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Design & Analysis of Multistratum Randomized Experiments

Ching-Shui Cheng

Nov. 30, 2006

National Tsing Hua University

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Schedule

Nov. 30Introduction, treatment and block structures, examples

Dec. 1Randomization models, null ANOVA, orthogonal designs

Dec. 7More on orthogonal designs, non-orthogonal designs

Dec. 8More complex treatment and block structures, factorial experiments

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Nelder (1965a, b)

The analysis of randomized experiments with

orthogonal block structure, Proceedings of the

Royal Society of London, Series A

Fundamental work on the analysis of randomized experiments with orthogonal block structures

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Bailey (1981) JRSS, Ser. A

“Although Nelder (1965a, b) gave a unified treatment of what he called ‘simple’ block structures over ten years ago, his ideas do not seem to have gained wide acceptance. It is a pity, because they are useful and, I believe, simplifying. However, there seems to be a widespread belief that his ideas are too difficult to be understood or used by practical statisticians or students.”

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Experimental Design

Planning of experiments to produce valid information as efficiently as possible

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Comparative Experiments

Treatments 處理 Varieties of grain, fertilizers, drugs, ….

Experimental Units

Plots, patients, ….

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Design: How to assign the treatments to the experimental units

Fundamental difficulty: variability among the units; no two units are exactly the same.

Different responses may be observed even if the same

treatment is assigned to the units.

Systematic assignments may lead to bias.

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Suppose is an observation on the th unit, and is the treatment assigned to that unit. Assume treatment-unit additivity:

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R. A. Fisher worked at the Rothamsted Experimental Station in the United Kingdom to evaluate the success of various fertilizer treatments.

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Fisher found the data from experiments going on for decades to be basically worthless because of poor experimental design.

Fertilizer had been applied to a field one year and not in another in order to compare the yield of grain produced in the two years. BUT

It may have rained more, or been sunnier, in different years. The seeds used may have differed between years as well.

Or fertilizer was applied to one field and not to a nearby field in the same year. BUT

The fields might have different soil, water, drainage, and history of previous use.

Too many factors affecting the results were “uncontrolled.”

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Fisher’s solution: Randomization 隨機化 In the same field and same year,

apply fertilizer to randomly spaced

plots within the field.

This averages out the effect of

variation within the field in

drainage and soil composition on

yield, as well as controlling for

weather, etc.

F F F F F F

F F F F F F F F

F F F F F

F F F F F F F F

F F F F F

F F F F

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Randomization prevents any particular treatment from

receiving more than its fair share of better units, thereby

eliminating potential systematic bias. Some treatments may

still get lucky, but if we assign many units to each treatment,

then the effects of chance will average out.

In addition to guarding against potential systematic biases,

randomization also provides a basis for doing statistical

inference.

(Randomization model)

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F F F F F F F F F F F F

F F F F F F F F F F F F

F F F F F F F F F F F F

Start with an initial design

Randomly permute (labels of) the experimental units

Complete randomization: Pick one of the 72! Permutationsrandomly

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1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2 2 2 2

3 3 3 3 3 3 3 3 3 3 3 3

3 3 3 3 3 3 4 4 4 4 4 4

4 4 4 4 4 4 4 4 4 4 4 4

Pick one of the 72! Permutations randomly

4 treatments

Completely randomized design

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Assume treatment-unit additivity

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Randomization model for a completely randomized design

The ’s are identically distributed

is a constant for all

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Blocking: an effective method for

improving precision

Randomized complete block design

After randomization:

完全區集設計

區集化

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Incomplete block design

7 treatments

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Incomplete block design

Balanced incomplete block design

Optimality was shown by Kiefer (1958)

Randomization is performed independently within each block, and the block labels are also randomly permuted.

平衡不完全區集設計

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Incomplete block design

Randomize by randomly choosing one out of the (7!)(3!)7 permutations that preserve the block structure.

These permutations form a subgroup of the group of all 21! permutations of the 21 unit labels.

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Block what you can and randomize what you cannot.

The purpose of randomization is to average out those nuisance factors that we cannot predict or cannot control, not to destroy the relevant information we have.

Choose a permutation group that preserves any known relevant structure on the units. Usually take the group for randomization to be the largest possible group that preserves the structure to give the greatest possible simplification of the model.

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Two simple block (unit) structures Nesting

block/unit

Crossing

row ＊ column

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Two simple block structures

Nesting

block/unit

Crossing

row ＊ columnLatin square

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Treatment structures

No structure

Treatments vs. control

Factorial structure

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Unstructured treatments

(Treatment contrast)

The set of all treatment contrasts form a dimensional space (generated by all the pairwisecomparisons.

Might be interested in estimating pairwise comparisons

or

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Treatments vs control

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Factorial structureEach treatment is a combination of several factors

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S=2, n=3:

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Interested in contrasts representing main

effects and interactions of the factors

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Here each is coded by 1 and -1.

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Treatment structure

Block structure (unit structure)

Design

Randomization

Analysis

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Choice of design

Efficiency Combinatorial considerations Practical considerations

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Simple orthogonal block structures

Iterated crossing and nesting

cover most, but not all block structures encountered in practice

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Consumer testing

A consumer organization wishes to compare 8 brands of

vacuum cleaner. There is one sample for each brand.

Each of four housewives tests two cleaners in her home

for a week. To allow for housewife effects, each housewife

tests each cleaner and therefore takes part in the trial for 4

weeks.

8 unstructured treatments

Block structure:

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A α B β C γ D δ

B γ A δ D α C β

C δ D γ A β B α

D β C α B δ A γ

Trojan square

Optimality of Trojan squares was shown byCheng and Bailey (1991)

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t = 18

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McLeod and Brewster (2004) Technometrics

Blocked split plots (Split-split plots)Chrome-plating process

Block structure: 4 weeks/4 days/2 runs

block/wholeplot/subplot

Treatment structure: A ＊ B ＊ C ＊ p ＊ q ＊ r

Each of the six factors has two levels

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Hard-to-vary treatment factors

A: chrome concentration B: Chrome to sulfate ratio C: bath temperature

Easy-to-vary treatment factors

p: etching current density q: plating current density r: part geometry

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Miller (1997) TechnometricsStrip-Plots

Experimental objective: Investigate methods of

reducing the wrinkling of clothes being laundered

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Miller (1997)

The experiment is run in 2 blocks and employs

4 washers and 4 driers. Sets of cloth samples

are run through the washers and the samples

are divided into groups such that each group

contains exactly one sample from each washer.

Each group of samples is then assigned to one

of the driers. Once dried, the extent of wrinkling

on each sample is evaluated.

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Treatment structure:

A, B, C, D, E, F: configurations of washers

a,b,c,d: configurations of dryers

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Block structure:2 blocks/(4 washers ＊ 4 dryers)

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Block 1 Block 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 0 10 0 0 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 1 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 10 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 1 00 1 1 0 1 1 0 0 1 1 0 1 1 1 0 0 0 1 0 10 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 0 1 00 1 1 0 1 1 1 1 1 1 0 1 1 1 0 0 1 0 0 11 0 1 1 0 1 0 0 0 0 1 0 1 0 1 0 0 1 1 01 0 1 1 0 1 0 0 1 1 1 0 1 0 1 0 0 1 0 11 0 1 1 0 1 1 1 0 0 1 0 1 0 1 0 1 0 1 01 0 1 1 0 1 1 1 1 1 1 0 1 0 1 0 1 0 0 11 1 0 1 1 0 0 0 0 0 1 1 0 0 0 1 0 1 1 0 1 1 0 1 1 0 0 0 1 1 1 1 0 0 0 1 0 1 0 1 1 1 0 1 1 0 1 1 0 0 1 1 0 0 0 1 1 0 1 01 1 0 1 1 0 1 1 1 1 1 1 0 0 0 1 1 0 0 1

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GenStat code

factor [nvalue=32;levels=2] block,A,B,C,D,E,F,a,b,c,d

& [levels=4] wash, dryer

generate block,wash,dryer

blockstructure block/(wash*dryer)

treatmentstructure

(A+B+C+D+E+F)*(A+B+C+D+E+F)

+(a+b+c+d)*(a+b+c+d)

+(A+B+C+D+E+F)*(a+b+c+d)

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matrix [rows=10; columns=5; values=“ b r1 r2 c1 c2"

0, 0, 1, 0, 0,0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0,1, 1, 0, 0, 0,1, 1, 1, 0, 0, 0, 0, 0, 0, 1,1, 0, 0, 0, 1, 1, 0, 0, 1, 0,

0, 0, 0, 1, 0] Mkey

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Akey [blockfactors=block,wash,dryer; Key=Mkey;rowprimes=!(10(2));colprimes=!(5(2)); colmappings

=!(1,2,2,3,3)] Pdesign Arandom [blocks=block/(wash*dryer);seed=12345]PDESIGN ANOVA

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Source of variation d.f. block stratum

AD=BE=CF=ab=cd 1 block.wash stratum

A=BC=EF 1B=AC=DF 1 C=AB=DF 1D=BF=CE 1E=AF=CD 1F=BD=AE 1

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block.dryer stratum

a 1b 1c 1d 1ac=bd 1bc=ad 1

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block.wash.dryer stratum

Aa=Db 1Ba=Eb 1Ca=Fb 1Da=Ab 1Ea=Bb 1Fa=Cb 1Ac=Dd 1Bc=Ed 1Cc=Fd 1Dc=Ad 1Ec=Bd 1Fc=Cd 1Residual 6Total 31

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Seven Error Terms!! Are you kidding??T. M. Loughin et al.

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Treatment structure: 3*2*3*7

Block structure: 4/((3/2/3)*7)

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Factor [nvalues=504;levels=4] Block & [levels=3] Sv,  Sr, Var, Rate  & [levels=2] St, Time & [levels=7] Sw, WeedGenerate Block, Sv, St, Sr, SwMatrix [rows=4;columns=6; \values="b1 b2 Col St Sr Row"\1, 0, 1, 0, 0, 0,\0, 0, 1, 1, 0, 0,\0, 0, 1, 1, 1, 0,\1, 1, 0, 0, 0, 1] CkeyAkey [blockfactor=Block,Sv,St,Sr,Sw; \Colprimes=!(2,2,3,2,3,7);Colmappings=!(1,1,2,3,4,5);Key=Ckey] Var, Time, Rate, WeedBlocks Block/((Sv/St/Sr)*Sw)Treatments Var*Time*Rate*WeedANOVA

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Block stratum 3 Block.Sv stratumVar 2Residual 6 Block.Sw stratumWeed 6Residual 18

Block.Sv.St stratumTime 1Var.Time 2Residual 9  

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 Block.Sv.Sw stratumVar.Weed 12Residual 36

Block.Sv.St.Sr stratumRate 2Var.Rate 4Time.Rate 2Var.Time.Rate 4Residual 36

 

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Block.Sv.St.Sw stratumTime.Weed 6Var.Time.Weed 12Residual 54 Block.Sv.St.Sr.Sw stratumRate.Weed 12Var.Rate.Weed 24Time.Rate.Weed 12Var.Time.Rate. Weed 24Residual 216

 Total 503