oxford [R] Companion for Experimental Design and …clcl/Q560_OLD/R_comp4Abdi_ExperimentalDesign… · Argentina Austria Brazil Chile CzechRepublic France Greece ... Oxford is a registeredtrademarkofOxfordUniversityPress

Lynne J. Williams, Anjali Krishnan & Hervé Abdi

[R] Companion for

Experimental Design and Analysis for Psychology

oxford

OXFORD UNIVERSITYPRESS

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First published 2009

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1 3 5 7 9 10 8 6 4 2

Preface

You have successfully designed your first experiment, run the subjects,and you are faced with a mountain of data. What’s next?1 Does computingan analysis of variance by hand suddenly appear mysteriously attractive?Granted, writing an [R] program and actually getting it to run may appearto be quite an intimidating task for the novice, but fear not! There is notime like the present to overcome your phobias. Welcome to the wonder-ful world of [R]

The purpose of this book is to introduce you to relatively simple [R]programs. Each of the experimental designs introduced in ExperimentalDesign and Analysis for Psychology by Abdi, et al. are reprinted herein, fol-lowed by their [R] code and output. The first chapter covers correlation,followed by regression, multiple regression, and various analysis of vari-ance designs. We urge you to familiarize yourself with the [R] codes and[R] output, as they in their relative simplicity should alleviate many of youranxieties.

We would like to emphasize that this book is not written as the tuto-rial in the [R] programming language. For that there are several excellentbooks on the market. Rather, use this manual as your own cook book ofbasic recipies. As you become more comfortable with [R], you may wantto add some additional flavors to enhance your programs beyond what wehave suggested herein.

1Panic is not the answer!

ii 0.0

c! 2009 Williams, Krishnan & Abdi

Contents

Preface i

1 Correlation 11.1 Example: Word Length and Number of Meanings . . . . . . . 1

1.1.1 [R] code . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 [R] output . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Simple Regression Analysis 72.1 Example: Memory Set and Reaction Time . . . . . . . . . . . 7

2.1.1 [R] code . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 [R] output . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Multiple Regression Analysis: Orthogonal Independent Variables 133.1 Example: Retroactive Interference . . . . . . . . . . . . . . . . 13

3.1.1 [R] code . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.2 [R] output . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Multiple Regression Analysis: Non-orthogonal Independent Vari-ables 214.1 Example: Age, Speech Rate and Memory Span . . . . . . . . . 21

4.1.1 [R] code . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.1.2 [R] output . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5 ANOVA One Factor Between-Subjects, "!#" 275.1 Example: Imagery and Memory . . . . . . . . . . . . . . . . . 27

5.1.1 [R] code . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.1.2 [R] output . . . . . . . . . . . . . . . . . . . . . . . . . . 285.1.3 ANOVA table . . . . . . . . . . . . . . . . . . . . . . . 30

5.2 Example: Romeo and Juliet . . . . . . . . . . . . . . . . . . . . 305.2.1 [R] code . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.2.2 [R] output . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.3 Example: Face Perception, "!#" with # random . . . . . . . . 345.3.1 [R] code . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.3.2 [R] output . . . . . . . . . . . . . . . . . . . . . . . . . . 355.3.3 ANOVA table . . . . . . . . . . . . . . . . . . . . . . . 37

iv 0.0 CONTENTS

5.4 Example: Images ... . . . . . . . . . . . . . . . . . . . . . . . . 385.4.1 [R] code . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.4.2 [R] output . . . . . . . . . . . . . . . . . . . . . . . . . . 395.4.3 ANOVA table . . . . . . . . . . . . . . . . . . . . . . . 40

6 ANOVA One Factor Between-Subjects: Regression Approach 416.1 Example: Imagery and Memory revisited . . . . . . . . . . . . 42

6.1.1 [R] code . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.1.2 [R] output . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6.2 Example: Restaging Romeo and Juliet . . . . . . . . . . . . . . 456.2.1 [R] code . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.2.2 [R] output . . . . . . . . . . . . . . . . . . . . . . . . . . 47

7 Planned Orthogonal Comparisons 517.1 Context and Memory . . . . . . . . . . . . . . . . . . . . . . . 51


8 Planned Non-orthogonal Comparisons 618.1 Classical approach: Tests for non-orthogonal comparisons . 618.2 Romeo and Juliet, non-orthogonal contrasts . . . . . . . . . . 62

8.2.1 [R] code . . . . . . . . . . . . . . . . . . . . . . . . . . . 638.2.2 [R] output . . . . . . . . . . . . . . . . . . . . . . . . . . 65

8.3 Multiple Regression and Orthogonal Contrasts . . . . . . . . 708.3.1 [R] code . . . . . . . . . . . . . . . . . . . . . . . . . . . 718.3.2 [R] output . . . . . . . . . . . . . . . . . . . . . . . . . . 73

8.4 Multiple Regression and Non-orthogonal Contrasts . . . . . 788.4.1 [R] code . . . . . . . . . . . . . . . . . . . . . . . . . . . 798.4.2 [R] output . . . . . . . . . . . . . . . . . . . . . . . . . . 82

9 Post hoc or a-posteriori analyses 879.1 Scheffe’s test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

9.1.1 Romeo and Juliet . . . . . . . . . . . . . . . . . . . . 889.1.2 [R] code . . . . . . . . . . . . . . . . . . . . . . . . . . . 889.1.3 [R] output . . . . . . . . . . . . . . . . . . . . . . . . . . 91

9.2 Tukey’s test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969.2.1 The return of Romeo and Juliet . . . . . . . . . . . . 96

9.2.1.1 [R] code . . . . . . . . . . . . . . . . . . . . . 979.2.1.2 [R] output . . . . . . . . . . . . . . . . . . . . 100

9.3 Newman-Keuls’ test . . . . . . . . . . . . . . . . . . . . . . . . 1069.3.1 Taking off with Loftus. . . . . . . . . . . . . . . . . . . 106

9.3.1.1 [R] code . . . . . . . . . . . . . . . . . . . . . 1079.3.1.2 [R] output . . . . . . . . . . . . . . . . . . . . 112

9.3.2 Guess who? . . . . . . . . . . . . . . . . . . . . . . . 119


0.0 CONTENTS v

9.3.2.1 [R] code . . . . . . . . . . . . . . . . . . . . . 1199.3.2.2 [R] output . . . . . . . . . . . . . . . . . . . . 122

10 ANOVA Two Factors; !!#$ %" 12910.1 Cute Cued Recall . . . . . . . . . . . . . . . . . . . . . . . . . . 129


10.2 Projective Tests and Test Administrators . . . . . . . . . . . . 13910.2.1 [R] code . . . . . . . . . . . . . . . . . . . . . . . . . . . 14010.2.2 [R] output . . . . . . . . . . . . . . . . . . . . . . . . . . 14010.2.3 ANOVA table . . . . . . . . . . . . . . . . . . . . . . . 143

11 ANOVA One Factor Repeated Measures, " $# 14511.1 " $# design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

11.1.1 [R] code . . . . . . . . . . . . . . . . . . . . . . . . . . . 14511.1.2 [R] output . . . . . . . . . . . . . . . . . . . . . . . . . . 146

11.2 Drugs and reaction time . . . . . . . . . . . . . . . . . . . . . 14811.2.1 [R] code . . . . . . . . . . . . . . . . . . . . . . . . . . . 14811.2.2 [R] output . . . . . . . . . . . . . . . . . . . . . . . . . . 15011.2.3 ANOVA table . . . . . . . . . . . . . . . . . . . . . . . 152

11.3 Proactive Interference . . . . . . . . . . . . . . . . . . . . . . . 15211.3.1 [R] code . . . . . . . . . . . . . . . . . . . . . . . . . . . 15211.3.2 [R] output . . . . . . . . . . . . . . . . . . . . . . . . . . 15411.3.3 ANOVA table . . . . . . . . . . . . . . . . . . . . . . . 156

12 Two Factors Repeated Measures, " $#$ % 15712.1 Plungin’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157


13 Factorial Design, Partially Repeated Measures: "!#"$ % 16513.1 Bat and Hat.... . . . . . . . . . . . . . . . . . . . . . . . . . . . 165


14 Nested Factorial Design: " $#!%" 17314.1 Faces in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

14.1.1 [R] code . . . . . . . . . . . . . . . . . . . . . . . . . . . 17314.1.2 [R] output . . . . . . . . . . . . . . . . . . . . . . . . . . 17514.1.3 ! and Quasi-! ratios . . . . . . . . . . . . . . . . . . . . 17814.1.4 ANOVA table . . . . . . . . . . . . . . . . . . . . . . . 179

Index 183


vi 0.0 CONTENTS


1Correlation

1.1 Example:Word Length and Number of Meanings

If you are in the habit of perusing dictionaries as a way of leisurely passingtime, you may have come to the conclusion that longer words apparentlyhave fewer meanings attributed to them. Now, finally, through the mira-cle of statistics, or more precisely, the Pearson Correlation Coefficient, youneed no longer ponder this question.

We decided to run a small experiment. The data come from a sampleof #$ words taken randomly from the Oxford English Dictionary. Table 1.1on the following page gives the results of this survey.

A quick look at Table 1.1 on the next page does indeed give the impres-sion that longer words tend to have fewer meanings than shorter words(e.g., compare “by” with “tarantula”.) Correlation, or more specifically thePearson coefficient of correlation, is a tool used to evaluate the similar-ity of two sets of measurements (or dependent variables) obtained on thesame observations. In this example, the goal of the coefficient of corre-lation is to express in a quantitative way the relationship between lengthand number of meanings of words.

For a more detailed description, please refer to Chapter # on Correla-tion in the textbook.

1.1.1 [R] code! "#$$%&'()#* +,'-.&%/ 0#$1 2%*3(4 '*1 56-7%$ #8 9%'*)*3:

! 0% 8)$:( %*(%$ (4% 1'(' 6*1%$ (;# 1)88%$%*( <'$)'7&%: *'-%:2%*3(4=>?@ABACABACADABAEADAFAGAHHAEAFA@ADAHIAEAFAHIJ9%'*)*3:=>?KAFAHIAHAHHAHAFA@AHABACAHADA@AFAHA@A@A@ACJ1'('=1'('L8$'-%?2%*3(4A9%'*)*3:J

9%'*=-%'*?1'('JM(1NO%<=:1?1'('J

! 0% *#; .&#( (4% .#)*(: '*1 MPQ+ )( ': ' ROS! 9'T% :6$% (# '11 (4% RPUV (# (4% &#>'()#* ;4%$% (4% .&#( ):

2 1.1 Example: Word Length and Number of Meanings

Number ofWord Length Meanings

bag 3 8buckle 6 4on 2 10insane 6 1by 2 11monastery 9 1relief 6 4slope 5 3scoundrel 9 1loss 4 6holiday 7 2pretentious 11 1solid 5 9time 4 3gut 3 4tarantula 9 1generality 10 3arise 5 3blot 4 3infectious 10 2

TABLE 1.1 Length (i.e., number of letters) and number of meanings of a random sample of 20 words taken from

the Oxford English Dictionary.

! (# 7% :'<%1.18?WX4#-%X'*Y'&)XO%:T(#.XZN:>$).(:XIHN"#$$%&'()#*X>#$$N.&#(L.18WJ.&#(?2%*3(4A9%'*)*3:A-')*=[R&#( #8 2%*3(4 <: 9%'*)*3:[J1%<L#88?J

! 0% *#; .%$8#$- ' >#$$%&'()#* '*1 ' (%:( #* (4% 1'(' ;4)>4 3)<%:! >#*8)1%*>% )*(%$<'&:>#$H=>#$L(%:(?2%*3(4A 9%'*)*3:A-%(4#1 = >?[.%'$:#*[JJ

! 0% *#; .%$8#$- ' $%3$%::)#* '*'&\:): #* (4% 1'('$%3H=&-?2%*3(4]9%'*)*3:J

! 0% *#; .%$8#$- '* P5^QP #* (4% 1'(''#<H='#<?2%*3(4]9%'*)*3:J

! 0% *#; .$)*( (4% 1'(' '*1 '&& (4% $%:6&(:.$)*(?1'('J.$)*(?9%'*J.$)*(?M(1NO%<J.$)*(?>#$HJ:6--'$\?$%3HJ:6--'$\?'#<HJ


1.1 Example: Word Length and Number of Meanings 3

1.1.2 [R] output_ ! "#$$%&'()#* +,'-.&%/ 0#$1 2%*3(4 '*1 56-7%$ #8 9%'*)*3:

_ ! 0% 8)$:( %*(%$ (4% 1'(' 6*1%$ (;# 1)88%$%*( <'$)'7&%: *'-%:_ 2%*3(4=>?@ABACABACADABAEADAFAGAHHAEAFA@ADAHIAEAFAHIJ_ 9%'*)*3:=>?KAFAHIAHAHHAHAFA@AHABACAHADA@AFAHA@A@A@ACJ_ 1'('=1'('L8$'-%?2%*3(4A9%'*)*3:J

_ 9%'*=-%'*?1'('J_ M(1NO%<=:1?1'('J

_ ! 0% *#; .&#( (4% .#)*(: '*1 MPQ+ )( ': ' ROS_ ! 9'T% :6$% (# '11 (4% RPUV (# (4% &#>'()#* ;4%$% (4% .&#( ):_ ! (# 7% :'<%1_ .18?WX4#-%X'*Y'&)XO%:T(#.XZN:>$).(:XIHN"#$$%&'()#*X>#$$N.&#(L.18WJ_ .&#(?2%*3(4A9%'*)*3:A-')*=[R&#( #8 2%*3(4 <: 9%'*)*3:[J_ 1%<L#88?J

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2 4 6 8 10

24

68

10

Plot of Length vs Meanings

Length

Meanings

_ ! 0% *#; .%$8#$- ' >#$$%&'()#* '*1 ' (%:( #* (4% 1'(' ;4)>4 3)<%:_ ! >#*8)1%*>% )*(%$<'&:_ >#$H=>#$L(%:(?2%*3(4A 9%'*)*3:A-%(4#1 = >?[.%'$:#*[JJ

_ ! 0% *#; .%$8#$- ' $%3$%::)#* '*'&\:): #* (4% 1'('_ $%3H=&-?2%*3(4]9%'*)*3:J

_ ! 0% *#; .%$8#$- '* P5^QP #* (4% 1'('_ '#<H='#<?2%*3(4]9%'*)*3:J

_ ! 0% *#; .$)*( (4% 1'(' '*1 '&& (4% $%:6&(:_ .$)*(?1'('J



```````````````````2%*3(4 9%'*)*3:

```````````````````H @ KC B F@ C HIF B HE C HHB D HG B FK E @D D HHI F BHH G CHC HH HH@ E DHF F @HE @ FHB D HHG HI @HK E @HD F @CI HI C```````````````````

_ .$)*(?9%'*J

````````````````2%*3(4 9%'*)*3:````````````````

B F````````````````

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```````````````````2%*3(4 9%'*)*3:

```````````````````CLKIDGEG @LHFEEDI```````````````````

_ .$)*(?>#$HJ

R%'$:#*W: .$#16>(`-#-%*( >#$$%&'()#*1'('/ 2%*3(4 '*1 9%'*)*3:

( = `FLEBFFA 18 = HKA .`<'&6% = ILIIICFI@

'&(%$*'()<% 4\.#(4%:):/ ($6% >#$$%&'()#* ): *#( %a6'& (# I

DE .%$>%*( >#*8)1%*>% )*(%$<'&/ ÌLKKG@EKK ÌLFCKDGED

:'-.&% %:()-'(%:/


1.1 Example: Word Length and Number of Meanings 5

``````````>#$

``````````ÌLG@CFEF@``````````

_ :6--'$\?$%3HJ

"'&&/&-?8#$-6&' = 2%*3(4 ] 9%'*)*3:J

Z%:)16'&:/``````````````````````````````````````````````````

9)* Hb 9%1)'* @b 9',```````````````````````````````````````````````````@LIIIII `HLBEFCB ÌLI@GC@ HLI@GC@ @L@FEGF``````````````````````````````````````````````````

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+:()-'(% M(1L +$$#$ ( <'&6% R$?_c(cJ```````````````````````````````````````````````````?d*(%$>%.(J KLBHGI ILGCCF HHLDCK ELEB%`HI eee9%'*)*3: ÌLBEF@ ILHF@@ `FLEBF ILIIICFI eee``````````````````````````````````````````````````````M)3*)8L >#1%:/ I WeeeW ILIIH WeeW ILIH WeW ILIE WLW ILH W W H

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96&().&% Z`:a6'$%1/ ILE@BEAP1Y6:(%1 Z`:a6'$%1/ ILEHIGS`:('():()>/ CILK@ #* H '*1 HK OSA .`<'&6%/ ILIIICFI@

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2Simple Regression Analysis

2.1 Example:Memory Set and Reaction Time

In an experiment originally designed by Sternberg (1969), subjects wereasked to memorize a set of random letters (like lqwh) called the memoryset. The number of letters in the set was called the memory set size. Thesubjects were then presented with a probe letter (say q). Subjects then gavethe answer Yes if the probe is present in the memory set and No if the probewas not present in the memory set (here the answer should be Yes). Thetime it took the subjects to answer was recorded. The goal of this experi-ment was to find out if subjects were “scanning” material stored in shortterm memory.

In this replication, each subject was tested one hundred times witha constant memory set size. For half of the trials, the probe is present,whereas for the other half the probe is absent. Four different set sizes areused: %, &, ', and ( letters. Twenty (fictitious) subjects are tested (five percondition). For each subject we used the mean reaction time for the cor-rect Yes answers as the dependent variable. The research hypothesis wasthat subjects need to serially scan the letters in the memory set and thatthey need to compare each letter in turn with the probe. If this is the case,then each letter would add a given time to the reaction time. Hence theslope of the line would correspond to the time needed to process one let-ter of the memory set. The time needed to produce the answer and en-code the probe should be constant for all conditions of the memory setsize. Hence it should correspond to the intercept. The results of this ex-periment are given in Table 2.1 on the following page.

2.1.1 [R] code! Z%3$%::)#* +,'-.&%/ 9%-#$\ M%( '*1 Z%'>()#* ()-%

! 0% 8)$:( '$$'*3% (4% 1'(' )*(# (4% R$%1)>(#$: ?fJ '*1 Z%3$%::#$ ?gJ! d* (4): %,'-.&% (4% .$%1)>(#$: '$% (4% :)h%: #8 (4% -%-#$\ :%( '*1! (4% $%3$%::#$: '$% (4% $%'>()#* ()-% #8 (4% .'$()>).'*(:L

8 2.1 Example: Memory Set and Reaction Time

Memory Set Size

! ! " ! ! # ! ! $ ! ! %

433 519 598 666435 511 584 674434 513 606 683441 520 605 685457 537 607 692

TABLE 2.1 Data from a replication of a Sternberg (1969) experiment. Each data point represents the mean

reaction time for the Yes answers of a given subject. Subjects are tested in only one condition. Twenty (fictitious)

subjects participated in this experiment. For example the mean reaction time of subject one who was tested with

a memory set of 1 was 433 (!! ! "##, "! ! $.)

f=>?HAHAHAHAHA@A@A@A@A@AEAEAEAEAEAGAGAGAGAGJg=>?F@@AF@EAF@FAFFHAFEGAEHDAEHHAEH@AECIAE@GAEDKAEKFABIBA

BIEABIGA BBBABGFABK@ABKEABDCJ

! 0% *#; 3%( ' :6--'$\ #8 :)-.&% :('():()>: 8#$ (4% 1'('

9%'*=-%'*?1'('JM(1NO%<=:1?1'('J$=>#$?fAgJ

! 0% *#; .&#( (4% .#)*(: '*1 (4% $%3$%::)#* &)*% '*1 MPQ+ ': ' .18! 9'T% :6$% (# '11 (4% RPUV (# (4% &#>'()#* ;4%$% (4% .&#( ): (# 7% :'<%1

.18?WX4#-%X'*Y'&)XO%:T(#.XZN:>$).(:XICNZ%3$%::)#*X$%3N.&#(L.18WJ

.&#(?fAgA-')*=[R&#( #8 9%-#$\ M%( ?fJ <: Z%'>()#* U)-% ?gJ[J$%3L&)*%?$%3HJ1%<L#88?J

! 0% *#; .%$8#$- (4% $%3$%::)#* '*'&\:): #* (4% 1'('

$%3H=&-?g]fJ

! 0% *#; .%$8#$- '* P5^QP #* (4% 1'('

'#<H='#<?g]fJ

! 0% *#; .$)*( (4% 1'(' '*1 '&& (4% $%:6&(:

.$)*(?1'('J

.$)*(?9%'*J

.$)*(?M(1NO%<J

.$)*(?$J:6--'$\?$%3HJ:6--'$\?'#<HJ


2.1 Example: Memory Set and Reaction Time 9

2.1.2 [R] output_ ! Z%3$%::)#* +,'-.&%/ 9%-#$\ M%( '*1 Z%'>()#* ()-%

_ ! 0% 8)$:( '$$'*3% (4% 1'(' )*(# (4% R$%1)>(#$: ?fJ '*1 Z%3$%::#$ ?gJ_ ! d* (4): %,'-.&% (4% .$%1)>(#$: '$% (4% :)h%: #8 (4% -%-#$\ :%( '*1_ ! (4% $%3$%::#$: '$% (4% $%'>()#* ()-% #8 (4% .'$()>).'*(:L

_ f=>?HAHAHAHAHA@A@A@A@A@AEAEAEAEAEAGAGAGAGAGJ_ g=>?F@@AF@EAF@FAFFHAFEGAEHDAEHHAEH@AECIAE@GAEDKAEKFABIBA

BIEABIGA BBBABGFABK@ABKEABDCJ

_ ! 0% *#; 3%( ' :6--'$\ #8 :)-.&% :('():()>: 8#$ (4% 1'('

_ 9%'*=-%'*?1'('J_ M(1NO%<=:1?1'('J_ $=>#$?fAgJ

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1 2 3 4 5 6 7

450

500

550

600

650

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Plot of Memory Set (X) vs Reaction Time (Y)

X

Y

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2.1 Example: Memory Set and Reaction Time 11

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3Multiple Regression Analysis:Orthogonal IndependentVariables

3.1 Example:Retroactive Interference

To illustrate the use of Multiple Regression Analysis, we present a replica-tion of Slamecka’s (1960) experiment on retroactive interference. The termretroactive interference refers to the interfering effect of later learning onrecall. The general paradigm used to test the effect of retroactive interfer-ence is as follows. Subjects in the experimental group are first presentedwith a list of words to memorize. After the subjects have memorized thislist, they are asked to learn a second list of words. When they have learnedthe second list, they are asked to recall the first list they learned. The num-ber of words recalled by the experimental subjects is then compared withthe number of words recalled by control subjects who learned only thefirst list of words. Results, in general, show that having to learn a secondlist impairs the recall of the first list (i.e., experimental subjects recall fewerwords than control subjects.)

In Slamecka’s experiment subjects had to learn complex sentences. Thesentences were presented to the subjects two, four, or eight times (this isthe first independent variable.) We will refer to this variable as the numberof learning trials or ". The subjects were then asked to learn a secondseries of sentences. This second series was again presented two, four, oreight times (this is the second independent variable.) We will refer to thisvariable as the number of interpolated lists or # . After the second learningsession, the subjects were asked to recall the first sentences presented. Foreach subject, the number of words correctly recalled was recorded (this isthe dependent variable.) We will refer to the dependent variable as $ .

In this example, a total of 18 subjects (two in each of the nine exper-imental conditions), were used. How well do the two independent vari-ables “number of learning trials” and “number of interpolated lists” pre-dict the dependent variable “number of words correctly recalled”? The re-

14 3.1 Example: Retroactive Interference

sults of this hypothetical replication are presented in Table 3.1.

3.1.1 [R] code

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3.1 Example: Retroactive Interference 15

Number of Number oflearning trials (!) interpolated lists (" )

2 4 8

2 35 21 639 31 8

4 40 34 1852 42 26

8 61 58 4673 66 52

TABLE 3.1 Results of an hypothetical replication of Slamecka (1960)’s retroactive interference experiment.

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4Multiple Regression Analysis:Non-orthogonal IndependentVariables

4.1 Example:Age, Speech Rate and Memory Span

To illustrate an experiment with two quantitative independent variables,we replicated an experiment originally designed by Hulme, Thomson,Muir, and Lawrence (1984, as reported by Baddeley, 1990, p.78 ff.). Chil-dren aged 4, 7, or 10 years (hence “age” is the first independent variable inthis experiment, denoted "), were tested in 10 series of immediate serialrecall of 15 items. The dependent variable is the total number of words cor-rectly recalled (i.e., in the correct order). In addition to age, the speech rateof each child was obtained by asking the child to read aloud a list of words.Dividing the number of words read by the time needed to read them gavethe speech rate (expressed in words per second) of the child. Speech rate isthe second independent variable in this experiment (we will denote it # ).

The research hypothesis states that the age and the speech rate of thechildren are determinants of their memory performance. Because the in-dependent variable speech rate cannot be manipulated, the two indepen-dent variables are not orthogonal. In other words, one can expect speechrate to be partly correlated with age (on average, older children tend tospeak faster than younger children.) Speech rate should be the major de-terminant of performance and the effect of age reflects more the confound-ed effect of speech rate rather than age, per se.

The data obtained from a sample of 6 subjects are given in the Table 4.1on the next page.

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22 4.1 Example: Age, Speech Rate and Memory Span

The Independent Variables The Dependent Variable

! " #Age Speech Rate Memory Span

(in years) (words per second) (number of words recalled)

4 1 144 2 237 2 307 4 50

10 3 3910 6 67

TABLE 4.1 Data from a (fictitious) replication of an experiment of Hulme et al. (1984). The dependent variable is

the total number of words recalled in 10 series of immediate recall of items, it is a measure of the memory span.

The first independent variable is the age of the child, the second independent variable is the speech rate of the

child.

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4.1 Example: Age, Speech Rate and Memory Span 23

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4.1.2 [R] output

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0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

●

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X

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4.1 Example: Age, Speech Rate and Memory Span 25

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5ANOVA One FactorBetween-Subjects, "!#"

5.1 Example:Imagery and Memory

Our research hypothesis is that material processed with imagery will bemore resistant to forgetting than material processed without imagery. Inour experiment, we ask subjects to learn pairs of words (e.g., “beauty-carrots”).Then, after some delay, the subjects are asked to give the second wordof the pair (e.g., “carrot”) when prompted with the first word of the pair(e.g., “beauty”). Two groups took part in the experiment: the experimen-tal group (in which the subjects learn the word pairs using imagery), andthe control group (in which the subjects learn without using imagery). Thedependent variable is the number of word pairs correctly recalled by eachsubject. The performance of the subjects is measured by testing their mem-ory for 20 word pairs, 24 hours after learning.

The results of the experiment are listed in the following table:

Experimental group Control group

1 82 85 96 116 14

5.1.1 [R] code! P5^QP ^*%`8'>(#$ 7%(;%%* :67Y%>(:A M?PJ! d-'3%$\ '*1 9%-#$\

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28 5.1 Example: Imagery and Memory

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5.1.2 [R] output

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5.1 Example: Imagery and Memory 29

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30 5.2 Example: Romeo and Juliet

```M)3*)8L >#1%:/ I WeeeW ILIIH WeeW ILIH WeW ILIE WLW ILH W W H

5.1.3 ANOVA tableThe results from our experiment can be condensed in an analysis of vari-ance table.

Source !" ## $# %

Between " &'$'' &'$'' "$$''Within " ( )($'' *$''

Total & "#($''

5.2 Example:Romeo and Juliet

In an experiment on the effect of context on memory, Bransford and John-son (1972) read the following passage to their subjects:

“If the balloons popped, the sound would not be able to carrysince everything would be too far away from the correct floor.A closed window would also prevent the sound from carryingsince most buildings tend to be well insulated. Since the wholeoperation depends on a steady flow of electricity, a break in themiddle of the wire would also cause problems. Of course thefellow could shout, but the human voice is not loud enough tocarry that far. An additional problem is that a string could breakon the instrument. Then there could be no accompaniment tothe message. It is clear that the best situation would involveless distance. Then there would be fewer potential problems.With face to face contact, the least number of things could gowrong.”

To show the importance of the context on the memorization of texts,the authors assigned subjects to one of four experimental conditions:

& 1. “No context” condition: subjects listened to the passage and triedto remember it.

& 2. “Appropriate context before” condition: subjects were providedwith an appropriate context in the form of a picture and then listenedto the passage.


5.2 Example: Romeo and Juliet 31

& 3. “Appropriate context after” condition: subjects first listened to thepassage and then were provided with an appropriate context in theform of a picture.

& 4. “Partial context” condition: subjects are provided with a contextthat does not allow them to make sense of the text at the same timethat they listened to the passage.

Strictly speaking this experiment involves one experimental group (group 2:“appropriate context before”), and three control groups (groups 1, 3, and 4).The raison d’etre of the control groups is to eliminate rival theoretical hy-potheses (i.e., rival theories that would give the same experimental predic-tions as the theory advocated by the authors).

For the (fictitious) replication of this experiment, we have chosen tohave 20 subjects assigned randomly to 4 groups. Hence there is ! ) ' sub-jects per group. The dependent variable is the “number of ideas” recalled(of a maximum of 14). The results are presented below.

No Context Context PartialContext Before After Context

& ' # '& * + +# , ' &+ + + '&% * % +

$#$ %' &' %- #%&#$ & ( &'# +'#

The figures taken from our SAS listing can be presented in an analysisof variance table:

Source "# $$ %$ ! ()!!"

# & '$'*$ %$'*( ('##!! .00288"!#" %- &('-$ #'&'

Total %* ,,''$

For more details on this experiment, please consult your textbook.


32 5.2 Example: Romeo and Juliet

5.2.1 [R] code! P5^QP ^*%`8'>(#$ 7%(;%%* :67Y%>(:A M?PJ! Z#-%# '*1 o6&)%(

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5.2 Example: Romeo and Juliet 33

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34 5.3 Example: Face Perception, "+#, with # random

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5.3 Example:Face Perception, "!#" with # random

In a series of experiments on face perception we set out to see whether thedegree of attention devoted to each face varies across faces. In order toverify this hypothesis, we assigned 40 undergraduate students to five ex-perimental conditions. For each condition we have a man’s face drawn atrandom from a collection of several thousand faces. We use the subjects’pupil dilation when viewing the face as an index of the attentional inter-est evoked by the face. The results are presented in Table 5.1 (with pupildilation expressed in arbitrary units).

Experimental Groups

Group 1 Group 2 Group 3 Group 4 Group 5

)' $# )* $- $-)) )* )$ $' )&)$ $' )( $# )&)* )$ )( )& )$#& $$ $" )% $-)* $- )$ $# )$)- $' )) $$ $-)- )& )& )& )(

%#$ )# $' )% $" )&

TABLE 5.1 Results of a (fictitious) experiment on face perception.


5.3 Example: Face Perception, "+#, with # random 35

5.3.1 [R] code! P5^QP ^*%`8'>(#$ 7%(;%%* :67Y%>(:A M?PJ! S'>% R%$>%.()#*

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5.3.2 [R] output_ ! P5^QP ^*%`8'>(#$ 7%(;%%* :67Y%>(:A M?PJ_ ! S'>% R%$>%.()#*

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36 5.3 Example: Face Perception, "+#, with # random

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5.4 Example: Face Perception, "+#, with # random 37

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5.3.3 ANOVA tableThe results of our fictitious face perception experiment are presented inthe following ANOVA Table:

Source !" ## $# % Pr(%)

# ) #-'$'' ('$'' "'$'' .000020"+#, #$ -('$'' ($''

Total #& *''$''

From this table it is clear that the research hypothesis is supported bythe experimental results: All faces do not attract the same amount of at-tention.


38 5.4 Example: Images ...

5.4 Example:Images ...

In another experiment on mental imagery, we have three groups of 5 stu-dents each (psychology majors for a change!) learn a list of 40 concretenouns and recall them one hour later. The first group learns each wordwith its definition, and draws the object denoted by the word (the builtimage condition). The second group was treated just like the first, but hadsimply to copy a drawing of the object instead of making it up themselves(the given image condition). The third group simply read the words andtheir definitions (the control condition.) Table 5.2 shows the number ofwords recalled 1 hour later by each subject. The experimental design is"!#", with ! ) ', * ) &, and # as a fixed factor.

Experimental Condition

Built Image Given Image Control

-- "# &"% & %-) ") "'-# "( "#-) -" "*

!""' %$ $$

%#$ -- "$ ""

TABLE 5.2 Results of the mental imagery experiment.

5.4.1 [R] code! P5^QP ^*%`8'>(#$ 7%(;%%* :67Y%>(:A M?PJ! P*#(4%$ %,'-.&%/ d-'3%:LLL

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5.4 Example: Images ... 39

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5.4.2 [R] output

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40 5.4 Example: Images ...

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5.4.3 ANOVA table

Source !" ## $# % Pr(%)

# - #"'$'' "$$$'' "'$##!! .0026"+#, "- "('$'' "$$''

Total ") )&'$''

We can conclude that instructions had an effect on memorization. Us-ing APA style (cf. APA manual, 1994, p. 68), to write our conclusion: “Thetype of instructions has an effect on memorization, !!#+ %#" ) %+'%$, %$% )%&'$(, , - '$%”.


6ANOVA One FactorBetween-Subjects: RegressionApproach

In order to use regression to analyze data from an analysis of variance de-sign, we use a trick that has a lot of interesting consequences. The mainidea is to find a way of replacing the nominal independent variable (i.e.,the experimental factor) by a numerical independent variable (rememberthat the independent variable should be numerical to run a regression).One way of looking at analysis of variance is as a technique predicting sub-jects’ behavior from the experimental group in which they were. The trickis to find a way of coding those groups. Several choices are possible, aneasy one is to represent a given experimental group by its mean for the de-pendent variable. Remember from Chapter 4 in the textbook (on regres-sion), that the rationale behind regression analysis implies that the inde-pendent variable is under the control of the experimenter. Using the groupmean seems to go against this requirement, because we need to wait untilafter the experiment to know the values of the independent variable. Thisis why we call our procedure a trick. It works because it is equivalent tomore elaborate coding schemes using multiple regression analysis. It hasthe advantage of being simpler both from a conceptual and computationalpoint of view.

In this framework, the general idea is to try to predict the subjects’scores from the mean of the group to which they belong. The rationale isthat, if there is an experimental effect, then the mean of a subject’s groupshould predict the subject’s score better than the grand mean. In otherwords, the larger the experimental effect, the better the predictive qualityof the group mean. Using the group mean to predict the subjects’ perfor-mance has an interesting consequence that makes regression and analysisof variance identical: When we predict the performance of subjects from themean of their group, the predicted value turns out to be the group mean too!

42 6.1 Example: Imagery and Memory revisited

6.1 Example:Imagery and Memory revisited

As a first illustration of the relationship between ANOVA and regression wereintroduce the experiment on Imagery and Memory detailed in Chapter9 of your textbook. Remember that in this experiment two groups of sub-jects were asked to learn pairs of words (e.g., “beauty-carrot”). Subjectsin the first group (control group) were simply asked to learn the pairs ofwords the best they could. Subjects in the second group (experimentalgroup) were asked to picture each word in a pair and to make an imageof the interaction between the two objects. After, some delay, subjectsin both groups were asked to give the second word (e.g., “carrot”) whenprompted with with the first word in the pair (e.g., “beauty”). For each sub-ject, the number of words correctly recalled was recorded. The purpose ofthis experiment was to demonstrate an effect of the independent variable(i.e., learning with imagery versus learning without imagery) on the depen-dent variable (i.e., number of words correctly recalled). The results of thescaled-down version of the experiment are presented in Table 6.1.

In order to use the regression approach, we use the respective groupmeans as predictor. See Table 6.2.

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Control Experimental

Subject 1: 1 Subject 1: 8Subject 2: 2 Subject 2: 8Subject 3: 5 Subject 3: 9Subject 4: 6 Subject 4: 11Subject 5: 6 Subject 5: 14

%$$ ! %Control ! ) %%$ ! %Experimental ! "'Grand Mean ! %! ! %$$ ! %

TABLE 6.1 Results of the “Memory and Imagery” experiment.


6.1 Example: Imagery and Memory revisited 43

" ! &!" Predictor " " " " " $& $& $& $& $&

! (Value to be predicted) $ % ' ( ( ) ) * $$ $"

TABLE 6.2 The data from Table 6.1 presented as a regression problem. The predictor " is the value of the mean

of the subject’s group.

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6.1.2 [R] output_ ! Z%3$%::)#* P..$#'>4/ P5^QP ^*%`8'>(#$ 7%(;%%* :67Y%>(:A M?PJ_ ! d-'3%$\ '*1 9%-#$\

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44 6.1 Example: Imagery and Memory revisited

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6.2 Example: Restaging Romeo and Juliet 45

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6.2 Example:Restaging Romeo and Juliet

This second example is again the “Romeo and Juliet” example from a repli-cation of Bransford et al.’s (1972) experiment. The rationale and details ofthe experiment are given in Chapter 9 in the textbook.

To refresh your memory: The general idea when using the regressionapproach for an analysis of variance problem is to predict subject scoresfrom the mean of the group to which they belong. The rationale for do-ing so is to consider the group mean as representing the experimental ef-fect, and hence as a predictor of the subjects’ behavior. If the independentvariable has an effect, the group mean should be a better predictor of the


46 6.2 Example: Restaging Romeo and Juliet

" # # # # # + + + + + #$% #$% #$% #$% #$% "$% "$% "$% "$% "$%

! # # % " # ' * ) " * % " ' " $ ' " # ' "

TABLE 6.3 The data from the Romeo and Juliet experiment presented as a regression problem. The predictor "

is the value of the mean of the subject’s group.

subjects behavior than the grand mean. Formally, we want to predict thescore $#' of subject . in condition / from a quantitative variable " thatwill be equal to the mean of the group / in which the . observation wascollected. With an equation, we want to predict the observation by:

"$ ) /. 0" (6.1)

with " being equal to &#'. The particular choice of " has several interest-ing consequences. A first important one, is that the mean of the predictor&" is also the mean of the dependent variable &! . These two means arealso equal to the grand mean of the analysis of variance. With an equation:

&'' ) &" ) &! ' (6.2)

Table VI.3 gives the values needed to do the computation using the re-gression approach.

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6.2.2 [R] output

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7ANOVA One factor: PlannedOrthogonal Comparisons

The planned comparisons (also called a priori comparisons) are selectedbefore running the experiment. In general, they correspond to the researchhypothesis that is being tested. If the experiment has been designed toconfront two or more alternative theories (e.g., with the use of rival hy-potheses), the comparisons are derived from those theories. When the ex-periment is actually run, it is possible to see if the results support or elimi-nate one of the theories. Because these comparisons are planned they areusually few in number.

A set of comparisons is composed of orthogonal comparisons if thehypotheses corresponding to each comparison are independent of eachother. The maximum number of possible orthogonal comparisons onecan perform, is one less than the number of levels of the independent vari-able (i.e., *' %). *' % is also the number of degrees of freedom of the sumof squares for #.

All different types of comparisons can be performed following the sameprocedure:

& First. Formalization of the comparison, and expression of the com-parison as a set of weights for the means.

& Second. Computation of the !comp. ratio (this is the usual ! ratio adaptedfor the case of testing a comparison).

& Third. Evaluation of the probability associated with !comp..

7.1 Context and Memory

This example is inspired by an experiment by Smith (1979). The main pur-pose in this experiment was to show that to be in the same context forlearning and for test can give a better performance than being in differentcontexts. More specifically, Smith wants to explore the effect of puttingoneself mentally in the same context. The experiment is organized as fol-low. During the learning phase, subjects learn a list made of 80 words in a

52 7.1 Context and Memory

room painted with an orange color, decorated with posters, paintings anda decent amount of paraphernalia. A first test of learning is given then, es-sentially to give subjects the impression that the experiment is over. Oneday after, subjects are unexpectedly re-tested for their memory. An exper-imenter will ask them to write down all the words of the list they can re-member. The test takes place in 5 different experimental conditions. Fiftysubjects (10 per group) are randomly assigned to the experimental groups.The formula of the experimental design is "!#" or "$&!#'". The depen-dent variable measured is the number of words correctly recalled. The fiveexperimental conditions are:

& 1. Same context. Subjects are tested in the same room in which theylearned the list.

& 2. Different context. Subjects are tested in a room very different fromthe one in which they learned the list. The new room is located in adifferent part of the Campus, is painted grey, and looks very austere.

& 3. Imaginary context. Subjects are tested in the same room as sub-jects from group 2. In addition, they are told to try to remember theroom in which they learned the list. In order to help them, the exper-imenter asks them several questions about the room and the objectsin it.

& 4. Photographed context. Subjects are placed in the same conditionas group 3, and, in addition, they are shown photos of the orangeroom in which they learned the list.

& 5. Placebo context. Subjects are in the same condition as subjectsin group 2. In addition, before starting to try to recall the words, theyare asked first to perform a warm-up task, namely to try to remembertheir living room.

Several research hypotheses can be tested with those groups. Let usaccept that the experiment was designed to test the following research hy-potheses:

& Research Hypothesis 1. Groups for which the context at test matchesthe context during learning (i.e., is the same or is simulated by imag-ing or photography) will perform differently (precisely they are ex-pected to do better) than groups with a different context or than groupswith a Placebo context.

& Research Hypothesis 2. The group with the same context will differfrom the group with imaginary or photographed context.

& Research Hypothesis 3. The imaginary context group differs from thephotographed context group


7.1 Context and Memory 53

& Research Hypothesis 4. The different context group differs from theplacebo group.

The following Table gives the set of the four contrasts specified in theSAS program.

Comparison Gr.1 Gr.2 Gr.3 Gr.4 Gr.5

1$ .# '& .# .# '&1% .# $ '% '% $1# $ $ .% '% $1" $ .% $ $ '%

The data and results of the replication of Smith’s experiment are givenin the two following Tables (Tables 7.1, and 7.2).

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Experimental Context

Group 1 Group 2 Group 3 Group 4 Group 5Same Different Imagery Photo Placebo

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TABLE 7.1 Results of a replication of an experiment by Smith (1979). The dependent variable is the number of

words recalled.



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7.1.3 ANOVA table

Source "# $$ %$ ! Pr(!)

# + ($$'$$ %(''$$ ''+-*!! .00119"!#" +' %+ ++$'$$ &#'$$

Total +* #+ %+$'$$

TABLE 7.2 ANOVA Table for a replication of Smith’s (1979) experiment.




8Planned Non-orthogonalComparisons

Non-orthogonal comparisons are more complex than orthogonal com-parisons. The main problem lies in assessing the importance of a givencomparison independently of the other comparisons of the set. There arecurrently two (main) approaches to this problem. The classical approachcorrects for multiple statistical tests (i.e., using a Sidak or Bonferonni cor-rection), but essentially evaluates each contrast as if it were coming from aset of orthogonal contrasts. The multiple regression (or modern) approachevaluates each contrast as a predictor from a set of non-orthogonal pre-dictors and estimates its specific contribution to the explanation of the de-pendent variable.

8.1 Classical approach: Tests for non-orthogonalcomparisons

The Sidak or the Bonferonni, Boole, Dunn inequalityare used to find a cor-rection on 2/(30 in order to keep 2/(4 0 fixed. The general idea of the pro-cedure is to correct 2/(30 in order to obtain the overall 2/(4 0 for the ex-periment. By deciding that the family is the unit for evaluating Type I error,the inequalities give an approximation for each 2/(30. The formula usedto evaluate the alpha level for each comparison using the Sidak inequalityis:

2/(30 ( %' !%' 2/(4 0"$() '

This is a conservative approximation, because the following inequality holds:

2/(30 ) %' !%' 2/(4 0"$() '

The formula used to evaluate the alpha level for each comparison usingBonferonni, Boole, Dunn inequality would be:

2/(30 ( 2/(4 0

3'

62 8.2 Romeo and Juliet, non-orthogonal contrasts

By using these approximations, the statistical test will be a conserva-tive one. That is to say, the real value of 2/(4 0 will always be smaller thanthe approximation we use. For example, suppose you want to performfour non-orthogonal comparisons, and that you want to limit the risk ofmaking at least one Type I error to an overall value of 2/(4 0 ) '$'. Usingthe Sidak correction you will consider that any comparison of the familyreaches significance if the probability associated with it is smaller than:

2/(30 ) %' !%' 2/(4 0"$() ) %' !%' '$'"$(" ) '$%#(

Note, this is a change from the usual '$' and '$%.

8.2 Romeo and Juliet, non-orthogonal contrasts

An example will help to review this section. Again, let us return to Brans-ford’s “Romeo and Juliet”. The following Table gives the different experi-mental conditions:

Context Partial Context WithoutBefore Context After Context

Suppose that Bransford had build his experiment to test a priori fourresearch hypotheses:

& 1. The presence of any context has an effect.

& 2. The context given after the story has an effect.

& 3. The context given before has a stronger effect than any other con-dition.

& 4. The partial context condition differs from the “context before”condition.

These hypotheses can easily be translated into a set of contrasts givenin the following Table.


1$ % % % '&1% $ $ % '%1# & '% '% '%1" % '% $ $


8.2 Romeo and Juliet, non-orthogonal contrasts 63

If 2/(4 0 is set to the value '$', this will lead to testing each contrast withthe 2/(30 level:

2/(30 ) %' '*'$(" ) '$%#( '

If you want to use the critical values method, the Table gives for 5% ) %-(this is the number of degrees of freedom of %$*,+-), 2/(4 0 ) '$', and3 ) + the value !critical Sidak ) ('*% (This is simply the critical value of thestandard Fisher ! with 1 and 16 degrees of freedom and with 2 ) 2/(30 )'$%#().

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70 8.3 Multiple Regression and Orthogonal Contrasts

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8.3 Multiple Regression and OrthogonalContrasts

ANOVA and multiple regression are equivalent if we use as many predictorsfor the multiple regression analysis as the number of degrees of freedomof the independent variable. An obvious choice for the predictors is to usea set of contrasts. Doing so makes contrast analysis a particular case ofmultiple regression analysis. Regression analysis, in return, helps solvingsome of the problems associated with the use of non-orthogonal contrasts:It suffices to use multiple regression and semi-partial coefficients of corre-lation to analyze non-orthogonal contrasts.

In this section, we illustrate the multiple regression analysis approachof the Bransford experiment (i.e., “Romeo and Juliet”). We will first look ata set of orthogonal contrasts and then a set of non-orthogonal contrasts.

A set of data from a replication of this experiment is given in Table 8.1.

Experimental Condition

Context Partial Context Withoutbefore context after context

5 5 2 3

9 4 4 3

8 3 5 2

4 5 4 4

9 4 1 3

TABLE 8.1 The data from a replication of Bransford’s “Romeo and Juliet” experiment. &$$ ! "$#'.

In order to analyze these data with a multiple regression approach wecan use any arbitrary set of contrasts as long as they satisfy the followingconstraints:

1. there are as many contrasts as the independent variable has degreesof freedom,

2. the set of contrasts is not multicolinear. That is, no contrast can beobtained by combining the other contrasts1.

1The technical synonym for non-multicolinear is linearly independent.


8.3 Multiple Regression and Orthogonal Contrasts 71

Groups

Contrast 1 2 3 4

1$ % % '% '%

1% % '% 0 0

1# 0 0 1 '%

TABLE 8.2 An arbitrary set of orthogonal contrasts for analyzing “Romeo and Juliet.”

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78 8.4 Multiple Regression and Non-orthogonal Contrasts

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8.4 Multiple Regression and Non-orthogonalContrasts

Most of the time, when experimentalists are concerned with a priori non-orthogonal comparisons, each comparison represents a prediction from agiven theory. The goal of the experiment is, in general, to decide which one(or which ones) of the theories can explain the data best. In other words,the experiment is designed to eliminate some theories by showing thatthey cannot predict what the other theory (or theories) can predict. There-fore experimenters are interested in what each theory can specifically ex-plain. In other words, when dealing with a priori non-orthogonal compar-isons what the experimenter wants to evaluate are semi-partial coefficientsof correlation because they express the specific effect of a variable. Withinthis framework, the multiple regression approach for non-orthogonal pre-dictors fits naturally. The main idea, when analyzing non-orthogonal con-trasts is simply to consider each contrast as an independent variable in anon-orthogonal multiple regression analyzing the dependent variable.

Suppose (for the beauty of the argument) that the “Romeo and Juliet”experiment was, in fact, designed to test three theories. Each of these the-ories is expressed as a contrast.

1. Bransford’s theory implies that only the subjects from the context be-fore group should be able to integrate the story with their long termknowledge. Therefore this group should do better than all the other


8.4 Multiple Regression and Non-orthogonal Contrasts 79

groups, which should perform equivalently. This is equivalent to thefollowing contrast:

1$ ) &$ 6$ ' %$ 6% ' %$ 6# ' %$ 6"

2. the imagery theory would predict (at least at the time the experimentwas designed) that any concrete context presented during learningwill improve learning. Therefore groups 1 and 2 should do betterthan the other groups. This is equivalent to the following contrast:

1% ) %$ 6$ %$ 6% ' %$ 6# ' %$ 6"

3. The retrieval cue theory would predict that the context acts duringthe retrieval phase (as opposed to Bransford’s theory which statesthat the context acts during the encoding phase). Therefore group 1and 3 should do better than the other groups. This is equivalent tothe following contrast:

1# ) %$ 6$ ' %$ 6% %$ 6# ' %$ 6"

Contrast Groups1 2 3 4

1$ & '% '% '%

1% % % '% '%

1# % '% % '%

TABLE 8.3 A set of non-orthogonal contrasts for analyzing “Romeo and Juliet.” The first contrast corresponds to

Bransford’s theory. The second contrast corresponds to the imagery theory. The third contrast corresponds to the

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9Post hoc or a-posteriorianalyses

Post hoc analyses are performed after the data have been collected, orin other words, after the fact. When looking at the results, you may findan unexpected pattern. If that pattern of results suggests some interestinghypothesis, then you want to be sure that it is not a fluke. This is the aimof post hoc (also called a posteriori) comparisons.

The main problem with post hoc comparisons involves the size of thefamily of possible comparisons. The number of possible comparisons,grows very quickly as a function of * (the number of levels of the indepen-dent variable), making the use of procedures such as Sidak or Bonferonni,Boole, Dunn inequalities unrealistic.

Two main approaches will be examined:

& Evaluating all the possible contrasts; this is known as Scheffe’s test.

& The specific problem of pairwise comparisons. Here we will see threedifferent tests: Tukey, Newman-Keuls, and Duncan.

Note that by default, SAS evaluates the contrasts with the 2 level set at'$'. If a lower 2 is desired, this must be specified by following the post hocoption name with ALPHA=.01. For example, to specify an alpha level of .01for a Scheffe’s test, you would give the following command:

MEANS GROUP / SCHEFFE ALPHA=.01.

9.1 Scheffe’s test

Scheffe’s test was devised in order to be able to test all the possible con-trasts a posteriori while maintaining the overall Type I error for the familyat a reasonable level, as well as trying to have a relatively powerful test.Specifically, the Scheffe test is a conservative test. The critical value forthe Scheffe test is larger than the critical value for other, more powerful,tests. In every case where the Scheffe test rejects the null hypothesis, morepowerful tests also reject the null hypothesis.

88 9.1 Scheffe’s test

9.1.1 Romeo and JulietWe will use, once again, Bransford et al.’s “Romeo and Juliet” experiment.The following Table gives the different experimental conditions:


The “error mean square” is %$*,+- ) #'&'; and ! ) '. Here are the val-ues of the experimental means (note that the means have been reorderedfrom the largest to the smallest):


&#' ('$$ +'#$ &'#$ &'$$

Suppose now that the experimenters wanted to test the following con-trasts after having collected the data.


1$ % % % '&1% $ $ % '%1# & '% '% '%1" % '% $ $

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The results of the Scheffe procedure for the family can be summarizedin the following Table:

Comparison $$comp. !comp. Decision Pr(!Scheffe)

1$ 12.15 5.17 ns .2015991% 0.10 $'$+ ns ! - %1# 46.82 19.92 reject 7& .0040191" 19.60 8.34 ns .074800


96 9.2 Tukey’s test

9.2 Tukey’s test

The Tukey test uses a distribution derived by Gosset, who is better knownunder his pen name of Student (yes, like Student-8!). Gosset/Student de-rived a distribution called Student’s 9 or the Studentized range, which is thevalue reported by SAS. The value reported in your textbook is a slightlymodified version of his distribution called !-range or !range. !-range is de-rived from 9 by the transformation:

!range )9%

#'

9.2.1 The return of Romeo and JulietFor an example, we will use, once again, Bransford et al.’s “Romeo andJuliet.” Recall that

%$*,+- ) #'&'1 ! ) '

and that the experimental results were:


&#' ('$$ +'#$ &'#$ &'$$

The pairwise difference between means can be given in a Table:

&$' &%' &#' &"'

&$' #',$ &',$ +'$$&%' %'$$ %'#$&#' $'#$

The values for !critical,Tukey given by the table are

,'#$ for 2/(4 0 ) '$'

%&'+( for 2/(4 0 ) '$%

The results of the computation of the different ! ratios for the pairwisecomparisons are given in the following Table, the sign * indicates a differ-ence significant at the '$' level, and ** indicates a difference significant atthe '$% level.


9.2 Tukey’s test 97

Note in the SAS output, that the “Critical Value of Studentized Range= 4.046”. Remember the formula to derive the !range from 9

!range )9%

#'

For our example,

!range ) ,'%,' )+'$+-%

#'

The difference observed between the value reported in the book and thatobtained using SAS’s value is due to rounding errors.

&$' &%' &#' &"'

&$' ,'&+! %''&-!! %('$#!!

&%' %'$- %''&&#' $'+$

Tukey test is clearly a conservative test. Several approaches have beendevised in order to have a more sensitive test. The most popular alterna-tive (but not the “safest”) is the Newman-Keuls test.

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106 9.3 Newman-Keuls’ test

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9.3 Newman-Keuls’ test

Essentially, the Newman-Keuls test consists of a sequential test inwhich the critical value depends on the range of each pair of means. Tomake the explanation easier, we will suppose that the means are orderedfrom the smallest to the largest. Hence &$$ is the smallest mean, and &+$

is the largest mean.The Newman-Keuls test starts like the Tukey test. The largest differ-

ence between two means is selected. The range of the difference is *. Thenull hypothesis is tested for that mean using !range following exactly thesame procedure as for the Tukey test. If the null hypothesis cannot be re-jected the test stops here, because not rejecting the null hypothesis for thelargest difference implies not rejecting the null hypothesis for any otherdifference.

If the null hypothesis is rejected for the largest difference, then the twodifferences with a range of * ' % are examined. They will be tested with acritical value of !range selected for a range of*'%. When the null hypothesiscannot be rejected for a given difference, none of the differences includedin that difference will be tested. If the null hypothesis can be rejected fora difference, then the procedure is re-iterated for a range of * ' #. Theprocedure is used until all the differences have been tested or declarednonsignificant by implication.

9.3.1 Taking off with Loftus. . .In an experiment on eyewitness testimony, Loftus and Palmer (1974) testedthe influence of the wording of a question on the answers given by eyewit-nesses. They presented a film of a multiple car crash to 20 subjects. Afterseeing the film, subjects were asked to answer a number of specific ques-tions. Among these questions, one question about the speed of the car waspresented with five different versions:

& “HIT”: About how fast were the cars going when they hit each other?

& “SMASH”: About how fast were the cars going when they smashed intoeach other?

& “COLLIDE”: About how fast were the cars going when they collidedwith each other?

& “BUMP”: About how fast were the cars going when they bumped intoeach other?


9.3 Newman-Keuls’ test 107

& “CONTACT”: About how fast were the cars going when they contactedeach other?

The mean speed estimation by subjects for each version is given in thefollowing Table:

Experimental GroupContact Hit Bump Collide Smash

&$$ &%$ &#$ &"$ &'$

&#$ &$'$$ &''$$ &,'$$ +%'$$ +-'$$

! ) %$1 %$*,+- ) ,$'$$The obtained ! ratios are given in the following Table.

Experimental GroupContact Hit Bump Collide Smash

Contact — %''- +'$$ (''-! %-'$$!!

Hit — $''- #'#' (''-!

Bump — $''- +'$$Collide — %''-Smash —

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9.3.2 Guess who?Using the data from Bransford‘s Romeo and Juliet, we ran the post hoc con-trasts with the Newman-Keuls test.

9.3.2.1 [R] code

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10ANOVA Two factor design:#$ % or "!#$ %"

10.1 Cute Cued Recall

To illustrate the use of a two-factor design, consider a replication of an ex-periment by Tulving & Pearlstone (1966), in which 60 subjects were askedto learn lists of 12, 24 or 48 words (factor # with 3 levels). These wordscan be put in pairs by categories (for example, apple and orange can begrouped as “fruits”). Subjects were asked to learn these words, and thecategory name was shown at the same time as the words were presented.Subjects were told that they did not have to learn the category names. Aftera very short time, subjects were asked to recall the words. At that time halfof the subjects were given the list of the category names, and the other halfhad to recall the words without the list of categories (factor % with 2 levels).The dependent variable is the number of words recalled by each subject.Note that both factors are fixed. The results obtained in the six experimen-tal conditions used in this experiment are presented in Table 10.1.

Factor #Factor % &$: 12 words &%: 24 words &#: 48 words

"" '% "# "$ "% "*'& "- "( "# -' -#

'$ "# "" "& '& -- "&Free Recall '& "' "# '( "# -'

'( "' '( ") -" "&

"- "' "# ") #- #'"- "- -" "# #" ##

'% '% "' -' ") -% -$Cued Recall '& '% "$ "* #' -$

'& "- "% '% -& -(

TABLE 10.1 Results of a replication of Tulving and Pearlstone’s experiment (1966). The dependent variable is the

number of words recalled (see text for explanation).

130 10.1 Cute Cued Recall

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_ MMNb6'1$'()>=:6--'$\?'#<?:>#$%]&):(N&%*3(4l$%>'&&N(\.%l)*(%$'>()#*A >#*($':(:=&):(?&):(N&%*3(4 =-'T%L>#*($':(:?b6'1$'()>JJJA :.&)( = &):(?&):(N&%*3(4 =&):(?[b6'1$'()>[ = HJJJiiHjjtM6-

_ MMN'HN<:N'CN'@=:6--'$\?'#<?:>#$%]&):(N&%*3(4l$%>'&&N(\.%l)*(%$'>()#*A>#*($':(: = &):(?&):(N&%*3(4 =-'T%L>#*($':(:?'HN<:N'CN'@JJJA :.&)( = &):(?&):(N&%*3(4 =&):(?['HN<:N'CN'@[ = HJJJiiHjjtM6-

_ MMN'CN<:N'@ = :6--'$\?'#<?:>#$%]&):(N&%*3(4 l $%>'&&N(\.% l)*(%$'>()#*A>#*($':(: = &):(?&):(N&%*3(4 =-'T%L>#*($':(:?'CN<:N'@JJJA :.&)( = &):(?&):(N&%*3(4 =&):(?['CN<:N'@[ = HJJJiiHjjtM6-

_ MMNPpN>#*($':( = :6--'$\?'#<?:>#$%]&):(N&%*3(4 l $%>'&&N(\.% l)*(%$'>()#*A >#*($':(: = &):(?)*(%$'>()#* =-'T%L>#*($':(:?PpN>#*($':(JJJA :.&)( = &):(?)*(%$'>()#* =&):(?[PpN>#*($':([ = HJJJiiHjjtM6-

_ MMN"#*( = 1'('L8$'-%?$7)*1?MMN2)*%'$A MMNb6'1$'()>AMMN'HN<:N'CN'@A MMN'CN<:N'@A MMNPpN>#*($':(JJ

_ 9MN2)*%'$ = :6--'$\?'#<?:>#$%]&):(N&%*3(4 l $%>'&&N(\.% l)*(%$'>()#*A >#*($':(: = &):(?&):(N&%*3(4 =-'T%L>#*($':(:?2)*%'$JJJA :.&)( = &):(?&):(N&%*3(4 =&):(?[2)*%'$[ = HJJJiiHjjt9%'*

_ 9MNb6'1$'()> = :6--'$\?'#<?:>#$%]&):(N&%*3(4 l $%>'&&N(\.% l)*(%$'>()#*A >#*($':(: = &):(?&):(N&%*3(4 =-'T%L>#*($':(:?b6'1$'()>JJJA :.&)( = &):(?&):(N&%*3(4 =&):(?[b6'1$'()>[ = HJJJiiHjjt9%'*

_ 9MN'HN<:N'CN'@ = :6--'$\?'#<?:>#$%]&):(N&%*3(4 l $%>'&&N(\.% l)*(%$'>()#*A >#*($':(: = &):(?&):(N&%*3(4 =-'T%L>#*($':(:?'HN<:N'CN'@JJJA :.&)( = &):(?&):(N&%*3(4 =&):(?['HN<:N'CN'@[ = HJJJiiHjjt9%'*

_ 9MN'CN<:N'@ = :6--'$\?'#<?:>#$%]&):(N&%*3(4 l $%>'&&N(\.% l)*(%$'>()#*A >#*($':(: = &):(?&):(N&%*3(4 =-'T%L>#*($':(:?'CN<:N'@JJJA :.&)( = &):(?&):(N&%*3(4 =&):(?['CN<:N'@[ = HJJJiiHjjt9%'*

_ 9MNPpN>#*($':( = :6--'$\?'#<?:>#$%]&):(N&%*3(4 l $%>'&&N(\.% l)*(%$'>()#*A >#*($':(: = &):(?)*(%$'>()#* =-'T%L>#*($':(:?PpN>#*($':(JJJA:.&)( = &):(?)*(%$'>()#* =&):(?[PpN>#*($':([ = HJJJiiHjjt9%'*

_ 9MN"#*(=1'('L8$'-%?$7)*1?9MN2)*%'$A 9MNb6'1$'()>A 9MN'HN<:N'CN'@A9MN'CN<:N'@A 9MNPpN>#*($':(JJ

_ SN2)*%'$ = :6--'$\?'#<?:>#$%]&):(N&%*3(4 l $%>'&&N(\.% l)*(%$'>()#*A >#*($':(: = &):(?&):(N&%*3(4 =-'T%L>#*($':(:?2)*%'$JJJA :.&)( = &):(?&):(N&%*3(4 =&):(?[2)*%'$[ = HJJJiiHjjtS

_ SNb6'1$'()> = :6--'$\?'#<?:>#$%]&):(N&%*3(4 l $%>'&&N(\.% l)*(%$'>()#*A >#*($':(: = &):(?&):(N&%*3(4 =-'T%L>#*($':(:?b6'1$'()>JJJA :.&)( = &):(?&):(N&%*3(4 =&):(?[b6'1$'()>[ = HJJJiiHjjtS



_ SN'HN<:N'CN'@ = :6--'$\?'#<?:>#$%]&):(N&%*3(4 l $%>'&&N(\.% l)*(%$'>()#*A >#*($':(: = &):(?&):(N&%*3(4 =-'T%L>#*($':(:?'HN<:N'CN'@JJJA :.&)( = &):(?&):(N&%*3(4 =&):(?['HN<:N'CN'@[ = HJJJiiHjjtS

_ SN'CN<:N'@ = :6--'$\?'#<?:>#$%]&):(N&%*3(4l$%>'&&N(\.% l)*(%$'>()#*A>#*($':(: = &):(?&):(N&%*3(4 =-'T%L>#*($':(:?'CN<:N'@JJJA :.&)( = &):(?&):(N&%*3(4 =&):(?['CN<:N'@[ = HJJJiiHjjtS

_ SNPpN>#*($':( = :6--'$\?'#<?:>#$%]&):(N&%*3(4 l $%>'&&N(\.% l)*(%$'>()#*A >#*($':(: = &):(?)*(%$'>()#* =-'T%L>#*($':(:?PpN>#*($':(JJJA :.&)( = &):(?)*(%$'>()#* =&):(?[PpN>#*($':([ = HJJJiiHjjtS

_ SN"#*( = 1'('L8$'-%?$7)*1?SN2)*%'$A SNb6'1$'()>A SN'HN<:N'CN'@ASN'CN<:N'@A SNPpN>#*($':(JJ

_ R$N2)*%'$ = :6--'$\?'#<?:>#$%]&):(N&%*3(4 l $%>'&&N(\.% l)*(%$'>()#*A >#*($':(: = &):(?&):(N&%*3(4 =-'T%L>#*($':(:?2)*%'$JJJA :.&)( = &):(?&):(N&%*3(4 =&):(?[2)*%'$[ = HJJJiiHjjtR$

_ R$Nb6'1$'()> = :6--'$\?'#<?:>#$%]&):(N&%*3(4 l $%>'&&N(\.% l)*(%$'>()#*A >#*($':(: = &):(?&):(N&%*3(4 =-'T%L>#*($':(:?b6'1$'()>JJJA :.&)( = &):(?&):(N&%*3(4 =&):(?[b6'1$'()>[ = HJJJiiHjjtR$

_ R$N'HN<:N'CN'@ = :6--'$\?'#<?:>#$%]&):(N&%*3(4 l $%>'&&N(\.% l)*(%$'>()#*A >#*($':(: = &):(?&):(N&%*3(4 =-'T%L>#*($':(:?'HN<:N'CN'@JJJA :.&)( = &):(?&):(N&%*3(4 =&):(?['HN<:N'CN'@[ = HJJJiiHjjtR$

_ R$N'CN<:N'@ = :6--'$\?'#<?:>#$%]&):(N&%*3(4 l $%>'&&N(\.% l)*(%$'>()#*A >#*($':(: = &):(?&):(N&%*3(4 =-'T%L>#*($':(:?'CN<:N'@JJJA :.&)( = &):(?&):(N&%*3(4 =&):(?['CN<:N'@[ = HJJJiiHjjtR$

_ R$NPpN>#*($':( = :6--'$\?'#<?:>#$%]&):(N&%*3(4 l $%>'&&N(\.% l)*(%$'>()#*A >#*($':(: = &):(?)*(%$'>()#* =-'T%L>#*($':(:?PpN>#*($':(JJJA:.&)( = &):(?)*(%$'>()#* =&):(?[PpN>#*($':([ = HJJJiiHjjtR$

_ R$N"#*( = 1'('L8$'-%?$7)*1?R$N2)*%'$A R$Nb6'1$'()>AR$N'HN<:N'CN'@A R$N'CN<:N'@A R$NPpN>#*($':(JJ

_ "#*($':(N*'-%:=>?[2)*%'$[A [b6'1$'()>[A ['H <: 'C w'@[A ['C <:'@[A[Pp[J

_ "#*(N-'(=$7)*1?[2)*%'$[=2)*%'$A[b6'1$'()>[=b6'1$'()>A['H <: 'Cw'@[='HN<:N'CN'@A['C <: '@[='CN<:N'@A[Pp[=PpN>#*($':(J

_ "#*($':(:=1'('L8$'-%?nH="#*(N-'(iAHjA nC = "#*(N-'(iACjA n@ ="#*(N-'(iA@jA nF = "#*(N-'(iAFjA nE = "#*(N-'(iAEjA nB ="#*(N-'(iABjJ

_ "#*($':(NM6--'$\=1'('L8$'-%?"#*($':( = "#*($':(N*'-%:A OS =>?O8N"#*(iH/FACjA O8N"#*(iEAFjJA "#*($':(NMM =>?MMN"#*(iH/FACjA MMN"#*(iEAFjJA 9%'*NMa6'$% =>?9MN"#*(iH/FACjA 9MN"#*(iEAFjJA SNQ'&6% = >?SN"#*(iH/FACjASN"#*(iEAFjJA R$=>?R$N"#*(iH/FACjA R$N"#*(iEAFjJJ

_ ! 5#; ;% .$)*( (4% 1'(' '*1 '&& (4% $%:6&(:_ .$)*(?1'('J



`````````````````````````````````````M>#$% S'>(#$NP S'>(#$Np n$#6.

`````````````````````````````````````H HH HC 0#$1: S$%% Z%>'&& 'H7HC H@ CF 0#$1: S$%% Z%>'&& 'H7H@ HG FK 0#$1: S$%% Z%>'&& 'H7HF D HC 0#$1: S$%% Z%>'&& 'H7HE HK CF 0#$1: S$%% Z%>'&& 'H7HB CI FK 0#$1: S$%% Z%>'&& 'H7HG H@ HC 0#$1: S$%% Z%>'&& 'H7HK HD CF 0#$1: S$%% Z%>'&& 'H7HD CC FK 0#$1: S$%% Z%>'&& 'H7HHI D HC 0#$1: S$%% Z%>'&& 'H7HHH H@ CF 0#$1: S$%% Z%>'&& 'C7HHC H@ FK 0#$1: S$%% Z%>'&& 'C7HH@ K HC 0#$1: S$%% Z%>'&& 'C7HHF K CF 0#$1: S$%% Z%>'&& 'C7HHE CH FK 0#$1: S$%% Z%>'&& 'C7HHB G HC 0#$1: S$%% Z%>'&& 'C7HHG HE CF 0#$1: S$%% Z%>'&& 'C7HHK HB FK 0#$1: S$%% Z%>'&& 'C7HHD HC HC 0#$1: S$%% Z%>'&& 'C7HCI H@ CF 0#$1: S$%% Z%>'&& 'C7HCH C@ FK 0#$1: S$%% Z%>'&& '@7HCC HH HC 0#$1: S$%% Z%>'&& '@7HC@ D CF 0#$1: S$%% Z%>'&& '@7HCF HD FK 0#$1: S$%% Z%>'&& '@7HCE HI HC 0#$1: S$%% Z%>'&& '@7HCB K CF 0#$1: S$%% Z%>'&& '@7HCG CI FK 0#$1: S$%% Z%>'&& '@7HCK HI HC 0#$1: S$%% Z%>'&& '@7HCD HF CF 0#$1: S$%% Z%>'&& '@7H@I HD FK 0#$1: S$%% Z%>'&& '@7H@H HC HC 0#$1: "6%1 Z%>'&& 'H7C@C H@ CF 0#$1: "6%1 Z%>'&& 'H7C@@ @C FK 0#$1: "6%1 Z%>'&& 'H7C@F HC HC 0#$1: "6%1 Z%>'&& 'H7C@E CH CF 0#$1: "6%1 Z%>'&& 'H7C@B @H FK 0#$1: "6%1 Z%>'&& 'H7C@G G HC 0#$1: "6%1 Z%>'&& 'H7C@K CI CF 0#$1: "6%1 Z%>'&& 'H7C@D CG FK 0#$1: "6%1 Z%>'&& 'H7CFI D HC 0#$1: "6%1 Z%>'&& 'H7CFH HE CF 0#$1: "6%1 Z%>'&& 'C7CFC @I FK 0#$1: "6%1 Z%>'&& 'C7CF@ D HC 0#$1: "6%1 Z%>'&& 'C7CFF HG CF 0#$1: "6%1 Z%>'&& 'C7CFE CD FK 0#$1: "6%1 Z%>'&& 'C7CFB HI HC 0#$1: "6%1 Z%>'&& 'C7CFG HF CF 0#$1: "6%1 Z%>'&& 'C7CFK @I FK 0#$1: "6%1 Z%>'&& 'C7CFD HC HC 0#$1: "6%1 Z%>'&& 'C7CEI H@ CF 0#$1: "6%1 Z%>'&& 'C7C



EH @@ FK 0#$1: "6%1 Z%>'&& '@7CEC HI HC 0#$1: "6%1 Z%>'&& '@7CE@ HF CF 0#$1: "6%1 Z%>'&& '@7CEF CE FK 0#$1: "6%1 Z%>'&& '@7CEE G HC 0#$1: "6%1 Z%>'&& '@7CEB HB CF 0#$1: "6%1 Z%>'&& '@7CEG CE FK 0#$1: "6%1 Z%>'&& '@7CEK HC HC 0#$1: "6%1 Z%>'&& '@7CED G CF 0#$1: "6%1 Z%>'&& '@7CBI CK FK 0#$1: "6%1 Z%>'&& '@7C`````````````````````````````````````

_ .$)*(?-#1%&L('7&%:?'#<HA[-%'*:[JA1)3)(:=@J

U'7&%: #8 -%'*:n$'*1 -%'*HB

2):(N&%*3(4`````````````````````````````HC 0#$1: CF 0#$1: FK 0#$1:`````````````````````````````

HI HF CF`````````````````````````````

Z%>'&&N(\.%````````````````````````S$%% Z%>'&& "6%1 Z%>'&&````````````````````````

HF HK````````````````````````

2):(N&%*3(4/Z%>'&&N(\.%```````````````````````````````````

Z%>'&&N(\.%```````````````````````

2):(N&%*3(4 S$%% Z%>'&& "6%1 Z%>'&&```````````````````````````````````

HC 0#$1: HI HICF 0#$1: H@ HEFK 0#$1: HD CD

```````````````````````````````````

_ :6--'$\?'#<HJ

``````````````````````````````````````````````````````````Ò8 M6- Ma 9%'* Ma S <'&6% R$?_SJ

```````````````````````````````````````````````````````````2):(N&%*3(4 C CIKI HIFI HHELEEB k CLC%`HB eeeZ%>'&&N(\.% H CFI CFI CBLBBG @LEGG%ÌB eee2):(N&%*3(4/Z%>'&&N(\.% C CKI HFI HELEEB FLBCF%ÌB eeeZ%:)16'&: EF FKB D``````````````````````````````````````````````````````````````


10.2 Projective Tests and Test Administrators 139

M)3*)8L >#1%:/ I WeeeW ILIIH WeeW ILIH WeW ILIE WLW ILH W W H

_ .$)*(?"#*($':(:J

``````````````````````````````nH nC n@ nF nE nB

``````````````````````````````2)*%'$ `H I H `H I Hb6'1$'()> H `C H H `C H'H <: 'C w'@ `C H H `C H H'C <: '@ I H `H I H `HPp `C C H `H H `H``````````````````````````````

_ .$)*(?"#*($':(NM6--'$\J

````````````````````````````````````````````````````````````````"#*($':( OS "#*($':(NMM 9%'*NMa6'$% SNQ'&6% R$

````````````````````````````````````````````````````````````````H 2)*%'$ H HDBI HDBI CHGLGGGGK HL@EHFI@%`CIC b6'1$'()> H HCI HCI H@L@@@@@ ELKDBCHB%ÌF@ 'H <: 'C w'@ H HIKI HIKI HCILIIIII CLFEDHDK%`HEF 'C <: '@ H HIII HIII HHHLHHHHH HLICFDDH%`HFE Pp H HCI HCI H@L@@@@@ ELKDBCHB%ÌF````````````````````````````````````````````````````````````````

10.1.3 ANOVA table

Source !" ## $# % Pr(%)

# - -( '('$'' "( ')'$'' ""$$$* ) $'''( ''"% " -)'$'' -)'$'' -*$*% $'''( ''%#% - -('$'' ")'$'' "$$$* $'''( ''("+#%, $) )(*$'' &$''

Total $& #( '(*$''

TABLE 10.2 ANOVA Table for Tulving and Pearlstone’s (1966) experiment.

10.2 Projective Tests and Test Administrators

For purposes of illustrating the [R] code for an #$% design with both fac-tors random (Model II), consider the following results from a hypotheticalexperiment on projective testing. The researchers were interested in theeffects of using different test administrators.


140 10.2 Projective Tests and Test Administrators

Note that only the ! values obtained using the random option in the [R]code are valid.

Test Administrators (#)Order (%) " - # ) Means

I "-% ""% """ "'("-" "'& """ "'' ""#

II ""% ""# """ "''"'& ""# "'" &- "'%

III "'% "'( && &-"'" "') &" &' &&

IV &( &$ &$ (%&) &# (& %% &"

V &% &* (& (&(& &- (# ($ &'

Means "'* "') &( &- "''

10.2.1 [R] code

10.2.2 [R] output_ ! P5^QP U;#`8'>(#$ 7%(;%%* :67Y%>(:A M?P,pJ_ ! R$#Y%>()<% U%:(: '*1 U%:( P1-)*):($'(#$:

_ ! 0% 4'<% C S'>(#$:A P w pA ;)(4 F &%<%&: )* S'>(#$ P '*1_ ! E &%<%&: )* S'>(#$ pL_ ! U4%$%8#$% (4%$% '$% F , E = CI 3$#6.: ;)(4 C #7:%$<'()#*:_ ! ?:67Y%>(:J .%$ 3$#6.L

_ ! S'>(#$ P/ U%:( P1-)*):($'(#$_ ! ?'H=HA'C=CA'@=@A'F=FJ

_ ! S'>(#$ p/ ^$1%$_ ! ?7H=dA 7C= ddA 7@= dddA 7F=dQA 7E=QJ

_ ! 0% >#&&%>( (4% 1'(' 8#$ %'>4 &%<%& #8 S'>(#$ p '>$#:: '&& (4%_ ! &%<%&: #8 S'>(#$ PL_ d=>?HCGAHHGAHHHAHIKA HCHAHIDAHHHAHIIJ_ dd=>?HHGAHH@A HHHA HIIA HIDAHH@AHIHADCJ_ ddd=>?HIGAHIKADDADCA HIHAHIFADHADIJ_ dQ=>?DKADEADEAKGA DFAD@AKDAGGJ_ Q=>?DGADBAKDAKDA KDADCAK@AKEJ

_ ! 0% *#; >#-7)*% (4% #7:%$<'()#*: )*(# #*% &#*3 >#&6-* ?:>#$%JL_ :>#$%=>?dAddAdddAdQAQJ

_ ! 0% *#; .$%.'$% (4% &'7%&: 8#$ (4% F,E,C :>#$%: '>>#$1)*3 (#_ ! (4% 8'>(#$ &%<%&:/



_ ! P1-)*NH P1-)*NC P1-)*N@ P1-)*NFA P1-)*NH P1-)*NC P1-)*N@_ ! P1-)*NFLLLLLLLL%(> 8#$ S'>(#$ P_ U%:(NP1-)*=3&?FAHAEeFeCA &'7%&: = >?[P1-)*NH[A [P1-)*NC[A

[P1-)*N@[A [P1-)*NF[JJ

_ ! d d dLLLLLLLA dd dd LLLLLLAddd ddd LLLLAdQ dQ LLLLLA Q QLLLLL_ ! %(> 8#$ S'>(#$ pL

_ ^$1%$=3&?EAFeCAEeFeCA &'7%&:=>?[d[A[dd[A[ddd[A[dQ[A[Q[JJ

_ ! 0% *#; 8#$- ' 1'(' 8$'-% ;)(4 (4% 1%.%*1%*( <'$)'7&% '*1 (4%_ ! 8'>(#$:L_ 1'(' = 1'('L8$'-%?:>#$% = :>#$%A S'>(#$NP = 8'>(#$?U%:(NP1-)*JA

S'>(#$Np=8'>(#$?^$1%$JJ

_ ! 0% *#; .%$8#$- (4% P5^QP #* (4% 1'('L_ '#<H='#<?:>#$%]U%:(NP1-)*e^$1%$A 1'('=1'('J

_ ! 9#1%& ddd ;4%* 7#(4 P '*1 p '$% $'*1#-_ '#<C = '#<?:>#$%]U%:(NP1-)* l ^$1%$ l +$$#$?U%:(NP1-)*/^$1%$JA

1'(' = 1'('J

_ ! 0% *#; .$)*( (4% 1'(' '*1 '&& (4% $%:6&(:_ .$)*(?1'('J

``````````````````````````M>#$% S'>(#$NP S'>(#$Np

``````````````````````````H HCG P1-)*NH dC HHG P1-)*NC d@ HHH P1-)*N@ dF HIK P1-)*NF dE HCH P1-)*NH dB HID P1-)*NC dG HHH P1-)*N@ dK HII P1-)*NF dD HHG P1-)*NH ddHI HH@ P1-)*NC ddHH HHH P1-)*N@ ddHC HII P1-)*NF ddH@ HID P1-)*NH ddHF HH@ P1-)*NC ddHE HIH P1-)*N@ ddHB DC P1-)*NF ddHG HIG P1-)*NH dddHK HIK P1-)*NC dddHD DD P1-)*N@ dddCI DC P1-)*NF dddCH HIH P1-)*NH dddCC HIF P1-)*NC dddC@ DH P1-)*N@ dddCF DI P1-)*NF dddCE DK P1-)*NH dQCB DE P1-)*NC dQ



CG DE P1-)*N@ dQCK KG P1-)*NF dQCD DF P1-)*NH dQ@I D@ P1-)*NC dQ@H KD P1-)*N@ dQ@C GG P1-)*NF dQ@@ DG P1-)*NH Q@F DB P1-)*NC Q@E KD P1-)*N@ Q@B KD P1-)*NF Q@G KD P1-)*NH Q@K DC P1-)*NC Q@D K@ P1-)*N@ QFI KE P1-)*NF Q``````````````````````````

_ :6--'$\?'#<HJ

```````````````````````````````````````````````````Ò8 M6- Ma 9%'* Ma S <'&6% R$?_SJ

```````````````````````````````````````````````````Ù%:(NP1-)* @ HCII FII CI @LHIC%ÌB eee^$1%$ F @CII KII FI CLK@B%ÌD eeeU%:(NP1-)*/^$1%$ HC CFI CI H ILFKCGZ%:)16'&: CI FII CI```````````````````````````````````````````````````````M)3*)8L >#1%:/ I WeeeW ILIIH WeeW ILIH WeW ILIE WLWILH W W H

_ .$)*(?-#1%&L('7&%:?'#<HA[-%'*:[JA1)3)(:=@J

U'7&%: #8 -%'*:n$'*1 -%'*

HII

U%:(NP1-)*```````````````````````````````P1-)*NH P1-)*NC P1-)*N@ P1-)*NF```````````````````````````````

HIB HIF DK DC```````````````````````````````

^$1%$```````````````````d dd ddd dQ Q

```````````````````HH@ HIG DD DH DI```````````````````

U%:(NP1-)*/^$1%$`````````````````````````````````

^$1%$`````````````````````



U%:(NP1-)* d dd ddd dQ Q``````````````````````````````````

P1-)*NH HCF HH@ HIF DB D@P1-)*NC HH@ HH@ HIB DF DFP1-)*N@ HHH HIB DE DC KBP1-)*NF HIF DB DH KC KG

`````````````````````````````````

_ :6--'$\?'#<CJ

+$$#$/ U%:(NP1-)*/^$1%$``````````````````````````````````````````````

O8 M6- Ma 9%'* Ma S <'&6% R$?_SJ`````````````````````````````````````````````Ù%:(NP1-)* @ HCII FII CI ELKHD%ÌE eee^$1%$ F @CII KII FI GLEDI%ÌG eeeZ%:)16'&: HC CFI CI`````````````````````````````````````````````````M)3*)8L >#1%:/ I WeeeW ILIIH WeeW ILIH WeW ILIE WLWILH W W H

+$$#$/ 0)(4)*``````````````````````````````````````````

O8 M6- Ma 9%'* Ma S <'&6% R$?_SJ``````````````````````````````````````````Z%:)16'&: CI FII CI``````````````````````````````````````````

10.2.3 ANOVA table

Source !" ## $# % Pr(%)

# # "( -''$'' )''$'' -'$''!! $''''(% ) #( -''$'' (''$'' )'$''!! $'''''#% "- -)'$'' -'$'' "$''ns $)(-()"+#%, -' )''$'' -'$''

Total #& $( ')'$''




11ANOVA One Factor RepeatedMeasures, " $#

11.1 " $# design

For illustrative purposes we designed a hypothetical experiment using awithin-subjects design. The independent variable consists of 4 levels andthe size of the experimental group is 5 (i.e., 5 subjects participated in theexperiment, the results are presented in Table 11.1:

11.1.1 [R] code! P5^QP ^*%`8'>(#$ ;)(4)* :67Y%>(:A M,P! 56-%$)>'& +,'-.&%

! 0% 4'<% H S'>(#$A PA ;)(4 F &%<%&: )* S'>(#$ P '*1 F! :67Y%>(:L! U4% F &%<%&: #8 S'>(#$ P '$%/ 'HA 'CA '@A 'FL! U4%$%8#$% (4%$% '$% F 3$#6.: ;)(4 (4% :'-% E #7:%$<'()#*:! ?:67Y%>(:J .%$ 3$#6.L

! 0% >#&&%>( (4% 1'(' 8#$ %'>4 :67Y%>( 8#$ '&& &%<%&: #8! S'>(#$ PL:67NH=>?EAFAHAKJ:67NC=>?GAFAHAHIJ

Levels of the independent variableSubjects !$ !% !# !" %$'

*$ . . . . . . . . . . 5 4 1 8 4.50*% . . . . . . . . . . 7 4 1 10 5.50*# . . . . . . . . . . 12 9 8 16 11.25*" . . . . . . . . . . 4 9 6 9 7.00*' . . . . . . . . . . 8 9 5 13 8.75

%#$ . . . . . . . . . 7.20 7.00 4.20 11.20 %$$ ! %$)'

TABLE 11.1 A numerical example of an " #$ design.

146 11.1 " $# design

:67N@=>?HCADAKAHBJ:67NF=>?FADABADJ:67NE=>?KADAEAH@J

! 0% *#; >#-7)*% (4% #7:%$<'()#*: )*(# #*% &#*3 >#&6-* ?:>#$%JL:>#$%=>?:67NHA:67NCA:67N@A:67NFA:67NEJ

! 0% *#; .$%.'$% (4% &'7%&: 8#$ (4% F,E :>#$%: '>>#$1)*3 (# (4%! 8'>(#$ &%<%&:/! 'H 'C '@ 'FA 'H 'C '@ 'FLLLLLLLL%(> 8#$ S'>(#$ PS'>(NP=3&?FAHAEeFeHA &'7%&:=>?['H[A['C[A['@[A['F[JJ

! :67NH :67NHLLLLLLLA :67NC :67NCLLLLLLA:67N@ :67N@ LLLLA:67NF! :67NF LLLLLA :67NE :67NELLLLL%(> 8#$ M67Y%>(:M67Y%>(=3&?EAFeHAEeFeHA &'7%&:=>?[:67NH[A [:67NC[A [:67N@[A

[:67NF[A [:67NE[JJ

! 0% *#; 8#$- ' 1'(' 8$'-% ;)(4 (4% 1%.%*1%*( <'$)'7&% '*1 (4% 8'>(#$:L1'('=1'('L8$'-%?:>#$% = :>#$%AS'>(#$NP = 8'>(#$?S'>(NPJA

M67Y%>( = 8'>(#$?M67Y%>(JJ

! P*#<' ;4%* [M67Y%>([ ): >#*:)1%$%1 ': ' $'*1#- 8'>(#$L'#<H='#<?:>#$%]S'>(NPl+$$#$?M67Y%>(JA1'('=1'('J

! 0% *#; .$)*( (4% 1'(' '*1 '&& (4% $%:6&(:

.$)*(?1'('J:6--'$\?'#<HJ.$)*(?-#1%&L('7&%:?'#<?:>#$%]S'>(NPlM67Y%>(JA[-%'*:[JA

11.1.2 [R] output

_ ! P5^QP ^*%`8'>(#$ ;)(4)* :67Y%>(:A M,P_ ! 56-%$)>'& +,'-.&%

_ ! 0% 4'<% H S'>(#$A PA ;)(4 F &%<%&: )* S'>(#$ P '*1 F_ ! :67Y%>(:L_ ! U4% F &%<%&: #8 S'>(#$ P '$%/ 'HA 'CA '@A 'FL_ ! U4%$%8#$% (4%$% '$% F 3$#6.: ;)(4 (4% :'-% E #7:%$<'()#*:_ ! ?:67Y%>(:J .%$ 3$#6.L

_ ! 0% >#&&%>( (4% 1'(' 8#$ %'>4 :67Y%>( 8#$ '&& &%<%&: #8_ ! S'>(#$ PL_ :67NH=>?EAFAHAKJ_ :67NC=>?GAFAHAHIJ_ :67N@=>?HCADAKAHBJ_ :67NF=>?FADABADJ_ :67NE=>?KADAEAH@J

_ ! 0% *#; >#-7)*% (4% #7:%$<'()#*: )*(# #*% &#*3 >#&6-* ?:>#$%JL_ :>#$%=>?:67NHA:67NCA:67N@A:67NFA:67NEJ

_ ! 0% *#; .$%.'$% (4% &'7%&: 8#$ (4% F,E :>#$%: '>>#$1)*3 (# (4%


11.1 " $# design 147

_ ! 8'>(#$ &%<%&:/_ ! 'H 'C '@ 'FA 'H 'C '@ 'FLLLLLLLL%(> 8#$ S'>(#$ P_ S'>(NP=3&?FAHAEeFeHA &'7%&:=>?['H[A['C[A['@[A['F[JJ

_ ! :67NH :67NHLLLLLLLA :67NC :67NCLLLLLLA:67N@ :67N@ LLLLA:67NF_ ! :67NF LLLLLA :67NE :67NELLLLL%(> 8#$ M67Y%>(:_ M67Y%>(=3&?EAFeHAEeFeHA &'7%&:=>?[:67NH[A [:67NC[A [:67N@[A

[:67NF[A [:67NE[JJ

_ ! 0% *#; 8#$- ' 1'(' 8$'-% ;)(4 (4% 1%.%*1%*( <'$)'7&% '*1 (4% 8'>(#$:L_ 1'('=1'('L8$'-%?:>#$% = :>#$%AS'>(#$NP = 8'>(#$?S'>(NPJA

M67Y%>( = 8'>(#$?M67Y%>(JJ

_ ! P*#<' ;4%* [M67Y%>([ ): >#*:)1%$%1 ': ' $'*1#- 8'>(#$L_ '#<H='#<?:>#$%]S'>(NPl+$$#$?M67Y%>(JA1'('=1'('J

_ ! 0% *#; .$)*( (4% 1'(' '*1 '&& (4% $%:6&(:

_ .$)*(?1'('J

`````````````````````````:>#$% S'>(#$NP M67Y%>(

`````````````````````````H E 'H :67NHC F 'C :67NH@ H '@ :67NHF K 'F :67NHE G 'H :67NCB F 'C :67NCG H '@ :67NCK HI 'F :67NCD HC 'H :67N@HI D 'C :67N@HH K '@ :67N@HC HB 'F :67N@H@ F 'H :67NFHF D 'C :67NFHE B '@ :67NFHB D 'F :67NFHG K 'H :67NEHK D 'C :67NEHD E '@ :67NECI H@ 'F :67NE`````````````````````````

_ :6--'$\?'#<HJ

+$$#$/ M67Y%>(```````````````````````````````````````````

O8 M6- Ma 9%'* Ma S <'&6% R$?_SJ```````````````````````````````````````````Z%:)16'&: F HHEL@II CKLKCE```````````````````````````````````````````

+$$#$/ 0)(4)*


148 11.2 Drugs and reaction time

``````````````````````````````````````````Ò8 M6- Ma 9%'* Ma S <'&6% R$?_SJ

```````````````````````````````````````````S'>(NP @ HCFLFII FHLFBG HFLHGG @%ÌF eeeZ%:)16'&: HC @ELHII CLDCE``````````````````````````````````````````````M)3*)8L >#1%:/ I WeeeW ILIIH WeeW ILIH WeW ILIEWLW ILH W W H

_ .$)*(?-#1%&L('7&%:?'#<?:>#$%]S'>(NPlM67Y%>(JA[-%'*:[JA1)3)(:=@J

U'7&%: #8 -%'*:n$'*1 -%'*

GLF

S'>(NP```````````````````'H 'C '@ 'F

```````````````````GLC GLI FLC HHLC```````````````````

M67Y%>(`````````````````````````````:67NH :67NC :67N@ :67NF :67NE`````````````````````````````FLEI ELEI HHLCE GLII KLGE````````````````````````````

11.2 Drugs and reaction time

In a psychopharmalogical experiment, we want to test the effect of twotypes of amphetamine-like drugs on latency performing a motor task. Inorder to control for any potential sources of variation due to individualreactions to amphetamines, the same six subjects were used in the threeconditions of the experiment: Drug A, Drug B, and Placebo. The depen-dent variable is the reaction time measured in msec. The data from theexperiment are presented in Table 11.2 on the facing page:

11.2.1 [R] code! P5^QP ^*%`8'>(#$ ;)(4)* :67Y%>(:A M,P! O$63: '*1 Z%'>()#* U)-%

! 0% 4'<% H S'>(#$A PA ;)(4 @ &%<%&: )* S'>(#$ P '*1 B! :67Y%>(:L


11.2 Drugs and reaction time 149

Experimental ConditionsSubject Drug A Placebo Drug B Total

*$ "-)$'' "'($'' "')$'' ##*$''*% "'$$'' "'%$'' "''$'' #"-$''*# "'%$'' &'$'' "''$'' -&%$''*" "'&$'' (&$'' &#$'' -&"$''*' &)$'' "'$$'' (&$'' -(($''*( "-"$'' %"$'' ()$'' -%*$''

Total **'$'' $%'$'' $%'$'' "( (''$''

TABLE 11.2 Results of a fictitious hypothetical experiment illustrating the computational routine for a * #+ design.

! U4% F &%<%&: #8 S'>(#$ P '$%/ O$63 PA R&'>%7#A O$63 pL! U4%$%8#$% (4%$% '$% @ 3$#6.: ;)(4 (4% :'-% E #7:%$<'()#*:! ?:67Y%>(:J .%$ 3$#6.L

! 0% >#&&%>( (4% 1'(' 8#$ %'>4 :67Y%>(: 8#$ '&& &%<%&: #8! S'>(#$ PL:67NH=>?HCFAHIKAHIFJ:67NC=>?HIEAHIGAHIIJ:67N@=>?HIGADIAHIIJ:67NF=>?HIDAKDAD@J:67NE=>?DFAHIEAKDJ:67NB=>?HCHAGHAKFJ

! 0% *#; >#-7)*% (4% #7:%$<'()#*: )*(# #*% &#*3 >#&6-* ?:>#$%JL:>#$%=>?:67NHA:67NCA:67N@A:67NFA:67NEA:67NBJ

! 0% *#; .$%.'$% (4% &'7%&: 8#$ (4% F,E :>#$%: '>>#$1)*3 (# (4% 8'>(#$ &%<%&:/! O$63NP R&'>%7# O$67NpA O$63NP R&'>%7# O$63NpLLLLLLLL%(> 8#$! S'>(#$ PO$63=3&?@AHABe@eHA &'7%&:=>?[O$63NP[A[R&'>%7#[A[O$63Np[JJ

! :67NH :67NHLLLLLLLA :67NC :67NCLLLLLLA:67N@ :67N@ LLLLA:67NF! :67NF LLLLLA :67NE :67NELLLLLA :67NB :67NB %(> 8#$ S'>(#$ pLM67Y%>(=3&?BA@eHABe@eHA &'7%&:=>?[:67 NH[A [:67NC[A [:67N@[A

[:67NF[A [:67NE[A [:67NB[JJ

! 0% *#; 8#$- ' 1'(' 8$'-% ;)(4 (4% 1%.%*1%*( <'$)'7&% '*1 (4%! 8'>(#$:L1'(' = 1'('L8$'-%?:>#$% = :>#$%A O$63 = 8'>(#$?O$63JA M67Y%>( =

8'>(#$?M67Y%>(JJ

! P*#<' ;4%* [M67Y%>([ ): >#*:)1%$%1 ': ' $'*1#- 8'>(#$L'#<H='#<?:>#$%]O$63l+$$#$?M67Y%>(JA1'('=1'('J

! 0% *#; .$)*( (4% 1'(' '*1 '&& (4% $%:6&(:.$)*(?1'('J:6--'$\?'#<HJ


150 11.2 Drugs and reaction time

.$)*(?-#1%&L('7&%:?'#<?:>#$%]O$63lM67Y%>(JA[-%'*:[JA1)3)(:=@J

11.2.2 [R] output

_ ! P5^QP ^*%`8'>(#$ ;)(4)* :67Y%>(:A M,P_ ! O$63: '*1 Z%'>()#* U)-%

_ ! 0% 4'<% H S'>(#$A PA ;)(4 @ &%<%&: )* S'>(#$ P '*1 B_ ! :67Y%>(:L

_ ! U4% F &%<%&: #8 S'>(#$ P '$%/ O$63 PA R&'>%7#A O$63 pL_ ! U4%$%8#$% (4%$% '$% @ 3$#6.: ;)(4 (4% :'-% E #7:%$<'()#*:_ ! ?:67Y%>(:J .%$ 3$#6.L

_ ! 0% >#&&%>( (4% 1'(' 8#$ %'>4 :67Y%>(: 8#$ '&& &%<%&: #8_ ! S'>(#$ PL_ :67NH=>?HCFAHIKAHIFJ_ :67NC=>?HIEAHIGAHIIJ_ :67N@=>?HIGADIAHIIJ_ :67NF=>?HIDAKDAD@J_ :67NE=>?DFAHIEAKDJ_ :67NB=>?HCHAGHAKFJ

_ ! 0% *#; >#-7)*% (4% #7:%$<'()#*: )*(# #*% &#*3 >#&6-* ?:>#$%JL_ :>#$%=>?:67NHA:67NCA:67N@A:67NFA:67NEA:67NBJ

_ ! 0% *#; .$%.'$% (4% &'7%&: 8#$ (4% F,E :>#$%: '>>#$1)*3 (# (4% 8'>(#$ &%<%&:/_ ! O$63NP R&'>%7# O$67NpA O$63NP R&'>%7# O$63NpLLLLLLLL%(> 8#$_ ! S'>(#$ P_ O$63=3&?@AHABe@eHA &'7%&:=>?[O$63NP[A[R&'>%7#[A[O$63Np[JJ

_ ! :67NH :67NHLLLLLLLA :67NC :67NCLLLLLLA:67N@ :67N@ LLLLA:67NF_ ! :67NF LLLLLA :67NE :67NELLLLLA :67NB :67NB %(> 8#$ S'>(#$ pL_ M67Y%>(=3&?BA@eHABe@eHA &'7%&:=>?[:67 NH[A [:67NC[A [:67N@[A

[:67NF[A [:67NE[A [:67NB[JJ

_ ! 0% *#; 8#$- ' 1'(' 8$'-% ;)(4 (4% 1%.%*1%*( <'$)'7&% '*1 (4%_ ! 8'>(#$:L_ 1'(' = 1'('L8$'-%?:>#$% = :>#$%A O$63 = 8'>(#$?O$63JA M67Y%>( =

8'>(#$?M67Y%>(JJ

_ ! P*#<' ;4%* [M67Y%>([ ): >#*:)1%$%1 ': ' $'*1#- 8'>(#$L_ '#<H='#<?:>#$%]O$63l+$$#$?M67Y%>(JA1'('=1'('J

_ ! 0% *#; .$)*( (4% 1'(' '*1 '&& (4% $%:6&(:_ .$)*(?1'('J

````````````````````````:>#$% O$63 M67Y%>(

````````````````````````H HCF O$63NP :67 NHC HIK R&'>%7# :67 NH@ HIF O$63Np :67 NH


11.2 Drugs and reaction time 151

F HIE O$63NP :67NCE HIG R&'>%7# :67NCB HII O$63Np :67NCG HIG O$63NP :67N@K DI R&'>%7# :67N@D HII O$63Np :67N@HI HID O$63NP :67NFHH KD R&'>%7# :67NFHC D@ O$63Np :67NFH@ DF O$63NP :67NEHF HIE R&'>%7# :67NEHE KD O$63Np :67NEHB HCH O$63NP :67NBHG GH R&'>%7# :67NBHK KF O$63Np :67NB````````````````````````

_ :6--'$\?'#<HJ

+$$#$/ M67Y%>(``````````````````````````````````````````

O8 M6- Ma 9%'* Ma S <'&6% R$?_SJ``````````````````````````````````````````Z%:)16'&: E GEI HEI``````````````````````````````````````````

+$$#$/ 0)(4)*```````````````````````````````````````````

O8 M6- Ma 9%'* Ma S <'&6% R$?_SJ``````````````````````````````````````````Ò$63 C DII FEI @LGE ILIBID@ LZ%:)16'&: HI HCII HCI``````````````````````````````````````````````M)3*)8L >#1%:/ I WeeeW ILIIH WeeW ILIH WeW ILIE WLW ILHW W H_ .$)*(?-#1%&L('7&%:?'#<?:>#$%]O$63lM67Y%>(JA[-%'*:[JA1)3)(:=@J

U'7&%: #8 -%'*:n$'*1 -%'*

HII

O$63``````````````````````Ò$63NP R&'>%7# O$63Np```````````````````````

HHI DE DE```````````````````````

M67Y%>(`````````````````````````````````````````:67 NH :67NC :67N@ :67NF :67NE :67NB`````````````````````````````````````````


152 11.3 Proactive Interference

HHC HIF DD DG DB DC`````````````````````````````````````````

11.2.3 ANOVA tableThe final results are presented in the analysis of variance table:

Source "# $$ %$ ! Pr(!)

# # *$$'$$ +'$'$$ &'(' '$-$" ' ('$'$$ %'$'$$#" %$ %+ #$$'$$ %#$'$$

Total %( #+ ,'$'$$

TABLE 11.3 ANOVA Table for the Drugs and Reaction Time experiment.

11.3 Proactive Interference

In an experiment on proactive interference, subjects were asked to learna list of ten pairs of words. Two days later they were asked to recall thesewords. Once they finished recalling this first list, they were asked to learna second list of ten pairs of words which they will be asked to recall after anew delay of two days. Recall of the second list was followed by a third listand so on until they learned and recalled six lists. The independent vari-able is the rank of the list in the learning sequence (first list, second list, ..., sixth list). The dependent variable is the number of words correctly re-called. The authors of this experiment predict that recall performance willdecrease as a function of the rank of the lists (this effect is called “proactiveinterference”). The data are presented in Table 11.4.

11.3.1 [R] code! P5^QP ^*%`8'>(#$ ;)(4)* :67Y%>(:A M,P! R$#'>()<% d*(%$8%$%*>%

! 0% 4'<% H S'>(#$A P ?Z'*TJA ;)(4 B &%<%&: )* S'>(#$ P '*1 K! :67Y%>(:L! U4% B &%<%&: #8 S'>(#$ P '$%/ $'*TNHA $'*TNCA $'*TN@A $'*TNFA! $'*TNEA $'*TNB! U4%$%8#$% (4%$% '$% B 3$#6.: ;)(4 (4% :'-% K #7:%$<'()#*:! ?:67Y%>(:J .%$ 3$#6.L

! 0% >#&&%>( (4% 1'(' 8#$ %'>4 :67Y%>(: 8#$ '&& &%<%&: #8


11.3 Proactive Interference 153

Rank of the listSubjects 1 2 3 4 5 6 Total*$ . . . . . . 17 13 12 12 11 11 76*% . . . . . . . 14 18 13 18 11 12 86*# . . . . . . . 17 16 13 11 15 14 86*" . . . . . . . 18 16 11 10 12 10 77*' . . . . . . . 17 12 13 10 11 13 76*( . . . . . . . 16 13 13 11 11 11 75*+ . . . . . . . 14 12 10 10 10 10 66*) . . . . . . . 16 17 15 11 13 11 83

Total 129 117 100 93 94 92 625

TABLE 11.4 Results of an experiment on the effects of proactive interference on memory.

! S'>(#$ PL:67NH=>?HGAH@AHCAHCAHHAHHJ:67NC=>?HFAHKAH@AHKAHHAHCJ:67N@=>?HGAHBAH@AHHAHEAHFJ:67NF=>?HKAHBAHHAHIAHCAHIJ:67NE=>?HGAHCAH@AHIAHHAH@J:67NB=>?HBAH@AH@AHHAHHAHHJ:67NG=>?HFAHCAHIAHIAHIAHIJ:67NK=>?HBAHGAHEAHHAH@AHHJ

! 0% *#; >#-7)*% (4% #7:%$<'()#*: )*(# #*% &#*3 >#&6-* ?:>#$%JL:>#$%=>?:67NHA:67NCA:67N@A:67NFA:67NEA:67NBA:67NGA:67NKJ

! 0% *#; .$%.'$% (4% &'7%&: 8#$ (4% B,K :>#$%: '>>#$1)*3 (# (4%! 8'>(#$ &%<%&:/! $'*TNH $'*TNC $'*TN@ $'*TNF $'*TNE $'*TNBLLLLLLLL%(> 8#$! S'>(#$ PZ'*T=3&?BAHAKeBeHA &'7%&: = >?[$'*TNH[A [$'*TNC[A [$'*TN@[A [$'*TF[A [$'*TNE[A [$'*TNB[JJ

! :67NH :67NHLLLLLLLA :67NC :67NCLLLLLLA:67N@ :67N@ LLLLA:67NF! :67NF LLLLLA :67NE :67NELLLLL%(> 8#$ M67Y%>(:

M67Y%>(=3&?KABeHAKeBeHA &'7%&:=>?[:67 NH[A [:67NC[A [:67N@[A[:67NF[A [:67NE[A [:67NB[A [:67NG[A [:67NK[JJ

! 0% *#; 8#$- ' 1'(' 8$'-% ;)(4 (4% 1%.%*1%*( <'$)'7&% '*1 (4%! 8'>(#$:L1'(' = 1'('L8$'-%?:>#$% = :>#$%A Z'*T = 8'>(#$?Z'*TJA M67Y%>( =

8'>(#$?M67Y%>(JJ

! P*#<' ;4%* [M67Y%>([ ): >#*:)1%$%1 ': ' $'*1#- 8'>(#$L'#<H='#<?:>#$%]Z'*Tl+$$#$?M67Y%>(JA1'('=1'('J

! 0% *#; .$)*( (4% 1'(' '*1 '&& (4% $%:6&(:.$)*(?1'('J



:6--'$\?'#<HJ.$)*(?-#1%&L('7&%:?'#<?:>#$% ] Z'*T l M67Y%>(A 1'(' =

1'('JA[-%'*:[JA1)3)(:=@J

11.3.2 [R] output

_ ! P5^QP ^*%`8'>(#$ ;)(4)* :67Y%>(:A M,P_ ! R$#'>()<% d*(%$8%$%*>%

_ ! 0% 4'<% H S'>(#$A P ?Z'*TJA ;)(4 B &%<%&: )* S'>(#$ P '*1 K_ ! :67Y%>(:L_ ! U4% B &%<%&: #8 S'>(#$ P '$%/ $'*TNHA $'*TNCA $'*TN@A $'*TNFA_ ! $'*TNEA $'*TNB_ ! U4%$%8#$% (4%$% '$% B 3$#6.: ;)(4 (4% :'-% K #7:%$<'()#*:_ ! ?:67Y%>(:J .%$ 3$#6.L

_ ! 0% >#&&%>( (4% 1'(' 8#$ %'>4 :67Y%>(: 8#$ '&& &%<%&: #8_ ! S'>(#$ PL_ :67NH=>?HGAH@AHCAHCAHHAHHJ_ :67NC=>?HFAHKAH@AHKAHHAHCJ_ :67N@=>?HGAHBAH@AHHAHEAHFJ_ :67NF=>?HKAHBAHHAHIAHCAHIJ_ :67NE=>?HGAHCAH@AHIAHHAH@J_ :67NB=>?HBAH@AH@AHHAHHAHHJ_ :67NG=>?HFAHCAHIAHIAHIAHIJ_ :67NK=>?HBAHGAHEAHHAH@AHHJ

_ ! 0% *#; >#-7)*% (4% #7:%$<'()#*: )*(# #*% &#*3 >#&6-* ?:>#$%JL_ :>#$%=>?:67NHA:67NCA:67N@A:67NFA:67NEA:67NBA:67NGA:67NKJ

_ ! 0% *#; .$%.'$% (4% &'7%&: 8#$ (4% B,K :>#$%: '>>#$1)*3 (# (4%_ ! 8'>(#$ &%<%&:/_ ! $'*TNH $'*TNC $'*TN@ $'*TNF $'*TNE $'*TNBLLLLLLLL%(> 8#$_ ! S'>(#$ P_ Z'*T=3&?BAHAKeBeHA &'7%&: = >?[$'*TNH[A [$'*TNC[A [$'*TN@[A [$'*TF[A [$'*TNE[A [$'*TNB[JJ

_ ! :67NH :67NHLLLLLLLA :67NC :67NCLLLLLLA:67N@ :67N@ LLLLA:67NF_ ! :67NF LLLLLA :67NE :67NELLLLL%(> 8#$ M67Y%>(:

_ M67Y%>(=3&?KABeHAKeBeHA &'7%&:=>?[:67 NH[A [:67NC[A [:67N@[A[:67NF[A [:67NE[A [:67NB[A [:67NG[A [:67NK[JJ

_ ! 0% *#; 8#$- ' 1'(' 8$'-% ;)(4 (4% 1%.%*1%*( <'$)'7&% '*1 (4%_ ! 8'>(#$:L_ 1'(' = 1'('L8$'-%?:>#$% = :>#$%A Z'*T = 8'>(#$?Z'*TJA M67Y%>( =

8'>(#$?M67Y%>(JJ

_ ! P*#<' ;4%* [M67Y%>([ ): >#*:)1%$%1 ': ' $'*1#- 8'>(#$L_ '#<H='#<?:>#$%]Z'*Tl+$$#$?M67Y%>(JA1'('=1'('J

_ ! 0% *#; .$)*( (4% 1'(' '*1 '&& (4% $%:6&(:_ .$)*(?1'('J


11.3 Proactive Interference 155

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11.3.3 ANOVA tableThe results from this experiment are presented in the analysis of variancetable.

Source !" ## $# % P(%)

# $ ")*$($ -&$#% "'$#-!! $'''( ''$" % $-$)( %$$'#" #$ &&$*$ -$($

Total )% -"( ('*$$'


12Two factors repeatedmeasures, " $#$ %

12.1 Plungin’

What follows is a replication of Godden and Baddeley’s (1975) experi-ment to show the effects of context on memory. Godden and Baddeley’shypothesis was that memory should be better when the conditions at testare more similar to the conditions experienced during learning. To oper-ationalize this idea, Godden and Baddeley decided to use a very particu-lar population: deep-sea divers. The divers were asked to learn a list of40 unrelated words either on the beach or under about 10 feet of water,The divers were then tested either on the beach or undersea. The diversneeded to be tested in both environments in order to make sure that anyeffect observed could not be attributed to a global effect of one of the en-vironments. The rationale behind using divers was twofold. The first rea-son was practical: is it worth designing training programs on dry land fordivers if they are not able to recall undersea what they have learned? Thereis strong evidence, incidently, that the problem is real. The second rea-son was more akin to good principles of experimental design, The differ-ence between contexts undersea and on the beach seems quite important,hence a context effect should be easier to demonstrate in this experiment.

Because it is not very easy to find deep-sea divers (willing in additionto participate in a memory experiment) it was decided to use the smallnumber of divers in all possible conditions of the design. The list of wordswere randomly created and assigned to each subject. The order of testingwas randomized in order to eliminate any possible carry-over effects byconfounding them with the experimental error.

The first independent variable is the place of learning. It has 2 levels (onthe beach and undersea), and it is denoted #. The second independentvariable is the place of testing. It has 2 levels (on the beach and undersea,like #), and it is denoted %. Crossing these 2 independent variables gives4 experimental conditions:

& 1. Learning on the beach and recalling on the beach.

& 2. Learning on the beach and recalling undersea.

158 12.1 Plungin’

& 3. Learning undersea and recalling on the beach.

& 4. Learning undersea and recalling undersea.

Because each subject in this experiment participates in all four experi-mental conditions, the factor " is crossed with the 2 experimental factors.Hence, the design can be symbolized as a " $#$ % design. For this (fic-titious) replication of Godden and Baddeley’s (1975) experiment we havebeen able to convince ! ) ' (fictitious) subjects to take part in this experi-ment (the original experiment had 16 subjects).

The subjects to learn lists made of 40 short words each. Each list hasbeen made by drawing randomly words from a dictionary. Each list is usedjust once (hence, because we have ! ) ' subjects and * $ : ) # $ # ) +experimental conditions, we have '$+ ) #$ lists). The dependent variableis the number of words recalled 10 minutes after learning (in order to haveenough time to plunge or to come back to the beach).

The results of the (fictitious) replication are given in Table 12.1. Pleasetake a careful look at it and make sure you understand the way the resultsare laid out.

Recall that the prediction of the authors was that memory should bebetter when the context of encoding and testing is the same than whenthe context of encoding and testing are different. This means that theyhave a very specific shape of effects (a so-called "-shaped interaction) in

#Learning Place

&$ On Land &% Underwater!

Means#$$' %$$'

*$ 34 14 48 24'$ *% 37 21 58 29Testing place *# 27 31 58 29On Land *" 43 27 70 35

*' 44 32 76 38#$$$ ! "($ #%$$ ! "-$ #$$$ ! #"'%$$$ ! #% %%$$ ! -$ %$$$ ! #"

*$ 18 22 40 20'% *% 21 25 46 23Testing place *# 25 33 58 29Underwater *" 37 33 70 35

*' 34 42 76 38#$%$ ! "#$ #%%$ ! "$$ #$$$ ! -&'%$%$ ! -% %%%$ ! #" %$$$ ! -&

TABLE 12.1 Result of a (fictitious) replication of Godden and Baddeley’s (1975) experiment with deep sea divers

(see text for explanation).


12.1 Plungin’ 159

mind. As a consequence, they predict that all of the experimental sums ofsquares should correspond to the sum of squares of interaction.

12.1.1 [R] code

! P5^QP U;#`8'>(#$ ;)(4)* :67Y%>(:A M,P! R&6*3)*W ` O%%. M%' O)<)*3 +,'-.&%

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! 7H 7CA 7H 7CLLLLLLL %(> 8#$ S'>(#$ pU%:()*3=3&?CACeEeHAEeFeHA&'7%&:=>?[7H[A[7C[JJ


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! 0% *#; 8#$- ' 1'(' 8$'-% ;)(4 (4% 1%.%*1%*( <'$)'7&% '*1 (4%! 8'>(#$:L1'(' = 1'('L8$'-%?:>#$% = :>#$%A 2%'$*)*3 = 8'>(#$?2%'$*)*3JA U%:()*3 = 8'>(#$?U%:()*3JA M67Y%>( = 8'>(#$?M67Y%>(JJ

! 0% *#; .%$8#$- '* '*#<' ;4%* [M67Y%>([ ): >#*:)1%$%1 ': ' $'*1#- 8'>(#$L'#<H = '#<?:>#$% ] ?2%'$*)*3 e U%:()*3J l +$$#$?M67Y%>( X

?2%'$*)*3 e U%:()*3JJA 1'(' = 1'('J

! 0% *#; .$)*( (4% 1'(' '*1 '&& (4% $%:6&(::6--'$\?'#<?:>#$%]2%'$*)*3eU%:()*3eM67Y%>(JJ:6--'$\?'#<HJ.$)*(?-#1%&L('7&%:?'#<?:>#$% ] 2%'$*)*3 e U%:()*3 e M67Y%>(A

1'(' = 1'('JA [-%'*:[JA 1)3)(: = @J


160 12.1 Plungin’

12.1.2 [R] output

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_ ! 0% *#; >#-7)*% (4% #7:%$<'()#*: )*(# #*% &#*3 >#&6-* ?:>#$%JL_ :>#$%=>?7HA7CJ

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_ M67Y%>(=3&?EAHAEeFeHA &'7%&:=>?[:67NH[A [:67NC[A [:67N@[A[:67NF[A [:67NE[JJ

_ ! 0% *#; 8#$- ' 1'(' 8$'-% ;)(4 (4% 1%.%*1%*( <'$)'7&% '*1 (4%_ ! 8'>(#$:L_ 1'(' = 1'('L8$'-%?:>#$% = :>#$%A 2%'$*)*3 = 8'>(#$?2%'$*)*3JA U%:()*3 = 8'>(#$?U%:()*3JA M67Y%>( = 8'>(#$?M67Y%>(JJ

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12.1 Plungin’ 161

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162 12.1 Plungin’

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```````````````

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12.1 Plungin’ 163

U%:()*3/M67Y%>(`````````````````````````````````````

M67Y%>(`````````````````````````````

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`````````````````````````````````````

2%'$*)*3/U%:()*3/M67Y%>(

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U%:()*3``````

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```````````````

A A M67Y%>( = :67NC```````````````

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```````````````

A A M67Y%>( = :67N@```````````````

U%:()*3``````

2%'$*)*3 7H 7C```````````````

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```````````````

A A M67Y%>( = :67NF```````````````

U%:()*3``````

2%'$*)*3 7H 7C```````````````

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164 12.1 Plungin’

```````````````

A A M67Y%>( = :67NE```````````````

U%:()*3``````

2%'$*)*3 7H 7C```````````````

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```````````````

12.1.3 ANOVA tableHere are the final results of the Godden and Baddeley’s experiment pre-sented in an ANOVA table.

Source +% !" ## $# % P(%)

# '$'$&'' " ('$'' ('$'' -$'' $--&%#% '$'")%$ " -'$'' -'$'' -$$' $"(("$" '$$'")% ) *('$'' "%'$'' —#% '$-#$&& " #-'$'' #-'$'' -'$'' $'"-#"#" '$""%&& ) "*'$'' )'$'' —%" '$'-#*' ) #-$'' ($'' —#%" '$')%-' ) *)$'' "*$'' —

Total "$'' "& "( #$*$''


13Factorial Designs:Partially Repeated Measures,"!#"$ %

13.1 Bat and Hat....

To illustrate a partially repeated measures or split-plot design, our exam-ple will be a (fictitious) replication of an experiment by Conrad (1971). Thegeneral idea was to explore the hypothesis that young children do not usephonological coding in short term memory. In order to do this, we se-lect 10 children: 5 five year olds and 5 twelve year olds. This constitutesthe first independent variable (# or age with 2 levels), which happens alsoto be what we have called a “tag” or “classificatory” variable. Because asubject is either five years old or twelve years old, the subject factor (") isnested in the (#) age factor.

The second independent variable deals with phonological similarity,and we will use the letter % to symbolize it. But before describing it, weneed to delve a bit more into the experiment. Each child was shown 100pairs of pictures of objects. A pilot study had made sure that children willalways use the same name for these pictures (i.e., the cat picture was al-ways called “a cat”, never “a pet” or “an animal”).

After the children had looked at the pictures, the pictures were turnedover so that the children could only see their backs. Then the experimentergives an identical pair of pictures to the children and asks them to positioneach new picture on top of the old ones (that are hidden by now) such thatthe new pictures match the hidden ones. For half of the pairs of pictures,the sound of the name of the objects was similar (i.e., hat and cat), whereasfor the other half of the pairs, the sound of the names of the objects in apair was dissimilar (i.e., horse and chair). This manipulation constitutesthe second experimental factor % or “phonological similarity.” It has twolevels: 0$ phonologically similar and 0% phonologically dissimilar. The de-pendent variable will be the number of pairs of pictures correctly posi-tioned by the child.

Conrad reasoned that if the older children use a phonological codeto rehearse information, then it would be more difficult for them to re-

166 13.1 Bat and Hat....

member the phonologically similar pairs than the phonologically dissim-ilar pairs. This should happen because of an interference effect. If theyoung children do not use a phonological code to rehearse the materialthey want to learn, then their performance should be unaffected by phono-logical similarity, and they should perform at the same level for both con-ditions of phonological similarity. In addition, because of the usual ageeffect, one can expect the old children to perform on the whole betterthan the young ones. Could you draw the graph corresponding to the ex-pected pattern of results? Could you express these predictions in terms ofthe analysis of variance model?

We expect a main effect of age (which is rather trivial), and also (and,this is the crucial point) we expect an interaction effect. This interactionwill be the really important test of Conrad’s theoretical prediction.

The results of this replication are given in Table 13.1.

13.1.1 [R] code! P5^QP U;#`8'>(#$ R'$()'&&\ Z%.%'(%1 9%':6$%:A M?PJ,p! p'( '*1 V'( +,'-.&%

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! U4% C &%<%&: #8 S'>(#$ P '$%/ S)<% g%'$: '*1 U;%&<% g%'$:

%Phonological Similarity

'$ Similar '% Dissimilar!

Means#$$' %$$'

*$ 15 13 28 14&$ *% 23 19 42 21Age: *# 12 10 22 11Five Years *" 16 16 32 16

*' 14 12 26 13#$$$ ! (' #$%$ ! %' #$$$ ! "$'%$$$ ! "* %$%$ ! ") %$$$ ! "$

#%$' %%$'*( 39 29 68 34

&% *+ 31 15 46 23Age: *) 40 30 70 35Twelve Years ** 32 26 58 29

*$& 38 30 68 34#%$$ ! "(' #%%$ ! "#' #%$$ ! #"'%%$$ ! #* %%%$ ! -* %%$$ ! #"

TABLE 13.1 Results of a replication of Conrad’s (1971) experiment.


13.1 Bat and Hat.... 167

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M67Y%>(=3&?HIAHAEeFeHA &'7%&: = >?[:67NH[A [:67NC[A [:67N@[A[:67NF[A [:67NE[A [:67NB[A [:67NG[A [:67NK[A [:67ND[A[:67NHI[JJ

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8'>(#$?R4#*#NM)-JA M67Y%>(=8'>(#$?M67Y%>(JJ

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R4#*#NM)-J l P3%JA 1'('=1'('J

! 0% *#; .$)*( (4% 1'(' '*1 '&& (4% $%:6&(:

.$)*(?1'('J:6--'$\?'#<HJ.$)*(?-#1%&L('7&%:?'#<?:>#$% ] P3% e R4#*#NM)- e M67Y%>(A 1'('

= 1'('JA [-%'*:[JA 1)3)(: = @J

13.1.2 [R] output

_ ! P5^QP U;#`8'>(#$ R'$()'&&\ Z%.%'(%1 9%':6$%:A M?PJ,p_ ! p'( '*1 V'( +,'-.&%

_ ! 0% 4'<% C S'>(#$:A P ?P3%JA ;)(4 C &%<%&: '*1 S'>(#$ p_ ! ?R4#*#&#3)>'& M)-)&'$)(\J ;)(4 C &%<%&: '*1 HI :67Y%>(:L


168 13.1 Bat and Hat....

_ ! U4% C &%<%&: #8 S'>(#$ P '$%/ S)<% g%'$: '*1 U;%&<% g%'$:_ ! U4% C &%<%&: #8 S'>(#$ p '$%/ M)-)&'$ '*1 O)::)-)&'$

_ ! U4% M67Y%>(: '$% *%:(%1 )* P3% '*1 >$#::%1 ;)(4 R4#*#&#3)>'&_ ! M)-)&'$)(\L_ ! U4%$%8#$% (4%$% '$% F 3$#6.: ;)(4 E #7:%$<'()#*: ?:67Y%>(:J_ ! .%$ 3$#6.L

_ ! 0% >#&&%>( (4% 1'(' 8#$ %'>4 :67Y%>(: 8#$ '&& &%<%&: #8_ ! S'>(#$ P '*1 S'>(#$ p 8#$ %'>4 :67Y%>(L_ 7H=>?HEAC@AHCAHBAHFA @DA@HAFIA@CA@KJ_ 7C=>?H@AHDAHIAHBAHCA CDAHEA@IACBA@IJ

_ ! 0% *#; >#-7)*% (4% #7:%$<'()#*: )*(# #*% &#*3 >#&6-* ?:>#$%JL_ :>#$%=>?7HA7CJ

_ ! 0% *#; .$%.'$% (4% &'7%&: 8#$ (4% F,E :>#$%: '>>#$1)*3 (# (4%_ ! 8'>(#$ &%<%&:/_ ! 'H 'CA 'H 'CLLLLLLLL%(> 8#$ S'>(#$ P_ P3%=3&?CAEeHAEeFeHA &'7%&:=>?['H[A['C[JJ

_ ! 7H 7CA 7H 7CLLLLLLL %(> 8#$ S'>(#$ p_ R4#*#NM)-=3&?CACeEeHAEeFeHA&'7%&:=>?[7H[A[7C[JJ


_ M67Y%>(=3&?HIAHAEeFeHA &'7%&: = >?[:67NH[A [:67NC[A [:67N@[A[:67NF[A [:67NE[A [:67NB[A [:67NG[A [:67NK[A [:67ND[A[:67NHI[JJ

_ ! 0% *#; 8#$- ' 1'(' 8$'-% ;)(4 (4% 1%.%*1%*( <'$)'7&% '*1 (4%_ ! 8'>(#$:L_ 1'(' = 1'('L8$'-%?:>#$% = :>#$%A P3% = 8'>(#$?P3%JA R4#*#NM)- =

8'>(#$?R4#*#NM)-JA M67Y%>(=8'>(#$?M67Y%>(JJ

_ ! P*#<' ;4%* [M67Y%>([ ): >#*:)1%$%1 ': ' $'*1#- 8'>(#$L_ '#<H = '#<?:>#$% ] ?P3% e R4#*#NM)-J l +$$#$?M67Y%>( X ?P3% e

R4#*#NM)-J l P3%JA 1'('=1'('J

_ ! 0% *#; .$)*( (4% 1'(' '*1 '&& (4% $%:6&(:

_ .$)*(?1'('J

``````````````````````````````:>#$% P3% R4#*#NM)- M67Y%>(

``````````````````````````````H HE 'H 7H :67NHC C@ 'H 7H :67NC@ HC 'H 7H :67N@F HB 'H 7H :67NFE HF 'H 7H :67NEB @D 'C 7H :67NB


13.1 Bat and Hat.... 169

G @H 'C 7H :67NGK FI 'C 7H :67NKD @C 'C 7H :67NDHI @K 'C 7H :67NHIHH H@ 'H 7C :67NHHC HD 'H 7C :67NCH@ HI 'H 7C :67N@HF HB 'H 7C :67NFHE HC 'H 7C :67NEHB CD 'C 7C :67NBHG HE 'C 7C :67NGHK @I 'C 7C :67NKHD CB 'C 7C :67NDCI @I 'C 7C :67NHI``````````````````````````````

_ :6--'$\?'#<HJ

+$$#$/ M67Y%>(`````````````````````````````````````````````

O8 M6- Ma 9%'* Ma S <'&6% R$?_SJ`````````````````````````````````````````````P3% H HCKI HCKI @C ILIIIFGGB eeeZ%:)16'&: K @CI FI````````````````````````````````````````````````M)3*)8L >#1%:/ I WeeeW ILIIH WeeW ILIH WeW ILIE WLWILH W W H

+$$#$/ M67Y%>(/R4#*#NM)-`````````````````````````````````````````````````

O8 M6- Ma 9%'* Ma S <'&6% R$?_SJ`````````````````````````````````````````````````R4#*#NM)- H HKI HKI FE ILIIIHEHF eeeP3%/R4#*#NM)- H KI KI CI ILIICIGG@ eeZ%:)16'&: K @C F````````````````````````````````````````````````````M)3*)8L >#1%:/ I WeeeW ILIIH WeeW ILIH WeW ILIE WLWILH W W H

_ .$)*(?-#1%&L('7&%:?'#<?:>#$% ] P3% e R4#*#NM)- e M67Y%>(A 1'('= 1'('JA [-%'*:[JA 1)3)(: = @J

U'7&%: #8 -%'*:n$'*1 -%'*

C@

P3%`````'H 'C`````HE @H`````


170 13.1 Bat and Hat....

R4#*#NM)-`````7H 7C`````CB CI`````

M67Y%>(```````````````````````````````````````````````````````:67NH :67NC :67N@ :67NF :67NE :67NB :67NG :67NK```````````````````````````````````````````````````````

CC CD HD CF CH CB HE CG```````````````````````````````````````````````````````

````````````:67ND :67NHI````````````

CH CB````````````

P3%/R4#*#NM)-`````````````

R4#*#NM)-````````

P3% 7H 7C`````````````'H HB HF'C @B CB

`````````````

R4#*#NM)-/M67Y%>(`````````````````````````````````````````````````````````

M67Y%>(```````````````````````````````````````````````

R4#*#NM)- :67NH :67NC :67N@ :67NF :67NE :67NB :67NG :67NK`````````````````````````````````````````````````````````

7H CE @@ CC CB CF CD CH @I7C HD CE HB CC HK C@ D CF

``````````````````````M67Y%>(

````````````R4#*#NM)- :67ND :67NHI``````````````````````

7H CC CK7C CI CF

`````````````````````


13.1 Bat and Hat.... 171

13.1.3 ANOVA tableWe can now fill in the ANOVA Table as shown in Table 13.2.

Source !" ## $# % ,-+%,

between subjects# . . . . . . . . . . . . . . . 1 1,280.00 1,280.00 32.00 .00056"+#, . . . . . . . . . . . . 8 320.00 40.00 -----within subjects% . . . . . . . . . . . . . . . 1 180.00 180.00 45.00 .00020#% . . . . . . . . . . . . . 1 80.00 80.00 20.00 .00220%"+#, . . . . . . . . . . 8 32.00 4.00 -----

Total . . . . . . . . . . . . 19 1,892.00

TABLE 13.2 The analysis of variance Table for a replication of Conrad’s (1971) experiment (data from Table 13.1).

As you can see from the results of the analysis of variance, the experi-mental predictions are supported by the experimental results. The resultssection of an APA style paper would indicate the following information:

The results were treated as an Age $ Phonological similarityanalysis of variance design with Age (5 year olds versus 12 yearolds) being a between-subject factor and phonological simi-larity (similar versus dissimilar) being a within-subject factor.There was a very clear effect of age, !!%+ ," ) &%'', %$% ) &&'&,,, - $%. The expected interaction of age by phonological sim-ilarity was also very reliable !!%+ ," ) &''#, %$% ) #',-, , -$%. A main effect of phonological similarity was also detected!!%+ ," ) '#'-, %$% ) #',-, , - $%, but its interpretation asa main effect is delicate because of the strong interaction be-tween phonological similarity and age.


172 13.1 Bat and Hat....


14Nested Factorial Design:" $#!%"

14.1 Faces in Space

Some faces give the impression of being original or bizarre. Some otherfaces, by contrast, give the impression of being average or common. Wesay that original faces are atypical; and that common faces are typical. Interms of design factors, we say that: Faces vary on the Typicality factor(which has 2 levels: typical vs. atypical).

In this example, we are interested by the effect of typicality on reac-tion time. Presumably, typical faces should be easier to process as facesthan atypical faces. In this1 example, we measured the reaction time of 4subjects in a face identification task. Ten faces, mixed with ten “distractorfaces,” were presented on the screen of a computer. The distractor faceswere jumbled faces (e.g., with the nose at the place of the mouth). Fiveof the ten faces were typical faces, and the other five faces were atypicalfaces. Subjects were asked to respond as quickly as they can. Only the datarecorded from the normal faces (i.e., not jumbled) were kept for furtheranalysis. All the subjects identified correctly the faces as faces. The data(made nice for the circumstances) are given in Table 14.1 on page 180.As usual, make sure that you understand its layout, and try to figure outwhether there is some effect of Typicality.

Here, like in most " $#!%" designs, we are mainly interested in thenesting factor (i.e., %). The nested factor [i.e., #!%"] is not, however, with-out interest. If it is statistically significant, this may indicate that the pat-tern of effects, which we see in the results, depends upon the specific sam-ple of items used in this experiment.

14.1.1 [R] code! P5^QP U;#`8'>(#$ 5%:(%1 8'>(#$)'& O%:)3*A M,P?pJ! S'>%: )* M.'>%

! S'>(#$ p ): U\.)>'&)(\L S'>(#$ P?pJ ): S'>%: *%:(%1 )*

1Somewhat fictitious, but close to some standard experiments in face recognition.

174 14.1 Faces in Space

! U\.)>'&)(\L U4%$% '$% F :67Y%>(: )* (4% %,.%$)-%*(L

! U4%$%8#$% (4%$% '$% F 3$#6.: ;)(4 E #7:%$<'()#*: ?:67Y%>(:J! .%$ 3$#6.L

! 0% >#&&%>( (4% 1'(' 8#$ %'>4 :67Y%>(: 8#$ '&& &%<%&: #8! S'>(#$ P '*1 S'>(#$ p 8#$ %'>4 :67Y%>(L'N7H=>?CIACCACEACFAHDA DAKACHACHACHA HKACIAHKACHA@@A

EAHFAHBACCAC@J'N7C=>?@GA@GAF@AFKAFEA @FA@EA@EA@GA@DA @EA@DA@DA@GAFIA

@KAFDAEHAEIAECJ

! 0% *#; >#-7)*% (4% #7:%$<'()#*: )*(# #*% &#*3 >#&6-* ?:>#$%JL:>#$%=>?'N7HA'N7CJ

! 0% *#; .$%.'$% (4% &'7%&: 8#$ (4% F,E,C :>#$%: '>>#$1)*3 (#(4% 8'>(#$ &%<%&:/

! 'H 'C '@ 'F 'EA 'H 'C '@ 'F 'ELLLLLLLL%(> 8#$ S'>(#$ PS'>%=3&?EAHAEeFeCA &'7%&:=>?['H[A['C[A['@[A['F[A['E[JJ

! 7H 7CA 7H 7CLLLLLLL %(> 8#$ S'>(#$ pU\.)>'&)(\=3&?CAFeEeHAEeFeCA&'7%&:=>?[P(\.)>'&[A[U\.)>'&[JJ

! :67NH :67NHLLLLLLLA :67NC :67NCLLLLLLA:67N@ :67N@ LLLLA:67NF! :67NF LLLLLA :67NE :67NELLLLL%(> 8#$ M67Y%>(:M67Y%>(=3&?FA EeHA EeFeCA &'7%&: = >?[:67NH[A [:67NC[A [:67N@[A

[:67NF[JJ

! 0% *#; 8#$- ' 1'(' 8$'-% ;)(4 (4% 1%.%*1%*( <'$)'7&% '*1 (4% ! 8'>(#$:L1'(' = 1'('L8$'-%?:>#$% = :>#$%A S'>% = 8'>(#$?S'>%JA

U\.)>'&)(\ = 8'>(#$?U\.)>'&)(\JJ

! P*#<' ;4%* [M67Y%>([ ): >#*:)1%$%1 ': ' $'*1#- 8'>(#$L'#<H = '#<?:>#$% ] ?M67Y%>( l S'>%x)*xU\.)>'&)(\ l U\.)>'&)(\ l

U\.)>'&)(\/M67Y%>(JJ

O8 = :6--'$\?'#<?:>#$% ] ?M67Y%>( l S'>%x)*xU\.)>'&)(\ lU\.)>'&)(\ l U\.)>'&)(\/M67Y%>(JJJiiHjjtO8

M6-NMa = :6--'$\?'#<?:>#$% ] ?M67Y%>( l S'>%x)*xU\.)>'&)(\ lU\.)>'&)(\ l U\.)>'&)(\/M67Y%>(JJJiiHjjtM6-

9M = :6--'$\?'#<?:>#$% ] ?M67Y%>( l S'>%x)*xU\.)>'&)(\ lU\.)>'&)(\ l U\.)>'&)(\/M67Y%>(JJJiiHjjt9%'*

S = :6--'$\?'#<?:>#$% ] ?M67Y%>( l S'>%x)*xU\.)>'&)(\ lU\.)>'&)(\ l U\.)>'&)(\/M67Y%>(JJJiiHjjtS

SiCj=5PR$ = :6--'$\?'#<?:>#$% ] ?M67Y%>( l S'>%x)*xU\.)>'&)(\ l

U\.)>'&)(\ l U\.)>'&)(\/M67Y%>(JJJiiHjjtR$R$iCj=5P

M#6$>%N*'-%: = >?[M67Y%>([A [U\.)>'&)(\[A [S'>%?U\.)>'&)(\J[A[M67Y%>( e U\.)>'&)(\[A [+$$#$/S'>% e M67Y%>(?U\.)>'&)(\J[J


14.1 Faces in Space 175

P*#<'N('7&% = 1'('L8$'-%?[O8[ = O8A [M6- Ma[ = M6-NMaA [9%'*Ma[ = 9MA [S Q'&6%[ = SA [R$ S[ = R$A$#;L 5'-%: =M#6$>%N*'-%:J

! 0% *#; .$)*( (4% 1'(' '*1 '&& (4% $%:6&(:.$)*(?1'('J.$)*(?P*#<'N('7&%J.$)*(?[U4% WU\.)>'&)(\W 8'>(#$ 4': ' b6':) S #$ SWL U4): SW 4':

*#( 7%%* 1):.&'\%1 )* (4% P*#<' ('7&% '*1 4': (# 7%>'&>6&'(%1 :%.'$'(%&\[J

.$)*(?-#1%&L('7&%:?'#<HA[-%'*:[JA1)3)(:=@J

14.1.2 [R] output

_ ! P5^QP U;#`8'>(#$ 5%:(%1 8'>(#$)'& O%:)3*A M,P?pJ_ ! S'>%: )* M.'>%

_ ! S'>(#$ p ): U\.)>'&)(\L S'>(#$ P?pJ ): S'>%: *%:(%1 )*_ ! U\.)>'&)(\L U4%$% '$% F :67Y%>(: )* (4% %,.%$)-%*(L

_ ! U4%$%8#$% (4%$% '$% F 3$#6.: ;)(4 E #7:%$<'()#*: ?:67Y%>(:J_ ! .%$ 3$#6.L

_ ! 0% >#&&%>( (4% 1'(' 8#$ %'>4 :67Y%>(: 8#$ '&& &%<%&: #8_ ! S'>(#$ P '*1 S'>(#$ p 8#$ %'>4 :67Y%>(L_ 'N7H=>?CIACCACEACFAHDA DAKACHACHACHA HKACIAHKACHA@@A

EAHFAHBACCAC@J_ 'N7C=>?@GA@GAF@AFKAFEA @FA@EA@EA@GA@DA @EA@DA@DA@GAFIA

@KAFDAEHAEIAECJ

_ ! 0% *#; >#-7)*% (4% #7:%$<'()#*: )*(# #*% &#*3 >#&6-* ?:>#$%JL_ :>#$%=>?'N7HA'N7CJ

_ ! 0% *#; .$%.'$% (4% &'7%&: 8#$ (4% F,E,C :>#$%: '>>#$1)*3 (#(4% 8'>(#$ &%<%&:/

_ ! 'H 'C '@ 'F 'EA 'H 'C '@ 'F 'ELLLLLLLL%(> 8#$ S'>(#$ P_ S'>%=3&?EAHAEeFeCA &'7%&:=>?['H[A['C[A['@[A['F[A['E[JJ

_ ! 7H 7CA 7H 7CLLLLLLL %(> 8#$ S'>(#$ p_ U\.)>'&)(\=3&?CAFeEeHAEeFeCA&'7%&:=>?[P(\.)>'&[A[U\.)>'&[JJ

_ ! :67NH :67NHLLLLLLLA :67NC :67NCLLLLLLA:67N@ :67N@ LLLLA:67NF_ ! :67NF LLLLLA :67NE :67NELLLLL%(> 8#$ M67Y%>(:_ M67Y%>(=3&?FA EeHA EeFeCA &'7%&: = >?[:67NH[A [:67NC[A [:67N@[A

[:67NF[JJ

_ ! 0% *#; 8#$- ' 1'(' 8$'-% ;)(4 (4% 1%.%*1%*( <'$)'7&% '*1 (4% _ ! 8'>(#$:L_ 1'(' = 1'('L8$'-%?:>#$% = :>#$%A S'>% = 8'>(#$?S'>%JA

U\.)>'&)(\ = 8'>(#$?U\.)>'&)(\JJ

_ ! P*#<' ;4%* [M67Y%>([ ): >#*:)1%$%1 ': ' $'*1#- 8'>(#$L_ '#<H = '#<?:>#$% ] ?M67Y%>( l S'>%x)*xU\.)>'&)(\ l U\.)>'&)(\ l



U\.)>'&)(\/M67Y%>(JJ

_ O8 = :6--'$\?'#<?:>#$% ] ?M67Y%>( l S'>%x)*xU\.)>'&)(\ lU\.)>'&)(\ l U\.)>'&)(\/M67Y%>(JJJiiHjjtO8

_ M6-NMa = :6--'$\?'#<?:>#$% ] ?M67Y%>( l S'>%x)*xU\.)>'&)(\ lU\.)>'&)(\ l U\.)>'&)(\/M67Y%>(JJJiiHjjtM6-

_ 9M = :6--'$\?'#<?:>#$% ] ?M67Y%>( l S'>%x)*xU\.)>'&)(\ lU\.)>'&)(\ l U\.)>'&)(\/M67Y%>(JJJiiHjjt9%'*

_ S = :6--'$\?'#<?:>#$% ] ?M67Y%>( l S'>%x)*xU\.)>'&)(\ lU\.)>'&)(\ l U\.)>'&)(\/M67Y%>(JJJiiHjjtS

_ SiCj=5P_ R$ = :6--'$\?'#<?:>#$% ] ?M67Y%>( l S'>%x)*xU\.)>'&)(\ l

U\.)>'&)(\ l U\.)>'&)(\/M67Y%>(JJJiiHjjtR$_ R$iCj=5P

_ M#6$>%N*'-%: = >?[M67Y%>([A [U\.)>'&)(\[A [S'>%?U\.)>'&)(\J[A[M67Y%>( e U\.)>'&)(\[A [+$$#$/S'>% e M67Y%>(?U\.)>'&)(\J[J

_ P*#<'N('7&% = 1'('L8$'-%?[O8[ = O8A [M6- Ma[ = M6-NMaA [9%'*Ma[ = 9MA [S Q'&6%[ = SA [R$ _ S[ = R$A$#;L 5'-%: =M#6$>%N*'-%:J

_ ! 0% *#; .$)*( (4% 1'(' '*1 '&& (4% $%:6&(:_ .$)*(?1'('J

````````````````````````:>#$% S'>% U\.)>'&)(\

````````````````````````H CI 'H P(\.)>'&C CC 'C P(\.)>'&@ CE '@ P(\.)>'&F CF 'F P(\.)>'&E HD 'E P(\.)>'&B D 'H P(\.)>'&G K 'C P(\.)>'&K CH '@ P(\.)>'&D CH 'F P(\.)>'&HI CH 'E P(\.)>'&HH HK 'H P(\.)>'&HC CI 'C P(\.)>'&H@ HK '@ P(\.)>'&HF CH 'F P(\.)>'&HE @@ 'E P(\.)>'&HB E 'H P(\.)>'&HG HF 'C P(\.)>'&HK HB '@ P(\.)>'&HD CC 'F P(\.)>'&CI C@ 'E P(\.)>'&CH @G 'H U\.)>'&CC @G 'C U\.)>'&C@ F@ '@ U\.)>'&



CF FK 'F U\.)>'&CE FE 'E U\.)>'&CB @F 'H U\.)>'&CG @E 'C U\.)>'&CK @E '@ U\.)>'&CD @G 'F U\.)>'&@I @D 'E U\.)>'&@H @E 'H U\.)>'&@C @D 'C U\.)>'&@@ @D '@ U\.)>'&@F @G 'F U\.)>'&@E FI 'E U\.)>'&@B @K 'H U\.)>'&@G FD 'C U\.)>'&@K EH '@ U\.)>'&@D EI 'F U\.)>'&FI EC 'E U\.)>'&````````````````````````

_ .$)*(?P*#<'N('7&%J

````````````````````````````````````````````````````````Ò8 M6-LMa 9%'*LMa SLQ'&6%

`````````````````````````````````````````````````````````M67Y%>( @ CFI KI EL@@@@@@U\.)>'&)(\ H FKFI FKFI 5PS'>%?U\.)>'&)(\J K FKI BI FLIIIIIIM67Y%>(eU\.)>'&)(\ @ @BI HCI KLIIIIII+$$#$/S'>%eM67Y%>(?U\.)>'&)(\J CF @BI HE 5P`````````````````````````````````````````````````````````

```````````````````````````````````````````R$LS

```````````````````````````````````````````M67Y%>( ILIIEKECEICEU\.)>'&)(\ 5PS'>%?U\.)>'&)(\J ILII@KKG@@CKM67Y%>(eU\.)>'&)(\ ILIIIGCHEHIC+$$#$/S'>%eM67Y%>(?U\.)>'&)(\J 5P```````````````````````````````````````````

_ .$)*(?[U4% WU\.)>'&)(\W 8'>(#$ 4': ' b6':) S #$ SWL U4): SW 4':*#( 7%%* 1):.&'\%1 )* (4% P*#<' ('7&% '*1 4': (# 7%>'&>6&'(%1 :%.'$'(%&\[J

iHj [U4% WU\.)>'&)(\W 8'>(#$ 4': ' b6':) S #$ SWL U4): SW4': *#( 7%%* 1):.&'\%1 )* (4% P*#<' ('7&% '*1 4': (# 7%>'&>6&'(%1 :%.'$'(%&\[

_ .$)*(?-#1%&L('7&%:?'#<HA[-%'*:[JA1)3)(:=@J

U'7&%: #8 -%'*:



n$'*1 -%'*@I

M67Y%>(```````````````````````:67NH :67NC :67N@ :67NF```````````````````````

@C CB @I @C```````````````````````

U\.)>'&)(\`````````````````P(\.)>'& U\.)>'&`````````````````

HD FH`````````````````

S'>%/U\.)>'&)(\`````````````````````

U\.)>'&)(\````````````````

S'>% P(\.)>'& U\.)>'&`````````````````````'H H@ @B'C HB FI'@ CI FC'F CC F@'E CF FF

`````````````````````

M67Y%>(/U\.)>'&)(\````````````````````````

U\.)>'&)(\````````````````

M67Y%>( P(\.)>'& U\.)>'&````````````````````````:67NH CC FC:67NC HB @B:67N@ CC @K:67NF HB FK

````````````````````````

14.1.3 ! and Quasi-! ratios

Remember the standard procedure? In order to evaluate the reliability of asource of variation, we need to find the expected value of its mean square.Then we assume that the null hypothesis is true, and we try to find an-other source whose mean square has the same expected value. This mean



square is the test mean square. Dividing the first mean square (the ef-fect mean square) by the test mean square gives an ! ratio. When thereis no test mean square, there is no way to compute an ! ratio, especiallywith SAS. However, as you know, combining several mean squares gives atest mean square called a test “quasi-mean square” or a “test mean squareprime.” The ratio of the effect mean square by its “quasi-mean square” givea “quasi-! ratio” (or ! %). The expected values of the mean squares for a" $#!%" design with #!%" random and % fixed are given in Table 14.2 onpage 181.

From Table 14.2 on page 181, we find that most sources of variationmay be evaluated by computing ! ratios using the #"!%" mean square.Unfortunately, the experimental factor of prime interest (i.e., %), cannotbe tested this way, but requires the use of a quasi-! %. The test mean squarefor the main effect of % is obtained as

%$ %test,& ) %$+,,- .%$,* '%$+*,,- ' (14.1)

The number of degrees of freedom of the mean square of test is approxi-mated by the following formula (Eeek!):

5 %% )

!%$+,,- .%$,* '%$+*,,-"%

%$%+,,-

"# +,,-.

%$%,*

"# ,*.

%$%+*,,-

"# +*,,-

' (14.2)

14.1.4 ANOVA tableWe can now fill in the ANOVA Table as shown in Table 14.3 on page 181.



Factor%(Typ

icality:Atyp

icalvs.Typical)

'$ :(A

typical)

'% :(Typ

ical)#

#,-

! - :#

#,-

" - :(A

typicalFaces)

(TypicalFaces)

&$

&%

&#

&"

&'

%$$'

&$

&%

&#

&"

&'

%$%'

%$$'

*$

2022

2524

1922

3737

4348

4542

32*%

98

2121

2116

3435

3537

3936

26*#

1820

1821

3322

3539

3937

4038

30*"

514

1622

2316

3849

5150

5248

32

%$$$

%%$$

%#$$

%"$$

%'$$

%$%$

%%%$

%#%$

%"%$

%'%$

1316

2022

2436

4042

4344

%$$$

!#'

%$$$

!"&

%$%$

!)"

TAB

LE14.1

Data

froma

fictitiousexperim

entwith

a"#

$,&

-design.

Factor&is

Typicality.Factor$

,&-

isFaces

(nestedin

Typicality).There

are4

subjectsin

thisexperim

ent.The

dependent

variableis

measured

incentiseconds

(incase

youw

onder:1centisecond

equals10

milliseconds);and

itisthe

time

takenby

asubjectto

respondthata

givenface

was

aface.



Source Expected Mean Squares $# test

% .%% . .%

#',-- ./.%-' . 0.%

#,-- ./01%- $#+,,- .$#,* '$#+*,,-

" .%% . .%

#',-- ./2.%' $#+*,,-

#+%, .%% . .%

#',-- . 0.%#,-- $#+*,,-

%" .%% . .%

#',-- ./.%-' $#+*,,-

#"+%, .%% . .%

#',--

TABLE 14.2 The expected mean squares when $ is random and & is fixed for an " #$,&- design.

Source &% !" ## $# % Pr(%) 3$ 3%

Face #+%, '$'( ( )('$'' *'$'' )$'' $'')' ( -)Typicality % '$%% " )( ()'$'' )( ()'$'' -&$##' $''#"' " $'

Subject " '$') # -)'$'' ('$'' $$## $''*' # -)Subject by Face #"+%, '$'* -) #*'$'' "$$''Subject by Typicality %" '$'* # #*'$'' "-'$'' ($'' $'''( # -)

Total "$'' #& "#( -$)$''

' This value has been obtained using a Quasi-% approach. See text forexplanation.

TABLE 14.3 The ANOVATable for the data from Table 14.1.




Index

analysis of variance, see ANOVA

ANOVA

Nested factorial designANOVA " $#!%", 173

One factorANOVA "!#", 27

Partially repeated measuresANOVA "!#"$ %, 165

Repeated MeasuresANOVA " $#, 145

Two factorANOVA !!#$ %", 129

BADDELEY, 21, 157Bonferonni inequality, see Bonfer-

onni, Boole, Dunn inequal-ity

Bonferonni, Boole, Dunn inequal-ity, 61

BONFERONNI, 61BONFERONNI, 61BOOLE, 61Boole inequality, see Bonferonni,

Boole, Dunn inequalityBRANSFORD, 30, 45, 62, 70, 78, 88,

96, 119

Comparisonsa posteriori, see Post hoca priori, see PlannedPlanned

Non-orthogonal, 61Orthogonal, 51

Post hoc, 87CONRAD, 165Correlation, 1

Pearson Correlation Coefficient,1

Duncan test, 87DUNN, 61Dunn inequality, see Bonferonni,

Boole, Dunn inequality

F-ratio, 178

GODDEN, 157GOSSET, 96

HULME, 21

JOHNSON, 30

LAWRENCE, 21LOFTUS, 106

Mean square, 179MUIR, 21

Newman-Keuls test, 87, 106

PALMER, 106PEARLSTONE, 129PEARSON, 1

quasi-!, 178, 179quasi-mean square, 179

&& code, 1, 7, 14, 21, 27, 32, 35,

38, 42, 46, 53, 63, 71, 79,145, 148, 152, 159, 166, 173

& output, 3, 9, 15, 23, 28, 32,35, 39, 43, 47, 55, 65, 73,82, 88, 91, 146, 150, 154, 160,167, 175

184 14.1 Index

RegressionMultiple Regression

Non-orthogonal, 21Orthogonal, 13

Simple Regression, 7Retroactive interference, 13

Scheffe test, 87SIDAK, 61Sidak inequality, 61SLAMECKA, 13SMITH, 51STERNBERG, 7STUDENT, see Gosset

THOMSON, 21Tukey test, 87, 96TULVING, 129


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