n e l s used f o r peak and

b a s e l i n e e s t i m a t i o n ; assumed numSer and type of i n t e r f e r i n g s p e c i e s , e t c . )

d e t e r m i n e s oo and OD, and t h e r e f o r e t h e d e t e c t i o n 1 i R i t .

e ) Con t inuous M o n i t o r s . 3 0 t h a n a l o g and d i g i t a l n o n i t o r s a r e used f o r

c o n t i n u o u s m o n i t o r i n g i n n o c l e a r p l a n t s . As no ted a l r e a d y i n s e c t i o n I I . D . l , I

one m u s t be c a u t i o u s i n a p p l y i n g ? o i s s o n s t a t i s t i c s i n u n c o n t r o l l e d

env i ronmen t s . Some b a s i c i n f o r m a t i o n on t h e s t a t i s t i c s of such coun t r a t e

m e t e r s is g i v e n , however , i n Evans ( R e f . 74) and more r e c e n t p u b l i c a t i o n s

such a s Ref. 73. Some of t h i s h a s been covered a l s o i n s e c t i o n 111 of t h i s

r e p o r t . Two b a s i c l i m i t i n g r e l a t i o n s , f o r example , a r e :

(29) 3 0 8 ~ = R / t i f t >> T

O R 2 = R / ~ T i f t << T

where R r e f e r s t o coun t r a t e , - t t o t h e a v e r a g i n g t i m e , and - T t o t h e t ime

c o n s t a n t f o r an a n a l o g c i r c u i t . A p p l i c a t i o n s of t h e r e l a t i o n s f o r l ong- t e rm

( 3 0 )

(Eq. 29) and i n s t a n t a n e o u s ( E q . 3 0 ) measurements a r e t r e a t e d i.n s e c t i o n 111.

(See a l s o n o t e B7.)

f ) Extreme Low-Level Coun t ing . When t h e expec ted number of b l ank c o u n t s

f o r a sample measurement i s l e s s t h a n a b o u t 5 i t is a d v i s a b l e t o u s e t h e

e x a c t P o i s s o n d i s t r i b u t i o n f o r making d e t e c t i o n d e c i s i o n s and s e t t i n g

d e t e c t i o n limits.

f o r s i m p l e c o u n t i n g [ E q . 25’1 , t h i s g i v e s a r e a s o n a b l e a p p r o x i m a t i o n even

down t o E ( y 1 ) = 1 . c o u n t -- s e e s e c t i o n III.D.l.) Although t r e a t m e n t s have

been g iven where bo th g r o s s s i g n a l ( y 1 ) a n d b l ank ( y 2 ) o b s e r v a t i o n s c o n t a i n

(So l o n g as t h e c o n s t a n t term z2 i s kep t i n t h e e x p r e s s i o n

few i f any c o u n t s ( R e f . 35, 751, we recommend t h e MP be d e s i g n e d s o t h a t a

51

reasonably precise estimate be available for B. The expected number of blank

counts in the 'blank' experiment ( y 2 = bB), for example, should exceed 100,

if possible.

In that case, a simple reduced activity diagram (Fig. 7 ) can be used to

instantly determine SC and the detection limit (in units of BEA) [Ref. 191.

A complete treatment of this subject is given in section III.C.l.

3. Design and Optimization

We consider briefly the question of experiment (i.e., MP) design because

this is the very question one faces when attempting to alter the adjustable

experimental variables in order to meet RETS requirements. The task is to

bring about the condition,

XD XR (9)

Optimization differs from design (in general) in that we adjust the variables

to minimize XD rather than simply to satisfy the inequality, Eq. (9). Design

and optimization are literally multidimensional operations when one treats ,a

multicomponent system with interfering spectra and/or decay curves and the

possibility of different schemes of multiple time and multiple energy band

observation. It is well beyond the scope of this manual.

For rather simpler systems, however, we can consider design from the

perspective of Equations 15, 18, and 26'.

2.71 + 3.29 -+ 0.10 RBt XD =(2:2:YEV) ( T [l-e-t/Tl

That is,

52

E q . ( 3 2 ) has been c a s t , of course, t o highl ight the cont ro l lab le var iables:

Y , E , V and t . (Note t h a t T = l / A = mean l i f e . ) Since the e f f e c t s of these

var iables f a l l i n two categories we s h a l l t r e a t each of the two main f a c t o r s

i n E q . 32 separately.

a ) Proportionate Factors, YEV. XD decreases d i r e c t l y w i t h each of these

f a c t o r s , so a r e q u i s i t e proportionate decrease t o meet the prescribed LLD I

I

( i . e . , XR) can be achieved ( i n p r inc ip le ) by a corresponding reduction i n any

one of them or i n t h e i r product.

I

I

The fac tor most readi ly avai lable is - V, f o r t h i s is a measure of the

sample s i z e taken. I n c e r t a i n s i t u a t i o n s , i t may have reached an upper l i m i t

fo r various p r a c t i c a l reasons, the most common of which is the s i z e tha t the

nuclear detector can accomodate. I f the amount of sample (or disappearance

through rapid decay) is not l i m i t i n g , V may be e f f e c t i v e l y increased fur ther

through concentration and/or radiochemical separation. If such s t e p s a r e too

labor in tens ive , a l t e r n a t i v e approaches may be preferred. I n general ,

however, because of i ts c o n t r o l l a b i l i t y and the inverse proport ional i ty

between XD and V , t h i s quantity provides the g r e a t e s t leverage.

Y cannot exceed u n i t y . I n the absence of sample preparation s t e p s , i t is

not even a re levant var iable . The most important circumstarices a r i s e when Y

is qui te small; major improvements i n procedures hav ing poor recovery could

have some impact.

The detect ion eff ic iency - E is a complex f a c t o r . Changes possibly a t our

disposal include geometry, external or self-absorption (or quenching i n the

case of l i q u i d s c i n t i l l a t i o n counting), and the se lec t ion of nuclear p a r t i c l e

or Y-ray t o be measured.

53

Some e f f e c t s are d ic ta ted by N a t u r e , however. Most no tewor thy is the

decay scheme, e s p e c i a l l y b r a n c h i n g r a t io s (or Y-abundances, e t c . ) . Other

t h i n g s b e i n g e q u a l , the LLD a c h i e v a b l e -- i . e . , XD -- w i l l v a r y i n v e r s e l y

w i t h the par t ic le o r Y-abundance of the r a d i a t i o n b e i n g measured. If

n u c l i d e s h a v i n g low Y-abundances are t o a c h i e v e t he same L L D ' s as t h o s e w i t h

high a b u n d a n c e s , o t h e r factors w i l l have t o be a c c o r d i n g l y a d j u s t e d .

Note t h a t t he e f f e c t i v e d e t e c t i o n e f f i c i e n c y may depend a l so on the data

r e d u c t i o n a l g o r i t h m -- e . g . , f r a c t i o n o f a Y-spectrum used f o r r a d i o n u c l i d e

e s t i m a t i o n . More e f f i c i e n t n u m e r i c a l i n f o r m a t i o n e x t r a c t i o n schemes may t h u s

be b e n e f i c i a l .

b ) Background ( B l a n k ) Rate; Count ing T ime . I t is clear f rom the

numera to r of t h e second factor i n E q . 32 t h a t d e c r e a s i n g t h e background r a t e

w i l l decrease LLD up t o a p o i n t . If t is f i x e d ( s a y , t h e maximum feas ib l e )

t h e n once the f irst ( e x t r e m e P o i s s o n ) term C1 p r e d o m i n a t e s , f u r t h e r r e d u c t i o n

i n t h e background (or b l a n k or i n t e r f e r e n c e ) w i l l have l i t t l e effect . I n

c o n t r a s t i f B is so large t h a t t h e t h i r d ( s y s t e m a t i c - e r r o r ) term C 3 B predomi-

n a t e s , t h e n B - r e d u c t i o n s w i l l have as l a r g e a n effect as p r o p o r t i o n a t e

i n c r e a s e s i n V and E . I n s e c t i o n I I . D . l , w e saw ( f o r t y p i c a l MP p a r a m e t e r

v a l u e s ) t h a t t h e B - t r a n s i t i o n p o i n t s o c c u r r e d a t a b o u t 1 c o u n t and 1000

c o u n t s . Pe rhaps the most i m p o r t a n t o p p o r t u n i t y f o r €3-reduction o c c u r s when

i t is d u e t o large amounts o f i n t e r f e r i n g n u c l i d e s which c a n be e l i m i n a t e d by

d e c a y o r r a d i o c h e m i c a l s e p a r a t i o n .

A second q u a n t i t y a t o u r d i s p o s a l is q. This depends on t h e amount o f

time or c h a n n e l s ( for a s i m p l e p e a k ) used for e s t i m a t i n g B f o r s i m p l e

c o u n t i n g . I n more complex (mul t i componen t ) s i t u a t i o n s , t h e data r e d u c t i o n

algorithm (as embodied i n 11) w i l l have some ef-fect on XD.

54

I

The major and most commonly c o n s i d e r e d v a r i a b l e i s c o u n t i n g time. I t is

i n t e r e s t i n g here t o c o n s i d e r two e x t r e m e s f o r the f a c t o r i n t he d e n o m i n a t o r ,

( 1 - e - t / T ) . If t < T t h i s f a c t o r a t . CT r e p r e s e n t s t h e mean l i f e , t1/2/!~n21.

A t ' t h e o the r e x t r e m e ( t k ) i t a p p r o a c h e s a c o n s t a n t ( o n e ) . We c a n r e p r e s e n t

the s i t t i a t i o n i n two d i m e n s i o n s as follows:

-- I_-

Table 5. LLD ( X D ) V a r i a t i o n s w i t h B and t ( a )

a ) U n i t s f o r B are c o u n t s . b ) For 41 = 0 . 0 1 , t h e upper c r o s s i n g p o i n t c h a n g e s from -1000 t o -27000

T e q u a l s t h e mean l i f e ( t l / z / k n 2 ) .

c o u n t s .

Depending on which domain of B and t we are i n , i t is clear t h a t i n c r e a s e s i n

c o u n t i n g time may decrease X D , h a v e n o e f f e c t , o r a t worst i n c r e a s e X D .

Also, i t i s i n t e r e s t i n g t h a t i n t n e r e g i o n of e x t r e m e l y small B , a l l

i n c r e a s e s i n t w i l l be b e n e f i c i a l ; i n f a c t , t h e i n i t i a l v a r i a t i o n ( i f t < T )

w i l l be p r o p o r t i o n a t e . ( A d m i t t e d l y , f o r f i x e d RB, i n c r e a s e d t w i l l t e n d t o

move B o u t of t he e x t r e m e P o i s s o n r e g i o n . However, i f t h e e x p e c t e d v a l u e of

B is s i g n i f i c a n t l y smaller t h a n 1 c o u n t , i n c r e a s e s i n t c a n be o f major

a d v a n t a g e if o n e is measur ing l o n g - l i v e d a c t i v i t y . )

When B is a l r e a d y q u i t e l a r g e , i n c r e a s e i n t c a n o n l y make matters worse.

The i n t e r m e d i a t e r e g i o n is i n t r i g u i n g . Here ( l<B<1000 -- c o u n t s ) " c o n v e n t i o n a l t 1

c o u n t i n g s t a t i s t i c s p r e d o m i n a t e ; and f o r f i x e d RD, XD decreases w i t h

i n c r e a s e d c o u n t i n g time f o r l o n g - l i v e d a c t i v i t y b u t reverses i t s e l f f o r

55

short-lived activity. Obviously there mdst be an optimum. Differentiating

the appropriate term in Eq. (32) shows that optimum to be the solution of the

transcendental equation.

where t is in units of the mean life T. The solution to equation (33) gives

the optimum counting time as -1.8 times the half-life.

It is worthy of re-statement that (Eq. 32, Table 5):

o Knowing the time and B-domains, one can quickly scale XD according to

the expected variation with time.

o Diminishing returns for background reduction set in when the term C1

begins to dominate.

o Diminjshed returns for LLD (XD) reduction by extended counting set in

once (a) t > 1.8 t112 or (b) B >

counts for the default values taken for blank and baseline relative

systematic error bounds. (This latter statement is equivalent to

indjcating -2% and - 1 1 % of the BEA as minimum achievable bounds for XD.)

( z / ~ B ) ~ which equals 1082 and 27060

o A rapid graphical approach for experiment planning, for all 3 B-domains

can be given in the form of the "Reduced Activity Diagram." Space does

not permit an exposition on this topic, but one such diagram (for extreme

low-level counting) is included as Fig. 7. Other diagrams for higher

activity levels and including the effects of non-Poisson error may be

found in Ref's 62 and 80.

56

4 . Q u a l i t y

a ) Communicat ion. F r e e and a c c u r a t e exchange o f i n f o r m a t i o n i s one

c r u c i a l l i n k f o r a s s u r i n g the q u a l i t y of a n MP and t h e e v a l u a t i o n of t h e

consequen t d a t a . A few h i g h l i g h t s i n t h i s a r e a , r e l e v a n t t o L L D and RETS

f o l l o w .

[ i ] Mixed Nuc l ide Measurements . I n t e r p r e t a t i o n o f n o n - s p e c i f i c

r a d i o n u c l i d e measurements is seldom p o s s i b l e u n l e s s t h e a v e r a g e t e m p o r a l and

de tec tor r e s p o n s e s a re f i x e d . C a l i b r a t i o n s and measurements of g r o s s n u c l i d e

m i x t u r e s r e q u i r e c o n t r o l s on the r e l a t i v e amounts of n u c l i d e s h a v i n g

d i f f e r e n t h a l f - l i v e s and d i f f e r e n t de tec tor r e s p o n s e s f o r mean ingfu l

i n t e r p r e t a t i o n .

[ i i ] "Black boxesf1 and Automat ic Data Reduc t ion . One o f t h e d i s -

b e n e f i t s of au tomated da ta a c q u i s i t i o n and e v a l u a t i o n i s l ack of i n f o r m a t i o n

on s o u r c e code or d e t a i l e d a lgo r i thms employed, s p e c i f i c n u c l e a r p a r a m e t e r

v a l u e s s t o r e d , and a r t i f i c i a l t h r e s h o l d s and i n t e r n a l "data massaging"

r o u t i n e s . A number of s u r p r i s e s and b l u n d e r s c o u l d be p r e v e n t e d i f there

were a d e q u a t e open communicat ion i n t h i s area. One problem of h i d d e n

a l g o r i t h m s which can be e s p e c i a l l y t roub le some for d e t e c t i o n d e c i s i o n s and

l i m i t s ( as w e l l as for q u a n t i f i c a t i o n ) i s i n t e n t i o n a l ( b u t unknown t o t h e

u s e r ) a l g o r i t h m s w i t c h i n g . A p o t e n t i a l means of c o n t r o l for these k i n d s of

problems is the u s e of a r t i f i c i a l (known) r e f e r e n c e data s e t s as d i s t r i b u t e d

by t h e I A E A [Ref 811. ( F u r t h e r comments on t h i s are g i v e n be low.)

[ i i i ] R e p o r t i n g Wi thou t Loss of I n f o r m a t i o n . The f o l l o w i n g p a r a g r a p h s

and F i g u r e s are a d a p t e d from [Ref. 381.

" Q u a l i t y d a t a , p o o r l y r e p o r t e d , leads t o n e e d l e s s i n f o r m a t i o n l o s s . T h i s

is e s p e c i a l l y t r u e a t t h e t race l e v e l , where r e s u l t s f r e q u e n t l y hover a b o u t

t h e l i m j t o f d e t e c t i o n . I n p a r t i c u l a r , r e p o r t s of upper limits or " n o t

57

detected" can mask important information, make intercomparison impossible,

and even produce bias in an overall data set. An example is given in Fig. 4

which relates to a very difficult radioanalytic problem involving fission

products in seawater (97). In this example, only six of the fifteen results

could be compared and only eight could be used to calculate a mean. Since

negative estimates were concealed by llND" and t r < r r , the mean was necessarily

positively biased. (The true value T in this exercise was, in fact, essen-

tially zero; and the use of a robust estimator, the median [m] does give a

consistent estimate.) Although upper limits convey more information than

f l N D 1 l , authors choose conventions ranging from the (possibly negative)

estimated mean ( i ) plus one standard error to some sort of fixed "detection lirnit.If Such differences are manifest when one finds variable upper limits

from one laboratory but constant upper limits from another (98).

The solution to the trace level reporting dilemma is to record all

relevant information, including as a minimm: the number of observations

(when pertinent), the estimated value x (even if it is negative!) and its A

standard deviation, and meaningful bounds for systematic error. More

thorough treatments of this issue may be found in Eisenhart (99) and Fennel

and West ( 1 00) .I1

When information is not fully preserved for a set of marginally detected

results, distributional information and parameters may be recovered by

statistical techniques (probability plots; maximum-likelihood estimates)

which have been developed for Ifcensoredf1 data. [Ref. 48,69,82,91 I. By

7fcensoredf1 we mean that although numerical results of some of the data may

not be preserved, the number of such results is recorded. Though such

techniques permit the partial recovery of information from censored data

sets, they cannot fully compensate for such information loss.

58

3 n I

0.2 t . 0.1

1 2 5 10 20 50 100

DEGREES OF FREEDOM ( v )

DATA X k SE HISTOGRAM < . l o ND 1 80

<7.30 ND ; 70 <40.00 9.50 I 60 ~ 2 0 . 9 0 2 <26.80 50

2.20 2 0.30 ; 40 9.20+ 8.60 30

77.00 f 11 .OO I 0.382 0.23 f

84.002 7.00 I

I

t ;: 1 0 I

I -T-m-

1 7

b2

0 '

014 2 9 a O

I i ND,UNU[NUUU 14.1 I

0.442 0.06 I I I

Fig. 4a. X / & V S degrees of freedom. interval for x / 6 . sinale or multiDle parameter models, and they give a direct

.The curves enclose the 95% confidence They may be used for assessing the fit of

. . . . . . . . . v

indication of the precision of standard deviation estimates.

4b. Reporting deficiencies. International comparison of 95Zr-95Nb in sample SW-1-1 of seawater (pCi/kg). The symbols have the following meanings: T = true value, 2 = arithmetic mean (positive results), m = median (all results), and b2 = a ffdouble blunder" - i.e., inconsistent result 77 1 1 was originally reported as 24. upper limits, respectively.

N and U indicate not detected, and

59

So l o n g as t h e f u l l i n i t i a l data are recorded and accessible , however , i t

may of c o u r s e be r e a s o n a b l e t o p r o v i d e summary r e p o r t s f o r s p e c i a l p u r p o s e s

which e x c l u d e t a b u l a t i o n s of n o n - s i g n i f i c a n t C I S .

ei ther z e r o or t o LLD g u a r a n t e e s c o n f u s i o n and biased a v e r a g i n g . The

q u e s t i o n of au tomated i n s t r u m e n t a t i o n and data r e d u c t i o n may a g a i n be

i n v o l v e d here, i f t h e " b l a c k box" does the c e n s o r i n g ra ther t h a n the u s e r .

But t o s e t them a l l t o

b ) Moni to r ing ( c o n t r o l ) . Three classes of c o n t r o l are c o n s i d e r e d

i m p o r t a n t f o r r e l i a b l e d e t e c t i o n d e c i s i o n s and measurements i n t h e r e g i o n of

t h e LLD. A t the i n t e r n a l l e v e l i t is c r u c i a l t ha t b l a n k v a r i a b i l i t y be

mon i to red by p e r i o d i c measurements of r e p l i c a t e s ; s i m i l a r l y , complex f i t t i n g

a n d / o r i n t e r f e r e n c e ( b a s e l i n e ) r o u t i n e s need t o be r e g u l a r l y mon i to red by

g o o d n e s s - o f - f i t tests and r e s i d u a l a n a l y s i s . If s u c h t e s t s do n o t i n d i c a t e

c o n s i s t e n c y w i t h P o i s s o n c o u n t i n g s t a t i s t i c s , t h e s i m p l e s u b s t i t u t i o n of s2

or r n u t l i p l i c a t i o n by X2/v i n p l a c e of t h e P o i s s o n s t a n d a r d e r ror is n o t

g e n e r a l l y recommended. I t c o u l d mask a s sumpt ion or model error u n l e s s t h a t

p o s s i b i l i t y has been c a r e f u l l y r u l e d o u t [Ref. 631. R e s u l t i n g LLD estimates

c o u l d t h e r e b y be q u i t e i n e r ror .

R e f e r e n c e samples, i n t e r n a l and e x t e r n a l , b l i n d and known, a re c r u c i a l

for m a i n t a i n i n g a c c u r a c y and e x p o s i n g u n s u s p e c t e d MP problems. "B l ind

s p l i t s f t and t h e EPA Cross-Check samples t h u s s e r v e a v e r y i m p o r t a n t need .

The u t i l i t y of e x t e r n a l q u a l i t y c o n t r o l s amples is h i g h e s t , of c o u r s e , when

such samples resemble f l rea l l r samples as c l o s e l y as p o s s i b l e i n t h e i r n u c l e a r

and chemical p r o p e r t i e s , when t h e i r t r u e v a l u e s are known ( t o t h e

d i s t r i b u t o r s ) , and when t h e y are r e a l l y f l b l i n d T f from t h e p e r s p e c t i v e o f t he

l a b o r a t o r y w i s h i n g t o m a i n t a i n i t s q u a l i t y . I n c o n n e c t i o n w i t h t h e LLD i t

might r e a l l y be v a l u a b l e t o p u r p o s e l y mon i to r ( i n t e r n a l l y a n d / o r e x t e r n a l l y )

60

performance a t t h i s l e v e l -- i .e . , t o p r o v i d e b l i n d samples c o n t a i n i n g b l a n k s

and r a d i o n u c l i d e s i n t h e ne ighborhood of the p r e s c r i b e d L L D s .

A t h i r d c lass of c o n t r o l re la tes t o t h e da ta e v a l u a t i o n phase of t h e MP.

The p r e s u m p t i o n t h a t c o n t r o l is q u i t e u n n e c e s s a r y f o r t h i s s t e p was b e l i e d by

t h e I A E A Y-ray s p e c t r u m i n t e r c o m p a r i s o n s t u d y referred t o ea r l i e r . A summary

of t h e s t r u c t u r e and outcome of t h a t e x e r c i s e ( a d a p t e d from Ref. 38) fol lows.

"One o f t h e most r e v e a l i n g tes ts of Y-ray peak e v a l u a t i o n a l g o r i t h m s was

u n d e r t a k e n by t h e I n t e r n a t i o n a l Atomic Energy Agency ( I A E A ) i n 1977. I n t h i s

e x e r c i s e , some 200 p a r t i c i p a n t s i n c l u d i n g t h i s a u t h o r were i n v i t e d t o a p p l y

t h e i r methods for p e a k e s t i m a t i o n , d e t e c t i o n and r e s o l u t i o n t o a s i m u l a t e d

da ta s e t c o n s t r u c t e d by t h e I A E A . The bas i s f o r the da ta were a c t u a l G e ( L i )

Y-ray o b s e r v a t i o n s made a t h i g h p r e c i s i o n . Fol lowing t h i s , t h e

i n t e r c o m p a r i s o n o r g a n i z e r s s y s t e m a t i c a l l y al tered peak p o s i t i o n s and

i n t e n s i t i e s , added known r e p l i c a t e P o i s s o n random e r r o r s , created a s e t of

m a r g i n a l l y detectable p e a k s , and p r e p a r e d o n e s p e c t r u m c o m p r i s i n g n i n e

d o u b l e t s . The a d v a n t a g e was t h a t the " t r u t h was knownff ( t o t h e I A E A ) , s o the

e x e r c i s e p r o v i d e d a n a u t h e n t i c t e s t of p r e c i s i o n and a c c u r a c y of t he c r u c i a l

e v a l u a t i o n steD of t h e CMP.

" S t a n d a r d , d o u b l e t and p e a k . d e t e c t i o n s p e c t r a ( F i g . 5 ) were p r o v i d e d ;

F i g . 6 summarizes t he r e s u l t s (81,921. While most p a r t i c i p a n t s were a b l e t o

produce r e s u l t s for t h e s i x r e p l i c a t e s of 22 e a s i l y de t ec t ab le s i n g l e p e a k s ,

less t h a n ha l f of them p r o v i d e d r e l i ab le u n c e r t a i n t y estimates. Two- th i rds

of the p a r t i c i p a n t s at tacked t h e problem of d o u b l e t r e s o l u t i o n , b u t o n l y 23%

were ab le t o p r o v i d e a r e s u l t f o r t h e most d i f f i c u l t case. (Accuracy

a s s e s s m e n t f o r t h e d o u b l e t results was n o t even a t t e m p t e d by t h e I A E A b e c a u s e

of t h e u n r e l i a b i l i t y of p a r t i c i p a n t s ' u n c e r t a i n t y estimates!) O f s p e c i a l

61

200

10

- - -

I I I 1 1 1 I 1 1 1 1 I I I I I I 1 I I

CHANNEL NUMBER

Fig. 5. I A E A test spectrum for peak detection

62

Peaks

DATA EVALU AT10 N -4AEA y-RAY INTERCOM PARISON

[Parr, Houtermans, Schaerf - 1979) Participants Observations

22 - Singlets (m = 6)

9 - Doublets

22 - Subliminal

205/212

144121 2

192121 2

uncertainties: 41 O/O (none), + 17% (inaccurate)

most difficult (l:lO, 1 ch.) -49 results

correctly detected: 2 to 19 peaks false positives: 0 to 23 peaks best methods: visual (19), 2nd deriv. (1 8), cross correl. (1 7)

Fig. 6. Data evaluation - I A E A Y-ray intercomparison. Column two indicates the fraction of the participants reporting on the six replicates for 22 single peaks, 9 overlapping peaks, and 22 barely detectable peaks. Column three summarizes the results, showing (a) the percent of participants giving inadequate uncertainty estimates, ( b ) the number of results for the doublet having a 1 : l O peak ratio with a 1 channel separation, and (c) the results of the peak detection exercise.

63

impor t from t h e p o i n t of view of trace a n a l y s i s , however , was t h e outcome f o r

the peak d e t e c t i o n e x e r c i s e . The r e s u l t s were s u r p r i s i n g : of the 22

s u b l i m i n a l p e a k s , t h e number c o r r e c t l y detected ranged from 2 t o 19 . Most

p a r t i c i p a n t s r e p o r t e d a t most one s p u r i o u s peak , b u t l a r g e numbers o f f a l s e

p o s i t i v e s d i d o c c u r , r a n g i n g up t o 23! C o n s i d e r i n g t h e model ing and

c o m p u t a t i o n a l power a v a i l a b l e t o d a y , i t was most i n t e r e s t i n g t h a t t h e bes t

peak d e t e c t i o n per formance was g i v e n by t h e ' t r a i n e d e y e ' ( v i s u a l method)."

5. M u l t i p l e D e t e c t i o n D e c i s i o n s

I t f o l l o w s o b v i o u s l y t h a t i f a l l r a d i o n u c l i d e s measured a r e p r e s e n t

e i t he r n o t a t a l l (Ho) or a t t h e LLD ( H D ) and t h e errors ci and f3 are each se t

a t 5%, t h e n 5% of the d e t e c t i o n d e c i s i o n s w i l l b e wrong " i n t h e l o n g run . "

Thus , f o r example , i n a Y-ray s p e c t r u m c o n t a i n i n g - no r a d i o n u c l i d e s , i f one

were t o examine s a y 200 l o c a t i o n s f o r the p o s s i b l e p r e s e n c e of r a d i o n u c l i d e s ,

10 f a l se p o s i t i v e s (on t h e a v e r a g e ) would r e s u l t . T h i s ca r r ies some c u r i o u s

i m p l i c a t i o n s f o r any i n s t r u c t i o n s t o " r e p o r t any a c t i v i t y detected" --

e s p e c i a l l y i f one m u l t i p l i e s t h e I O f a l se p o s i t i v e s by t h e number of s p e c t r a

examined , f o r example i n a Q u a r t e r . (One may f i n d a n a p p a r e n t l y t i g h t e r

c o n s t r a i n t i n a p h r a s e s u c h as "detected and i d e n t i f i e d , " b u t t h i s would

r e q u i r e a second manual t o s t r u g g l e w i t h a r i g o r o u s meaning f o r t h e term

t l i d e n t i f i e d ' l i n such a c o n t e x t ! )

If t h e number of n u c l i d e s s o u g h t is res t r ic ted p u r e l y t o t h e l f p r i n c i p a l

r a d i o n u c l i d e s , " t h e s i t u a t i o n is a l te red n u m e r i c a l l y b u t n o t q u a l i t a t i v e l y .

If there were j u s t one sample pe r month and 10 n u c l i d e s s o u g h t i n each

s a m p l e , w e would e x p e c t a f t e r 1 year ( o r 12 s a m p l e s ) - 6 f a l se p o s i t i v e s ( if

the re were i n f ac t no a c t i v i t y ) .

64

I

S o l u t i o n s t o t h i s dilemma a re e i ther t o a c c e p t a n e r r o r r a t e of 5% f a l s e

p o s i t i v e o r f a l s e n e g a t i v e r e s u l t s , o r t o r e d e f i n e the g o a l s u c h t h a t there

is o n l y a 5% chance of g e t t i n g a s i n g l e f a l se p o s i t i v e g i v e n t h e e n t i r e s e t

o f measurements . ( T h i s seems t h e o n l y r a t i o n a l a l t e r n a t i v e when s c a n n i n g a

h i g h r e s o l u t i o n s p e c t r u m f o r t h e u n s u s p e c t e d t i n y p e a k s . ) The c r i t i c a l l e v e l

must be c o r r e s p o n d i n g l y i n c r e a s e d and w i t h i t , t h e d e t e c t i o n l i m i t . ( I f one

were t o ass:ime some p r i o r unequa l appor t ionmen t of t h e samples t o h y p o t h e s e s

H o and H D , t h e i n c r e a s e s i n S c and SD cos ld d i f f e r s x b s t a n t i a l l y from one

a n o t h e r , b u t we s h a l l n o t t r e a t t h i s case . )

To address t h i s matter e x p l i c i t l y , l e t ’2s assume t h a t N d e c i s i o n s ( e r g o ,

meas- i rements ) a re made a l l a t r i s k - l e v e l a‘ . The p r o b a b i l i t y t h a t none is

i n c o r r e c t c a n be g i v e n by t h e Binomial d i s t r i b * i t i o n :

The p r o b a b i l i t y t h a t PO d e c i s i o n is i n c o r r e c t is by d e f i n i t i o n 1-a, where a

i s t h e r i s k o r p r o b a b i l i t y t h a t 1 or more is i n c o r r e c t . Therefore, t h e a‘ we

need t o impose on each d e c i s i o n is

a‘ = 1 - ( l - a V = a/N (35 1

f o r small a. I f N=100, f o r example , and a r ema ins 0.05, t h e n

CY‘ = 1-(0.95)0*01 = 0.000513

I f Norma l i ty c o u l d be assumed s o f a r o u t on t h e t a i l o f t h e d i s t r i b , i t i o n ,

~ 1 ~ ~ ‘ = 3.27. T r e a t i n g B ’ i n t h e same way, we would c o n c l u d e t h a t d e c i s i o n

l e v e l s and d e t e c t i o n limits would each need t o be i n c r e a s e d by a b o u t a f a c t o r

of two (from 1.645).

A somewhat re la ted i s s u e i n v o l v i n g t h e q u e s t i o n of r e p o r t i n g non-

p r i n c i p a l r a d i o n u c l i d e s i f de tec ted is i l l ! i s t r a t ed by r e s u l t i b i n F i g . 3.

Here a n o b s e r v a t i o n b r i n g s t h e d e c i s i o n f fde tec tedTf and t h e act i ia l LLD ( X D ) is

below t h e p r e s c r i b e d LLD ( x R ) . (Also, as shown, i ts uppe r l i m i t as wel l l i e s

65

below xR.) What fol lows i s t h a t u n l e s s there is t r u l y z e r o a c t i v i t y i n a s e t

of s a m p l e s examined, t h a t t h e more s e n s i t i v e MP's (lower X D ' S ) w i l l "detect"

more r a d i o n u c l i d e s even though t h e y may be well below t h e p r e s c r i b e d LLD ( x R )

i f any .

66

~ 111. PROPOSED APPLICATION TO R A D I O L O G I C A L EFFLUENT T E C H N I C A L SPECIFICATIONS I \ - I

A . Lower L i m i t of D e t e c t i o n - Basic Formula t ion

1 . D e f i n i t i o n

The LLD is d e f i n e d , f o r p x p o s e s of these s p e c i f i c a t i o n s , as t h e smallest

c o n c e n t r a t i o n of r a d i o a c t i v e material i n a sample t h a t w i l l y i e l d a n e t

c o m t , above t h e rneaswement p r o c e s s (MP) b l a n k , t h a t w i l l be d e t e c t e d w i t h

a t l eas t 95% p r o b a b i l i t y w i t h no g r e a t e r t h a n a 5% p r o b a b i l i t y of f a l s e l y

conclLtding t h a t a b l ank o b s e r v a t i o n r e p r e s e n t s a l lreall l s i g n a l . l lBlankl l i n

t h i s c o n t e x t means ( t h e e f f e c t s o f ) e v e r y t h i n g a p a r t f rom t h e s i g n a l s o g g h t

-- i . e . , background, c o n t a m i n a t i o n , and a l l i n t e r f e r i n g r a d i o n x l i d e s .

For a p a r t i c u l a r measurement s y s t e m , which may inclLtde r a d i o c h e m i c a l

1 s e p a r a t i o n :

The Lower L i m i t of D e t e c t i o n is e x p r e s s e d i n terms of r a d i o a c t i v i t y

c o n c e n t r a t i o n (pCi pe r gram o r l i t e r [A3]) ; i t re fers t o t h e a p r i o r i [ A 4 1

d e t e c t i o n c a p a b i l i t y

The d e t e c t i o n d e c i s i o n is based on t h e obse rved n e t sigr?al 5 (a p o s t e r i o r i [ A Q ] ) i n compar ison t o t h e c r i t i c a l l e v e l ( c o u n t s ) :

S c = A + ~ 1 - ~ o ~ ( 2 )

where t h e l l s t a t i s t i c a l l l p a r t of t h e . d e f i n i t i o n s (when f = 1 , A = 0) s e t s t h e

f a l s e p o s i t i v e and fa l se n e g a t i v e r i s k s a t a and 8 , r e s p e c t i v e l y [AS].

Meanings of t h e symbols f o l l o w s . (See a l so App. A ) .

--I--I---------

IPar t s A and B of S e c t i o n I11 r e p r e s e n t proposed s g b s t i t u t e RETS p a g e s . Part A is t h e more comprehens ive , and i t i s f ramed i n a manner t h a t s h o u l d be a p p l i c a b l e t o most c o m t i n g s i t g a t i o n s . Par t B is o f f e r e d as a s i m p l i f i e d v e r s i o n , which w i l l s u f f i c e f o r l l s imple l l g r o s s s igna l -minas -backgromd measgrements .

67

- A is the overall calibration factor, transforming counts to pCi/g (or

pCi/L).

E is the overall counting efficiency, as counts per disintegration; it -

comprises factors for solid angle, absorption and scattering, detector

efficiency, branching ratios and even data reduction algorithm [A6, A71,

V is the sample size in units of mass or volume,

2.22 is the number of disintegrations per minute per picocurie [A3],

Y is the fractional radiochemical yield, when applicable.

a

-

-

is the Poisson standard deviation of the estimated net counts (5) when the true value of S equals zero (i.e., a blank). (The relation of a, to the

background or baseline depends upon the exact mode of data reduction [see

section III.C.31.)

~ 1 - ~ , ~ 1 - ~ = the critical values of the standard normal variate -- taking on

the value 1.645 for 5% risks (one-sided) of false positives (a) and false

negatives ( B ) , when single detection decisions are made. (Multiple detection

decisions require inflated values for Z I - ~ to prevent significant occurrence of

spurious peaks -- as in high resolution Y-ray spectroscopy.) When a, B risks

are equal, and systematic error negligible, the detection limit for net

counts, SD, is just twice S c . (Assumes the Normal Limit for Poisson counts.)

(When subscripts are omitted in the following text z will denote 20.95 =

1.645).

T = the effective counting tj.me, or decay function, to convert counts to -

initial counting rate (time "zero!': end of sampling) [A9]. It is numeri-

cally equal to - e-htt,)/h, where ta and tb are the i-nitial and final

times (of the measurement interval) and A , the decay constant. For ht<<l,

T+At = tb-ta. [Multicomponent decay curve analysis yields a more complicated

68

expression for T -- and generally ao /T , the standard deviation of the

estimated initial rate is given directly.] (T must have units of minutes,

for LLD to be expressed in pCi.1 [A3,A6,A7,A93.

f and A are propartionate and additive parameters which represent bounds - -

I

for systematic and non-Poisson random error. (The only totally acceptable

Alternative to thjs is complete replication of the entire measurement process

(including recalibration, e.g., for every sample measured) and making several

replicate measurements of the blank for each mixture of interfering nuclides

and counting time under consideration [AIO]. )

I f will be set equal to 1 . 1 , to make allowance for up to a 10% systematic -

error in the denominator A of Eq. ( 1 ) --- viz., in the estimate of the

product EVY [All]. [If there are large random variations in A then full

replication should be considered together with the use of (radioactivity

concentration) and a,.] Note that - A is equivalent to the slope of the

calibration curve. If the curve deviates from linearity (e.g., -- due to

saturation effects, algorithm deficiencies or changes with counting rate,

-

signal amplitude, etc.) a more complex model and expression for LLD may be

required.

A will be set equal to 5% of the blank counts plus 1 % of the total - interference counts (baseline minus blank) in order to give some protection

against non-Poisson random or systematic .error in the (assumed) magnitude of

the blank (Ref's 2 0 , 7 2 ) [All].

% is the detection limit expressed in terms of counts.

& (Eq. 2) is the critical number of counts for making the (a posteriori)

Detection Decision, with false positive risk-a.

69

LLD ( E q . 1 ) is the lower limit of detection (radioactivity - concentration), given the decision criterion of E q . 2 (and risk-a), where the

false negative risk (failing to deteqt a real signal) is B, [a priori]. The

symbol XD is used synonymously with LLD for later algebraic convenience.

SC is applied to the observed net signal (units are counts) [A12];

whereas LLD refers to the smallest observable (detectable) concentration

(units are disintegrations per unit time per unit volume or mass). LLD is

without meaning unless the decision rule (Sc) is defined and applied [A13].

Bounds for systematic error in the blank ( A , counts) and (relative)

systematic error in the proportionate calibration factor ( f ) are included to

prevent overly optimistic estimates of 5~ or LLD based on extended counting

times. Also, they take into account the possibility of systematic errors

arising from the common practice of assuming simple models for peak baselines

(linear or flat) and repeatedly using average values for blanks and calibra-

tion factors (Y,E). (Random calibration errors of course become systematic

unless the system is recalibrated for each sample.) Inclusion of A and f in

the equations for S c and LLD converts the probability statements into

inequalities. That is, a 5 0.05 and 6 6 0.05.

2. Tutorial Extensions and Notes

[AI]. Alternative Formulation in Terms of sQ. If the measurement

process (including counting time, nature and levels of all interfering

radionuclides, data reduction algorithm) is rigorously defined and under

control, then it would be appropriate to replace Z I - ~ O ~ in E q . (2) by tsO,

where t is Student's-t at the selected levels of signjficance (a, B ) accord-

ing to the number of degrees of freedom (df) accompanying the estimate so2 of

2 0 0 *

70

In this case, however, a small complication arises in calculating L L D ,

because SD (detection limit in terms of counts) is approximately 2too (for

a=B) not 2ts0. A conservative approach would be to uye the upper (95%) limit

for oo -- i.e. s o / K , where FL is the lower ( 5 % ) limit for the distribution,

X2/df.

validity of the Poisson assumption through replication. (Ref. 20) [A2].

-

The recommended procedure is to use zoo (Poisson) but to test the

I

CA21. Uncertainty in the Detection Limit. For reasonably well behaved - - - -

systems, the critical level (SC)-which tests net signals for statistical

significance can be fairly rigorously defined. (One need2 d controlled MP and

reliable functional and random error models.) The detection limit (radioac-

tivity, trace element concentration, ... 1, however, requires knowledge of

additional quantities which can only be estimated -- e.g., standard deviation Lhe blank, calibration factors, chemical recoveries, etc. Thus, although

Lhere exists a definite detection limit corresponding to the decision

criterion (SC or a) and the false negative error ( 6 = 0 . 0 5 ) , its exact

magnitude may be unknown because of systematic and/or random error in these

add i t i ondl factors . Two approaches may be taken to deal with-this problem: (a) give an

uncertainty interval for LLD, knowing that its true value (at B = 0.05)

probably lies somewhere wjthin (49) or (b) state Liie iupper limit of the uncer-

tain1,y interval as LLD, such that the false.negative risk becomes an inequal-

ity -- i.e., B 5 0.05. We prefer the latter procedure, because it provides a

definite and conservative bound. Also, this is in keeping with the spirit of

RETS, which simply requires that the actual LLD (XD) not exceed the

prescribed maximum (xR).

I

I 71

1

I

One v e r y i m p o r t a n t i l l u s t r a t i o n of t h i s matter ar ises i n c o n n e c t i o n w i t h

I

s i g n a l d e t e c t i o n limits based on r e p l i c a t i o n . If t h e estimated n e t s i g n a l

(when S=O) i s n o r m a l l y d i s t r i b u t e d and sampled n- t imes ( e . g . , - v i a p a i r e d

compar i sons of a p p r o p r i a t e l y selected b l a n k p a i r s ) , t h e c r i t i c a l l e v e l is

g i v e n by t n - l s / J i , where s is t h e s q u a r e r o o t of t h e estimated v a r i a n c e and

tn-1 is S t u d e n t ' s - t based on n-1 d e g r e e s o f freedom. The minimum de tec t ab le

s i g n a l i s g i v e n by t h e n o n - c e n t r a l - t times t h e t r u e (unknown) s t a n d a r d error .

T h i s is a p p r o x i m a t e l y 2 tn- l o/&.

t i o n :

d e t e c t i o n l i m i t ( B 2 0.05) would t h u s be

-

Bounds f o r o o b t a i n from t h e x2 d i s t r i b u -

( x 2 / n - l ) - 0 5 < 52/02 - < ( x2/n-1 ) -95 . The uppe r bound f o r t h e s i g n a l -

C ~ ~ ~ - ~ S / J ; ; I / [ ( X ~ / ~ - I ) .0511/2 ( 3 )

For example , suppose t h a t 10 r e p l i c a t e p a i r e d b l ank measurements were

made, y i e l d i n g a s t a n d a r d e r r o r (s/v'%) --- f o r t h e n e t s i g n a l ( B i - B j ) of 30 cpm.

Then t g = 1.83 ( f o r c1 = 0 .05) and RC = t q - S E = 54.9 cpm.

= 0.607, t h e uppe r bound fo r t h e d e t e c t i o n l i m i t would be h i g h e r by a f a c t o r

of 2/0.607, or RD = 181. cpm. ( 6 2 0.05) .

u n c e r t a i n t y f o r t h e s t a n d a r d e r r o r and hence RD ( B = 0 .05) is g i v e n by t h e

r a t i o of t h e uppe r and lower ( . 9 5 , . 05 ) bounds f o r e, i n t h i s case ( n = 1 0 )

e q u i v a l e n t t o a f a c t o r o f 2.26.

of 2.00 ( u p p e r l imit / lower l i m i t ) would r e q u i r e a t l ea s t 13 r e p l i c a t e s f o r

S i n c e

The t o t a l (90% CI) r e l a t i v e

To r e d u c e t h e u n c e r t a i n t y i n RD t o a fac tor

t h e e s t i m a t i o n of o. [See Table 6 and Note B2 .1

I € , r a t h e r t h a n p a i r e d r e p l i c a t e s , a s i n g l e sample measurement i s to be

compared w i t h t h e estimated b l a n k , and t h e l a t t e r i s d e r i v e d t h r o u g h

r e p l ica t i o n ,

Thus , t h e uppe r l i m i t f o r SD becomes

72

CA31. S.I. Units. The preferred (S.1.) m i t for radioactivity is the

Becquerel (Bq) which is defined as 1 disintegration/second (s-l). To express

LLD in units of Bq, the conversion factor 2.22 (dpm/pCi) in the denominator

of E q ( 1 ) would be replaced by 1. (dps/Bq) and the factor T wogld have >nits

of seconds.

[AII]. A priori (before the fact) and a posteriori (after the fact) refer

to the estimate 5 or fact in that it does not depend on the specific (random) outcome of the MP.

However, all parameters of the MP (including interference levels) must be

known or estimated before

lated. (Such parameters may be estimated from the resxlts of the MP, itself,

or they may be determined from a preliminary or "screeningff experiment with

the sample in question.)

or decision process as the LLD is before the

priori" valxes for SC or LLD (XD) can be calm-

[A5]. Poisson Limit. Equations (1 ) and (2) are valid only in the limit

of large nmbers of background or baseline counts. If fewer than -70 counts

are obtained, special formulations are required to take into account devia-

tions from normality. (See section III.C.l note B9, and Ref. 19). The

simple sm in E q ( 1 ) -- (z~-~+z~-B) -- is an approximation; strictly valid only when o ( S ) is constant.

level counting and fpr certain other measurement situations involving

artificial thresholds ( 7 6 ) .

This is a bad approximation for extreme low-

[AS]. Mixed Nuclides, Gross Comting. For mixed, non-resolved

radionuclides, where ffgrossll radiation measurements are made, the factors E

and T are meaningful only if the particular mix (relative ~ amounts and

energies or half-lives) is specified. Common agreement on the radionuclides

selected for efficiency calibration for lfgrossll counting is likewise

mandatory.

73

C A 7 1 . For multicomponent spectroscopy and decay curve analysis, the

factors E and/or T are generally subsumed into the (computer-generated)

expression for o o , where oo then has dimensions of disintegrations or

(initial) counting rate or radioactivity (pCi or Bq). Both factors may thus

depend upon the algorithm selected for data reduction -- i.e., the Itinforma-

tion utilization efficiency" (see section III.C.3).

[A8]. Formulation of the Basic Equations. The expressions given for LLD

and SC are perfectly general, with one exception CA51, and intended to avert

many pitfalls associated with errors in assmptions (non-Poisson random

error, model error, systematic error, non-Normality from non-linear estima-

tion) which can subvert the more familiar formglation. By formylating Eqls

( 1 ) and ( 2 ) in terms of ao, we are able to apply them to all facets of

radioactivity measurement, including the most intricate Y-ray spectrum

deconvolution algorithms.

Use of z l - a o o in place of tl-aso was a hard choice. I made it because

LLD (as opposed to SC) reqJires knowledge (or assumption) of o o , as was noted

in the discussion on replicate blanks CA21; and Y-spectrum algorithms, for

example, seldom are really applied to replicate baselines! Also, there is

serious danger in so being estimated at one activity and interference level

(and counting time!

to Poisson statistics. The formally simple approach of adding the term A to

and assumed equivalent [or a: l / f i ] for changes according

Eq ( 2 ) limits both misuse and ignorance of a tso formulation. [To my

knowledge, an all-encompassing rigorous solution to the problem (non-Poisson

random and systematic error effects on detection capabilities) does not

exist. 3

74

C A 9 1 . Time Factor. Obviomly, T cogld be factored into an initial decay

- e-htb)/A = eVAta(l - e-xAt)/A. correction and decay during counting:

I

I

I Explicit expressions will not be given for decay during sampling or for

mltistep counting schemes, became they depend xpon the exact design (and

input function) for the sampling or counting process.

C A 1 0 1 . Excess (Non-Poisson) Random Error. In place of a massive

replication stxdy (to replace A + zoo [Eq. 2 1 by ts,) one coxld asszme a

two-component variance model and fit the non-Poisson parameter for approxi-

mate estimates of detection limit variations with comting time and

interference level ( 2 0 ) . This could become crucial when B >> 1.

[ A 1 11. Systematic Error. - A and - f have been set at "reasonable" values

to represent the routine state of the art. These may be subject to more

careful evaluation by the NRC or specific estimation by the licensee. This

is crucial for instruments in mcontrolled environments where these "rea-

sonable" values may be too small; see footnote, p. 96. Similarly, if demon-

strated smaller bounds of, say, 2% B limits could be sgbstituted for the

defaglt bound of 5% B. A most important consequence of including reasonable

bounds for systematic error is that LLD cannot be arbitrarily decreased by

increasing T.

rA121. Multicomponent Analysis, A - Uncertainties. In cases of

multicomponent decay cxve analysis or (a, B , V > spectroscopy, SC may be

transformed to a critical level (decision level) for an initial rate or

activity due to spectrum or decay curve shape differences among the compo-

nents. Common factor transformations (Y,E,V) applied, with their uncer-

tainties, to SC would simply needlessly increase the detection limit. A s

75

shown in E q . (l), s x h common (calibration) factors and their mcertainties

mtlst, however, be inclLtded to calcxlate the value of the a priori performance

characteristic, LLD.

[A13]. Decisions and Reporting of Data. SC (or LLDl2.20) is x e d for

testing (a posteriori) each experimental resglt ( S ) for statistical signifi-

cance. If S > S C , the decision is Itdetected;lt otherwise, not. Regardless of

the otltcome of this process, the experimental result and its estimated mcer-

tainty should be recorded, even if it shoJld be a negative nmber. (Proper

averaging is otherwise impossible, except with certain techniqdes devised for

lightly ttcensoredTt [bgt not tttrmcatedtt] data [Ref. 21, pp 7-16fl.) The

decision otltcome, of cogrse, shoxld Se noted and for non-significant reslllts,

the actaal detection limit (for those particular samples) s h o J l d be given. If

desired, a second decision level of significance ssing 1.9 SC, may be

noted, in view of the effects of mxltiple decisions on ct and 3 . (See section

II.D.5 on the treatment of multiple detection decisions and the origin of the

coefficient 1.9.) Obviously, changes in S c (i.e., in Z I - ~ ) alter the

detection limit, because of the sum, ( z I - ~ + zl-~), in E q . (1).

[A14]. Variance of the Blank. Estimation of oo2 by sg = s g v is

completely valid only if the entire rigorossly defined, Yeasrement Process

can be replicated. This is rarely achievable if there are significant levels

of interference (BI), for BI will doubtless be unique for each sample. A

suggested alternative, therefore, - if the sg approach is to be applied, is to

estimate S& for the Blank (non-baseline) and to combine this (necessarily as

an approximation) with the Poisson C I ~ from the spectrm fitting.

caztion:

and lack-of-fit (model error), btlt it shosld never be x e d as an arbitrary

correction factor for the Poisson variance ( 6 1 , 6 3 1 .

One

x 2 i s appropriate to estimate bomds for non-Poisson variance ( 2 0 )

76

-

I

. C A 1 5 1 . sn v s XD and E r r o r P r o p a g a t i o n . The f o r m u l a t i o n g i v e n here i s

based on s i g n a l d e t e c t i o n ( s c , S D ) . T r a n s f o r m a t i o n t o a c o n c e n t r a t i o n

d e t e c t i o n l i m i t (XD which is t h e L L D ) i n v o l v e s u n c e r t a i n t i e s i n t h e e s t i n a t e d

d e n o m i n a t o r , A . I n t h i s r e p o r t , we do n o t "p ropaga te" s u c h u n c e r t a i n t i e s

d i r e c t l y , b u t r a t h e r u s e them t o e s t a b l i s h a c o r r e s p o n d i n g m c e r t a i n t y

i n t e r v a l f o r XD, g i v e n SD. If @A (RSD) is small, and eA random, t h e n

XD = S D / A has t h e same R S D ( @ A ) .

f o r XD c a n be d e r i v e d d i r e c t l y from t h e lower and : ipper bounds f o r A . We

take a c o n s e r v a t i v e p o s i t i o n , s e t t i n g LLD e q J a l t o t h e uppe r bound f o r XD.

T h i s c a n be f u r t h e r i n t e r p r e t e d as a dua l i sm: i . e . , LLD [Eq. 11 i s t h e uppe r

( 9 5 % ) l i m i t f o r XD, and B = 0.05; o r , LLD [ E q . 11 is XD, b u t B < 0.05 ( u p p e r

95% l i m i t f o r B ) . (Eq. ( 1 1 , where f = 1 . 1 , takes t h e r e l a t i v e u n c e r t a i n t y i n

- A t o be + l o % . ) S c , of c o ' i r s e , is u n a f f e c t e d by @ A . An a l t e r n a t i v e t r e a t m e n t

( fTx-basedl f ra ther "S-based") is g i v e n Ref. 7 6 , where X D i s estimated from

f u l l e r ror p r o p a g a t i o n , b u t where one is l e f t w i t h u n c e r t a i n t y i n t e r v a l s f o r

bo th a and B . The b e s t s o l u t i o n c l e a r l y is t o a l l b u t e l i m i n a t e @ A , b u t i n

any case i t s h o u l d be k e p t w i t h i n t h e bounds g i v e n by t h e d e f a u l t v a l u e of

factor f i f a t a l l p o s s i b l e .

-

I f @A 7 0 . 1 , t h e n t h e * i n c e r t a i n t y i n t e r v a l

-

-

-

C A 1 6 1 . C a l i b r a t i o n Factor (A) V a r i a t i o n s . I f , f o r a g i v e n measurement

p r o c e s s A a c t u a l l y v a r i e s -- e . g . , . i f y i e l d s o r e f f i c i e n c i e s , e t c . , f l u c t u a t e

a b o u t t h e i r mean v a l u e s f rom sample t o sample -- t h e n t h e L L D i tself v a r i e s .

If t h i s v a r i a t i o n i s s i g n i f i c a n t . ( i n a p r a c t i c a l s e n s e ) and a mean v a l u e i s

used f o r A , t h e n XD would b e s t b e d e s c r i b e d by a t o l e r a n c e i n t e r v a l f o r t h e

v a r y i n g p o p u l a t i o n sampled. F a r be t te r , i n t h i s case, is t h e u s e of d i r e c t

o r i n d i r e c t measu res for - A (o r i ts component f a c t o r s -- Y,E,V) f o r each

sample ; s u c h methods i n c l u d e i s o t o p e d i l u t i o n ( f o r Y) and i n t e r n a l and

-

-

77

e x t e r n a l e f f i c i e n c y c a l i b r a t i o n ( f o r E ) . Sampling e r r o r s , which can be v e r y

l a r g e i n d e e d , come under t h i s same t o p i c ; b u t f u r t h e r d i s c u s s i o n i s beyond

t h e scope o f t h i s r e p o r t .

5 . Proposed S i m p l i f i e d RETS Page f o r "Simpleff Count ing1

(See f o o t n o t e a t b e g i n n i n g of s e c t i o n 111.)

1 . The LLD is Def ined f o r p u r p o s e s of these s p e c i f i c a t i o n s , a s the smallest

c o n c e n t r a t i o n of r a d i o a c t i v e material i n a sample t h a t w i l l y i e l d a n e t

c o u n t , above s y s t e m b l a n k , t h a t w i l l be d e t e c t e d w i t h a t l e a s t 95% p r o b a b i l -

i t y w i t h no g r e a t e r t h a n a 5% p r o b a b i l i t y of f a l se ly c o n c l u d i n g t h a t a b l ank

o b s e r v a t i o n r e p r e s e n t s a l f real t l s i g n a l . "Blank" i n t h i s c o n t e x t means ( t h e

e f f e c t s o f ) e v e r y t h i n g apa r t from t h e s i g n a l s o x g h t -- i . e . , b a c k g r o m d ,

c o n t a m i n a t i o n , and a l l i n t - e r f e r i n g r a d i o n u c l i d e s . 2

For a p a r t i c u l a r measurement s y s t e m , which may i n c l u d e r a d i o c h e m i c a l

s e p a r a t i o n :

3.29 OB^ ( YEVT ) LLD E ( 0 . 1 1 ) BEA + ( 0 . 5 0 )

The above e q u a t i o n g i v e s a c o n s e r v a t i v e estimate f o r LLD ( i n p c i p e r a n i t

mass o r volume ( V ) ) , i n c l u d i n g bounds f o r r e l a t i v e s y s t e m a t i c e r r o r f o r t he

b l a n k o f 5%, f o r b a s e l i n e ( i n t e r f e r e n c e ) of 1 % , and f o r t h e c a l i b r a t i o n

q u a n t i t i e s (Y,E,V) of 10% [ B 5 ] .

u sed above ; f o r b a s e l i n e e r r o r , s u b s t i t u t e 41 as i n d i c a t e d under 'BEA'

( A 5% b l a n k systematic e r r o r b o m d [ 4 ~ 1 was

be low.) The f l s t a t i s t i c a l f T p a r t -- numera tor o f t h e second term is based on

--------------- l f f S i m p l e l f , a s used h e r e , means t h a t the n e t s i g n a l is estimated f rom j u s t two

o b s e r v a t i o n s ( n o t n e c e s s a r i l y of e q u a l times o r number o f c h a n n e l s ) . One o b s e r v a t i o n i n c l u d e s t he s i g n a l + b l a n k ( o r i n t e r f e r e n c e b a s e l i n e ) ; t h e o t h e r b e i n g a "PLlre" b l ank ( o r b a s e l i n e ) o b s e r v a t i o n . A l s o , t h e " e x p e c t e d f f ( a v e r a g e ) number of b l a n k c o u n t s m u s t exceed -70 c o u n t s , f o r Eq. 6 t o be a d e q u a t e l y v a l i d [ B S l .

b r a c k e t s - - e . g . , [BSI. 2 R e f e r e n c e s t o n o t e s which f o l l o w ( i n s e c t i o n I I I . B . 2 ) a re i n d i c a t e d i n

78

1

. 5% f a l s e p o s i t i v e and f a l s e n e g a t i v e risks and t h e s t a n d a r d d e v i a t i o n o f t h e

Dlank o r b a s e l i n e ( i n t e r f e r e n c e ) (03) i n : ;n i t s of co:;nts, f o r t h e sample

meas'xernect t ime a t . [See a l s o Eq. (71, pg. 81. 3

. .

?leanings of t h e o t h e r a x a n t i t i e s a r e :

3EP. = Flank S a s i v a l e n t .tictiv:ty (pCi /mass o r vo1:;ne). I f t h e b a s e l i n e

:,;nderneath a? i s o l a t e d Y-ray p e a k ) is l a r a e compared to t h e b l a n k ,

s : ; S s t i t s t e " 3 a s e l i n e " f o r "31ar!ktt i n t h e f i r s t t e r n ~f EC. (6), and : i se a

c o e f f i c i e n t D f 9.3220 ;n ? l z c e of 0 . 1 1 ,

'i = Radicchemica l r e c o v e r y

- = Clverall C o i n t i n g e f f i c i e n c y !co:;ntc/c:;sintegratior! 1352 1 - T

= e - h t 3 : 1 - e - h A t ) / h , t h e " e f f e c t i v e t ' co!;nting time !rr . in: ; tes) ; where i is

t h e decay " o n s t a n t , t, i s t h e t i r !e s;.?ce s z r n p l i ~ g , 2nd h t is t h e l e n g t h o f

ne coo::nting i n t e r v a l ;For- i t < < i , T = ~ t j [ 3 7 , .46, 49:

og = , m f o r Po i s son col int ing s t a t i s t i c s ( 3 e q i a l s t h e expec ted n:iirber of

3 l ank Dr s a s e l i n e co: ;nts) . DO n o t ::se ~ q . (6) : i n l e s s 3 7 70 c o m t s .

Note t h a t : ise of t h e obse rved n:;rnber o f S lank c o m t s , 3 i n p l a c e of t h e

- i n o b s e r v a b l e t r : i e va1:;e 2 i n t r o d : j c e s a r e l a t i v e m c e r t z i n t y ( 1 8 ) of <6%

( P o i s s o n ) i n t h e e s t i m a t e d OE, i f 2 > 7C co!;nts [393 .

..

r! = 1 + (k) gh, &ere At is t h e rneas:irement time f o r t h e s a m p l e , and

C t B i s t h e meas*;rernent t ime f3r t h e backgro*ind. T:?e 3 l r n e n s i o n s l e s s f a c t o r gA

t a k e s i n t o a c c o * m t p o s s i b l e inf1:;ences of cnanges i n t h e c a l i b r a t i o n f a c t o r A

on t h e b lank -- due t o b lank i n t e rac t ions / co r re l a t ;ons w i t h y i e l d , e f f i c i e n c y

o r s a n p l e vol7;me ( m a s s ) . G e n e r a l l y , g~ w i l l have t h e v a l , i e , ' i n i t y (77,78).

-

The D e t e c t i o n D e c i s i o n : ( a p o s t e r i o r i ) is made .;sing a s t h e c r i t i c a l

l e v e l LLDl2.20. Un les s such a va1:ie i s : i sed i n c o n j u n c t i o n wi th E q . (6), t h e

p r o b a b i l i s t i c meaning ( 5 % f a l s e - p o s i t t v e , n e g a t i v e - r i s k s ) is n o n - e x i s t e n t ( 5 ) !

73

2. T u t o r i a l E x t e n s i o n s and Notes

I [ B l ] S imple S p e c t r o s c o p y : Eq. (6) may be used w i t h i s o l a t e d a- o r Y-ray

peaks by s u b s t i t u t i n g : ( a ) b a s e l i n e h e i g h t ( c o u n t s unde r the s e l e c t e d sample

peak c h a n n e l s ) f o r B i n o r d e r t o c a l c u l a t e BEA and O B ; and ( b ) t he e x p r e s s i o n

( 1 + n l / n 2 ) f o r n , where nl = number o f peak c h a n n e l s t a k e n and n2 = t o t a l

number of c h a n n e l s u sed t o estimate the p u r e ( l i n e a r o r f l a t ) b a s e l i n e . (For

a l i n e a r b a s e l i n e , n2 s h o u l d be s y m m e t r i c a l l y d i s t r i b u t e d a b o u t t h e peak

i n t e g r a t i o n r e g i o n .

CB21. R e p l i c a t i o n : The v a r i a b i l i t y of t h e b l a n k s h o u l d a l w a y s be t e s t ed

by r e p l i c a t i o n , u s i n g s 2 and x2. (See a s o n o t e s - - A2, A 1 4 . ) If t h e

r e p l i c a t i o n - e s t i m a t e d s t a n d a r d d e v i a t i o n s i g n i f i c a n t l y e x c e e d s t he P o i s s o n

v a l u e (&), t h e c a u s e should be de te rmined . ' If e x c e s s v a r i a b i l i t y is random

and s tab le the f a c t o r s 3.29 OR i n Eq. (6) may be r e p l a c e d b y 2 t OUL as

d e f i n e d i n n o t e A 2 . - Some v a l u e s o f t and O U L / S ( b o t h a t ct = 0 .05) follow:

Table 6 . LLD E s t i m a t i o n by R e p l i c a t i o n : S t u d e n t ' s - t and ( o / s ) - Bounds v s Number of O b s e r v a t i o n s

no. o f reDlicates: 5 10 1 3 20 120 03

S t u d e n t ' s - t : 2.13 1.83 1 .78 1 .73 1.66 1.645 1.12 1.100 or , r , / s : 2.37 1.65 1.51 1.37

[B3]. S y s t e m a t i c E r r o r Bounds. The p r e s e n c e o f systematic e r r o r bounds

limits u n r e a l i s t i c r e d u c t i o n of t h e LLD t h rough e x t e n d e d c o u n t i n g . The

v a l u e s ( I % , 5% and 10% f o r b l a n k , b a s e l i n e and c a l i b r a t i o n f a c t o r s , r e sp . )

a r e b e l i e v e d r e a s o n a b l e [Ref. 7 2 1 , b u t i f d e m o n s t r a t e d l o w e r bounds a r e

a c h i e v e d , t h e y s h o u l d be a c c o r d i n g l y , s u b s t i t u t e d .

' A t least 1 3 r e p l i c a t e s are n e c e s s a r y t o " a s s u r e f f (90% c o n f i d e n c e ) t h a t - s be w i t h i n -50% of t h e t r u e o. [A21

80

c B 4 1 . Some I n e q u a l i t i e s f o r Rapid D e c i s i o n Making and LLD E s t i m a t i o n .

Equa t ion ( 6 ) can be w r i t t e n : '

LLD x = XD = 1 .I (2xc) = 1 . I (2CQBEA + 1.645 0x01) ( 7 )

where Xc, BEA and 0x0 have d imens ions of a c t i v i t y p e r u n i t mass o r volume.

I n t h e a b s e n c e of s y s t e m a t i c error bounds ; XD = ~ X C , 4+O, and l . l + l .

s t a n d a r d d e v i a t i o n of the estimated c o n c e n t r a t i o n when i ts t r u e v a l u e is

The

z e r o , is oxo which e q u a l s & /[2.22 ( Y E V T ) ] f o r ' fsimple'f c o u n t i n g .

One r e s u l t which is n o r m a l l y a v a i l a b l e f o l l o w i n g a l l r a d i o n u c l i d e

measurements is t h e estimate of t h e r a d i o a c t i v i t y c o n c e n t r a t i o n , ;, and i t s

P o i s s o n s t a n d a r d d e v i a t i o n ox. S i n c e ox t oxo n e c e s s a r i l y ( t h e e q u a l i t y

a p p l y i n g o n l y when x=O -- i .e . , a b l a n k ) ,

XC' = 4BEA + 1.645 O X L XC ( 8 )

and

x;>f = 1 . i (2 xc') L X D ( 9 )

w i t h these two i n e q u a l i t i e s , u s i n g t h e r e s u l t which is a v a i l a b l e w i t h e v e r y

e x p e r i m e n t ( a , ) , we can i n s t a n t l y c a l c u l a t e q u a n t i t i e s f o r c o n s e r v a t i v e u s e

f o r D e t e c t i o n D e c i s i o n s and for s e t t i n g a bound f o r LLD.

E q u a t i o n ( 8 ) s h o u l d be c o n s i d e r e d as a new ( q u i t e l e g i t i m a t e ) d e c i s i o n

t h r e s h o l d , f o r which a 5 0.05. S i m i l a r l y , u s i n g XC' f o r d e t e c t i o n d e c i s i o n s ,

X D I (Eq. 9 ) may be c o n s i d e r e d a d e t e c t i o n l i m i t f o r which B S 0.05.

l i t t l e more work, one c o u l d c a l c u l a t e the ( 6 = 0.05) LLD, which would be

(With a

--------------- 1 F o r conven ience of a l g e b r a i c s t a t e m e n t , XD w i l l be u s e d here t o symbolize t he

a c t u a l LLD. (See App. A . ) Also, when u n i t s a re c o n c e n t r a t i o n , f ' ~ O I 1 w i l l be t r a n s f o r m e d a c c o r d i n g l y : i . e . , o x o = oo/A, t h u s , oxo i s ox f o r x=O. -

81

smaller than xD1, using x c l . ) If, then, X D ) is less than the prescribed

regulatory value XR for LLD, the requirements will have been met; and actual

calculation of o o , and LLD using Eq ( l ) , would be unnecessary. Obviously,

this approach cannot be applied completely a priori, in the absence of any

experimental results. Operationally, however, it i s straightforward,

conservative, and satisfies the goals of RETS.

Limits for the ratios of x ~ ' / x ~ , which are necessarily the same for

xc1/xc, are readily given for simple counting.

counts (S) is not zero, then the quantity fi is replaced with where

r = S/B, the ratio of sample to blank counts (llreduced activityf1 [Ref. 193).

Thus, for S = B, for example, and n=1 (well-known blank), a. would be

increased by a factor of

ratio xb'/xD would likewise be f i , if there were no systematic error.

systematic error dominates (4BEA in E q . 81, then XD'/XD -1 showing no change.

If the true valxe of sample

= 0, and this woxld be reflected in ox. The

When

[B5]. Calibration Factor Variations. If there are large random varia-

tions in Y, E, or V, the full replication of x (radioactivity concentration)

and a x should be considered in place of the f-systematic error bound

approach.

[BS]. Branching Ratios (or absolxte radiation -- a, 6 , Y, eK, --

fractions) may be shown explicitly by factoring the efficiency. Thus, for

example, E = Eye&, where Ey represents the counting efficiency for a Y-ray

of the energy in question, and Ck represents the branching ratio for that

energy Y-ray from radionuclide-k. All else being equal, then LLD 0: 1/ck.

CB7-j. Continuous (Monitoring) Observations [See also footnote: p.511.

When a digital count rate meter is employed (Ref. 731, or when a trlongtr

average estimate with an analog rate meter is made, the standard deviation of

the background rate is unchanged -- i.e., OBIT = (for Xt<<l ) . When an

82

i . . - - "instantaneousff analog reaging is made, however, T + ~ T ( T = resolving time of

the circuit), so OB/T+ [Ref. 741. Changes in analog ratemeter

readings are governed by the instrumental time constant, just as they are in

exponential radioactivity growth and decay, by the nuclear time constant.

[B8]. Decisions and Reporting of Data. Sc (or LLD/2.20) is used for

testing each experimental (a posteriori) resalt ( $ ) for statistical signifi-

ance. Regardless of

the outcome of this process, the experimental resslt and its estimated uncer-

tainty should be recorded, even if it should be a negative namber. (Proper

averaging is otherwise impossible, except with certain techniques devised for

lightly "censored" [but not fftrancatedfl] data [Ref. 21, pp 7-16fI.) The

decision outcome, of course, shoxld be noted and for non-significant results,

the actual detection limit (for those particular samples) should be given. If

desired, a second level of significance, using 1.9 x SC, may be noted, in

view of the effects of multiple decisions on a and B . (See Section II.D.5 on

the treatment of mltiple detection decisions.)

If 5 > SC, the decision i s "detected"; otherwise, not.

CB91. Counts Required for Adequate Approximation of O B and sD. When B

is large, the approximations r

( i ) O B = JR and (11) SD = 2sy: = 22

become qaite acceptable. They are, in fact, asymptotically correct, just as

the Poisson distribution is asymptotically Normal. Regions of validity can

be set by requiring, for example, that each approximate expression deviate no

more than 10% from the correct expression.

For Case (I), where the observed number of counts is x e d as an estimate

for the Poisson parameter, we require:

0.90 - < $B / m< - 1.10

83

Taking the u p p e r l i m i t 6 = B + z1 -y &, we have

For ' 1 0 ' ( z = 1 ) , t h i s means B - > 22.7 c o u n t s ; f o r t h e '958 C I ' ( z = 1 . 9 6 ) , t he

l i m i t is B>87.1 - c o u n t s . A most i m p o r t a n t p o i n t is t h a t t h e B re fe r red t o is

t h a t a s s o c i a t e d w i t h t h e Blank e x p e r i m e n t , because t h a t is t h e s o u r c e o f t h e

estimate 6 .

b l a n k " / ( s i g n a l + b l a n k ) 1 , t he R S D o f 6 i s g i v e n by 1 /a. number o f c o u n t s bB is s t i l l ( z / 0 . 2 1 ) 2 , b u t B i t s e l f is reduced t o

( z l - ~ / o . 2 1 ) 2 / b [ b - > 1 3 .

Thus , if b = Atg/At e q u a l s t h e r a t i o of c o u n t i n g times [ "pure

The r e q u i s i t e

I f , f o r example , t h e b l ank is measured t w i c e a s long

a s t h e s a m p l e , t h e ' l a ' ( z = 1 ) l i m i t f o r a p p r o x i m a t i o n ( I ) is B - > 11.3 c o u n t s

( expec ted ) .

t ha t is ,

( 2 2 + 22 f i q ) / ( 2 z 6) 5 1 .10

t h i s r e d u c e s t o ( f o r Z I - ~ = Z I - B = 1 . 6 4 5 )

B > ( ~ z ) ~ / Q = ( 5 * 1 . 6 4 5 ) 2 / q = 6 7 . 6 / ~ Counts -

Taking t h e u s u a l limits f o r q , we have

B > 67.6 c o u n t s ( q = 1 , llwell-knownll b l a n k )

B > 33.8 c o u n t s ( n = 2 , " p a i r e d compar i son" )

-

- S i n c e q = 1 + l / b , t h i s second a p p r o x i m a t i o n (11) is t h e more s t r i n g e n t .

C. LLD f o r S p e c i f i c Types of Coun t ing

1 . Extreme Low-Level Coun t ing

When fewer t h a n -70 background o r b a s e l i n e c o u n t s (B) a r e o b s e r v e d , t h e

11SimplefI c o u n t i n g fo rmula f o r SD must have added t h e term z2 = 2.71 ( f o r

84

a=f3=0.05) t o a c c o u n t f o r minor d e v i a t i o n s of t he P o i s s o n d i s t r i b u t i o n from

Normal i ty . [Ref. 5.1 ( O b v i o u s l y , t h i s term may be r e t a i n e d f o r B > 7 0 , b u t

i t s c o n t r i b u t i o n is t h e n r e l a t i v e l y minor . )

When t h e mean ( e x p e c t e d ) number of background c o u n t s is fewer t h a n a b o u t

5 , s u c h as may o c c u r i n l o w - l e v e l a - c o u n t i n g , f u r t h e r c a u t i o n is n e c e s s a r y

b e c a u s e of t he ra ther l a r g e d e v i a t i o n s from Normal i ty . T h i s i s s u e has been

t reated i n some d e t a i l i n Ref ' s 1 9 and 75. The e x t r e m e case o c c u r s , of

c o u r s e , when B-0 where t h e a s y m p t o t i c f o r m u l a (SD = 3.29 f i ) would g i v e a I

d e t e c t i o n l i m i t ( c o u n t s ) of z e r o , and t h e i n t e r m e d i a t e f o r m u l a , 2.71. I n

f a c t , as w i l l b e shown below, t h e t r u e d e t e c t i o n l i m i t (a=B=0.05) , i n t he

case of n e g l i g i b l e background, is 3.00 c o u n t s . Though the i n t e r m e d i a t e

f o r m u l a is n o t so bad i n t h i s case ( w i t h i n -10% f o r S D ) , t h e a c c u r a c y f o r SC

and SD f l u c t u a t e s a s B i n c r e a s e s from z e r o t o -5 c o u n t s ; b u t above t h i s p o i n t

( B = 5 c o u n t s ) t h e d e v i a t i o n s are g e n e r a l l y w i t h i n 10% r e l a t i v e . (Note t h a t

t h e symbol B re fe rs t o t h e t r u e or e x p e c t e d v a l s e of t h e b l a n k ; 6 refers t o

a n e x p e r i m e n t a l estimate.)

For a c c u r a t e s e t t i n g of c r i t i c a l l e v e l s ( f o r d e t e c t i o n d e c i s i o n s ) and

d e t e c t i o n limits, when B < 5 c o u n t s , we therefore recommend u s i n g t h e e x a c t

P o i s s o n d i s t r i b u t i o n . I n t h e f o l l o w i n g t e x t w e s h a l l m e t h e development

g i v e n i n Ref. 1 9 and make e x p l i c i t m e of F i g . 1 from t h a t r e f e r e n c e -- which

a p p e a r s here as Fig . , 7. Before f u l l y d i s c u s s i n g t h e u s e of t h i s f i g u r e ,

l e t u s make some c r i t i c a l o b s e r v a t i o n s :

o The mean number of background c o u n t s is assmed known.. Such a n assump-

t i o n is both r e a s o n a b l e and n e c e s s a r y . I t is r e a s o n a b l e i n t h a t , e v e n

f o r t h e lowest l e v e l c o u n t i n g a r r a n g e m e n t s , long- te rm background measure-

ments s h o u l d be made y i e l d i n g , ' s a y , a t l ea s t 100 c o u n t s . (An RSD of 10%

is t r i v i a l i n t h e p r e s e n t c o n t e x t . ) The a s s u m p t i o n is more or l e s s

85

n e c e s s a r y , i n t h a t a r i g o r o u s d e t e c t i o n l i m i t c anno t be s ta ted f o r t he

d i f f e r e n c e between two estimated P o i s s o n v a r i a b l e s , a l t h o u g h r i g o r o u s

d e t e c t i o n d e c i s i o n s and r e l a t i v e limits can be g i v e n . (See r e f e r e n c e s

19, 36, and 75 f o r f u r t h e r d e t a i l s . )

F i g . 7 g i v e s t h e d e t e c t i o n limits i n u n i t s o f BEA (background e q i l i v a l e n t

a c t i v i t y ) as a f u n c t i o n of B. For r e l a t i v e l y small u n c e r t a i n t i e s i n B ,

one can deduce l i m i t i n g v a l u e s from the ct l rve.

The i n t e g e r s above the c u r v e e n v e l o p e i n d i c a t e t he c r i t i c a l number of

g r o s s c o u n t s ( y c = Sc + B).

e x p e c t e d v a l g e s are rea l numbers, t h e c r i t i c a l l e v e l f o r y ( y c ) as w e l l

as a l l o b s e r v e d g r o s s c o u n t s are n e c e s s a r i l y i n t e g e r s . )

The l l s awtoo th l t s t r u c t u r e o f t he e n v e l o p e r e f l e c t s t h e d i scre te ( d i g i t a l )

n a t u r e of t h e P o i s s o n d i s t r i b u t i o n . A conseqgence is t h a t t h e f a l se

(Though B and S and y -- i . e . , t r u e o r

p o s i t i v e r i s k becomes an i n e q u a l i t y -- i . e . , Q - < 0.05.

0.05, and t h e n i t is g r a d u a l l y decreases u n t i l t h e n e x t i n t e g e r s a t i s f i e s

t h e Q = 0.05 c o n d i t i o n .

The dashed c u r v e r e p r e s e n t s t h e l o c t l s of t he i n t e r m e d i a t e e x p r e s s i o n

(SD -- 2.71 + 1.645 6) .

I t is s e e n t h a t t he extreme l o w - l e v e l s i t u a t i o n g e n e r a l l y a p p l i e s t o t he

case where the P o i s s o n d e t e c t i o n l i m i t exceeds t h e BEA. I n f a c t , t h i s

A t e ach peak Q =

o c c u r s once B is less t h a n -16 c o u n t s . I t is recommended t h a t F i g . 7

be used f o r d e t e c t i o n d e c i s i o n s (SC + B = i n t e g e r s above t h e c u r v e

e n v e l o p e ) and estimated d e t e c t i o n limits ( o r d i n a t e = d e t e c t i o n l i m i t , i n

BEA u n i t s ) . I n a d d i t i o n , t h e f i g u r e can be u s e f u l f o r d e s i g n i n g ( p l a n -

n i n g ) t h e measurement p r o c e s s . For example , i f t h e BEA f o r a p a r t i c u l a r

I

n u c l i d e is 1 pCi/L and one wishes t o be able t o de t ec t 5 pCi /L , i t i s

clear t h a t t h e e x p e c t e d number o f background c o u n t s must be a t l eas t

86

102

10'

100

10-1

0.0 10-2 10-1 1 00 10' 102

MEAN BG - COUNTS

- 100 ? '" > 10 e 2 0 1 a n

n E 0.01

I-

W 2 0.1

W

Figtlre 7a.

7b.

0.6%

1.29'0

2.5% 5 O/O

0

- 100 ? 2 > 10 k 2 0 1 a a 0 0.1 n

k

W

3

W 0.01 i o 102 103 104 105 i o6 j'C

BGCOUNTS(B) C D BG COUNTS (B)

Reduced A c t i v i t y ( p ) v s Mean Background C o u n t s ( p ~ ) and Observed Gross Coun t s ( n ) . Each of the s o l i d c u r v e s r e w e s e n t s t he xppe r _. _ _ _ - . . _ _ l i m i t s ' f o r p v s p ~ , g i v e n n. The e n v e l o p e of the c u r v e s , c o n n e c t e d by a d o t t e d l i n e , r e p r e s e n t s t h e d e t e c t i o n l i m i t ( p ~ ) and c r i t i c a l c o u n t s (n,) as a f u n c t i o n of p ~ . ( a = 8 = 0.05)

Reduced a c t i v i t y c u r v e s . Contour p l o t s are p r e s e n t e d for r e d u c e d a c t i v i t y (S/B = p ) v e r s u s background c o u n t s ( B ) and c o u n t i n g p r e c i s i o n ( e ) . Part ( a > i n c l s d e s P o i s s o n e r r o r s o n l y ; p a r t ( b ) i n c o r p o r a t e s a d d i t i o n a l random e r r o r (0 .50% f o r c o u n t i n g e f f i c i e n c y , 1.0% f o r background v a r i a b i l i t y ) .

87

I -1.3. If the background ra te is, e .g . , 3 c o u n t s / h o u r , t h i s means a 26

min measurement is n e c e s s a r y (assuming the mean background r a t e t o be I

I r e a s o n a b l y w e l l known).

o A f u r t h e r u s e f o r F i g . 7 is the s e t t i n g o f the upper limits when y <

yc. Tha t is, t h e s e q u e n c e o f c u r v e s below t h e d e t e c t i o n l i m i t e n v e l o p e ,

which have i n t e g e r s less t h a n y c , r e p r e s e n t a l l p o s s i b l e outcomes when

a c t i v i t y is n o t detected. For example , i f B ( e x p e c t e d v a l a e ) = 1.0

c o u n t , yc = 3 ( so S c = 2.0) and the n o r m a l i z e d d e t e c t i o n l i m i t is 6.75

BEA. If an e x p e r i m e n t a l r e s u l t were y = 1 c o u n t , t h e second c u r v e

below ( labeled 11111> i n t e r s e c t s w i t h B = 1.0 and t h e o r d i n a t e a t the ( 5 % )

uppe r l i m i t o f 3.74 BEA.

o T a b l e 7 is offered a s an a l t e r n a t i v e t o F i g . 7. Again , t h e mean

background ra te is assumed well-known, and a - < 0.05 whi le 6 = 0.05. For

the case ear l ie r d i s c u s s e d ( B = 1 . 3 c o u n t s ) , w e see t h a t t h e n e t c r i t i c a l

number o f c o u n t s is 1 .7 [ i . e . , 3 - 1.31 where y c is n e c e s s a r i l y a n i n t e g e r ;

and the d e t e c t i o n l i m i t is 7.75 - 1.30 = 6.45 c o u n t s , which is indeed

- 5 BEA. (Though = 0.050, f o r t h i s p a r t i c u l a r case i t can be shown

t h a t a = 0.043. ) The i n t e r m e d i a t e f o r m u l a would have g i v e n 1.88 c o u n t s

(1.645 a) f o r SC and 6.46 c o u n t s f o r SD -- resu l t s t h a t are f o r t u i t o u s l y

c l o s e t o the c o r r e c t v a l u e s . (The f o r t u i t o u s n e s s becomes c lear when one

c a l c u l a t e s SC and SD f o r B = 2 . 0 , f o r example . )

I

88

Table 7 . Cr i t ica l L e v e l and D e t e c t i o n L i m i t s f o r Extreme Low-Level C o u n t i n g (Assumes B , known)

Background Counts Gross Counts B - Range Y_c. = SC + B YD = SD + B

( i n t e g e r 1

0 - 0.051 0.052 - 0.35 0.36 - 0.81 0.82 - 1.36 1 .37 - 1.96 1 .97 - 2.60 2.61 - 3.28 3.29 - 3.97 3.98 - 4.69 4.70 - 5.42

0 1 2 3 4 5 6 7 8 9

3.00 4.74 6.30 7.75 9.15

10.51 11 .84 13.15 14.43 15.71

2. R e d u c t i o n s of t h e G e n e r a l E q u a t i o n s .

For d i r e c t a p p l i c a t i o n of E q ' s ( 1 ) and (2) we take t h e

f o l l o w i n g p a r a m e t e r v a l u e s ,

f = 1.10 (10% Y E V - " c a l i b r a t i o n " systematic e r ror bound)

21-a = 21-8 = 1.645 ( 5 % fa l se p o s i t i v e and n e g a t i v e r i s k s )

A = AK + AI = 4 K BK + 41 B I = 0.05 BK + 0.01 B I

where: A K , A I r e p r e s e n t s y s t e m a t i c e r ror bounds ( c o u n t s ) from t h e b l a n k

and i n t e r f e r e n c e ( e . g . , non-blank component of a b a s e l i n e ) ,

r e s p e c t i v e l y . 1

~ K , I d e n o t e r e l a t i v e s y s t e m a t i c error bounds of t h e Blank ( c o u n t s ,

BK) and of t h e I n t e r f e r e n c e ( B I ) . 5% and 1 % v a l u e s a re t a k e n as

r e a s o n a b l e f o r r o u t i n e measurement , b u t these may be r e p l a c e d by

--------------- 'Note t h a t t h e Blank :and B a s e l i n e (non-blank p o r t i o n ) a re p r o p e r l y t rea ted a p a r t ( a ) b e c a u s e t h e Blank may c o n t r i b u t e d i r e c t l y t o a peak ( a , Y-ray) d u e t o c o n t a m i n a t i o n by the v e r y n u c l i d e s o u g h t , and ( b ) b e c a u s e of d i f f e r e n c e i n bo th t h e o r i g i n s o f t h e i r s y s t e m a t i c e r rors , and t h e i r ( e x t e r n a l ) v a r i a b i l i t y .

89

[Symbols without subscripts will denote summation, e.g.,

Thus, Eq. (1) takes the form,

1.1 ( 2 S C ) LLD = = XD

2.22 (YEV)T (10)

and

S c = 0.05 BK + 0.01 BI + 1.645 o0 (12)

where: BEA = Blank (or Interference) Equivalent Activity

i.e., BEA = B/[2.22 (YEVIT] = B/A (13)

From the above equations it is clear also that the critical level,

expressed in the same units as LLD, is just LLD/2.2. Use of this is equiva-

lent to applying SC to test net counts for significance; and the form of data

oxtput available may make it (LLD/2.2) more convenient to m e than SC. In the

------

absence of systematic calibration error, of course, this equals LLD/2.

3. Derivation and Application of Expressions for 0, -- The Poisson Standard Deviation of the Estimated Net Signal, Under the Null Hypothesis [Blank11

A. ffSimpleff counting (gross signal minus blank)

i) Derivation

When two (sets of) observations (y1,y2) are made, one of the sample and

one of the pure Blank (or Interference), we have

y1 = S + B + el (counts) [observed] (1 4)

-_------------- 11n the following text, A b , and B Will be used without subscripts, in order to simplify the presentation. The context will indicate whether the Blank (BK) or interference (BI) predominates. As noted elsewhere, if the number of background (or interference) counts exceeds -70, the normal approximation of (Poisson statistics) is adequate, and the relative uncertainty in estimating oo (or OB) will be less than 6%.

90

1

I

Y2 = bB + e2 ( c o m t s ) [obse rved 1

(where e l , e2 a re e r r o r terms) A

Then, S = y1 - y2/b

( 1 5 )

* The c p p r o x i m a t i o n ( t o be Ssed t h r o u g h o u t t h i s s e c t i o n ) i n v o l v e s t a k i n g y ( o r B ) , ra ther t h a n t h e expec ted v a l u e E ( y ) ( o r B), t o estimate the P o i s s o n v a r i a n c e CB91.

For t he n x l l h y p o t h e s i s (S = O ) ,

i . e . oo = OB&= fi, where rl = ( 1 + l / b )

The c r i t i c a l l e v e l Sc t h u s e q u a l s

Taking a = B , t h i s l eads t o ,

SD = z 2 + 2200 2sc

Since for o = 6 = 0 . 0 5 , z2 = ( 1 .64512 = 2.71,

( 1 9 )

SD = 2.71 + 3.29 ( coc ln t s ) ( 2 0 )

The f i r s t term is n o t c o m p l e t e l y n e g l i g i b l e i f B is small. For a p p r o x i -

mate n o r m a l i t y , B 7 9 coclnts (Ref. 1 9 ) ; b u t t o make t h e f irst term above

( 2 . 7 1 ) n e g l i g i b l e -- i . e . , l ess t h a n 10% of SD, we r e q u i r e a t l ea s t 67

c o u n t s , s i n c e T-I 2 1 . . [ B e l o w B = 5 t o 10 c o u n t s , t h e "ex t r eme Po i s son"

t e c h n i q u e s f o r d e t e c t i o n l i m i t s , d i s c u s s e d i n Ref. 19 and s e c t i o n I I I . C . l ,

s h o x l d be employed; and f o r 5 - - < B < 70 c o u n t s the f d l e q u a t i o n above s h o u l d

be used (See a l s o Ref. 3 6 . ) . 1

91

i i ) Two Special Cases

[I.] Gross Signal - blank

[RANDOM PART]

If the sample is measxed for time tl, yielding y1 comts; and the blank

for time t2, yielding y2 comts, then

A

B = ~ 2 / b

and 5 = y1 - y2/b = y1 - y2 (tl/t2)

(Note that if t2 - > tl, then the limits for T? are obvioasly, 1 and 2.)

This is to be compared with the critical nmber of cotlnts SC,

If S < SC, we concltlde ND; otherwise - D - - SD = 2.71 + 3.29 00

and,

or, xing Eq. ( 1 1 ) directly [last term divided by 1.11

(21 )

(22)

XD - (2.22)(YEVT) 2.22(YEVT)

where the first approximation comes from dropping the term 2.71 in the

numerator, and the second approximation comes from asing B for the mobser-

vable trxe valxe [Bg]. (Both approximations are adequate so long as B 5 70

counts, and t2 > tl . ) -

92

If t i is small compared t o t h e h a l f - l i f e , t h e n T - t l ( c a l l e d A t , e a r l i e r )

S i n c e B = R B t i , oo and Sc scale as t1112, and XD a t i l l 2 .

t 2 / t l o r fo r t 2 >> t l .I

(For f i x e d

When decay d u r i n g c o u n t i n g is n o t n e g l i g i b l e t h e n XD decreases l e s s

r a p i d l y w i t h i n c r e a s i n g t l ; and e v e n t a a l l y ( t l > > t 1 / 2 ) T assmes t h e form

e-Ata/A which i s i n d e p e n d e n t of A t ( i . e . t l ) , so XD a s y m p t o t i c a l l y i n c r e a s e s

a s t l 1 l 2 . [See s e c t i o n

II.D.3 on Des ign . ]

O b v i o x s l y , there is a n optimum (minimum LLD or X D ) .

[+SYSTEMATIC PART1

Eq's ( 1 1 ) a n d ( 1 2 ) i n c l u d e terms f o r s y s t e m a t i c error bounds f o r B (e, ABK and A B I ) , where fo r t h e Blank ( a l l t h a t ' s b e i n g c o n s i d e r e d he re ) , t h e

r e l a t i v e error 4 is t a k e n as 0.05 .

Sc = 0.05 BK + 1.645 oo

= 0.05 B +. 1.645J (r) t l + t 2 [ c o u n t s ]

and

- - 2.22( YEVT )

S i n c e t h e f i r s t term i n t h e numera tor v a r i e s more r a p i d l y w i t h B t h a n the

s e c o n d , t h e s y s t e m a t i c error bound w i l l p r e d o m i n a t e above a c , e r t a i n number o f \

Blank c o u n t s ;

93

BLt! = (sr ~1 = 1082.0 = 1082 (9) comts (25)

Again, for long-lived radion'xlides, (tl << t1/2), T = tl, and since B =

RBtl 9

The asymptotic constant value for XD is determined therefore by the Blank

rate, as indicated in the first term.

For tl >> t1/2, T + emhta/h = constant; so, from equation (24)

thgs, XD asymptotically increases with tl.

As stated elsewhere, the xse of systematic error bounds converts the

statistical risks into inequalities: a 5 0.05, B 5 0.05.

[REPLICATION]

Let LIS szppose that 11-observations were made of the Blank; all for the

same time, tl. (Otherwise, the simple replication model is invalid.) Then,

following the common estimation procedwe, C (Bi-B) 2

n 2 -

1 2 = y1 - g, where B = Z Bi/n, SB = og2 = -

n- 1

and

SI?(;) = s / f i ( 2 8 ) 2

is cy1 + sBO11/2

NOW, in place of zOB, we use tsg, so zoo -, t s B J K where now r, = (n+l >/n

because t2 has been replaced with C ti = noti. n

1 In the absence of systematic error, the critical number of counts is

given by

94

where ta,V = t . 0 5 , ~ - 1 is S t u d e n t ' s - t a t t h e 5% s i g n i f i c a n c e l e v e l w i t h n-1

d e g r e e s of freedom ( v ) ,

I

The i n e q u a l i t y g i v e s an t lpper l i m i t f o r XD, t a k i n g i n t o a c c o u n t t h e

m c e r t a i n t y of OB t h r o s g h t h e u s e of t h e x 2 . ( F I - ~ , " , ~ is e q u a l t o x2/v f o r

I

I

v-degrees of f reedom a t t h e a t h p e r c e n t a g e p o i n t . )

An a l t e r n a t i v e t r e a t m e n t , where in a non-Poisson (o r l l ex t r aneo t l s l ' )

v a r i a n c e component is estimated and combined w i t h t h e P o i s s o n e s t i m a t e , 5,

I S c o r n o t ) . T h i s is v i t a l b o t h f o r u n b i a s e d a v e r a g i n g , and f o r t h e p o s s i b i l -

I

is d e s c r i b e d i n Ref. 20.

[ R E P O R T I N G ]

Recommendations fo r r e p o r t i n g t h e r e s u l t s f o l l o w i n g the above t e s t s : t h e

estimate

[ f & ( B E A ) ] , and t h e s t a n d a r d error 0 ~ / ( 2 . 2 2 Y E V T ) , s h o u l d a l l be r e c o r d e d

regardless of the outcome of t h e d e t e c t i o n t e s t fo r s i g n i f i c a n c e (whe the r 2 >

= ,?/(2.22 Y E V T ) , t h e estimated bound f o r s y s t e m a t i c error

i t y o f f u t u r e tests a t d i f f e r e n t l e v e l s of s i g n i f i c a n c e o r w i t h d i f f e r e n t

estimates o f s y s t e m a t i c e r ror . For "ND" restllts, t h e c o r r e s p o n d i n g estimate

of XD s h o x l d be p rov ided . For t h e sake of t lniform r e p o r t i n g p r a c t i c e and t o

a v o i d s t r a i n i n g t h e d i s t r i b s t i o n a l a s s x n p t i o n s ( P o i s s o n = Normal) one -

s t a n d a r d d e v i a t i o n ( n o t a m t l l t i p l e thereof ) shotlld be r e p o r t e d .

----___------__ IBecazse o f t h e l a r g e u n c e r t a i n t y i n t e r v a l f o r s / o mless v is v e r y l a r g e ,

t h e u s e of a n t lpper l i m i t f o r XD is p r e f e r r e d t o t h e s i m p l e s u b s t i t u t i o n o f s fo r OB i n t h e p r e v i o u s eq t l a t ion . [A21

95

[CONTINUOUS MEASUREMENT]

A l ong- t e rm ( t 1 , 2 > > ~ ) measurement w i t h a n a n a l o g c o u n t - r a t e meter or a

d i g i t a l c o u n t ra te meter measurement f o l l o w e s s e n t i a l l y t he same s t a t i s t i c s

as above .

For a n T 1 i n s t a n t a n e o u s l ' measurement w i t h a n a n a l o g meter, however , the

u n c e r t a i n t y i n t he r a t e is g i v e n by - o = / R / ( ~ T )

where T is the RC t i m e c o n s t a n t .

The p r o d u c t OB /;;/T i n Eq ( 1 B ) is t h e r e f o r e replaced by /RBr/e-Ata, where

now rl = =. 1 / ( 2 ~ ) (assuming t 2 >> T and t 1 / 2 >> T ) . Fo r a n -

1 1 i n s t a n t a n e o u s 7 T o b s e r v a t i o n o f a sample, we c o r r e s p o n d i n g l y f i n d :

( 3 2 ) Rgr oss-R B

; O R n e t Rnet =

e- Ata e- Ata

The c o r r e s p o n d i n g r a d i o a c t i v i t y c o n c e n t r a t i o n s are found by d i v i d i n g the

respective R ' s by ( 2 . 2 2 ) ( Y E V ) ; and t h e f a c t o r s 1 .645 and 3.29 are u s e d ,

r e s p e c t i v e l y , t o c a l c u l a t e c r i t i c a l l e v e l s and d e t e c t i o n limits (LLD). 1

A f u r t h e r c o m p l i c a t i o n wi th r a t e meters is t h e e q u i l i b r a t i o n time ( R C f o r

a n a l o g i n s t r u m e n t s ) which must be t a k e n i n t o c o n s i d e r a t i o n ( 7 4 ) .

---_I_---------

1The reader s h o u l d be a le r ted t o t h e f a c t t h a t a n i n s t r u m e n t i n a r e l a t i v e l y u n c o n t r o l l e d e n v i r o n m e n t , such as a c o u n t r a t e meter, may be s u b j e c t t o ra ther s i g n i f i c a n t non-Poisson "backgroundf1 v a r i a t i o n s . T h e r e f o r e , i t is u r g e n t t ha t t h e x2 t es t f o r background r e p r o d u c i b i l i t y be carried o u t , and i f non-Poisson random v a r i a b i l i t y is i m p l i e d , s2 s h o u l d be u s e d i n p l a c e o f t h e P o i s s o n v a r i a n c e estimate. (See t h e e a r l i e r s e c t i o n on the u s e o f S t u d e n t ' s - t w i t h r e p l i c a t i o n p r o c e d u r e s . )

Worse s t i l l , s u c h f l u c t u a t i o n s may be non-Normal or even non-random i n character. I n t h i s case a s y s t e m - s p e c i f i c estimate s h o u l d be made f o r t h e r e l a t i v e u n c e r t a i n t y bounds -- i . e . , 4 ~ . (One s h o u l d n o t s i m p l y a d o p t t h e " r e a s o n a b l e f f v a l u e of 58, s u g g e s t e d f o r c o n t r o l l e d env i ronmen t [well- s h i e l d e d ] c o u n t i n g systems.)

96

. ' [INSTRUMENTAL THRESHOLD 1

On o c c a s i o n , when there is t f s e n s i t i v i t y t o s p a r e " a f i x e d , p o s s i b l y

a r b i t r a r y t h r e s h o l d ( K ) w i l l be se t i n P l a c e of Sc.

number of c o u n t s i s t h e n g i v e n by:

The minimum detectable

SD = K + Z JSD + oo2

T h i s e q u a t i o n has the a p p r o x i m a t e s o l u t i o n ,

(33)

2 1 / 2 SD = K + z2 + Z [ K + 00 + z 2 / 2 ] ( 3 4 a )

o r , i f K >> 0 0 2 = B ~ ,

SD = K + 1.645 JK For s u c h a s o l u t i o n , a < < 0 . 0 5 , b u t B = 0.05. Also, s i n c e K is a f i x e d

number ( l i k e 103 c o u n t s , or i n x - u n i t s 30 pCi/g f o r e x a m p l e ) , SD is no l o n g e r

much i n f l u e n c e d by t h e s t a t i s t i c a l u n c e r t a i n t y i n B . On the o t h e r h a n d , the

d e t e c t i o n l i m i t is i n c r e a s e d by a n amount K o r more.

( 3 4 b )

C21 Simple S p e c t r o s c o p y

[ l i n e a r o r f l a t b a s e l i n e ]

If a b a s e l i n e u n d e r l y i n g a s p e c t r a l peak (a-,Y-) is estimated from a

r e g i o n well removed from t h a t peak , t h e n t h e d e c i s i o n and d e t e c t i o n e q u a t i o n s

are f o r m a l l y i d e n t i c a l t o t h o s e p r e s e n t e d above . One s i m p l y s u b s t i t u t e s ( f o r

t l and t 2 ) n1 and "2, t h e r e s p e c t i v e number of c h a n n e l s u s e d f o r e s t i m a t i n g

t h e peak and the b a s e l i n e . The o n l y o t h e r d i f f e r e n c e is t h a t the f u l l

e x p r e s s i o n f o r A -- ( A K + A I ) -- must be u s e d , when o n e i n c l u d e s bounds f o r

s y s t e m a t i c error .

[RANDOM PART 1

If two e q u i v a l e n t , p u r e b a s e l i n e r e g i o n s l i e s y m m e t r i c a l l y a b o u t t h e

p e a k , as shown i n F i g . 8 , each h a v i n g n2/2 c h a n n e l s , t h e n

97

I

cn t- Z 3 ~

0 0

n2/2 "1

CHANNEL

n212

Fig. 8. Simple Counting: Detection Limit for a Spectrum Peak

98

SI = C S i e q u a l s t h e number of n e t sample c o u n t s i n t h e peak r e g i o n . 1 n

Under t h e a s s a m p t i o n of l i n e a r i t y fo r B i ( b a s e l i n e c o s n t s i n c h a n n e l - i ) ,

Thus ,

"1 "1 + "2

"2 "2 where q = 1 + - =

Thus , t h e f o r m u l a t i o n is i d e n t i c a l t o t h e p r e c e d i n g one for g r o s s s i g n a l

minus b l a n k , e x c e p t t h a t n i t s r e p l a c e t h e t i ' s .

[+SYSTEMATIC PART]

The formal s t r u c t u r e a g a i n is unchanged. However, s i n c e we now t rea t

b a s e l i n e error botlnds rather t h a n b l a n k s y s t e m a t i c e r ror bounds, 4 + 0.01

r a t h e r t h a n 0.05. (The common, l i m i t i n g case when o n e has b a s e l i n e i n t e r f e r -

e n c e is assumed h e r e : t h a t BK << B I , so A = 0.01 B I , w i t h BI = b a s e l i n e i n

reg ion-1 ( p e a k ) . T h i s q u a n t i t y is estimated as y 2 ( n , / n 2 ) .

99

Thus ,

a n d ,

( 2 4 ' )

0.022 B1 + 3.624- - - 2.22(YEVT)

where rl = ("1 + n 2 ) / n 2

The p o i n t a t which the s y s t e m a t i c b a s e l i n e e r r o r term domina te s t he

e x p r e s s i o n f o r XD is,

I 3.62 2 Beq =[-I n = (2 .70 x 1 0 4 h counts ( 2 5 ' )

0.022

B. Mutual I n t e r f e r e n c e ( 2 components)

i ) Zero d e g r e e s o f freedom - 2 o b s e r v a t i o n s

I n b o t h t h e e v a l u a t i o n of decay c u r v e s and s i m p l e s p e c t r o s c o p y , o n e o f t e n

e n c o u n t e r s t h e s i t u a t i o n where there is l lmutual i n t e r f e r e n c e " -- i . e . , where

r a d i a t i o n s from two components c o n t r i b u t e t o e a c h o f t he o b s e r v a t i o n s t a k e n ,

o r t o e a c h o f t h e two classes of o b s e r v a t i o n s . I f t h e r e l a t i v e c o n t r i b u t i o n s

d i f f e r , t h e two components may be r e s o l v a b l e (depend ing upon s t a t i s t i c s ) .

[For the f o l l o w i n g d i s c u s s i o n , re fe r t o F i g . 9 f o r s i m p l e decay c u r v e

r e s o l u t i o n , and F i g . 1 0 f o r s i m p l e spec t rum peak a n a l y s i s . ]

100

t 1 t 2

TIME

F i g u r e 9.

101

n212 "1 n212

CHANNEL

Figure 10.

102

y1 = 1 y i = S1 + B1 + el (38 ) "1

y 2 = 1 y i = a S1 + bB1 + e 2 ( 3 9 ) "2

For t h e decay c u r v e , t h e p a r a m e t e r s - a and - b a re u n i q u e l y d e t e r m i n e d by

t h e t l / z l s (or A's) of t h e 2 components , t h e s p a c i n g ( t ime) of t h e two

o b s e r v a t i o n s , and t h e m e a s x e m e n t i n t e r v a l s t l and t 2 . l .

t h e 2nd component is e q u i v a l e n t t o a b l a n k a n d / o r l o n g - l i v e d i n t e r f e r i n g

n u c l i d e . ] For t h e s p e c t r u m p e a k , n1 and n2 r e p r e s e n t t h e r e s p e c t i v e numbers

of c h a n n e l s as before; and the n 2 ' s are s y m m e t r i c a l l y p l a c e d a b o u t t h e peak

r e g i o n (symmetr ic w i t h r e s p e c t t o t h e mid-n l -channel ) f o r a b a s e l i n e model

which is l i n e a r o r f l a t . The same formalism a p p l i e s a l s o f o r t h e case of two

o v e r l a p p i n g s p e c t r a ( p r o v i d e d the b l a n k i s n e g l i g i b l e o r c o r r e c t e d ) , s x h as

Y-ray d o u b l e t s . ( I t s h o u l d n o t be o v e r l o o k e d t h a t , f o r the Y-peak, t h e

e f f e c t i v e d e t e c t i o n e f f i c i e n c y [ E ] here depends upon t h e a l g o r i t h m -- i . e . ,

t h e l o c a t i o n s , w i d t h s and s e p a r a t i o n s o f r e g i o n s -1 and -2.)

[If A2 = 0 , t h e n

Simply t o s o l v e t h e s e e q u a t i o n s , we must assme t h a t a and b -- i . e . , t h e - -

d e c a y c u r v e or s p e c t r u m s h a p e s -- are known. When B (component-2) is a

l i n e a r b a s e l i n e o r a c o n s t a n t b l a n k or i n t e r f e r e n c e ( d e c a y c u r v e ) , - b is

d i c t a t e d by t h e model, t h e n a < b , and

d e c a y c u r v e : b = t 2 / t l

s p e c t r u m : b = n2/n1

__------------- l T h u s , a (and b , i f A2 f 0 ) subsgmes the p a r a m e t e r T i n Eq. ( 1 1 ) [Good

approximation-if to is se t a t t h e m i d p o i n t of t h e f irst i n t e r v a l ( t l ) ] .

103

[RANDOM PART]

The s o l u t i o n s ( P o i s s o n s t a t i s t i c s p a r t ) f o l l o w :

a n d , r e p l a c i n g SI by z e r o ,

b ( b + l ) where n =

( b - a ) 2

[When a + 0 , a s i n " s i m p l e c o u n t i n g " , we g e t t he p r e v i o u s r e s u l t , t h a t +

( b + l ) /b ]

A s b e f o r e ,

Sc = Z I - ~ O ~ = 1.645 JBlrl ( 1 7 ' )

However, t h e minimum de tec t ab le S1 - c o u n t s takes t h e form,

( 2 0 ' ) SD = 2 1.1 + 2Sc = ( 2 . 7 1 ) 1 ~ - + 3.29

b2+a

( b - a ) 2 where p = > 1

Some g e n e r a l i z a t i o n s f o l l o w :

( a ) If a = b , bo th Sc and SD d i v e r g e (SD more r a p i d l y )

( b ) The term z2p which comes a b o u t because o f P o i s s o n c o u n t i n g s t a t i s t i c s

h a s g r e a t e r i n f l u e n c e t h a n t h e term z2 which we f i n d i n " s imple c o u n t i n g " .

( c ) I n f ac t t h e p r e v i o u s a p p r o x i m a t i o n SD = 2Sc is p o o r e r , e s p e c i a l l y

when a a p p r o a c h e s - b , -

1 0 4

Asymptotic forms for K :

l o when b >> 1 and b >> a (e.g., n2 >> "1, or t2 >> tl or for barely b

b-a overlapping peaks), K -* = (-) = 1 [a1 so, p and n = 1 1

(l+a)/J-T

(1-a) o when b = 1 (e.g., for blank or linear baseline), K +

I

For the first asymptote, SD = 2Sc (within 10%) when B > 68 counts, as

before (f'simpleff comting). For the second asymptote, K ranges from 0.707

[a=Ol to m [a=11. Taking for example, a=1/2 [ K = 2.121, we find that SD =

2Sc once B > 304 counts. \

Thus, the extra Poisson term (z2p) cannot be so

readily ignored as in the case of "simpleff counting.

Once again, XD = S~4.22 (YEVT)] where T will already have been included

in the coefficients a and b for the decay curve example, and E will be

inflgenced by the normalization of the coefficients for the spectrum peak

example. (Here, E=E1, the total efficiency corresponding to the fraction of

the peak contained in region-1.)

That is, for the decay-curve mutual interference example, XD =

R~/[(yEV)(2.22)] because R D (initial counting rate of the 'signal' 0 0

radionuclide) depends on the equations including T:

where

105

[SYSTEMATIC PART]

Let us next consider bounds for systematic error in B. At this point, a

new problem presents itself: shoald we assume that the relative uncertainty

4~ applies to B1 in y1, or to B2 = bB, in y2, or both?

question as posed is inappropriate. The systematic error in fitting is due

to model or shape error (in the baseline) rather than a discrete shift from a

signal-free blank observation as in "simple counting."

In fact, the

In order to simply present the systematic (shape) error contribution to

xD,

region-I to the entire portion of the spectrum or decay curve involved in'the

it will be convenient first to change the normalization basis from

fitting. We accomplish this by re-writing Eq's (38) and (39) to read,

y1 = a1 S + bl B + el ( 4 5 )

y2 = a2 S + b2 B + e2 ( 4 6 )

where the a's and the b's are normalized to unity (Ca=l, Cb=l). Thus S and B

represent the contributions of the net signal and blank to the total peak

area that we analyze ( S + B = y1 + y2).

The solution is formally identical to that obtained before, A

s = c1 Y1 + c2 Y2

2 2 2 00 C1 (blB) + ~2 (b2B) = Bn

where now

and c1 = b2/D, c2 = -bl /D, D = (a1 b2 - a2bl )

2 n = C Ci bi

The relation,

( 5 0 1

(51 1

106

where

2 p = C c i a i ( 5 2 )

is s t i l l v a l i d , and i t can be shown

and u = v*/al where t h e a s t e r i s k r e fe r s t o t h e p r e v i o u s n o r m a l i z a t i o n (where

t h a t c = c * / a l , o0 = a o * / a l ,

a1, b l + 1 ) . I t f o l l o w s tha t

s; * - = XD X D = - = ~ - SD stf

(2 .22 Y V T ) E (2 .22 YVT)Eal (2 .22 YVT)E*

Thus , t h e ( P o i s s o n p a r t o f > t h e d e t e c t i o n l i m i t does n o t depend on t h e a ,

5 n o r m a l i z a t i o n .

With t h i s r e - n o r m a l i z a t i o n i t becomes s t r a i g h t f o r w a r d t o t r e a t s y s t e m a t i c

error . S u b s t i t u t i n g Ayi f o r y i i n Eq. ( 4 7 ) we o b t a i n A

As = c i AYi + c 2 A Y ~ ( 5 3 )

I f t h e A y i ' s a re d:ie t o s y s t e m a t i c s h a p e e r r o r s i n t h e b a s e l i n e , we have A

AS = ClB Ab1 + c2i3 Ab2 = B C C i Abi ( 5 4 )

where t h e A b i ' s are t h e d e v i a t i o n s of t h e a c t u a l b a s e l i n e s h a p e from t h e

assumed s h a p e and B r e p r e s e n t s t h e b a s e l i n e area ( c o u n t s ) unde r t h e f i t t e d

r e g i o n . Thus t h e q u a n t i t y c i Abi r e p l a c e s t h e which o c c u r r e d i n t h e

e x p r e s s i o n fo r " s i m p l e c o u n t i n g f T sys temat ic e r ror , s o e x a c t l y the same

e q u a t i o n may be used f o r c a l c u l a t i n g the d e t e c t i o n l i m i t . (Because of

o r t h o g o n a l i t y be tween t h e { c i } and t h e t r u e b a s e l i n e { b i ] , AB c a n be a l s o

c a l c u l a t e d d i r e c t l y from t h e a l t e r n a t i v e b a s e l i n e - s h a p e b i ) .

4~ = Z c i b i (55 1

A s i g n i f i c a n t change i n c o n c e p t h a s e n t e r e d , however , i n t h a t t h e Abi

r e p r e s e n t s y s t e m a t i c b a s e l i n e s h a p e a l t e r n a t i v e s r a t h e r t h a n s i m p l y a

b a s e l i n e l e v e l s h i f t . (Thus , t h e Ab i ' s r e p r e s e n t g e n e r a l l y a smooth

t r a n s i t i o n i n f u n c t i o n -- as from a l i n e a r t o a q u a d r a t i c b a s e l i n e , e t c . )

107

The formalism developed here can be extended quite directly to the

estimation of systematic model error even for multicomponent least squares

fitting of spectra and decay curves. Some of the basic theory and details

have been developed in Ref. 72 ("bias matrix").

(ii) Finite Degrees of Freedom - Least Squares

For just two components (as baseline and spectral peak, etc.) it is

relatively simple to extend the above considerations to many observations

such as one finds with multichannel spectrum analysis or mxltiobservation

decay curve analysis. (This is because it is trivial to write down the

expression for the inversion of the 2 x 2 vfnormal-equationsTf matrix.)

same basic matrix formulation applies, however, for any number of components.

The

[ 1 ] General WLS Formulation

In this case ( P > 2, n > P ) , the observations (counts) yi can be written:

o r , in matrix notation,

where

M

a2

an

Y = M 0 + e

OT = ( S B) and

The weighted least-squares (WLS) solxtion to Eq. (57) is, A A

S = 01 = [(MT wM)'~ MT WY]~

(56 )

(57)

(58)

108

and

where t h e w e i g h t s w a r e , - W i i = 1/Vy = l / ( M B ) i = l / y i

i where the second e q u a l i t y a p p l i e s f o r P o i s s o n s t a t i s t i c s .

( 6 0 )

( I f the

o b s e r v a t i o n s a re i n d e p e n d e n t , - w is a d i a g o n a l m a t r i x -- i . e . , w i j = 0 f o r

i # j . )

D e f i n i n g ,

c i z [ ( M T w ~ 1 - 1 MT ~ I l i (61

we c a n a l t e r n a t e l y express V i by means o f e r r o r p r o p a g a t i o n , t h a t i s , A A

S = 01 = C C i y i (58')

Thus , f o r t h e case o f P o i s s o n c o u n t i n g s t a t i s t i c s , A

V s = S p + B n ( 6 2 )

where

2 2 - 1-1 = C C i a i f C C i b i

Beyond thi-s, t he development is i d e n t i c a l t o t h a t g i v e n above for z e r o

degrees of f reedom ( P = n ) . Thus ,

sc = ZOO mere oo = J X ( 6 3 )

SD 2sC z2 ( 6 4 )

A; = ~4 = B Z C i b; ( 6 5

where b i is a n a l t e r n a t i v e b a s e l i n e s h a p e , u sed f o r e s t i m a t i n g p o s s i b l e

s y s t e m a t i c error.

109

I

The development t h u s fa r has been pe r fec t ly g e n e r a l ; t h a t is, n e i t h e r t h e

n m b e r o f components ( P I n o r t h e number of o b s e r v a t i o n s have been res t r ic ted .

Components o t h e r t h a n the one of i n t e r e s t (S = 0 1 ) h a v e , however , been

coalesced t o form a compos i t e i n t e r f e r e n c e or b a s e l i n e , B.

[2] E x p l i c i t S o l u t i o n for P=2

I f we t r ea t the b a s e l i n e ( o r any o t h e r s i n g l e component) as a t r p u r e t t

s econd component, h a v i n g f i x e d shape, t h e n the e x p l i c i t s o l u t i o n f o r 5 , Vi and t h e c i may e a s i l y be s t a t ed . The r e s u l t s f o l l o w from the i n v e r s i o n of

t he 2 x 2 m a t r i x

where

and

~1 = c w a2 C2 = C wb2 C12 = C wab

Taking t h e n u l l case (S = 01, t h e weights e q u a l ,

W i i = l / ( B b i ) ( 6 7 )

u s i n g the above e x p r e s s i o n f o r W i i and the p r e v i o u s d e f i n j t i o n f o r c i ,

t o g e t h e r w i t h t h e e x p l i c i t e x p r e s s i o n f o r t he m a t r i x M and i t s i n v e r s e , i t

c a n be shown t h a t ,

-

2 C i = [ a i / b i - 1 ) I / C C a i / b i - 1 1 (68 )

A l l o t h e r q u a n t i t i e s o f i n t e r e s t -- p, n, o o , SC, SD and

f o l l o w d i r e c t l y as i n d i c a t e d above . (A r e m i n d e r : we have n o r m a l i z e d a l l

r t spec t rumf t components f o r t he f o r e g o i n g d e r i v a t i o n s . That is ,

( g i v e n b’) --

Cai = Cbi = 1.)

I

1 1 0

. - . Equation (68) yields specific solutions once the peak (ai) and baseline

(bi) shapes are given. For a flat baseline (bi = l/n), for example, Eq (68)

reduces to 2

ci =[ai - l/nl / [Zai - 1/1-11

It follows that

Gaussian Peak

( 6 9 )

If the peak shape (ai) is symmetric, then 3 and ci are even functions,

which means that if alternative are odd (and share the same center of

symmetry) then the systematic, baseline-model error vanishes.

4 = 1 ci bi = even vector odd vector = 0

This suggests that for a symmetric isolated peak, one can treat the baseline

as flat--even tho3gh it may be linear or otherwise odd (about the peak

center) -- without introducing bias.

Passing beyond jgst the assmption of symmetry, and specifying the peak

to be gaussian, we can calculate explicit valxes for the ci once - n is known. It is interesting to examine this case as a fanction of channel density

(number of channels per FWHM or per +3 standard deviations, (SD), etc.). The

results of such a calcxlation are illustrated below.

1 1 1

2 2 2 Channel D e n s i t y p = C C i a i = C C i b i = C c i / n

n = ch /peak peak f f 3 SD

3

6

2.80

2.03

1.83

1 .60

m 1 .924 1 . 4 4 4

The above v a l u e s f o r 1-1 and q may be used t o estimate t h e s e v e r a l

q u a n t i t i e s o f i n t e r e s t f o r the d e t e c t i o n o f a n i s o l a t e d peak . Note t ha t if

t h e o b s e r v a t i o n s are ex tended well beyond t h e peak (beyond ?3 SD), p and TI

c a n be r e d u c e d s u b s t a n t i a l l y . The l i m i t i n g v a l u e s ( n = m ) become 1 . I 6 and

0.591, r e s p e c t i v e l y .

C31 Some F i n a l Comments

The immedia te ly p r e c e d i n g d i s c u s s i o n was g i v e n from t h e p e r s p e c t i v e o f

The same fo rma l i sm would f o l l o w ( u p t o t h e Y-ray ( o r a - p a r t i c l e ) s p e c t r a .

s p e c i f i c a t i o n o f a g a u s s i a n o r symmetr ic p e a k ) f o r d e t e c t i o n i n d e c a y c u r v e

a n a l y s i s , o r B-spectrum a n a l y s i s , e t c . Excep t for t h e g e n e r a l m a t r i x f o r m u l a t i o n and t r e a t m e n t as compos i t e

i n t e r f e r e n c e ( b a s e l i n e ) , the f u l l mul t icomponent decay o r s p e c t r u m a n a l y s i s

d e t e c t i o n i s s u e w i l l n o t be t reated here . F u r t h e r d i s c u s s i o n would r e q u i r e

e x p l i c i t assumed models ( i n t e r f e r i n g r a d i o n u c l i d e s ) ; b u t the basic p r i n c i p l e s

and basic e q u a t i o n s would be unchanged.

Regard ing t h i s more c o m p l i c a t e d s i t u a t i o n , however , three p r o c e d u r a l

comments, and th ree n o t e s o f c a u t i o n may be g i v e n :

112

1

~ [PROCEDURAL COMMENTS]:

o peak s e a r c h i n g e f f i c i e n c y and d e t e c t i o n power depend on t h e e x a c t

n a t u r e o f the a l g o r i t h m employed. For t h e I A E A t es t s p e c t r u m f o r peak

d e t e c t i o n , f o r example , a t l ea s t s i x i n d e p e n d e n t p r i n c i p l e s were used

by 212 p a r t i c i p a n t s t o detect peaks i n t he same d i g i t i z e d , s y n t h e t i c

Y-ray s p e c t r u m [ F i g . 6 and Ref. E l ] . Yet, f a l se p o s i t i v e s r anged f rom

0 t o 2 3 p e a k s , and fa lse n e g a t i v e s r a n g e d from 3 t o 20 peaks . (The

number of a c t u a l peaks i n the spec t rum was 22.)

o Eq. ( 6 4 ) is a p p r o x i m a t e o n l y because o f chang ing s t a t i s t i c a l w e i g h t s as

S i n c r e a s e s f rom z e r o t o SD. An e x a c t s o l u t i o n may be o b t a i n e d by

i t e r a t i o n (Ref. 61 1.

o S y s t e m a t i c model e r r o r f o r t he mut l icomponent s i t u a t i o n may be d e r i v e d

w i t h t he u s e of a "Bias M a t r i x , " which can be d e r i v e d f rom t h e l ea s t

s q u a r e s s o l u t i o n for S , --- t o g e t h e r w i t h a l t e r n a t i v e models (Ref .

7 2 ) .

n

[CAUTIONS] :

o Searches f o r m u l t i p l e components o f t e n lead t o m y l t i p l e d e t e c t i o n

decisions. T h e overall probability of a false positive (a) in

s e a r c h i n g a s i n g l e spectrum c a n t h u s be s u b s t a n t i a l l y more than the

s i n g l e - d e c i s i o n r i s k . (See Ref. 53 and S e c t i o n II.D.5 for more on

t h i s t o p i c . )

o If n o n - l i n e a r searches ( i n v o l v i n g , f o r example , e s t i m a t i o n o f h a l f -

l i v e s a n d / o r Y-energ ies as w e l l as a m p l i t u d e s ) are made, t h e estimated

s i g n a l d i s t r i b u t i o n (5) is no l o n g e r normal .

d e v i a t i o n s from p r e s m e d v a l u e s of - a may be t h e resgl t (Ref. 9 0 ) .

Again , s u b s t a n t i a l

1 1 3

o Bad models and experimental blunders may inflate x2 because of poor ' ,

fit.

yield misleading random error estimates, and erode detection

capability. (See note [ A 1 4 1 and Ref. 63.)

Multiplication of Poisson standard errors by mis-fit x / d E will

APPEND1 X

Appendix A . N o t a t i c n and Terminology

Response

E ( y ) = B + AX = B + S y - = B + Ax + ey = E ( y ) + ey [ o b s e r v a t i o n ]

y = B + A x [estimate] A A A

;; = ( y - 6 ) / i E ( y ) = r e s p o n s e or g r o s s s i g n a l ( c o u n t s ) , t r u e or "expec ted f f v a l u e [ y i d e n o t e s

Y

6

0

A

4

4 A

Y

- Y

S

B

BEA

X

A

t h e i th sample or time p e r i o d or e n e r g y b i n , e t c ]

= o b s e r v e d ( " sampled f f ) v a l u e of y , c h a r a c t e r i z e d by a n error ey

= random e r r o r

= s t a n d a r d d e v i a t i o n (SD); o / h = SD of the mean ( s t a n d a r d e r r o r , S E ) ;

R S D = r e l a t i v e s t a n d a r d d e v i a t i o n

= s y s t e m a t i c e r ror (bound)

= r e l a t i v e - o ( R S D )

= r e l a t i v e - A

= s t a t i s t i c a l l y estimated v a l u e f o r y ( e . g . , we igh ted mean, ... ) ( s i m i l a r l y f o r i, 6 , i, 4)

= assumed or t f s c i e n t i f i c a l l y f f ' estimated v a l u e f o r y

= t r u e n e t s i g n a l ( c o u n t s ) [ "expec ted v a l u e f f ]

= t r u e background o r b l a n k or b a s e l i n e ( c o u n t s ) ( B K = b l a n k ; BI =

i n t e r f e r e n c e c o u n t s )

= Background E q u i v a l e n t A c t i v i t y = B/A

= t r u e r a d i o a c t i v i t y c o n c e n t r a t i o n , p e r u n i t mass or volume [pCi o r Bq/g

o r L]. To be referred t o i n t he t e x t s i m p l y a s " c o n c e n t r a t i o n f f

= g e n e r a l i z e d c a l i b r a t i o n fac tor ; f o r s i m p l e c o u n t i n g , w i t h x i n > C i / ( g

o r L), A = 2.22 ( Y E V T ) , where

Y = ( radio)chemical y i e l d or r e c o v e r y

115

E = detection efficiency (overall, including branching ratio)

V = volume or mass of sample

T = appropriate time factor or function (minutes)

Vs, V,, etc = variance of the sgbscripted quantity

a s , a x , etc = SD of the subscripted quantity = f l

oo = SD of 5 (at S = O ) [counts]; oxo = SD of Z (at X=O) [concentrationl

rl = a multiplier which converts og to ao: a. = 0 g 6 - Its value depends

on the design of the Measurement Process.

b 3: ratio of counting times (or channels) blank/(signal + blank); then

rl = 1 + l / b

S c , xc = critical or decision levels for judging whether radioactivity is

present, with false positive risk-a

SD, xD = corresponding detection limits, with false negative risk -8

ZI-a, 21-6 - - percentiles of the standardized Normal distribution, equal to

1.645 for a, B = 0.05

LLD = Lower Limit of Detection (for radioactivity concentration) = XD

xR = prescribed regulatory LLD -- i.e., limiting valge which licensee is This is in contrast to the actual LLD (XD) which supposed to meet.

is achieved under specific experimental circumstances. (Thus,

generally, XD - < XR)

v or df = degrees of freedom

Appendix B. Guide to Tutorial Extensions and Notes

Section 1II.A and 1II.B were prepared as proposed substitute RETS pages

-- the former cast as a more or less comprehensive statement, and the

latter, for ttsimplett gross signal-minus-blank counting. A-series and

' I

B-series notes, respectively, were appended to these sections, so that each

116

1 c o u l d be t o a l a r g e e x t e n t s e l f - c o n t a i n e d . The f o l l o w i n g g u i d e ( o r i n d e x ) t o

these n o t e s is g i v e n b e c a u s e of t he i r p o s s i b l e g e n e r a l u t i l i t y , and b e c a u s e

the two s e t s of n o t e s a re n o t o n l y r e d u n d a n t (as i n t e n d e d ) b u t a l s o

complementary.

% \

1 . Basic I s s u e s

a ) Use of S . I . u n i t s - Note A 3

b ) G e n e r a l f o r m u l a t i o n - Note A 8

Eq. ( 1 ) was d e v e l o p e d f o r a p p l i c a t i o n t o most c o u n t i n g s i t u a t i o n s ,

t h r o u g h t h e i n t r o d u c t i o n of p a r a m e t e r s oo, f and A which c a n be eva1 , la ted f o r

the s p e c i f i c c o u n t i n g method and data r e d u c t i o n a l g o r i t h m i n u s e . The

e q u a t i o n must be m o d i f i e d , however , when small numbers of c o u n t s a re

i n v o l v e d . Normal v a r i a t e p e r c e n t i l e s ( z I - ~ , z1-6) are i n c l u d e d as p a r a m e t e r s

which may be mod i f i ed as a p p r o p r i a t e ( e . g . , m u l t i p l e d e t e c t i o n d e c i s i o n s ) .

c ) A p r i o r i v s a p o s t e r i o r i - Note A 4

Measurement p r o c e s s charac te r i s t ics must be known i n a d v a n c e before a n Ita

-

p r i o r i " d e t e c t i o n l i m i t c a n be s p e c i f i e d -- may c a l l f o r a p r e l i m i n a r y

e x p e r i m e n t .

d ) D e c i s i o n s and r e p o r t i n g - Notes A 1 3 , B 8 ( i d e n t i c a l )

The c r i t i c a l l e v e l (Sc) may need t o be i n c r e a s e d i n t h e case of m u l t i p l e

d e t e c t i o n d e c i s i o n s ; LLD t h e n a u t o m a t i c a l l y i n c r e a s e s . Non-detected and

n e g a t i v e r e s u l t s s h o u l d be recorded; re la ted t o p i c s ; a v e r a g i n g , t r u n c a t i o n .

2. LLD F o r m u l a t i o n -- C o n v e n t i o n a l ( P o i s s o n ) Count inn S t a t i s t i c s

a )

b ) E x t e n s i o n of the s i m p l i f i e d e x p r e s s i o n (Ea . 6 ) t o i s o l a t e d s p e c t r u m

Rapid d e t e c t i o n d e c i s i o n s , LLD bounds - v i a i n e q u a l i t i e s - Note B4

peaks . - Note B1

1 1 7

c) Continuous monitors - Note B7 d) Mixed nuclides, tlgrossct radioactivity - Note A6 e) Factors for detection efficiency (E), counting time (TI. - Notes A7,

Branching ratios, spectrum shapes, decay curves and sampling designs all

affect LLD beyond just the matter of counting statistics. Interpretation of

E q . (1) (mixing of factors o o , E, T) varies accordingly.

3. Non Poisson (P) - Normal (N) Errors a) Extreme low-level counting (P f N); limits of validity for

approximate expressions - Notes A5, B9

b ) Replication, lack of fit, use and misuse of s2, x2 - Notes AI, A2, A8, A10, A14, B2

c) Uncertainty in and variability of the LLD. Blank variations;

multiplicative parameters: Y,E,V - Notes A2, A12, A15, A16, B5 d ) Systematic error bounds - Notes A8, All, B3 Additive and nultiplicative components; default valxes; limiting effect

on LLD reduction.

118

DETECTION CAPABILITIES OF CHEMICAL AND RADIOCHEMICAL MEASUREMENT SYSTEMS:

A Survey of the Literature (1923-1982+)

L. A. Carrie Center for Analytical Chemistry National BJreau of Standards

Washington, DC 20234

Introduction

The twin issues of the detection capability of a Chemical Measurement

Process (CMP) and the detection decision regarding the outcome of a specific

measurement are fundamental in the practice of Nuclear and Analytical Chem-

istry, yet the literature on the topic is extremely diverse, and common

understanding has yet to be achieved. Besides their importance to the

fundamentals of chemical and radiochemical measurement these issues have

great practical importance in application, ranging from the detection of

impurities in industrial materials, to the detection of chemical signals of

pathological conditions in humans, to the detection of hazardous chemical and

radioactive species in the environment. It is in connection with this last

area, as related to the regulation of nuclear effluents and environmental

radioactive contamination, and at the request of the Nuclear Regglatory

Commission (NRC), that this report has been prepared. Highlights from our

extensive search of the literature are given in the following text.

Scope of the Survey

The focus of the literature survey was directed toward two points: (1)

basic principles, terminology and formulations relating to detection in

Analytical Chemistry; and ( 2 ) basic, but more detailed or specialized studies

119

I

I

relating to detection limits in the measurement of radionuclides, as well as

important practical applications in this area. The search was conducted with

the aid of five computer data bases, complemented by the examination of major

reviews and books treating mathematical and statistical aspects of Analytical

Chemistry.

Carefully constructed patterns of keywords led to a total of 1711 titles

(1964-1982) which were scanned. From these, 700 were identified as important

to our purpose, so abstracts were copied and studied. A final catalog of 387

articles from the computer literature search was prepared, and from this

about 100 were marked as having special relevance. Discovering so extensive

a literature on so esoteric a topic was somewhat surprising; also surprising,

or at least noteworthy, i s the fact that a very large fraction of the work on

this topic has originated in foreign institutions with major contributions

coming from Western and Eastern Europe, the Soviet Union, and Japan.

I

Basic References and Key Issues

For the purposes of this appendix-report our discussion of the literature

must be highly selective; thus only a few of the most critical sources are

discussed. We have given primary emphasis to the Ifarchived" literature (e.g.,

journal articles as opposed to reports); and more general publications

treating mathematics, statistics, radioactivity measurement, and quality

assurance have been cited only if detection limits were given major focus. A

slightly expanded, classified bibliography appears in appendix C.2.d.

The key issues which were addressed or cited in the literature included,

as noted above, terminology and formulation (definitions) resulting from

exposition of the basic principles of statistical estimation and hypothesis

testing in chemical analysis. Special (but basic) topics treated by several

120

1

I

, % a u t h o r s i n c l u d e d : t h e e f f ec t s o f c o u n t i n g s t a t i s t i c s , non-coun t ing and

non-normal random e r r o r s , random and s y s t e m a t i c v a r i a t i o n s i n t h e b l a n k ,

r e p o r t i n g and a v e r a g i n g p r a c t i c e s , m u l t i p l e d e t e c t i o i i d e c i s i o n s , Bayes i an

I

a p p r o a c h e s , t h e i n f l u e n c e of t h e number of d e g r e e s o f f r eedom, i n t e r l a b o r a -

t o r y e r r o r s , c o n t r o l and s t a b i l i t y , o p t i m i z a t i o n o f d e t e c t i o n l i m i t s ,

i n t e r f e r e n c e e f f e c t s , da t a t r u n c a t i o n , and d e c i s i o n s v s d e t e c t i o n v s determi-

n a t i o n v s i d e n t i f i c a t i o n l i m i t s . Major t o p i c s which re la ted s p e c i f i c a l l y t o

r a d i o a c t i v i t y measurements i n c l u d e d the i n f l u e n c e o f a l t e r n a t i v e (Y-, 6 - )

s p e c t r u m d e c o n v o l u t i o n t e c h n i q u e s , c o m p a r i s o n / s e l e c t i o n o f a l t e r n a t i v e

i n s t r u m e n t s and r a d i o c h e m i c a l schemes o f a n a l y s i s ( e s p e c i a l l y i n t he area o f

a c t i v a t i o n a n a l y s i s ) , t he t r e a t m e n t o f v e r y low- leve l a c t i v i t y and t h e

t r e a t m e n t o f v e r y s h o r t - l i v e d r a d i o n u c l i d e s . T i t l e s i n t h e h i g h l y s e l e c t e d

- - -

b i b l i o g r a p h y r e f l e c t a number o f these s p e c i f i c i s s u e s .

To c o n c l u d e t h i s summary r e p o r t , I s h o u l d l i k e t o c i t e j u s t a few

s o u r c e s which I b e l i e v e e i t h e r s e t f o r t h o r r e v i e w some o f t h e more basic

i s s u e s . The groundwork ( w i t h i n t h e p r e s e n t time frame) was l a i d by Kaiser

( 2 1 , who a d o p t e d the basic s t a t i s t i c a l p r i n c i p l e s of h y p o t h e s i s t e s t i n g (and

t y p e - I , t y p e - I 1 e r r o r s ) t o d e t e c t i o n i n s p e c t r o g r a p h i c a n a l y s i s . O the r

f r e q u e n t l y - c i t e d w o r k s from t h e 6 0 ' s are papers by St. J o h n , McCarthy and

Wine fo rdne r (31 , A l t s h u l e r and P a s t e r n a c k ( 4 1 , and C u r r i e (51, t h e l a t t e r two

t r e a t i n g t h e q u e s t i o n o f r a d i o a c t i v i t y . Later i m p o r t a n t works which s p e c i f -

i c a l l y t rea t r a d i o a c t i v i t y d e t e c t i o n are g i v e n i n r e f e r e n c e s ( 6 ) - ( 2 1 ) .

( F u r t h e r comments c a n n o t be g i v e n i n t h i s brief r e p o r t ; see the t i t l e s f o r

t h e f o c u s o f each p a p e r . )

F i n a l l y , some o f t he most u s e f u l e x p o s i t i o n s and summaries of LLD

t r e a t m e n t s and p r i n c i p l e s and unso1,ved p rob lems may be found i n t he books and

r e v i e w s b e g i n n i n g w i t h r e f e r e n c e ( 2 2 ) . S p e c i a l a t t e n t i o n s h o u l d be d i r ec t ed

121

., ,, to the IUPAC statement (22,23), the papers by Wilson (341, the chapter by Currie

(33), the review by Boumans (261, and the books by Winefordner (301, Kateman

and Pijpers (29), and Massart, Dijkstra and Kaufman (28).

ADDendiX c.2

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Donn, J. J.; Wolke, R. L. The statistical interpretation of counting

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tiver messung geringer aktivitsten. Atomkernenergie. 29: 266; 1977.

Robertson, R . ; Spyrou, N. M.; Kennett, T. J. Low level Y-ray spectrom-

etry; Nal(T1) vs. Ge(Li). Anal. Chem. 47: 6 5 ; 1975.

Zimmer, W. H. Limits of detection of isotopic activity. ARHCO Analyti-

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activity in Releases of Radioactive Materials in Liquid and Airborne

Effluents from Uranium Mills". 1977 June; Regulatory Guide 4.16 "Mea-

suring, Evaluating, and Reporting Radioactivity in Releases of Radio-

active Materials in Liquid and Airborne Effluents from Nuclear Fuel

Processing and Fabrication Plants.

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Nucl. Sci. NS-19: 1 1 9 ; 1972.

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Health

sections

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31

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Nomenclature, Symbols, Units and Their Usage in Spectrochemical

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Spectrochim. Acta 33b: 241 ; 1978.

Nalimov, V. V. The application of mathematical statistics to chemical

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Svoboda, V; Gerbatsch, R. Definition of limiting values for the sensi-

tivity of analytical methods. Fresenius' Z. Anal. Chem. 242(1): 1-13;

1968.

Boumans, P. W. J. M. A Tutorial review of some elementary concepts in

the statistical evaluation of trace element measurements. Spectrochim.

Acta 33B: 625; 1978.

Liteanu, C.; Rica, I. On the detection limit. Pure Appl. Chem. 44(3):

535-553 ; 1 975.

Massart, D. L.; Dijkstra, A.; Kaufman, L. Evaluation and optimization of

laboratory methods and analytical procedures. New Y o r k : Elsevier

Scientific Publishing Co.; 1978.

Kateman, C.; Pijpers, F. W. Quality control in analytical chemistry.

New York: John Wiley & Sons; 1981.

Winefordner, J. D., ed. Trace analysis. New York: Wiley; 1976. (See

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Liteanu, C.; Rica, I. Statistical theory and methodology of trace

analysis. New York: John Wiley & Sons; 1980.

125

32. Eckschlager, K . ; Stepanek, V. InFormation theory as applied to chemical 5 5

analysis. New York: John Wiley & Sons; 1980.

33. Currie, L . A. Sources of error and the approach to accuracy in

analytical chemistry, Chapter 4 in Treatise on Analytical Chemistry,

Vol. 1 , P. Elving and I. M. Kolthoff, eds. New York: J. Wiley & Sons;

1978.

34. Wilson, A . L. The performance-characteristics of analytjcal methods.

Talanta 17: 2 1 ; 1970; 1 7 ; 31: 1970; 20 : 725; 1973; and 21: 1109; 1974.

35. Hirschfeld, T. Limits of analysis. Anal. Chem. 48: I T A ; 1976.

(d) Further Classified References

Cil Basic Principles

36. Nicholson, W. L. "What Can Be Detected," Developments in Applied

Spectroscopy, v . 6 , Plenum Press, p. 101-113, 1968; Nicholson, W. L . ,

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bility of analytical chemical data used f o r public purposes. Subcommit-

tee dealing with the scientific aspects of Regulatory Measurements,

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38. Currie, L . A. Quality of analytical results, with special reference to

trace analysis and sociochemical problems. Pure & Appl. Chem. 5 4 ( 4 ) :

71 5-754 ; 1982.

39. Winefordner, J. D . ; Vickers, T. J. Calculation of the limit of

detectability in atomic absorption flame spectrometry. Anal. Chem. 36:

1947; 1964.

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Buchanan, J. D. Sensitivity and detection limit. Health Physics Vol.

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Ann. N. Y. Acad. Sci. 320: 43-7; 1979.

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2 6 ( 7 ) ; 20-24; 1975.

Grinzaid, E. L . ; Zil'bershtein, Kh. I.; Nadezhina, L. S.; Yufa, B. Ya.

Terms and methods of estimating detection limits in various analytical

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analytical chemistry. Analytical Letters 13(A14) : 1293-1297; 1980.

46. Liteanu, C.; Rica, I.; Hopirtean, E. On the determination limit.

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limit. J. Anal. Chem. - USSR 34: 1 ; 1979.

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Mikrochim. Acta 2 ( 3 ) : 311-323; 1975.

50. Tanaka, N. Calculation for detection limit in routine chemical analyses.

Kyoto-fu Eisei-Kogai Kenkyusho Nempo 22: 121-122; 1978.

51. Ingle, J. D., Jr. Sensitivity and limit of detection in quantitative

spectrometric methods. J. Chem. Educ. 5 1 ( 2 ) : 100-5; 1974.

127

52. Pan tony , D. A . ; H u r l e y , P. W . S t a t i s t i c a l and p r a c t i c a l c o n s i d e r a t i o n s z *,

o f l i m i t s o f d e t e c t i o n i n x-ray s p e c t r o m e t r y . A n a l y s t 97: 497; 1972.

[i i] A p p l i c a t i o n s t o R a d i o a c t i v i t y

53. Head, J. H. Ninimuc: detectable photopeak areas i n G e ( L i ) s p e c t r a .

Nucl. I n s t r u m . & Meth. 98: 419; 1972.

54. F i s e n n e , I. M . ; O ' T o l l e , A . ; C u t l e r , R . Least s q u a r e s a n a l y s i s and

minimum detectis: ; l e v e l s a p p l i e d t o multi-component a lpha e m i t t i n g

samples. Radiochem. Rad ioana l . Letters 16(1): 5-16; 1973.

55. Obrusn ik , I.; Kucera , J . The d i g i t a l methods of peak area c o m p u t a t i o n

and d e t e c t i o n l i m i t i n gamma-ray s p e c t r o m e t r y . Radiochem. R a d i o a n a l .

Le t t . 32(3-4): 149-159; 1978.

56. Heydorn, K . ; Wanscher, B. A p p l i c a t i o n of s t a t i s t i c a l methods t o ac t i -

v a t i o n a n a l y t i c a l r e s u l t s n e a r t he l i m i t of d e t e c t i o n . F r e s e n i u s '

Z. Anal. Chem. 292(1): 34-38; 1978.

57. Hearn , R . A . ; McFarland, R . C . ; McLain, M . E . , J r . High s e n s i t i v i t y

sampl ing and a n a l y s i s of gaseous radwaste e f f l u e n t s . I E E E T r a n s . Nucl.

SCI. NS-23(1): 354; 1976.

58. T s c h u r l o v i t s , M.; N i e s n e r , R . On t h e o p t i m i z a t i o n of l i q u i d s c i n t i l l a -

t i o n c o u n t i n g of aqueous s o l u t i o n s . I n t . J. Appl . Rad. and I s o t o p e s

30: 1-2; 1979.

59. Bowman, W . B. 11; S w i n d l e , D. L. P r o c e d u r e o p t i m i z a t i o n and e r r o r a n a l y -

sis f o r s e v e r a l l a b o r a t o r y r o u t i n e s i n d e t e r m i n i n g s p e c i f i c a c t i v i t y .

Health P h y s i c s , Vol. 31 : 355-361 ; 1976.

128

Mundschenk, H. S e n s i t i v i t y o f a l o w - l e v e l G e ( L i ) s p e c t r o m e t e r a p p l i e d t o

e n v i r o n m e n t a l a q u a t i c s t u d i e s . Nucl. I n s t r u m e n t s & Methods 177:

563-575 ; 1 980.

C u r r i e , L. A. The d i s c o v e r y of e r r o r s i n t h e d e t e c t i o n o f trace compon-

e n t s i n gamma s p e c t r a l a n a l y s i s , i n Modern t r e n d s i n a c t i v a t i o n a n a l y -

s is , Vol. 11. J. R. DeVoe; P. D . L a F l e u r , e d s . Nat. Bur. S t a n d (U.S.)

Spec . Publ . 312; p. 1215 , 1968.

C u r r i e , L. A . The e v a l u a t i o n o f r a d i o c a r b o n measurements and i n h e r e n t

s t a t i s t i c a l l i m i t a t i o n s i n a g e r e s o l u t i o n , i n P r o c e e d i n g s of t h e E i g h t h

I n t e r n a t i o n a l Confe rence on Radiocarbon D a t i n g , Royal S o c i e t y of N e w

Z e a l a n d ; p. 5 9 7 , 1973.

C u r r i e , L. A. On t h e i n t e r p r e t a t i o n of e r r o r s i n c o u n t i n g e x p e r i m e n t s .

Anal. Lett . 4: 873; 1971.

P a z d u r , M . F. Coun t ing s t a t i s t i c s i n low l e v e l r a d i o a c t i v i t y measure-

ments w i t h f l u c t u a t i n g c o u n t i n g e f f i c i e n c y . I n t . J. Appl. Rad. &

I s o t o p e s 27 : 179-1 84 ; 1 976.

Roger s , V. C . D e t e c t i o n limits f o r gamma-ray spec t ra l a n a l y s i s . Anal.

Chem. 42: 807 ; 1970.

Bosch, J . ; H a r t w i g , C . ; T a u r i t , R . ; Wehinger , H . ; W i l l e n b e c h e r , H.

S t u d i e s on d e t e c t i o n l i m i t s of r a d i a t i o n p r o t e c t i o n i n s t r u m e n t s . GIT

Fachz . Lab 2 5 ( 3 ) : 202, 205-8; 1981.

S t e v e n s o n , P. C. P r o c e s s i n g of c o u n t i n g data , N a t i o n a l Academy of

S c i e n c e s - N a t i o n a l Research C o u n c i l , Nuc lea r S c i e n c e Series

NAS-NS-3109: S p r i n g f i e l d , Va.: Dept . o f Commerce; 1965.

S t e r l i n s k i , S. The lower l i m i t of d e t e c t i o n f o r v e r y s h o r t - l i v e d r a d i o -

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mean and variance from radionuclide data sets containing negative,

unreported or less-than values. Health Phys. 40(3): 377-390; 1981.

Tschurlovits, M. Definition of the detection limit for non-selective

measurement of low activities. Atomkernenergie 29(4): 266-269; 1977.

Plato, P.; Oller, W. L. Minimum-detectable activity of beta-particle

emitters by spectrum analysis. Nuclear Instruments and Methods 125:

177-1 81 ; 1975

(e) Extension of References (non-classified)

Currie, L. A. J. of Radioanalytical Chemistry 39, 223-237 (1977).

Swanston, E.; Jefford, J. D.; Krull, R.; and Malm, H. L. "A L i q d i d

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Interpretation of Counting Experiments Designed to Detect Low Levels o f

Radioactivity,11 (National Radiological. Protection Board, Harwell,

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C u r r i e , L. A. f lAccuracy and Merit i n L i q u i d S c i n t i l l a t i o n Coun t ing ,1 t

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- - -

433 e

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' I '

I ., 31. Zoohen, 4. C. J r . ' 'Tables f o r ?lax<.!n*ir? L,;kellhood Zs t i rna t e s : Singly

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A n a l y s i s , VBS Spec . ?zQl. 422, Vol. 11, U.S. Governnent ?r int : .ng c ~ f f i c e ,

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( J u n e 1980)

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100. F e n n e l , R . 'd. and J e s t , T . S. P*ire Appl. Chem. 1 8 , 437 ( 1 9 6 9 ) . -- - -

Appendix D . Nmerical Examples1

1 . I so l a t ed Y-Ray Peak

. .

C o n s i d e r a S e ( L i ) measurement o f a n i s o l a t e d Y-ray, i n w h i c h a 500 mL -

H20 sample is coun ted f o r 200 min, and f o r which t h e d e t e c t i o n e f f i c i e n c y

(cpm-peakldpm) i s 2% a b s o l u t e . Let u s assume t h a t t h e e x p e c t e d b l ank r a t e

f o r t h e peak r e g i o n is 2 .0 cpm, and t h a t e q u a l n m b e r s o f c h a n n e l s x e d s e d

t o estimate t h e b a s e l i n e as are used t o estimate t h e g r o s s peak c o u n t s . T h i s

makes t h e n e t peak area e s t i m a t i o n c a l c u l a t i o n e x a c t l y e q u i v a l e n t t o t h e

"s imple1 ' g ros s - s igna l -minus -background measurement , w i t h e q u a l c o u n t i n g times.

R e f e r r i n g t o F i g u r e 8 and E q ' s ( 3 5 ) - ( 3 7 ) , we see t h a t n1 = n2 (he re 6 c h a n n e l s

e a c h ) , so rl = 2 and oo = OB^

a ) S i m p l e s t Case

I g n o r i n g p o s s i b l e s y s t e m a t i c e r r o r components , t h e c a l c u l a t i o n s a re as

f o l l o w s :

Y = 1 , E = 0 . 0 2 , V = 0 . 5 L., T = 200 min

R B = 2 .0 cpm,

SC = 1 .645 oo = 1.645 O B K= 1.645 m) = 46.5 c o u n t s

B = RBT = 400 c o u n t s

Thus , i f t h e n e t peak exceeded 46.5 c o m t s one would c o n c l a d e t h a t a s i g n a l

. h a d been de tec ted . ' ( O b v i o z s l y any o b s e r v e d n e t s i g n a l mxst be a n i n t e g e r ,

t hough SC i t s e l f can be a real n m b e r . ) The d e t e c t i o n l i m i t ( i n c o u n t s ) is

SD = 2.71 + 2Sc = 95 .8 c o m t s

The c o n c e n t r a t i o n d e t e c t i o n l i m i t XD is

95.8 - = 21.6 ( p C i / L )

SD LLD = X D =

2.22(YEVT) (2.22)(1)(0.02)(0.5)(200) I

-----_-______-- l A 1 1 e q g a t i o n numbers refer t o S e c t i o n I11 of t h i s r e p o r t , e x c e p t f o r example l g which refers t o S e c t i o n 11.

133

I f t h e mandated LLD ExR] were a t y p i c a l 30 pCi /L , t h e e x p e r i m e n t a l

" s e n s i t i v i t y " would be c o n s i d e r e d a d e q u a t e .

b ) I n t e r f e r e n c e

l'he above c a l c u l a t i o n was " p u r e a p r i o r i . " Let u s s u p p o s e , however ,

that , t h e ac';ual sample b e i n g measured e x h i b i t e d a Compton b a s e l i n e of 30 cpm

o v e r t h e peak r e g i o n (6 c h a n n e l s ) . E v e r y t h i n g t h e n becomes sca led by a

f a c t o r of ( b e c a u s e o o a fi). Thus ,

B = RBT = 6000 c o u n t s

s c .= 1.645 J&= 1.645 (109.5) = 180.2 c o u n t s

2.71 + 2Sc 363.1

2.22(YEVT) 4.44 X D = - - = 81.1 pCi/L

T h i s e x c e e d s t h e h y p o t h e t i c a l mandated v a l u e (30 p C i / L ) , so we n e x t face t h e

i s s u e o f Des ign -- i . e . , change o f t h e Measurement P r o c e s s , t o a t t a i n t h e

d e s i r e d l i m i t .

For l o n g - l i v e d a c t i v i t y i n t h e a b s e n c e of non-Poisson e r ror , SC and XD

b o t h decrease as ( Y E V ) - l and as Z I T = m. a c h i e v e d t h e r e f o r e by ( 1 ) d e c r e a s i n g t h e b l a n k r a t e o r i n c r e a s i n g t h e

A lowered LLD ( X D ) c o u l d be

c o u n t i n g time by a f a c t o r of (81.8/3012 = 7.43, o r , (2) i n c r e a s i n g the

p r o d u c t ( Y E V ) by (81.8130) = 2.73. For t h e p r e s e n t example , n e i t h e r Y n o r R B

may be a l t e r e d ( u n l e s s r a d i o c h e m i c a l s e p a r , a t i o n c o u l d be a p p l i e d t o remove

t h e i n t e r f e r i n g a c t i v i t y ) ; and we s h a l l assume t h a t E is f i x e d . I n c r e a s e of

t he e f f e c t i v e volume ( p o s s i b l y - v i a c o n c e n t r a t i o n ) would p r o b a b l y be t h e most

e f f i c i e n t p r o c e d u r e , b u t , f a i l i n g t h a t , t h e c o u n t i n g time might be e x t e n d e d

t o 1487 min ( - 1 d a y ) .

* . . c ) 31ank V a r i a S i l i t y ( s s )

To i l 1 , ; s t r a t e a n o t h e r p o i n t , l e t : is ass:;me t h a t a s e r i e s of 2C r e p l i c a t e

I

2 2 Th7;s, sg/ag = (135/77.L1)2 = 1 . 8 4 , x ? ~ i c h exceeds t h e 25 p e r c e n t ' l l e o f t h e x 2 / d f

I

d i s t r i b , i t : . o n (j:;st s l i g h t l y - s e e 7 i g . 4 . 4 ) . Iv'e m i g h t concl,;de t n a t t h i s I s

d, le t o bad L,;ck ( c h a n c e ) , o r t h a t t h e r e is non-random s t r . ; c t > l r e a s s o c i a t e d

w'l th t h e s e r : e s o f S lanks, o r t h a t t h e r e is a c t x a l l y a d d i t i o n a l ( n o n - P o i s s o n )

v a r i a b i l i t y . For t h i s l a s t ?ss:imed case , we co:i ld sse t s g J F a n d 2 t O U L J ~ f o r

5~ and SD ( b o ; n d ) , r e s ? . ( s e e eq:;2-t'lons 3-5, and n o t e 3 2 ) . That i s

-

- - -

Sc = ts3dr; = 1 . 7 3 ( 1 0 5 ) / 2 = 2'56.9 co:;nts

and

X ~ J = S3/(2.22 Y3VT) = 158.5 p C i / L

[The f a c t o r O ; ~ L / S nay be fo,;nd 'In Tab le 6 accompanying n o t e B2.1 Th,;s, t h e

c r i t i c a l l e v e l is i n f l a t e d b y ro);ghly 4 0 % ) compared t o t h e e a r l i e r ( P o i s s o n )

-

e s t i m a t e [ S c ( P o i s s o n ) = 133.2 coS;nts] ; and t'?e D e t e c t i o n L i m L t I s n e a r l y

d o , i b l e d . (Note t h a t SD a n d x3 a r e both s p p e r l imits.)

d ) R a p i d E s t i m a t i o n of L L D , Us ing 1 n e q : i a l i t y R e l a t i o n s

Fol lowlng E q ' s ( 8 ) a n d (9) i n n o t e 34, we can s e t a l i m i t f o r LLD

d i r e c t l y from an e x p e r i m e n t a l r e s : i l t -- f o r example , from a we'Ighted l e a s t

s c p a r e s (WLS) spectr,;rn d e c o n v o l - i t i o n . C o n t i n u i n g t h e same exam?le , l e t 2s

s';ppose t h a t t h e r e s : ; l t from 'VJLS f i t t i n g was ,. A

x g x = 95.6 32 .3 p c i / ~

I

135

I g n o r i n g s y s t e m a t i c error € o r t h e moment, we would take ;x - > oxo.

T h e r e f o r e ,

X c ’ = 1.645 ox = 53.0 pCi/L - > xc

X D ‘ = 2xc’ = 106 pCi/L - > XD

The r e su l t 2 would t h u s be judged s i g n i f i c a n t ( d e t e c t e d ) , and 106 pCi/L

c o u l d be taken as a n uppe r l i m i t f o r LLD.

e ) C a l i b r a t i o n and S y s t e m a t i c Blank E r r o r

Con t in t l i ng w i t h t he same example , w i t h i n t e r f e r e n c e : B = 6000 c o u n t s ,

T = 200 min , rl = 2, Y = 1 , E = 0.02, and V = 0.5 L , we can u s e Eq. (6) f o r a

d i r e c t estimate of XD.

3.29 OB/:-

YEVT LLD = XD = (0 .0220) BEA + (0.50)

(0.11 has been r e p l a c e d w i t h 0.0220 because w e a re t r e a t i n g a b a s e l i n e rather

t h a n a b l a n k f o r t h e pu rpose of t h i s i l l u s t r a t i o n . )

a c t i v i t y is R~/(2.22 Y E V ) , o r 30 cpm/0.0222 = 1351. pCf/L. Thus , t he LLD,

t a k i n g a l i m i t o f 1 % f o r b a s e l i n e s y s t e m a t i c e r r o r ( e .g . -- d e v i a t i o n from

the assumed s h a p e ) and 10% f o r p o s s i b l e r e l a t i v e e r r o r i n ( Y E V ) , we o b t a i n

The b a s e l i n e e q u i v a l e n t

I (3.29)J(6000) (2) I (1)(0.02)(0.5)(200) LLD = (0.0220)(1351.) + (0.50)

LLD = 29.7 + 90.1

Thus, the P o i s s o n p a r t (90.l/f = 90.1/1.1 = 81.9 pCi /L) i s i n c r e a s e d by 10%

t o a c c o u n t fo r u n c e r t a i n t y i n t h e m u l t i p l i c a t i v e f a c t o r s , p l u s a v e r y

s i g n i f i c a n t 33% (29.7/90.1) t o a c c o u n t f o r p o s s i b l e B u n c e r t a i n t y -- u s i n g

41 = 0.01.

136

-

g ) M u l t i p l e D e t e c t i o n D e c i s i o n s

If we wished t o compensa te f o r t he number o f n u c l i d e s s o u g h t b u t n o t

*

-. I -

, \

f ) L i m i t s f o r LLD Reduc t ion

A f i n i t e h a l f - l i f e ( s u c h as t h e 8.05 d a y s f o r l 3 l I ) and t h e s y s t e m a t i c

e r r o r bounds ( f , 41) b o t h l i m i t the amount o f LLD r e d u c t i o n t h a t can be

accomplished t h r o u g h i n c r e a s e d c o u n t i n g time.

8 1 . 9 p C i / L f o r t = 200 m i n ) , t a k i n g t 1 / 2 = 8.05 d and 41 = 0.01 , f = 1 . 1 0 ,

i t can be shown t h a t w i t h the optimum c o u n t i n g i n t e r v a l (1.8 x t 1 / 2 , o r - 2

w e e k s ) , t h e P o i s s o n component o f LLD is reduced o n l y t o 13 .9 pCi /L , and t h e

added c o n t r i b u t i o n from the systematic e r r o r bound (41) i n the b a s e l i n e

t h e n e q u a l s 47.3 pCi/L. ( S e t t i n g f + 1 . 1 g i v e s a f u r t h e r i n c r e a s e o f 101.1

Thus , fo r t h i s example , i n c r e a s i n g the c o u n t i n g time by a b o u t a f a c t o r o f 100

r e s u l t s i n an o v e r a l l LLD r e d u c t i o n of o n l y - 25%!

I n t he above example ( X D =

found i n a mul t icomponent s p e c t r u m search, w e s h o u l d i n c r e a s e SC (and

t h e r e f o r e n e c e s s a r i l y LLD) from the above v a l u e s . For example , i f j u s t 1 0

spec i f ic peaks were s o u g h t i n a g i v e n s p e c t r u m , and we wished t o m a i n t a i n an

o v e r a l l 5% r i s k of a ( s i n g l e ) fa l se p o s i t i v e , we c o u l d employ Eq. 2-35 t o

c a l c u l a t e t h e needed a d j u s t m e n t i n a and Z I - ~ .

a ' = 1 - ( 1 - 0 . 0 5 ) 0 * 1 = 0.00512

Tha t would be :

Z I - ~ ' is t h u s 2.57. If we were t o s i m i l a r l y decrease t h e fa l se n e g a t i v e r i s k

( 0 1 , b o t h SC and SD (and therefore LLD) would be i n c r e a s e d by the same f a c t o r

2.57/1.645 = 1.56. The r e s u l t i n g XD f o r t h e peak under d i s c u s s i o n would b e ,

XD + 1.56 ( 8 1 . 9 ) = 128 p C i / L

I

137

' i

2. S imple Beta Count ing

C o n s i d e r t h e measurement of 90Sr , where R B = 0.50 cpm, y = 0.85, E =

0.40, and t = 1000 min. ( V is i r re levant f o r t h i s example . ) We must

c o n s i d e r decay d u r i n g c o u n t i n g f o r t h e 64 h r ( t 1 / 2 ) 90)' a c t u a l l y measured;

and we s h a l l take 4~ = 0.05, and f = 1.10 as b e f o r e .

The LLD is g i v e n by Eq. ( 6 ) :

LLD = (0 .11 ) BEA + (0 .50) YEVT

For t h i s example w e s h a l l assume a v e r y l o n g a v e r a g e d background ( A rl = I ) ,

BEA = ( R ~ t ) / ( 2 . 2 2 YEVT), and T = ( l -e -At) /A = 915 min. Thus , YEVT =

( 0 . 8 5 ) ( 0 . 4 0 ) ( 1 ) ( 9 1 5 ) = 311 min, and

500 ] + (0 .50 ) [ 311 1 (2 .22) (31 1 ) ( 3 .29 ) I'%

LLD = (0 .11 ) [ = 0.080 + 0.118 = 0.198 pCi ,

where t h e s y s t e m a t i c e r ror bounds i n t h e b l a n k and m u l t i p l i c a t i v e f a c t o r s (5%

and 101, r e s p . ) a c c o u n t f o r -46% o f t h e t o t a l . Tha t is , w i t h f .+ 1 and -

A .+ 0 , LLD = 3.29 J 5 0 0 / C ( 2 . 2 2 ) ( 3 1 1 ) 1 = 0.106 pCi. The c o r r e s p o n d i n g d e c i s i o n

p o i n t xc is XD/(2f) or 0 .198/2 .20 = 0.090 pCi.

3. Low-Level a-Counting

Assume t h a t a Measurement P r o c e s s f o r 239Pu had t h e f o l l o w i n g

c h a r a c t e r i s t i c s .

RB = 0.01 cpm, E = 0.30, Y = 0.80, t = 1 h r

R e f e r r i n g t o T a b l e 7 , and t a k i n g B = 0.60 counts , w e f i n d yc = 2 counts and

YD = 6.30 counts.

( g r o s s ) were o b s e r v e d , t h e 239Pu would be c o n s i d e r e d I v d e t e c t e d v v .

Tha t i s , i f i n a 60 mln o b s e r v a t i o n more t h a n 2 counts

The LLD is

g i v e n by

(YD - B ) (6.30 - 0.60) X D = - - = 0.18 pCi

2.22(YEVT) 2.22(0.80)(0.30)(60)

I f R B were known t o o n l y 10% (i.e., based on 100 c o u n t s o b s e r v e d ) , we - c o u l d set l i m i t s : B = 0.60 f 0.06 comts , so yc and YD remain unchanged, b u t

6.30 - (0.60 f 0.06)

2.22(0.80)(0.30)(60) X D = = 0.178 f 0.0019

The c o n s e r v a t i v e ( u p p e r ) l i m i t f o r XD t h s s e q u a l s 0.18, pCi .

The above estimates c o u l d , of course, have been o b t a i n e d u s i n g F i g . 7A.

139