A THEORY OF SIGNAL DETECTIONBASED UPON HYPOTHESIS ANALYSIS

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Authors Fobes, James L.

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FOBES, James Lewis, 1946-A THEORY OF SIGNAL DETECTION BASED UPON HYPOTHESIS ANALYSIS.

The University of Arizona, Ph.D., 1975 Psychology, experimental

Xerox University Microfilms , Ann Arbor, Michigan 48106

THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED.

A THEORY OF SIGNAL DETECTION BASED

UPON HYPOTHESIS ANALYSIS

by

James Lewis Fobes

A Dissertation Submitted to the Faculty of the

DEPARTMENT OF PSYCHOLOGY

In Partial Fulfillment of the Requirements For the Degree of

DOCTOR OF PHILOSOPHY

In the Graduate College

THE UNIVERSITY OF ARIZONA

19 7 5

THE UNIVERSITY OF ARIZONA

GRADUATE COLLEGE

I hereby recommend that this dissertation prepared under my

direction by James Lewis Fobes .

entitled A THEORY OF SIGNAL DETECTION BASED UPON

"HYPOTHESIS ANALYSIS

be accepted as fulfilling the dissertation requirement of the

degree of DOCTOR OF PHILOSOPHY

n ̂ f . pL, , H /1 A S~~ f Dissertation Directory Date

\

After inspection of the final copy of the dissertation, the

follovring members of the Final Examination Committee concur in

its approval and recommend its acceptance-.-'*

Hh hf

4/ ihf -y/g/76

This approval and acceptance is contingent on the candidate's

adequate performance and defense of this dissertation at the

final oral examination. The inclusion of this sheet bound into

the library copy of the dissertation is evidence of satisfactory

performance at the final examination.

STATEMENT BY AUTHOR

This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.

Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.

SIGNED:

This dissertation is dedicated to my wife

Jacqueline, and to my major professor James King, in

appreciation of their continuous encouragement and support.

iii

ACKNOWLEDGMENTS

I would like to acknowledge the contributions of:

Dr. James E. King whose assistance was invaluable in all

stages of this dissertation; other major commitee members

Drs. Sigmund Hsiao and Ronald H. Pool; minor committee

members Drs. Terry C. Daniel and Dennis L. Clark; Theresa

Burton and Beth Wenzel who assisted in data collection;

Evan Thomas who made conceptual contributions; Charles

Davison who aided with equipment maintenance; and Robert

Dylan who cautioned that nothing is revealed. This research

was supported by Training Grant MH-112 86 from the United

States Public Health Service.

iv

TABLE OF CONTENTS

Page

LIST OF TABLES vi

LIST OF ILLUSTRATIONS viii

ABSTRACT lx

INTRODUCTION 1

THEORETICAL CONCEPTUALIZATION AND ANALYTICAL

PROCEDURE 9

METHOD 25 Subjects . . . 25 Apparatus . , 25 Procedure 26

RESULTS 29

DISCUSSION 39

APPENDIX A. FORMULAS FOR ESTIMATION OF HYPOTHESES . . 52

APPENDIX B. FORMULAS FOR ESTIMATION OF K 61

REFERENCES 67

y

LIST OF TABLES

Table Page

1. Probability of Detection and Nondetection States for Stimuli Intersected with Attention and Nonattention States, as a Function of Procedure 11

2. Probability of Response Outcome for Detection and Nondetection States as a Function of Procedure 11

3. Probability of Response Outcome for Stimuli Intersected with Attention and Non-attention States, as a Function of Procedure . 12

4. Characteristics of Hypotheses Involving Detection States , . . . . 14

5. Characteristics of Hypotheses not Involving Detection States 16

6. Thirty-Two Unique Problem Sequences 17

7. Signal Level Presentation Order 28

8. Percentage of Correct Responses as a Function of Procedure and Signal Level , 29

9. Proportion of Variance Explained as a Function of Procedure and Signal Level , « , , 34

10. Sensitivity as a Function of Procedure and Signal Level 35

11. Probability of Detection and Nondetection States per Trial Given Attention, as a Function of Procedure and Stimulus 35

12. Sensitivity Estimates Obtained with Two-Alternative Forced-Choice, and Values Predicted from Yes-No, as a Function of Signal Level 37

vi

vii

LIST OF TABLES—Continued

Table Page

13. Probability of Entering an Attention State per Trial, and of Zero, One, Two, and Three Attention States per Problem, as a Function o f Procedure a nd Signal Level . . . . 37

LIST OF ILLUSTRATIONS

Figure Page

1. Proportional Strengths of Hypotheses Involving Detection States, as a Function of Procedure and Signal Level 30

2. Proportional Strengths of Hypotheses not Involving Detection States, as a Function of Procedure and Signal Level 31

viii

ABSTRACT

This dissertation consisted of the description and

^application of a new theory of detection processes. The

sensory portion featured detection of noise as well as

signal stimuli and an absolute threshold for recognition of

these stimuli. This recognition threshold for noise or

signal was never exceeded by the other stimulus alone. It

was also assumed that an observer either attended or failed

to attend to the stimulation presented on each trial. When

the observer entered an attention state, either a detection

state or a nondetection state resulted. If a nonattention

state occurred, a detection state was precluded.

The theory's response portion assumed that the com

bination of an attention and detection state resulted in

correct responding, the combination of an attention and non-

detection state resulted in random responding, and that a

nonattention state resulted in nonrandom responding that was

determined by a response bias. The false alarms that

occurred were considered to be independent of the sensory

mechanism and to have resulted from either random responding

accompanying the combination of a nondetection and attention

state or from a response bias.

Therefore for the yes-no procedure model, an

attention state in the presence of a signal resulted in

ix

X

either a detection of the signal with correct responding or

a nondetection state accompanied by random responding.

Likewise, when an attention state occurred in the presence

of noise, the observer entered a detect noise state or a

nondetect state. For the two-alternative forced-choice pro

cedure model, an attention state resulted in either a detec

tion of the difference between signal and noise with correct

responding or a nondetection state accompanied by random

responding. For both psychophysical procedures, a non-

attention state always resulted in responding determined by

a response bias.

This theory was applied to data obtained by pre

senting three-trial brightness discrimination problems of

varying difficulty to capuchin monkeys. The outcomes of

these problems were assumed to be combinations of states

attention and detection, attention and nondetection, or non-

attention that occurred on each trial of a problem. These

states resulted in manifestations that reflected various

modes of systematic responding which permitted estimation of

their relative occurrence with an adaptation of the hypoth

esis and analysis technique.

This analytical procedure was used to determine the

strength of three modes of detection and four sequentially

dependent response biases. These values were then used to

determine the sensitivity index that was expressed in terms

of the probability of detection given attention. This

sensitivity index was considered to be independent of non-

sensory response factors and its method of calculation

included estimation of the probability of an attention state

per trial as well as the probabilities of the various

possible combinations of attention and nonattention states

on three-trial problems.

Support was provided for this theory which featured

a variety of information that was not included in previous

approaches. Specifically, a more detailed description of

detection processes was provided that included estimation of

the amount of responding due to specific nonsensory response

biases. This conceptualization formed the basis for non-

parametric estimation of sequential biases whose effects

were isolated from the nonparametric sensitivity index.

Further, the theory was not tied to a specific family of

receiver operating characteristic curves. A concept of

sensory thresholds was also included and a procedure was

suggested for determination of absolute recognition

thresholds that are not confounded with the effects of

response bias.

INTRODUCTION

Various classical psychophysical methods have

afforded considerable information concerning the relation

ship between perception and physical characteristics of

stimulation (Guilford, 1954). Typically, however, the usage

of these techniques has emphasized only the percentage of

correct responses with each stimulus value. With such pro

cedures, the reputed sensitivity of the subject, based upon

this single variable, confounded detection with response

biases. While variability and reduced discrimination

accuracy due to nonsensory response biasing factors have

been recognized for some time, attempts to deal with such

biases have generally been inadequate, particularly so with

single stimulus psychometric methods. Frequently, the

attempt to control response biases consisted of either

extended training of subjects or experimental designs such

as counterbalancing (Engen, 1971).

A common psychophysical approach to estimating

sensitivity involved detection tasks that favored almost

exclusive presentation of a stimulus to be detected, which

was designated as signal (S), while trials containing only a

stimulus designated as noise (N) were infrequently presented.

When subjects incorrectly reported the presence of a S on

the few catch trials containing only N that were presented,

1

subjects were typically advised to pay increased attention.

Such false alarms were typically not included in data

analysis. A more recent procedure involved more frequent

presentation of trials containing only a N stimulus and the

proportion of false alarms on these trials was utilized in

an attempt to obtain the proportion of detections corrected

for guessing (Blackwell, 1953, 1963) . However, the pro

cedure for correction resulted in sensitivity estimates that

were not independent of response biases; the assumption of

statistical independence between the proportion of detec

tions corrected for guessing and false alarms was unjusti

fied (Green and Swets, 1966; Swets, 1973).

Techniques for the separation of certain response

biases and the sensitivity estimate were considerably ad

vanced by a theory of decision making based upon the trans

lation of that portion of statistical decision theory

dealing with situations in which a choice is made between

two alternatives on the basis of an observed event (Smith

and Wilson, 1953; Tanner and Swets, 1954). This approach

conceptualized the detection process as a problem of S-to-N

ratio (Tanner and Swets, 1954; Swets, Tanner, and Birdsall,

1961; Green and Swets, 1966; Swets, 1973). When a sensory

event occurred that was due to either N or to S, the observer

was thought to choose between normal distributions of N and

S along which the effects of stimulation continuously varied

as a single dimension of increasing magnitude. Therefore,

3

the sensory effect of an observation was thought to arise,

with a specific probability, from either the N or the S

probability density function.

It was further assumed that observers assigned con

ditional probabilities that a particular sensory effect

arose from N and that it arose from S. The continuum of

sensory effects was thus characterized as a continuum of

likelihood ratio values, each of which expressed the likeli

hood that a particular observation arose from the S distri

bution relative to the likelihood that it arose from the N

distribution. Thus, on the basis of the likelihood ratios

that varied in magnitude on the basis of their sensory

effect, or some monotonic function of the ratios, observers

were thought to decide which of two populations was sampled.

The actual choice between populations was considered to be

determined by a decision rule involving whether or not a

particular observation's likelihood ratio value exceeded

some fixed response criterion or response threshold value

along the continuum of increasing likelihood ratio values.

For example, if the likelihood ratio value of the sample was

below the criterion likelihood ratio value, the subject

indicated that the observation came from the N distribution;

samples with corresponding likelihood ratio values above the

critical value resulted in an indication that the observa

tion was from the S distribution.

4

The d' index of S detectability or sensitivity was

defined as the difference between the means of the N and S

density functions, expressed in terms of standard deviation.

Both this d1 sensitivity estimate and the response threshold

value (3) were estimated with the conditional probability of

saying S given (|) S and of saying S|N. For d1, a table of

normal curve functions gave the z score corresponding to the

conditional probability of saying S|N. This z value indi

cated the distance along the dimension of sensory effects

that the criterion was above the mean of the N density

function. Similarly, the z score corresponding to the con

ditional probability of saying s|s indicated the distance

that the criterion was below the mean of the S density

function. The value of d' equaled the combination of these

two z scores and was computable regardless of the location

of the criterion along the dimension of sensory effects; d'

was independent of the location of the criterion. The

measure 3 was the ratio of the ordinate of the S density

function to the ordinate of the N density function, at the

location of the response criterion. The numerator and de

nominator in this ratio were estimated from the two condi

tional probabilities (S|S and S|N) with a table of ordinates

of the normal curve.

Thus, d' was considered to be a function of de

tectability and monotonically related to S strength. For a

given S strength, the location of the variable decision

5

criterion along the continuum of likelihood ratio values has

been found to be affected by such things as the information

an observer has concerning relevant variables of the situa

tion and situational goals. However, the method of deter

mination of d' resulted in an estimate of sensitivity that

was relatively unaffected by certain types of biases; d' was

relatively independent of: (1) instructions (response

criteria from "be careful" to "be careless," or simultaneous

usage of multiple response criteria), (2) the a priori

probability of a S being presented (expectations), (3) the

values and costs associated with various decision outcomes

(motivation), and (4) the psychophysical procedure used to

estimate sensitivity (single stimulus or forced-choice).

However, the generality of these findings has been ques

tioned (Pike, 1973).

In addition to these biases, various types of

sequential response dependencies have been reported (Senders

and Soward, 1952; Verplank, Collier, and Cotton, 1952;

Howard and Bulmer, 1956; Speeth and Mathews, 1961;

Carterette and Wyman, 1963; Kinchla, 1964; Parducci, 1964;

Parducci and Sandusky, 1965; Haller, 1969; Baumstiler, 1970;

Sandusky, 1971; Sandusky and Ahumada, 1971). The lack of

response independence between trials can reasonably be

expected to affect the assumed stability of the response

criterion and accordingly the estimated value of d' (Pastore

and Scheirer, 1974). While signal detection theory (SDT)

6

has provided for an estimate of sensitivity that was rela

tively unaffected by certain response biases that affect the

response criterion, it did not include a technique for

dealing with sequential dependencies nor did it estimate the

strength of specific response biases (Green and Swets, 1966).

Further, this approach to detection systematically

excluded a sensory threshold with a lower limit on sensi

tivity. Thus, rather than assisting in the determination of

sensory thresholds unaffected by response biases, some

exponents of the SDT under discussion concluded that a

determination of the level at which a threshold may possibly

exist is neither critical nor useful. Such a view ignores

the intent of a considerable body of literature.

This dissertation consists of the description of a

proposed theory of sensitivity as well as its application

with two psychophysical procedures. The sensory portion of

the theory includes detecting or not detecting N stimuli as

well as S stimuli, and the occurrences of an attention (A)

state is though to be accompanied by either a detection (D)

state or a nondetection (D) state. If a nonattention (A)

state is entered, a D state cannot occur. A sensory

threshold for recognition of both N and S is assumed, with

continuous gradation of sensation above the threshold and

no gradation below. The recognition threshold for N or S

is thought to be never exceeded by the other stimulus alone.

7

The response portion of the theory assumes that the

intersection (A) of an A and D state results in correct

responding, A A D states result in random responding, and a

A state results in responding that is determined by a

response bias. False alarms are considered to be independent

of the sensory mechanism and result from either random re

sponding accompanying A A D states or from a .A state with

its ensuing response bias responding.

As an illustration of the proposed theory, data from

three-trial brightness discrimination problems, presented by

two psychophysical procedures, were analyzed by a hypothesis

(H) analysis technique similar to Levine's (1965) Method II.

This analytical technique was used to determine the relative

proportion of four response biases and three modes of D

states. The strengths of these hypotheses (Hs), in combina

tion with other values to be described, were then used to

estimate the sensitivity parameter K, the probability of

D|A. The calculation procedure to determine sensitivity

also resulted in estimation of the probabilities of partic

ular combinations of A, A, D, and D states on three-trial

problems. Further, estimation of all measures was accom

plished without assumptions concerning possible probability

distributions of the sensory effects of S or N.

An important advantage to the proposed approach

resulted from the inclusion of sequential responding infor

mation and nonsensory response .bias emerged as

8

multidimensional. These dimensions included: (a) the

tendency to perseverate with yes (Y) or no (No) responses

in yes-no (Y-N) tasks, or with a particular position in two-

alternative forced-choice (2AFC) tasks, (b) the tendency to

alternate responses, as well as (c) the win-stay:lose-shift

and win-shift:lose-stay syndromes. These particular non-

sensory factors are considered to be especially appropriate

for subhuman observers. The type and amount of these

response biases were specifically estimated and their

effects were isolated from the sensitivity estimate. A

technique is also suggested for estimation of sensory recog

nition thresholds which are not confounded with the effects

of response bias.

The theory's empirically testable consequences that

have been investigated indicate support for the theory.

This is noteworthy in that the theory includes a high-

threshold concept of absolute sensory recognition thresholds.

THEORETICAL CONCEPTUALIZATION AND ANALYTICAL PROCEDURE

The proposed theory of sensitivity assumes that S as

well as N stimuli may be detected or not detected, and a

momentarily varying, absolute sensory threshold for the

recognition of S or N stimuli is proposed. The recognition

threshold for N and S stimuli is thought to feature con

tinuous gradation of sensation above the threshold and no

gradation below. The recognition threshold for S is thought

to be never exceeded by N alone and the recognition threshold

for N is thought to be never exceeded by S alone. Thus,

observers can detect or not detect only that which is

presented.

The sensory portion of the theory also assumes that

on each trial subjects enter a state of either A or A to

stimulation. For the Y-N model, the occurrence of an A

state in the presence of S is thought to result in either a

Dg state or a D state. Likewise, the occurrence of an A

state in the presence of N results in the subject entering

state DN or state D. It is assumed that the probability

(P) of (D IS A A) = P(D |N A A) = K,. In the model for ̂ IN -L

2AFC, a D state is entered when the subject detects the

difference between the S and N conditions, with the

P(D|A) - ̂ 2' For both models, the occurrence of an A state

9

10

precludes a D state. A proposed relationship between and

K2 will be discussed later.

These assumptions concerning the sensory portion of

the theory are presented in Table 1 which depicts the

probability of a D or D state on each trial given a

particular stimulus condition A A or A states. The Y-N

technique involves presentation of only one stimulus (S or

N) per trial. Therefore, subjects may evidence Dg A A or

D A A when S is presented and DN A A or D A A when N is

presented. Conversely, the A state always results in a D

state. The 2AFC procedure entails presentation of both

stimuli (S and N) per trial and, with the spatial 2AFC used

(King and Fobes, 1973), S occurred on either the right (R)

or left (L).side. With this procedure, subjects may or may

not enter a state of D relative to the difference between S

and N. The A state in 2AFC also results in a D state.

Table 2 depicts the response aspects of the theory.

It is assumed that a D state always results in a correct

response and that a D state always results in random

responding. An A state is thought to always result in a D

state that is accompanied by nonrandom responding which is

determined by a response bias. For both D A A and A

states, outcomes are uncorrelated with S and N and the

probability of a correct response is one-half when the a

priori probability of S is one half. False alarms (saying

S|N) are considered to be independent of the sensory

11

Table 1. Probability of Detection and Nondetection States for Stimuli Intersected with Attention and Non-attention States, as a Function of Procedure

Y-N 2AFC

State State

Event DS °N D Event Ddifference D

S A A K1 0 1 - K1 s (L) A A K2 1 K2

N A A 0 K1 1 - K1 s (R) A A k2 1 - K2

S A A 0 0 1 s (L) A A 0 1

N A A 0 0 1 s (R) A A 0 1

Table 2. Probability of Response Outcome for Detection and Nondetection States as a Function of Procedure

Y-N 2AFC

State Correct Incorrect State Correct Incorrect

DS 1 0 Ddifference 1 0

dn 1 0

D 1/2 1/2 D 1/2 1/2

12

mechanism and are assumed to arise from either random

responding accompanying states D A A or from a response bias

accompanying state A.

Table 3 depicts the probabilities of response out

comes for a particular stimulus A A as well as A. These

probabilities are based upon the results of matrix multi

plication of Table 1 times Table 2. When an A state occurs,

a correct response results with probability (K + l)/2 and an

incorrect response occurs with probability (1 - K)/2. Again,

an A state always results in a D state with outcomes un-

correlated with the presence or absence of S. The proba

bility of a correct or incorrect response is one-half when

the a priori probability of S equals one-half. It should

also be noted that the values of K determined with the Y-N

and 2AFC procedures will generally not be the same.

Table 3. Probability of Response Outcome for Stimuli Intersected with Attention and Nonattention States, as a Function of Procedure

Y-N 2AFC

Outcomes

Correct Incorrect

S A A S (L) A A (K + l)/2 (1 - K)/2

N A A S (R) A A (K + l)/2 (1 - K)/2

S A A s (L) A A 1/2 1/2

N A A S (R) A A 1/2 1/2

13

The results of three-trial problems are assumed to

be combinations of states D A A, D A A, and A that occur on

each trial. Further, these states result in manifestations

that reflect various modes of systematic responding which

can be referred to as Hs. The unfortunate nature of the

term H with respect to unintentional connotations is

acknowledged, but it is used in order to be consistent with

Levine (1959, 1965). Thus, Hs are considered to be

sequences of internal states that determine behavior and

are manifested by a specificable pattern of responses to

selected patterns of stimulation. As such, the H is <

regarded as a dependent variable.

The Hs that include at least one D state on a

three-trial problem are listed in Table 4. If D states

occur on all three trials of a problem, the only possible

manifestation is three correct responses (+++), as indi

cated for H A in Table 4. With a D state on exactly two

trials, two correct responses result on these trials. The

outcomes of H B will also include correct as well as

incorrect responses (-) for the one trial on which states

D A A or A occur. Therefore, outcomes to sequences involv

ing two D states include: +++, ++-, +-+, and - + +.

Likewise for H C, exactly one D state occurs and the out

comes include correct and incorrect responses for the two

trials on which states D A A or A occur. Possible

14

Table 4. Characteristics of Hypotheses Involving Detection States

State H Sequence Manifestation

A: 3 Ds D D D + + +

B: 2 Ds D D DAA + + + or + + —

or D D A.

D DAA D + + + or + - +

or DA D.

DAA D D + + + or - + +

or A D D

C: 1 D D DAA DAA, + + + , + — + ,

D A A , + + - or + - -

D A DAA,

or D DAA A.

DAA D DAA, + + + / - + + ,

A D A , + + - or - + -

A D DAA,

or DAA D A.

DAA DAA D, + + + , — + + ,

A A D, + - + or - - +

A DAA D,

or DAA A D

15

manifestations of C include all sequences except - - - ,

since at least one correct response results with the single

D state.

The Hs that do not involve a D state on any of a

problem's trials are depicted in Table 5. Response bias Hs

are defined when an A state is entered on all three trials

of a problem and the residual H is defined by all other

sequences of A and Astates not involving D. The manifesta

tions of these Hs are specified in terms of response

sequences, where I is the type of response (Y or No) on the

f irst trial of a problem and 0 is the other type of response.

The results of three-trial problems are entirely

specified by the stimulus, response, and outcome sequences

depicted in.Table 6. The eight possible stimulus sequences

can be grouped into symmetrical pairs, viz, SSS and NNN,

SSN and NNS, SNS and NSN, as well as SNN and NSS. In

Table 6 these pairs are represented by the sequences XXX,

XXV, XVX, and XW, where X is the stimulus (S or N) on the

f irst trial and-V is the other stimulus. The outcome of a

particular stimulus sequence, in terms of being correct or

incorrect, is determined by the response sequence (III, 110,

101, or 100) evidenced. The lower case letters in each cell

in Table 6 are the symbols that indicate the Hs that can

result in each particular outcome given those stimulus and

response sequences. The number in each cell will be used

for reference purposes.

Table 5. Characteristics of Hypotheses not Involving Detection States

H Definition State Sequence Manifestation

D: Triple Response Repetition

Type of Response Stays the Same for Three Consecutive Trials

A A A I I I

E: Double Response Repetition

Type of Response Stays the Same for Two Consecutive Trials

A A A I I 0 or I 0 0

F: Win-Stay: Lose-Shift

Response Type the Same as Preceding Rewarded Trial

A A A I+I+I, I+I-O

1-0+0 or I-O-I

G: Win-Shift: Lose-Stay

Response Type Opposite from Preceding Rewarded Trial

A A A I+O+I, 1+0-0,

I-I+O or I-I-I

R: Residual Sequences Not Defined Above

A DAA

DAA DAA DAA,

DAA, DAA A DAA,

All Sequences

r

DAA DAA A, A A DAA,

A DAA A or DAA A A

Table 6. Thirty-Two Unique Problem Sequences

17

Stimulus Sequence

Response Sequence Stimulus Sequence I I I I I 0 I 0 I 10 0

X X X 1) d g r

2) c e

- - + r

3) - + c r

- 4) — + + b 2c e f r

5) + + + a 3b 3c d f r

6) b 2c

+ + -e r

7) + -b 2c r

+ 8) + - -c e g r

X X V 9) - - + c d g r

10) e r

_ _ _ 11) - + b 2c r

+ 12) - + -c e f r

13) + + -b 2c d f r

14) a 3b r

+ + + , 3c e

15) + -c r

16) + - + b 2c e g r

X V X 17) - + -c d r

18) b 2c r

- + + e g

19) - -f r

- 20) + c e r

21) + - + b 2c d r

22) c e

+ - -f r

23) + + a 3b 3c r

+ g

2 4 ) + + — b 2c e r

X V V 25) - + + b 2c d r

26) c e

- + -g r

27) - -c f r

+ 28) e r

29) + - -c d r

30) b 2c

+ - + e f

31) + + -b 2c g r

32) + + + a 3b 3c e

18

Table 6 indicates that a given response sequence is

always consistent with more than a single H. For example,

the response sequence I-I-I- is consistent with Hs d, g, and

r in cell number one. Although a given response sequence is

always consistent with more than one H, estimation of the

strength of each H can be accomplished by a technique

similar to Levine's (1965) Method II. However, the particu

lar Hs presented here, as well as their solution equations,

differ from Levine's. An additional important difference

from Levine's procedure is the present use of weighting

coefficients in Table 6 which reflect the l ikelihoods of

different response sequences when one or two D states occur

on a problem. As indicated for H B in Table 4, when two D

states occur on a problem, either two or three correct out

comes can result depending upon whether a correct response

occurs on the trial involving states D A A or A. Therefore,

the possible manifestations are: +++, ++-, +-+, and

- + +. If all manifestations of two D states are assumed to

be equally l ikely, then + + + is three times as l ikely to

occur as any other manifestation of B. Therefore, b is

assigned a coefficient of three in those cells containing

+ + + and assigned a coefficient of one in all cells con

taining exactly two correct outcomes. The value of b is

equal to the probability that H B occurs and is manifested

by, for example, +•

19

Likewise for C, al l outcomes with exactly one D

state are possible and include: + + +, + + - , + - +, - + +,

- - + , + - - , a n d - + T h e + + + s e q u e n c e i s a s s i g n e d a

coefficient of three since it is three times as l ikely as

- - or + - given H C. Sequences ++-,+-+,

and - + + are similarly assigned a coefficient of two since

they are twice as l ikely as +, - + - , or + - - .

The analysis of systematic patterns of response

sequences begins with categorization of the frequencies

with which each of the 32 unique sequences in Table 6

occurs. The relationship between these resulting 32

frequencies and the probability of each H is determined by

the assumptions that: (a) the Hs are mutually exclusive,

(b) their effects are additive, and (c) al l Hs whose

probability exceeds zero are included. Summing across any

row in Table 6 results in the statement

a + 6b + 12c + 2d + 4e + 2f + 2g + 8r = 1, (1)

where a = A, 6b = B, 12c = C, 2d = D, 4e = E, 2f = F, 2g =

G, and 8r = R, Line (1) may also be stated as

A + B + C + D + E + F + G + R = l , ( 2 )

Therefore, each term in the two above equations may be

interpreted as a probability or proportion with the terms in

Equation (2) being the proportions of three*-trial sequences

accounted for by the corresponding H.

20

The relative frequencies with which each of the 32

three-trial sequences occur can be used to estimate the

probabilit ies in Equations (1) and (2). For example, the

theoretical probability of three incorrect outcomes given

the stimulus sequence XXX (cell number one) equals the

sum of the probabilit ies of the associated Hs since the Hs

are assumed to be mutually exclusive and additive.

P t(- - - |X X X) = d + g + r. (3)

The theoretical probability of all 32 sequences in Table 6

may similarly be expressed by a particular l inear combina

tion of certain H probabilit ies.

The obtained frequencies of each three-trial

sequence may then be used to estimate the probability of

each H. Continuing with the sequence from cell number one,

P (- - - | X X X) = n(X- X- X-)/n(X X X), (4)

where n(X-X-X-) is the frequence with which X~X*-X- occurs

and n(X X X) is the frequency with which XXX occurs. To

estimate the theoretical from the observed probability, i t

follows from Equations (3) and (4) that

d + g + r = n(X- X- X-)/n(X X X). (5)

Levine's (1965) Method II may then be used to

evaluate the probability of Hs in such a way as to minimize

the sum of the squared differences between the theoretical

and obtained cell frequencies (see Levine, 1959). While a

21

complete derivation of H analysis solution formulas is

presented in Appendix A, a brief description of the pro

cedure for obtaining solutions is i l lustrated by the solu

tion of H A.

From the 3 2 equations, those containing A include

(.cell #5) a+3b + 3c + d + f + r = Q^,

(cell #14) a+3b + 3c + e + r =

(cell #23) a+3b+3c+g+r= , and

(cell #32) a + 3b + 3c + e + r = (6)

where each Qa is the observed proportion of the particular

outcome sequence. It should be noted that in the present

case each stimulus sequence was presented equally often.

These equations in l ine (6) involving A totalled

EQ = 4a + 12b + 12c +d+2e+f+g+ 4r. (7) a.

Applying Equation CI) and solving for a = A results

in

2EQa = 8a + 24b + 24c + (1 - a - 6b - 12c) , ( .8)

which reduces to

2£Q - 1 = 7a + 18b + 12c a

(9)

which may be rewritten

a = (2EQa - 18b - 12c - l) /7 CIO)

22

The solution in l ine (10) for A is thus found to be a

function of Hs B and C. Applying Equation (1) and solving

for B and C in an analogous manner results in

b = (l /2EQb - a - 6c - l) /6 (11)

and

c = (2EQc - a - 6b - 7)/12. (12)

These three l inear equations (10, 11, 12) with three

unknowns may then be solved by Cramer's rule with these

resulting unique solutions

A = a = l /2ZQa - l/2IQb + l /2£Q c - 1, (13)

B = 6b = 6(l/4£Qb - l/12IQa - 5/12EQ c - 1), (14)

C = 12c = 12 (1/3EQ - 1/12XQ, - 1). (15) —~ C JD

The solution for D is a function of the EQ^ as well

as E. These two equations with two unknowns may be simul

taneously solved with the unique solutions

D = 2d = 2(l/4ZQd + l /8IQ e - 1/2), (16)

and

E = 4e = 4(3/16EQ e + l /8ZQd - 1/2). (17)

The solutions for F and G are similarly a function

of each other with the unique solutions

23

F = 2f = 2(3/16ZQ f + 1/16ZQ - 1/4) (18)

and

G = 2g = 2(3/16EQ + 1/16ZQ- - 1/4). y

(19)

Finally, from Equation (2), R is simply a residual,

Once the H strengths are obtained, they may then be

used to estimate the sensitivity parameter K, the P(D|A).

This is accomplished by solution of the f ive equations

l isted below which contain five unknowns (K, M^, t , and

T ) . V a l u e s f o r A, B, C, D, E, F, G, and R are H strengths

and the remaining terms are defined as:

T = P (Ah A An + 1 A An + 2) , K = P (D | A) , M1 = P (A A A)

+ P (A A A) + P (A A A) , M2 = P (A A A) + P (A A A)

+ P (A A A) , t = P(An) , (J) = P(A|D) = (t - tK) / (1 - tK) ,

0 = P(A|D) = (1 - t) /( l - tK) , hits = P(Y|S), and

false alarms = P(Y|N).

A = TK3 , (1)

B = 3TK2(1 - K) + M-^K2 , (2)

C = 3TK (1 - K) 2 + 2M ] ,K(1 - K) + M2K, (3)

R = L = ( A + B + C + D + E + F + G ) . ( 2 0 )

24

T = A + B ( p + C ( p 2 + R{{l-{ [30(j)]/ l - [ (D + E + F + G) /

( D + E + F + G + R ) ] } } } , ( 4 )

Hits - False Alarms = tK. (5)

The derivation of these equations is contained in Appendix B.

Since the equations containing K are l inear when K is

constant, a computer may be used to obtain solutions for

values of K from 0.0 to 1.0 for Equations (1), (2), (3), and (5) .

While this set has an infinite number of solutions, only

one particular solution gives a value of T in Equations

(1), (2), and (3) that is equal to the value of T in

Equation (4), for a particular value of K. This value of

K defined the solution selected.

METHOD

Subj ects

Four adult male capuchin monkeys (Cebus apella)

served as subjects. These feral animals had extensive prior

experience with sameness-difference learning-set (King and

Fobes, in press, 1975) and 2AFC brightness discriminations

( K i n g a n d F o b e s , 1 9 7 3 ) .

Apparatus

The apparatus featured stimulus presentation behind

a one-way-screen (7.5 by 6.5 cm) through which subjects

viewed stimuli only during i l lumination behind the screen

with a 4 0-watt frosted incandescent l ight. Stimuli con

sisted of block mounted pigmented papers (3.8 by 3.8 cm)

previously presented in the 2AFC investigation of brightness

discrimination (King and Fobes, 1973).

Stimuli became visible at the start of each trial

and i l lumination automatically ceased following: (a) a

correct Y response of interrupting a photocell beam, which

was recessed and centered in front of the screen, before

f ive seconds had elapsed (a "go" condition); (b) a correct

No response of not breaking the photocell beam for five

seconds (a "no-go" condition); or (c) an incorrect response.

Correct responses were followed by a one second tone and

dispensation of one-half raisin into a receptacle mounted

below the screen,

25

Procedure

Sixteen three-trial problems a day were presented

throughout pretraining, training, and testing. Eight

stimulus sequences of S and N were used throughout and each

of these stimulus sequences (viz. SSS, NNN, SSN, NNS, SNS,

NSN, SNN, and NSS) appeared twice daily in random order with

the restriction that no stimulus was presented for more than

three consecutive trials. The S condition consisted of a

grey stimulus and a white stimulus and the N condition con

dition consisted of a pair of white stimuli . Subjects were

pretrained by being rewarded for a Y response to a black

next to a white stimulus and for a No response to two side-

by-side white stimuli . This Y-N pretraining continued until

subjects achieved 90 per cent correct responding with both

types of responses for two consecutive days,

Subjects were then trained with a t itration pro

cedure that involved the same series of stimuli (white to

dark grey in 60 increments) previously presented in the

2AFC phase. This training (King and Fobes, 1973) began with

a brightness discrimination between the darkest grey-white

and white-white. After three consecutively correct

responses of a Y to grey-white with a particular intensity

of grey, the intensity of grey was increased by one step;

failure to achieve three consecutively correct responses

resulted in a one step decrease in the intensity of the

grey. Subjects were thus tested to determine the grey

27

stimulus intensity around which responding stabilized.

Three intensity steps below this point of stable responding

was arbitrarily selected as the base intensity value of

grey for each individual subject. The log^Q intensity

difference between each subject's base grey and white

was then divided by three in order to create three sub

jectively equal proportions (Guilford, 1954), and a third

of the intensity difference was selected as A grey for that

subject. Varying around a mean A intensity of 0.062 foot/

candle, the intensity values of three previously presented

greys that were included in the present investigation were:

(a) 1/2A, (b) A, and (c) 2A. Since the base value was

individually determined for each subject the actual inten

sity of A differed among subjects.

In the present investigation, this t itration pro

cedure was presented as a training task with a Y-N pro

cedure. To assure that Y-N performance had stabilized,

t itration continued until each subject responded correctly

with a Y to grey-white and a No to white-white on a problem

consisting of a discrimination between its 2A grey and white

versus white and white.

Testing consisted of three phases. In each phase,

subjects were tested with one of the three (1/2A, A, and 2A)

grey-white S levels and white-white N for 12 days. Each

phase was preceded by two days of training with the

particular intensity of grey to be tested, The order in

28

which each of these grey-white S conditions was presented is

depicted in Table 7 and was determined for each subject by

random assignment of one of the six permutations of three

quantities.

Table 7. Signal Level Presentation Order

Phase

Subject 1 2 3

1 1/2A 2A A

2 2A A 1/2A

3 A 2A 1/2A

4 A 1/2A 2A

Thus, on each trial subjects were presented with one

of two stimulus alternatives and could respond with one of

two response alternatives. The S consisted of a condition

wherein white was presented next to one of three grey

stimuli that varied in brightness. The appropriate response

to indicate S detection was a Y response of interrupting a

photocell beam within five seconds. The N condition con

sisted of white-white stimuli and a No response of not

breaking the beam before f ive seconds had elapsed indicated

N detection.

RESULTS

Table 8 depicts the percentage of correct responses

with each S intensity level of grey for both the present Y-N

procedure and the previous 2AFC technique (King and Fobes,

1973). Percentages of correct responses were virtually

identical for the two procedures and were higher for A and

2A than for the 1/2A condition.

Table 8. Percentage of Correct Responses as a Function of Procedure and Signal Level

Y-N 2AFC

1/2A ' A 2A 1/2A A 2A

74% 83% 83% 72% 80% 81%

Figures 1 and 2 depict the proportional strength of

Hs which was determined with the computational formulas

presented in the Theoretical Conceptualization and Analyti

cal Procedure Section as well as in Appendix A. Figure 1

depicts the obtained proportion of Hs that involved one or

more D states in a three-trial sequence, as a function of

procedure and S level. With both procedures, the proportion

of three D states per problem (A) increased as the task

30

Y-N 2AFC

.5 r

.4

x I— g 3 LU cr h~ CO

< .2 O I— QC O Q_

§ J Q_

0

0 O B D • • c ®

-•

1

1/2 A A

BRIGHTNESS

2A

Figure 1. Proportional Strengths of Hypotheses Involving Detection States, as a Function of Procedure and Signal Level

31

.5

4

X I— -2 o .O -Z. LLI or h-i f )

< 2 o i-a: o Q_ 0 01 Q_

. 2

Y-N 2 AFC • 9 D • • O O E • •

• — — p S

O -o G

A A R A -A

1 1/2 A A

BRIGHTNESS 2A

Figure 2. Proportional Strengths of Hypotheses not Involving Detection States, as a Function of Procedure and Signal Level

32

became easier with decreasing intensities of grey, the

proportion of two D states per problem (B) showed an

inverted U shaped curve and the proportion for a single D

state per problem (C) showed a U shaped curve.

Figure 2 depicts the obtained proportions of Hs not

involving any D states, as a function of procedure and S

level. The triple response repetition (D) and residual (R)

Hs accounted for the bulk of these D state manifestations.

The proportions for double response repetition (E), win-

stay : lose-shift (F) , and win-shift:lose-stay (G) were

negligible. The D H displayed a U shaped curve that was

more pronounced with the Y-N technique and the other

prominent H, R, decreased as the brightness of grey

decreased.

In Table 5, E was defined as response repetition for

two consecutive trials of a problem (I00 or 110). An addi

tional type of sequential dependency consists of response

alternation (101). While D and response repetition or

alternation (E2) can be estimated by H analysis, no unique

solution exists for D with both response repetition and

alternation. Therefore, D and response alternation were

solved assuming response repetition to be zero while D and

response repetition were solved assuming alternation to be

zero. Response repetition was found to be of greater

magnitude than alternation and is therefore presented here.

It should be further noted that formulas for D and

33

alternation differ from those presented for D and response

repetition.

To check the validity of the theoretical approach

underlying the H analysis, predicted frequencies were

calculated for each of the 3 2 unique combinations of

stimulus, response, and outcome sequences in Table 6. These

frequencies were based upon the overall strength of each H

during the testing by a given procedure with a particular

S level (see Levine, 1965) . A measure of the amount of

variability among the 32 obtained cell frequencies accounted

for by the Hs in the model was obtained from the following

formula, where CK and are the observed and predicted

proportions of cell i , and M is the mean of the observed

cell proportions.

3 2 1 - 2 ( 0 i - P i ) 2 / ( 0 i - M ) 2 .

i=l

This statistic is similar to Levine's (1965) Proportion of

Variance Explained (P.V.E.) but differs sl ightly since

Levine obtained the predicted frequencies by his Method I in

such a way that each predicted frequency was based upon data

which did not include the corresponding obtained frequency.

Both Levine's P,V,E, and this modified version are thought

2 to be fairly comparable to r_ and thus give an indication

of the amount of between cell variance predictable by the

a d d i t i v e e f f e c t s o f t h e H s m e a s u r e d ( L e v i n e , 1 9 5 9 ) .

34

Table 9 depicts the P.V.E. values determined. These high

values are evidence that all nonnegligible l is were included

in the analysis and that the Hs combined additively to

determine the frequency of each of the 32 possible combina

tions depicted in Table 6.

Table 9. Proportion of Variance Explained as a Function of Procedure and Signal Level

Y-N 2AFC

1/2 A A 2A 1/2A A 2A

. 9 7 . 9 9 . 9 9 . 9 7 . 9 9 . 9 8

The computed strengths of the Hs were then used in

the K estimation formulas presented in the Theoretical

Conceptualization and Analytical Procedure Section as well

as in Appendix B. The values for K determined as a

function of procedure and brightness level of S are

depicted in Table 10. For both procedures, K increased as

the discriminations became easier.

In order to relate K resulting from a 2AFC pro

cedure to K estimated by a Y-N task in such a way as to be

able to predict one from the other, all possible outcomes

for an attentional trial were considered as depicted in

Table 11, Values in this table are based upon the

35

Table 10. Sensitivity as a Function of Procedure and Signal Level

Y-N 2AFC

1/2A A 2A 1/2A A 2A

. 8 7 . 9 1 1 . 0 0 . 8 2 . 8 7 . 9 5

Table 11. Probability of Detection and Nondetection States per Trial Given Attention, as a Function of Procedure and Stimulus

y-n 2AFC

S n Probability S n Probability

DS |a K1 Ds |a °n 1a k2(l - K2^

dn |a K1 Ds 1a dn 1 a k2(i - k2)

°S |a 1 l

h

Ds 1a dn 1a (k2)2

°n |A 1 " K1 °s 1a 1a (i - k2)2

36

assumption that the probability of detecting S or N in a

2AFC task is equal to the probability of detecting these

same events in a Y-N task. It is further assumed that the

subject will enter a D state in the 2AFC task whenever S,

N, or both are detected. Therefore, i t follows that

K2 = 2K1(1 - K^) + (I

37

Table 12. Sensitivity Estimates Obtained with Two-Alternative Forced-Choice, and Values Predicted from Yes-No, as a Function of Signal Level

K 1/2A A 2A

Obtained .82 .87 .95

Predicted .83 .87 .96

Table 13. Probability of Entering an Attention State per Trial, and of Zero, One, Two, and Three Attention States per Problem, as a Function of Procedure and Signal Level

State Sequence

Y-N 2AFC State

Sequence 1/2A A 2A 1/2A A 2 A

A Per Trial . 54 . 71 . 65 . 55 . 69 . 6 6

3 As Per Problem . 4 0 , 5 6 . 4 9 . 4 0 . 4 9 . 4 6

2 As Per Problem . 0 6 - . 0 5 . 25 . 2 6 . 1 5 . 22

1 A Per Problem .19 . 2 5 . 1 2 . 08 . 22 .19

3 As Per Problem . 3 5 . 24 . 1 4 . 26 . 1 4 . 1 3

two, and three A states per problem for each procedure and

S level. These values were obtained with the solution of

the equations for K estimation presented in the Theoretical

Conceptualization and Analytical Procedure Section. The

values of these various A and A states across S levels were

comparable between procedures.

DISCUSSION

The data reported in the results section were pro

vided to i l lustrate an application of the proposed theory

with two different testing procedures, rather than to

provide an experimental comparison of these particular

experimental designs. Therefore, tests for statistical

significance of the differences were not included. The pro

posed theory contains a three-state discrete sensory model

which assumes that on each trial subjects either attend or

do not attend to the S or N stimulation presented. If sub

jects enter an A state on a given trial , they also enter

either a D or a D state. An A state is always assumed to

result in a D state. In addition, the sensory portion of

the theory is unique in applying the threshold concept to

the sensory effects of both N and S stimuli , with the assump

tion that P(D |s A A) equals P(D|N A A). This is considered

reasonable since the designation of what is S and what is N

is totally arbitrary and detecting one of the stimuli to be

discriminated is just as l ikely as detecting the other.

The response model is partly deterministic in that a

detect S state and a detect N state are assumed to always

result in a correct response. It is also partly probabil

istic since the probability of a correct response with a D

state equals the a priori probability of S being presented.

40

With Y-N presentation, an A state in the presence of S

results in either correct responding with P( + | s A A) =

( K + l ) / 2 o r i n c o r r e c t r e s p o n d i n g w i t h P ( - | s A A) =

( 1 - K ) / 2 . S i m i l a r l y , a n A s t a t e i n t h e p r e s e n c e o f N a l s o

results in either correct responding with P(+|N A A) =

( K + l ) / 2 o r i n c o r r e c t r e s p o n d i n g w i t h P ( - | N A A ) =

( 1 - K ) / 2 . W i t h 2 A F C p r e s e n t a t i o n , a n A s t a t e r e s u l t s i n

correct responding with P(+|s A A) = (K + l ) /2 or incorrect i_j

responding with P(- |S A A) = (1 - K)/2. A A state with J -L

both Y-N and 2AFC always results in a D state accompanied by

responding which is determined by a response bias.

The advantage of the present approach is thought to

be the variety of information that i t provides which was not

included in previous approaches (Blackwell , 1953, 1963;

Luce, 1963a, 1963b; Green and Swets, 1966). Specifically,

the proposed theory is thought to result in a highly de

tailed description of the detection process which includes

an estimation of the amount of responding due to specific

nonsensory response biases. This analysis forms the basis

for an estimate of sensitivity that is independent of

response bias including the effects of sequential response

dependencies. Both the sensitivity estimate and response

bias measures are nonparametric and a technique is sug

gested for the estimation of sensory thresholds that are not

confounded with the effects of response bias. Further, this

theory includes the concept of a high-threshold and is not

41

tied to a specific family of receiver operating character

istics (ROC) curves.

Initially, the probabilit ies of internal state sequences

assumed to be manifested by systematic responding during Y-N

and 2AFC problems are estimated by the relative proportions

of the various Hs analyzed. This determination is based upon

H analysis of the results of three-trial problems when

results are assumed to reflect combinations of states

D A A, D A A, and A that occur on each trial . The Hs that

involve various modes of D states provide a nonparametric

estimation of the proportion of one (A), two (B), and three

(C) D A A states per problem. Nonparametric estimation of

the type and amount of specific nonsensory response factors

is provided by the proportions of response bias Hs. The

lack of response independence between trials that is

examined is based upon a conceptualization of response bias

as multidimensional. These dimensions include (a) the

tendency to perseverate responses for two (E) or three (D)

trials, (b) the tendency to alternate responses (E^), as

well as (c) the win-stay:lose-shift (F) and win-shift:lose-

stay (G) syndromes.

The accuracy of prediction for the degree of

responding accounted for on the basis of H analysis was

quantitatively described by the P.V.E. estimations. The

P.V.E. values presented in Table 9 indicate that the H

analysis procedure quite accurately estimated the patterns

J

of frequencies observed. Changes in the occurrence of the

proportions of the various Hs as a function of such

variables as testing procedure and S level may then be

examined as depicted in Figures 1 and 2.

The H strengths were used to estimate the sensi

tivity index that is expressed in terms of the P(d|a).

Evidence that this index is independent of response biases

including those of sequential response dependencies will be

discussed shortly. In addition, the method of calculation

of the sensitivity index also results in determination of

the probability of an a state per trial as well as the prob

ability of one, two, and three a states and three a states

on three-trial problems. These combinations of a and a

states may also be examined as a function of such variables

as testing procedure and S intensity, as depicted in Table

13.

In l ine with the concept of a sensory recognition

threshold, the present procedure includes the following

technique for estimation of sensory thresholds in such a

way as to be unconfounded with a range of nonsensory

response biases. This procedure consists of replacing the

proportion of correct responses to various stimulus inten

sities with probability values of K. The threshold would

then be defined by the stimulus magnitude corresponding to a

value of K equal to .5. This technique is presently being

43

accessed by i ts use for the determination of Macaque visual

acuity thresholds (Fobes, in preparation, 1975) .

Nonparametric est imation of sensit ivity and response

bias avoids the diff icult ies with assumptions accompanying

parametric d' est imation with SDT procedures. The measures

of sensit ivity and response bias used with the theory pre

sented here do not depend upon any concept of underlying

density functions describing the effects of sensory events .

Nor do they depend upon accompanying assumptions concerning

normality and variance as does the frequently used d' tech

nique of Green and Swets (1966) . Their procedure for cal

culat ion of the sensit ivity est imate depends upon whether

the N and S density functions are assumed to be normal with

equal or unequal variances. Thus, the present approach is

in l ine with increasing interest in nonparametric est imators

of sensit ivity and response bias (Grier, 19 71; Hammerton and

Altham, 197i; Richardson, 1972) .

The nature of the variance assumption that i s made

in a given case is usual ly based upon the degree of symmetry

of an P.OC curve. Such a curve connects plots of the P (Y|S)

on the ordinate and the P(Y|N) on the abscissa as the P(Y)

responses i s varied with the S level held constant. How

ever with the present data, the empirical ROC curve can not

be determined s ince the P(Y) responses was varied between

rather than within S levels . Thus, distribution l inked d 1

can not be calculated on these data s ince neither the

44

distribution nor the appropriate variance assumption can be

determined. This does not pose a problem for the est imation

of nonparametric K.

The concept of a sensory threshold that i s rarely

or never exceeded in the absence of the S has been specif i

cal ly crit ic ized by some advocates of SDT. In their

analysis of the high-threshold concept, Green and Swets

(1966) compared SDT with a specif ic version of high-

threshold theory that was advanced by Blackwell (1953, 1963) .

Blackwell 's high-threshold theory included a sensory

threshold that was thought to be rarely i f ever exceeded in

the absence of S and below which sensory events were in-

discriminable from one another. While the theoretical pro

portion of "true" false alarms [(Y|N)*3 was assumed to

approximate zero, empirical proportions of false alarms

greater than zero were assumed to result from a Y response

to some sensory events that fai led to exceed the sensory

threshold. Since subthreshold sensory events were con

sidered to be indist inguishable from each other, these Y

responses were guesses and were correct by chance. There

fore, the obtained proportion of hits was assumed to con

s ist of the proportion of "true" hits [(Y |s)*], the value

of which depended upon S strength, plus a guessing factor

modif ied by the opportunity for guessing.

P(Y|S) = P(Y|S)* + P(Y|N) [ l - P(y|s)*3 ( l )

A procedure was therefore used to obtain the pro

portion of "true" hits corrected for guessing which adjusted

the obtained hits according to the obtained false alarms

that were taken as an index of the amount by which the hit

rate was inf lated by guessing. This correct ion for chance

success was a rearrangement of l ine (1) and took the form

P(Y|S)* = P(Y|S) - P(Y|N) / 1 - P(Y|N). (2)

This correct ion attempted to normalize the obtained psycho

metric function by e l iminating the proportion of false

alarms from the proportion of obtained hits . Upon deter

mination of the P(Y|S)* for each st imulus magnitude, a

psychometric function was plotted and the st imulus magnitude

corresponding to a .5 P(Y|S)* was selected as the threshold

value. Since the guessing mechanism which produced false

alarms was thought to operate only in the absence of a

sensory basis for a response, the procedure for correct ion

of chance success assumed ff iat the P(Y|S)* and the P(Y|N)

were independent.

This assumption of the stat ist ical independence

between "true" hits and false alarms has been shown to be

false by several l ines of evidence. One approach consisted

of a determination of the degree of correlat ion between the

p(y |S)* and the P(y|n ) . Stat ist ical ly s ignif icant correla

t ion coeff ic ients for these measures were found to be on the

order of .90 (Green and Swets , 1966) . Green and Swets also

noted other diff icult ies with Blackwell 's theory. Thresh

olds determined with Y-N and 2AFC procedures were not con

s istent with each other. In addit ion, the equation in l ine

(1) expressed the P(Y|S) as a linear function of the P(Y|N).

Thus, the theoretical ROC curve based upon Blackwell 's

version of high-threshold theory was a straight l ine from

P(Y|S)*, P(Y|N)* through P(Y|S ) = 1 , P(Y|N ) = 1 . The vast

majority of published ROC data are not adequately f i t ted by

a straight l ine. Although Green and Swets (1966) have

successful ly argued against Blackwell 's (1953, 1963) de

fect ive version of high-threshold theory, their evidence

does not automatical ly inval idate al l formulations that

include a high-threshold concept, in contrast to their

strong implication to the contrary.

While Blackwell 's (1953, 1963) theoretical ROC curve

resulted in a poor approximation of ROC data, the theo

ret ical ROC curve result ing from the low-threshold theory

proposed by Luce (1963a, 1963b) occasional ly provides a good

f i t for empirical curves (Luce, 1963a; Green and Swets , 1966;

King and Fobes, in preparation, 1975) . Luce's (1963a, 1963b)

theory specif ied a sensory threshold that was frequently

exceeded in the absence of S; the theoretical proportion of

false alarms was greater than zero. As in Blackwell 's

(1953, 1963) theory, Luce proposed a guessing mechanism that

resulted in an observed proportion of false alarms that was

greater than the "true" proportion due to random Y responses

47

to some subthreshold sensory events . In addit ion, a

guessing mechanism was also proposed that resulted in an

obtained proportion of hits that was less than the "true"

proportion due to random N responses to some suprathreshold

sensory events . Luce's theoretical ROC curve consisted of

two linear segments. One segment went from a zero P(Y|S)

and P(y|n ) at the lower left corner of the ROC graph to the

point representing the calculated "true" probabil i ty that S

results in detect ion and the calculated "true" probabil i ty

that N alone results in detect ion. From this point , the

other segment went to a P(yjs) and P(Y|N) of one at the

upper r ight corner of the graph.

One of the problems with this low-threshold concept

i s that i t had an observer frequently detect ing something

than was not presented, that i s , the theoretical P(Dg|N) was

greater than zero. An addit ional diff iculty with Luce's

theory i s that while ROC curves of the predicted shape are

sometimes obtained, the most frequently reported empirical

shape of the best f i t t ing l ine connecting ROC data i s

curvi l inear, the shape predicted by SDT's theoretical ROC

curve (Green and Swets , 1966) .

Two general points emerge from a consideration of

theoretical and empirical ROC curves. The proposed theory

involves more components than did previous formulations and

consequently, s l ight emphasis i s placed upon the two con

dit ional probabil i t ies (hits and false alarms) that form the

48

basis of ROC data. Rather than being t ied to ROC curves of

a particular form, the proposed theory i s consistent with

any ROC curve according to the relat ionship

tK = P(Y|S) - P(Y|N) , (3)

which may be rearranged

P(Y|S) = tK + P(Y|N) , (4)

This reduced emphasis upon ROC data gives the present theory

more general i ty than those l inked to ROC curves belonging to

a particular family. Data f i t ted by the curve predicted by

Luce's theory have provided some diff iculty for SDT and the

f inding of data f i t ted by the curve predicted by SDT are

diff icult to account for with Luce's theory. The second

point i s that not a l l formulations of high-threshold theory

necessari ly imply ROC curves of the type predicted by

Blackwell 's version of high-threshold theory.

In addit ion to the effects of sequential response

bias , the proposed approach i s also thought to provide a

sensit ivity index that i s independent of other nonsensory

biases . I t was noted in the introduction that the SDT

approach of Green and Swets (1966) a lso provided a measure

of sensit ivity that was independent of the effects of some

nonsensory factors , although not independent of the effects

of sequential response bias . Specif ical ly , d' was rela

t ively independent of varied response criteria, the a priori

probabil i ty of S, and psychophysical procedure.

49

While ongoing research at The University of Arizona

has yet to be completed on al l of these areas, a con

siderable amount of data have been col lected which are

directed at the quest ion of independence of the sensit ivity

index K and mult iple response criteria. King and Fobes ( in

preparation, 1975) presented humans with visual problems by

both 2AFC and Y-N rat ing scale procedures. Subjects used a

confidence index that featured f ive categories of confidence

levels which varied from absolutely certain to just guessing.

For data that have been analyzed, calculated values of K

varied only 2.5% as a function of decis ion criteria. This

indicates that the sensit ivity est imate i s independent of

response biases . I t is noteworthy that d' was calculated

with these data and d' was found to vary systematical ly as

much as 15% as a function of decis ion criteria. This

evidence in support of the proposed theory is in addit ion

to that provided by this dissertat ion. That i s , the theory

is conceptual ly sound, est imated sensit ivity increases as

the S level increases , a reasonable profi le of Hs emerges,

and a relat ionship exists between sensit ivity est imates

obtained by several psychophysical procedures.

An invest igation featuring various a priori prob

abi l i t ies of S i s currently being completed with pigeons

(Thomas and King, in preparation, 1975) . As reported for

the d' sensit ivity index (Green and Swets , 1966) , the value

50

of the sensit ivity index K i s expected to remain constant as

the a priori probabil i ty of S i s varied.

A theory of detect ion processes should have gener

al i ty and not be bound to any s ingle psychophysical method.

Therefore, a complete theory should predict performance in

a variety of test s i tuations. Green and Swets (1966)

derived a predicted relat ionship between performance with

two test ing procedures using a constant S level; the area

under the Y-N rat ing ROC curve approximated the percentage

of correct responses with the forced-choice technique. The

present approach includes a proposed relat ionship between

the sensit ivity est imate result ing from the differing pro

cedures, however the init ial predict ion had to be modif ied.

I t i s not known i f this modif icat ion was necessitated by the

within subject design that tested al l subjects init ial ly

with the 2AFC procedure, or i f the usage and s ize of the

constant correct ion is a necessary part of the formula. In

addit ion, the comparison between K^ a n t^ was based upon a

spatial rather than upon the temporal forced-choice pre

sentation which was used by Green and Swets (1966) . This

issue may be resolved after al l the human rat ing data are

analyzed (King and Fobes, in preparation, 1975) . The com

parison of the sensit ivity est imate obtained by the two

psychophysical procedures wil l then include the typical

between groups, temporal 2AFC design.

51

The trend in contemporary psychophysics i s to con

sider a threshold in terms of a response continuum. This

has resulted in part from SDT's separation of sensit ivity

est imation and factors affect ing a response threshold. The

present paper describes an approach which also permits

separation of nonsensory bias and sensit ivity, while re

taining the concept of a sensory threshold. This theory is

promising at this stage and warrants addit ional examination

of i ts empirical ly testable consequences.

APPENDIX A

FORMULAS FOR ESTIMATION OF HYPOTHESES

Table 6 indicated that a given response sequence to

a particular st imulus pattern was not synonymous with a

s ingle H. Therefore, formulas were developed to est imate

the proportion of the observed frequency due to each H,

Assumptions concerning the relat ionship between Hs' proba

bi l i t ies and the observed frequencies of the 32 events in

Table 6 permit the statement

4a + 24b + 48c + 8d + 16e + 8f + 8g + 32r = 4 , (A. l )

which reduced to

a + 6b + 12c + 2d + 4e + 2f + 2g + 8r = 1 , (A.2)

where a = A, 6b = B, 12c = C, 2d = D, 4e = E, 2f = F, 2g =

G, and 8r = R.

The solutions for "a" involved the equations that

included "a" (cel ls 5, 14, 23, and 32) from the 32 equations

in Table 6 , These total led

EQ = 4a + 12b + 12c +d+2e+f+g+ 4r. (A.3)

In order to apply Equation (A.2) , i t was rewritten

2d + 4e + 2f + 2g + 8r = 1 - (a + 6b + 12c) . (A.4)

52

53

The ZQ in l ine (A.3) was then mult ipl ied by two, ci

2£Q = 8a + 24b + 24c + (2d + 4e + 2f + 2g + 8r) , (A.5) ci

and the term on the r ight s ide of Equation (A.4) was

substituted for the identical expression in l ine (A.5)

2EQ = 8a + 24b + 24c + (1 - a - 6b - 12c) . (A.6) ci

Solving for "a,"

a = (2£Q= - 18b - 12c - l ) /7 . (A.7) CI

The solution for "a" was thus found to be a function of the

obtained proportion of "a" as wel l as variables "b" and

" c . "

The equations containing "b" (cel ls 4 , 5 , 6 , 1 , 11,

13, 14, 16, 18, 21, 23, 24, 25, 30, 31, and 32) total led

= 4a + 24b + 36c + 4d + 8e + 4f + 4g + 16r. (A.8)

Applying l ine (A.2) ,

l /2ZQb = 2a + 12b + 18c + (1 - a - 6b - 12c) . (A.9)

Solving for "b,"

b = ( l /2ZQb - a - 6c - l ) /6 . (A.10)

The equations containing "c" (al l cel ls except 1 ,

10, 19, and 28) total led

£Q c = 4a + 24b + 48c + 7d + 14e + 7f + 7g + 28r. (A.11)

54

Applying l ine (A.2) ,

2/7£Q c = 8/7a + 48/7b + 96/7c + (1 - a - 6b - 12c) .

(A.12)

Solving for "c,"

c = (2IQC - a - 6b - 7) /12 (A.13)

These three solution equations (A.7, A.10, and A.13)

may be rewritten

2£Q - 1 = 7a + 18b + 12c a

1/220^ ~* 1 — a + 6b + 6c

2£Q c - 7 = a + 6b + 12c, (A.14)

and solved by the method of determinates where

7 18 12

A = 1 6 6

1 , 6 12

504 + 108 +72-72 ~ 216 - 252

144. (A.15)

Substituting 2£Q - 1, 1/2EQ, - 1, and 2EQ - 7 for the Ct iJ

coeff ic ients of "a" in the matrix of l ine (A.15) ,

55

2EQ -1 a

18 12

Aa = l /2EQb - l 6 6

2ZQ -7 c

6 12

72(2EQ a - 1) + 108(2£Q c 7) + 72(l /2EQb - 1)

- 72(2EQC - 7) - 216(l /2EQb - 1) - 36(2EQ a - 1)

= 36(2£Q a - 1) - 144(l /2EQb - 1) + 36(2ZQ c - 7)

(A.16)

Where a = Aa/A,

a = [ 3 6 (2EQ - 1) - 144(1/2EQ. - 1) + 36(22Q„ - 7J/144 a. JJ C

(A.17)

which reduced to

a = l /4(2ZQ a - 1) - ( l /2ZQb - 1) + 1/4C2EQ c - 7) . (A.18)

Therefore,

A =-=a = , l /2EQ a - l /2EQb + l /2EQ c - 1 (A.19)

Substituting 22Q - 1, 1/2ZQ, - 1, and 2£Q - 7 for 'a ' ~~^b ' -"«c

the coeff ic ients of "b" in the matrix of l ine (A.15) ,

b =

2X0,-1 12 cL

1 l /2EQb - l

1 2EQ c -7 12

= 84Cl/2EQb - 1) + 6(2EQ a - 1) + 12(2£QC - 7)

- 12 ( l /2£Qb - 1) - 12(2EQ a - 1) - 42(2ZQ c - 7)

56

= 72(l /2ZQb - 1) - 6(2EQ a - 1) - 30(2EQ c - 7) . (A.20)

Where b = Ab/A,

b = [72(l /2£Qb - 1) - 6(2£Q a - 1) - 30(2ZQ c - 7)] /144,

(A. 21)

which reduced to

b = l /2( l /2ZQb - 1) - 1/24(2£Q a - 1) - 5/24(2EQ c - 7) .

(A.22)

Therefore,

B = 6b = 6( l /4EQb - l /12EQ a - 5/12EQ c + 1) . (A.23)

Final ly , substituting 2EQ a - 1, l /2EQb - 1, and

2EQ - 7 for the coeff ic ients of "c" in the matrix of l ine c

(A,15) ,

Ac =

7 18 2£Q a~l

1 6 l /2EQb - l

2ZQC~7 1 6

= 42 (2£QC - 7) + 18U/2EQb 1 ) + 6 ( 2 E Q - 1 )

- 6 (2ZQ_ - 1) - 18(2EQ - 7) - 4 2(1/2EQK - 1)

24C2ZQ c - 7) - 24(l /2ZQb - 1) . (A.24)

Where c = Ac/A,

c = [24(2EQC - 7) - 24(l /2£Qb - 1) ] /144, (A. 25)

57

which reduced to

c = 1/6(2EQC - 7) - l /6( l /2ZQb - 1) . (A.26)

Therefore,

C = 12c = 12(1/3ZQ - l /2EQb - 1. (A.27)

The solution for D included the equations containing

"d" (cel ls 1 , 5 , 9 , 13, 17, 21, 25, and 29) which total led

= a + 6b + 12c + 8d + 2f + 2g + 8r. (A.28)

Applying l ine (A.2) ,

= 1 - (2d + 4e + 2f + 2g + 8r)

+ 8d + 2f + 2g + 8r. (A.29)

Solving for."d,"

d = (SQd - 4e - l ) /6 . (A.30)

Thus, "d" was found to be a function of the obtained pro

portion of "d" as wel l as variable "e."

The equations containing "e" (al l even numbered

cel ls) total led

£Q e = 2a + 12b + 24c + 16e + 4f + 4g + 16r. (A.31)

Applying l ine (A.2) ,

l /2IQ e = 1 - (2d + 4e + 2f + 2g + 8r) + 8e + 2f

+ 2g + 8r, (A.32)

58

Solving for "e,"

e = ( l /2ZQ e + 2d - l ) /4 . (A.33)

Lines (A.30) and (A.33) may be rewritten

£Qd - 1 = 6d - 4d

l /2ZQ e - 1 = 4e - 2d, (A.34)

and these two equations with two unknowns can be s imul

taneously solved. For "d," the £Qd - 1 = 6d - 4e and

l /2£Q e - 1 = 4e - 2d combined to

4d = EQd - 1 + l /2EQ e - 1, (A.35)

which reduced to

D = 2d = 2( l /4£Qd + 1/8ZQ - 1/2) . (A.36)

For "e," l /2ZQ e - 1 was mult ipl ied by a posit ive

number ( three) to prevent cancel lat ion of "e" by combination

of - 1 and l /2£Q e - 1. Therefore, £Qd - 1 = 6d - 4e and

3( l /2EQ e - 1) = 12e - 6d combined to

8e = 3( l /2ZQ e - 1) + EQd - 1, (A.37)

which reduced to

E = 4e = 4(3/16EQ e + l /8EQd - 1/2) . (A.38)

To solve for F, the equations containing "f" (cel ls

4, 5 , 12, 13, 19, 22, 27, and 30) were total led

59

£Q f = a + 6b + 12c + 2d + 4e + 8f + 8r. (A.39)

Applying l ine (A.2) ,

EQ f = 1 - (2d + 4e + 2f + 2g + 8r) + 2d + 4e + 8f + 8r.

(A. 40)

Solving for "f ,"

f = (£Q f + 2g - l ) /6 . (A.41)

Thus, "f" was found to be a function of the obtained

proportion of "f" as wel l as variable "g."

The equations containing "g" (cel ls 1 , 8 , 9 , 16, 18,

23, 26, and 31) total led

EQ = a + 6b + 12c + 2d + 4e + 8g + 8r. (A.42) g

Applying l ine (A.2) ,

£Q g = 1 - (2d + 4e + 2f + 2g + 8r) + 2d + 4e + 8g + 8r.

(A.43)

Solving for "g,"

g = (£Q„ + 2f - l ) /6 . (A.44) y

Lines (A.41) and (A.44) may be rewritten

ZQf - 1 = 6f - 2g

EQ - 1 = 6g - 2f , (A. 45) 9

and these two equations with two unknowns can be s imul

taneously solved.

60

For "f", 3(ZQ^ - 1) and EQ^ - 1 combined to

16f = 3(EQ f - 1) + EQ g - 1, (A.46)

which reduced to

F = 2f = 2(3/16EQ f + 1/16EQ - 1/4) . (A.47) l. y

For "g," 3(EQg - 1) and EQ^ - 1 combined to

16g = 3 (EQ - 1) + EQ f - 1, (A.48) y

which reduced to

G = 2g = 2(3/16EQ + 1/16EQ- - 1/4) . (A.49) H J-

APPENDIX B

FORMULAS FOR ESTIMATION OF K

The formulas for the est imation of K presented in

the Theoretical Conceptual izat ion and Analytical Procedure

sect ion involved the fol lowing components .

A = P(3Ds) = P(3Ds A 3As) (B.1)

B = P (2Ds) = P(2Ds A 3As) + P(2Ds A 2As) (B.2)

C = P ( I D )

= P ( I D A 3 A s ) + P ( I D A 2 A s ) + P ( 1 D A 1 A ) ( B . 3 )

D = obtained value of H D (B.4)

E = obtained value of H E (B.5)

F = obtained value of H F (B.6)

G = obtained value of H G (B.7)

R = 1 - ( A + B + C + D + E + F + G ) ( B . 8 )

Mj^ = P (2As A 1A)

= P ( An A An+1 A V2 1 + P ( An A Vl A W

+ P ( An A An+1 A W ( B - 9 )

M , = P ( 1 A A 2 A s )

= P ( A„ A An+X A + P ( An A A„+ l A An+2>

+ P ( A f l A n + 1 A A n + 2 * ( B . 1 0 ) n

61

62

t = P ( A n ) ( B . l l )

T = P ( A A A A A „ ( B . 1 2 ) n n+1 n+2

K = P ( D n | A n ) ( B . 1 3 )

63

+ p ( D n A V l A V 2 I 3 A S ) + P ( D n A V l A V 2 I 3 A S ) I

+ p ( An A Vl A An+2' P ( D„ A Vl A V2 lAn A An+1

A V 2 > + P ( A n A A n + 1 A V 2 > P ( D n A V l A V 2 l A n

A A A A , „ ) + P ( A A A . A A ) P ( D A D , n+1 n+2 n n+1 n+2 n n+1

A D . J A A A - A A , 0 ) . ( B . 2 0 ) n + 2 ' n n + 1 n + 2

Substituting for def init ions in (B.12) and (B.13) ,

B = 3TK 2 (1-K) + [P(An A An + 1 A An + 2> + P(An A An + 1

A An+2> + P ( An A An+1 A An+ 2 ) I K 2 - ( B" 2 1 )

Substituting for the definit ion in (B.9) ,

B = 3 T K 2 ( ( 1 - K ) + M