factorial betweenfactorial between--ps anova ii:ps anova iiuqwloui1/stats/3010 for post... ·...

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psyc3010 lecture 3psyc3010 lecture 3

factorial betweenfactorial between--Ps ANOVA II:Ps ANOVA II:Following up significant effectsFollowing up significant effects

Effect sizesEffect sizes

last lecture: factorial between-Ps ANOVA II(omnibus tests)

next lecture: factorial between-Ps ANOVA III(higher-order ANOVA)

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announcementsannouncementsQUIZ 1 QUIZ 1 –– ANOVAANOVA

to be completed online 27 March 9am to be completed online 27 March 9am –– 28 March 9pm28 March 9pmpractice ?s and answers up on Blackboard this Fridaypractice ?s and answers up on Blackboard this Fridaypractice quiz up on Blackboard next weekpractice quiz up on Blackboard next week

ASSIGNMENT 1 ASSIGNMENT 1 –– ANOVAANOVATuteTute attendance is vitalattendance is vitalDue Monday 11 April Due Monday 11 April at 12:00 noon

OTHEROTHER•• There is a cool weekly survey function on BB where you There is a cool weekly survey function on BB where you

can provide anonymous feedback on the lecture can provide anonymous feedback on the lecture –– thank thank you to posters!you to posters!

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last lecture last lecture this lecturethis lecture

last lecture:last lecture:–– conceptual overview of 2conceptual overview of 2--way factorial ANOVAway factorial ANOVA–– omnibus tests: 2 main effects + interactionomnibus tests: 2 main effects + interaction–– Effect sizes: omega squared, eta squared, and the Effect sizes: omega squared, eta squared, and the

infamous partial eta squaredinfamous partial eta squared

this lecture:this lecture:–– following up significant effects in 2following up significant effects in 2--way ANOVAway ANOVA

•• main effectsmain effects•• InteractionInteraction

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topics for this lecturetopics for this lecturerere--cap from last lecturecap from last lecture

Omnibus tests for main effects & Omnibus tests for main effects & interactionsinteractions

rere--cap from 2cap from 2ndnd year statisticsyear statisticsfollowing up variables with > 2 levelsfollowing up variables with > 2 levels

following up a significant main effectfollowing up a significant main effectmain effect comparisonsmain effect comparisons

following up a significant interactionfollowing up a significant interactionsimple effectssimple effectssimple comparisonssimple comparisons

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recap from last lecturerecap from last lecture

main effects,main effects,interactions,interactions,

and theand themystery of themystery of the22--way ANOVAway ANOVA

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_____________

Source df SS MS F sig___

A (cons) 2 3332.3 1666.15 20.07 .000

B (dist) 1 168.75 168.75 2.03 .161

AB 2 1978.12 989.06 11.91 .000

Error 42 3487.5 83.02

Total 47 8966.7

summary table from last lecture

SS = sum of squares index of variability around a mean

MS = mean squares = SS / dfcorrected variance estimate used to calculate F ratio

df = degrees of freedom (observations minus number of constraints)

F = MStreat / MSerror

p values(significance levels)

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omnibus tests in 2omnibus tests in 2--way ANOVAway ANOVA•• any test resulting from the preliminary partitioning of any test resulting from the preliminary partitioning of

variance in ANOVA is called an variance in ANOVA is called an omnibus testomnibus test

•• an omnibus test looks for any and all possible differencesan omnibus test looks for any and all possible differencesamong the levels of a factor (among the levels of a factor (omniomni == all all in Latin)in Latin)

•• three omnibus tests in 2three omnibus tests in 2--way factorial ANOVA:way factorial ANOVA:

main effect of Factor 1main effect of Factor 1: : HH00: no difference between any marginal means for this factor: no difference between any marginal means for this factor

main effect of Factor 2main effect of Factor 2: : HH00: no difference between any marginal means for this factor: no difference between any marginal means for this factor

interaction effectinteraction effect: : HH00: no difference among cell means that aren’t explained by: no difference among cell means that aren’t explained by

main effectsmain effects

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mystery of the 2mystery of the 2--way ANOVAway ANOVA•• 2 possible outcomes of omnibus tests:2 possible outcomes of omnibus tests:

HH00 = no difference (anywhere) between the means= no difference (anywhere) between the meansHH11 = difference (somewhere) between the means= difference (somewhere) between the means

•• if significant main effect,if significant main effect, the next step is to interpret it the next step is to interpret it to establish the direction of the effect (i.e., its meaning)to establish the direction of the effect (i.e., its meaning)

if factor = 2 levels:if factor = 2 levels: can easily interpret direction can easily interpret direction

if factor > 2 levels:if factor > 2 levels: which means differ from each which means differ from each other???other???

•• if significant interaction,if significant interaction, the next step is to interpret it to the next step is to interpret it to establish the direction(s) of the simple effect(s):establish the direction(s) of the simple effect(s):

the differences between levels of which factor are the differences between levels of which factor are different at which levels of the other factor???different at which levels of the other factor???

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99

main effect of Factor A main effect of Factor A (alcohol consumption):(alcohol consumption):HH00: : μμ11 = = μμ2 2 = = μμ3 3

reject reject HH00 if MSif MSA A / / MSMSerrorerrorresults in a significant results in a significant obtained F valueobtained F value

FF (2, 42) = 20.06, (2, 42) = 20.06, pp < .001< .001iindicates that the 3 levels ndicates that the 3 levels of Factor A differ of Factor A differ (collapsed across factor B)(collapsed across factor B)indicates that the indicates that the marginal means of marginal means of Factor A differFactor A differ

0 2 4(means)

50 45 3055 60 3080 85 3065 65 55

Distraction 70 70 3575 70 2075 80 4565 60 40

Cell Totals 535 535 285 1355Cell Means 66.88 66.88 35.63 56.46

65 70 5570 65 6560 60 70

Controls 60 70 5560 65 5555 60 6060 60 5055 50 50


Marginal Totals (A ) 1020 1035 745 2800

Means 63.75 64.69 46.56 58.33

Participant Distraction

Marginal Totals (B)

Alcohol Consumption (pints)

? ? ?? ? ?

back to the example from Lecture 2:

finding the mystery

1010

interaction of A x Binteraction of A x BHH00: : μμ1111 -- μμ21 21 = = μμ12 12 -- μμ2222 = =

μμ13 13 -- μμ2323

reject reject HH00 if MSif MSAB AB / / MSMSerrorerrorresults in a significant results in a significant obtained Fobtained F

FF (2,42) = 11.91, (2,42) = 11.91, pp < .001< .001indicates that the effect indicates that the effect of factor B is not the of factor B is not the same at all levels of same at all levels of factor A (or vice versa)factor A (or vice versa)difference in cell means difference in cell means for levels of one factorfor levels of one factorchanges depending on changes depending on level of other factorlevel of other factor

0 2 4(means)

50 45 3055 60 3080 85 3065 65 55

Distraction 70 70 3575 70 2075 80 4565 60 40


65 70 5570 65 6560 60 70

Controls 60 70 5560 65 5555 60 6060 60 5055 50 50



Means 63.75 64.69 46.56 58.33

Distraction Marginal Totals (B)


??????

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1111

recap from 2recap from 2ndnd year statsyear stats

following following up effects up effects

for for variables variables with > 2 with > 2 levelslevels

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following up effects when > 2 levelsfollowing up effects when > 2 levels

thethe ““protected tprotected t--testtest”” is used to conduct is used to conduct pairwisepairwise comparisons (i.e., compare 2 means)comparisons (i.e., compare 2 means)

““protected”protected” against Type 1 error rate inflationagainst Type 1 error rate inflation

just the same as a normal just the same as a normal tt--test, test, but the error term used is but the error term used is MSMSerrorerror

nMSXXterror221 −= abdf error −=

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following up effects when > 2 levelsfollowing up effects when > 2 levels

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nMSa

Lterror

2j∑

= ∑= jjXaL

abdferror −=

the protected t-test is a special case of this technique

as an alternative, could use as an alternative, could use Linear ContrastsLinear Contrasts to determine to determine if one group if one group or set of groupsor set of groups is different from another group is different from another group or set of groupsor set of groups

a set of weights, a set of weights, aajj, is used to define the contrast, is used to define the contraste.g., e.g., XX11 & X& X22 vsvs XX3 3 [ 1 1 [ 1 1 --2 ]2 ]

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following up main effectsfollowing up main effectstt--tests & linear contraststests & linear contrasts

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following up the main following up the main effect of alcohol effect of alcohol

consumptionconsumption

questions we could ask:questions we could ask:is creativity lower at is creativity lower at 2 pints than at 0 pints?2 pints than at 0 pints?

is creativity lower at is creativity lower at 4 pints than at 0 pints?4 pints than at 0 pints?

is creativity lower at is creativity lower at 4 pints than at 2 pints?4 pints than at 2 pints?

0 2 4(means)

50 45 3055 60 3080 85 3065 65 55

Distraction 70 70 3575 70 2075 80 4565 60 40


65 70 5570 65 6560 60 70

Controls 60 70 5560 65 5555 60 6060 60 5055 50 50



Means 63.75 64.69 46.56 58.33



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following up main effects:following up main effects:(differences among (differences among marginalmarginal meansmeans))

thethe ““protected tprotected t--testtest”” is used to conduct is used to conduct pairwisepairwisecomparisons (i.e., compare 2 means at a time), comparisons (i.e., compare 2 means at a time), but only if the main effect is significantbut only if the main effect is significant

example: example: comparing 2 comparing 2 vsvs 4 pints of alcohol consumption4 pints of alcohol consumption

dnMS2XXterror

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×

−=

comparing means of 2 levels of the factor

number of observations in test = n x # of levels in other IV

d = # of levels in the distraction factor (the other IV)

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includes results for all pairwise comparisons -so lots of extra information provided!for calculations, see Lecture 3 Extra Materials

levels of IV being compared

followfollow--up tests for main effectup tests for main effectof alcohol consumption (SPSS)of alcohol consumption (SPSS)

1 = 0 pints2 = 2 pints3 = 4 pints

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followfollow--up tests for main effectup tests for main effectof alcohol consumption (SPSS)of alcohol consumption (SPSS)

• SPSS output gives you the mean difference and the significance level (p value)

• but it does not give you the calculated t-value

you will learn how to calculate this in Tutorial 3

significant difference or not? 0 vs 2 pints:0 vs 4 pints:2 vs 4 pints:

NONO

YES!YES!YES!YES!

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you may have hypotheses about you may have hypotheses about sets of groupssets of groups, e.g.:, e.g.:more creativity at 0 pints than at 2 or 4 pints (any alcohol)more creativity at 0 pints than at 2 or 4 pints (any alcohol)Linear ContrastsLinear Contrasts can determine if a set of groups is can determine if a set of groups is different from another set of groups different from another set of groups a set of weights, a set of weights, aajj, is used to define the contrast, is used to define the contraste.g., e.g., XX11 & X& X22 vsvs XX3 3 [1 1 [1 1 --2]2]

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following up main effects:following up main effects:(differences among (differences among marginalmarginal meansmeans))

IV.other.of.levels.nonMSa

Lterror

2j

×

=∑

∑= jjXaL

abdferror −=

THIS METHOD WILL BE COVERED IN TUTORIALS

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following up interactions following up interactions part 1part 1

simple effectssimple effects

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following up interactionsfollowing up interactions(differences among (differences among cell meanscell means))

a significant interaction must always be followed up with simple effects– Simple effects test the effects of one factor at each level of the

other factor

example with a 3 x 2 ANOVA:effect of Factor 1 at... Level 1 of Factor 2

Level 2 of Factor 2

effect of Factor 2 at... Level 1 of Factor 1Level 2 of Factor 1Level 3 of Factor 1

Simple effects of a factor compare cell means for that factor a row or column of the data table

simple effectsof Factor 1

simple effectsof Factor 2

2222

0

10

20

30

40

50

60

70

80

0 2 4

Alcohol consumed (pints)

Crea

tivity Distraction

Controls

the simple effects of distraction describe the differences in creativity between distracted and controls at each level of alcohol consumed

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0

10

20

30

40

50

60

70

80

0 2 4


Crea

tivity Distracted

Controls

the simple effects of alcohol consumeddescribe the differences in creativity after 0, 2, or 4 pints consumed at each level of distraction

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how we test the simple effectshow we test the simple effectssay we’re testing the simple effects of Factor A...1. Calculate SStreatment for Factor A at first level of Factor B

– SStreatment for simple effects = variability of cell means: Factor A in one level of Factor B

2. Calculate MStreatment, using the degrees of freedom (DF) for the omnibus main effect of Factor A– i.e., from original ANOVA summary table

3. Use MSerror from omnibus tests– i.e., from the original ANOVA summary table

4. Calculate F ratio: F = MStreatment / MSerror

5. Repeat for each remaining level of Factor B

to learn more about the SS formulae, look

at the extra materials for

Lecture 3

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0 2 4(means)

50 45 3055 60 3080 85 3065 65 55

Distraction 70 70 3575 70 2075 80 4565 60 40


65 70 5570 65 6560 60 70

Controls 60 70 5560 65 5555 60 6060 60 5055 50 50



Means 63.75 64.69 46.56 58.33



“what is the effect of distraction

at each level of consumption?”is there an effect of

distraction for participants who have consumed .

0 pints?

2 pints?

4 pints?

simple effects of simple effects of distractiondistraction

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summary table for summary table for simple effects of simple effects of distractiondistractionSource SS df MS F p

D at C1 156.25 1 156.25 1.88 0.177D at C2 76.56 1 76.56 0.92 0.342D at C3 1914.06 1 1914.06 23.05 0.000

Error 3487.5 42 83.04

critical F at alpha=.05 (1,42) = 4.08

if obtained F exceeds critical F reject the null hypothesis

includes F tests for all simple effects (i.e., the effect of the distraction factor at each level of the consumption factor)if you want to see how the SS values were calculated from the data table, see the extra materials for Lecture 3

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Source SS df MS F p

D at C1 156.25 1 156.25 1.88 0.177D at C2 76.56 1 76.56 0.92 0.342D at C3 1914.06 1 1914.06 23.05 0.000

Error 3487.5 42 83.04

critical F at alpha=.05 (1,42) = 4.08


these are the SS values for each simple effect

degrees of freedom for a simple effect are just the df for the

associated main effect

df = dfdistraction (2 - 1) = 1

SSerror term (and df) is taken from the omnibus ANOVA summary table

mean squares and F valuescalculated as SS / df and

MSeffect / MSerror

C1 = 0 pints, C2 = 2 pints, C3 = 4 pints

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0 2 4(means)

50 45 3055 60 3080 85 3065 65 55

Distraction 70 70 3575 70 2075 80 4565 60 40


65 70 5570 65 6560 60 70

Controls 60 70 5560 65 5555 60 6060 60 5055 50 50


Marginal Totals (C) 1020 1035 745 2800

Means 63.75 64.69 46.56 58.33

Distraction Marginal Totals (D)

Alcohol Consumption (pints)simple effects of simple effects of consumptionconsumption“what is the effect of consumption

at each level of distraction?”

is there an effect of consumption for .

distracted?

controls?

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summary table for summary table for simple effects of simple effects of consumptionconsumption

Source SS df MS F p

C at D1 5208.33 2 2604.17 31.36 0.000C at D2 102.08 2 51.04 0.61 0.546

Error 3487.5 42 83.04

critical F at alpha=.05 (242) = 3.23


includes F tests for all simple effects (i.e., the effects of the consumption factor at each level of the distraction factor)if you want to see how these SS values were calculated from the data table, see the extra materials for Lecture 3

3030

Source SS df MS F p

C at D1 5208.33 2 2604.17 31.36 0.000C at D2 102.08 2 51.04 0.61 0.546

Error 3487.5 42 83.04

critical F at alpha=.05 (242) = 3.23


these are the SS values for each simple effect

degrees of freedom for a simple effect are just the df for the

associated main effect

df = dfconsumption (3 - 1) = 2

SSerror term (and df) is taken from the omnibus ANOVA summary table

mean squares and F valuescalculated as SS / df and

MSeffect / MSerror

D1 = distracted D2 = control (not distracted)

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additivity of omnibus testsadditivity of omnibus testsremember: remember: in ANOVA what we are doing is in ANOVA what we are doing is partitioning variancepartitioning variance

a 2 x 3 betweena 2 x 3 between--subjects ANOVA partitions the total subjects ANOVA partitions the total variance into 4 parts:variance into 4 parts:–– effect due to first factoreffect due to first factor–– effect due to second factoreffect due to second factor–– effect due to interactioneffect due to interaction–– error / residual / error / residual /

withinwithin--group variancegroup variance

SSSStotaltotal = SS= SSCC + SS+ SSDD + SS+ SSCDCD + + SSSSerrorerror

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additivity of simple effects (i)additivity of simple effects (i)simple effects resimple effects re--partition the main effect and interaction partition the main effect and interaction variancevariance∑∑ simple effects of factor 1 = simple effects of factor 1 = ∑ main effect 1 + interaction∑ main effect 1 + interaction

in our examplein our example::SSSSdistractiondistraction + + SSSSinteractioninteraction= = 168.75 + + 1978.12= 2146.872146.87

SSSSdistractiondistraction at C1 at C1 + + SSSSdistractiondistraction at C2 at C2 + + SSSSdistractiondistraction at C3 at C3 = = 156.25 + + 76.56 + 1914.06= 2146.872146.87

∑∑ dfdf for simple effects of factor 1 = for simple effects of factor 1 = ∑ ∑ dfdf main effect 1 + main effect 1 + dfdf interactioninteraction

Dist

intx

main effect of Distraction + C x D interaction

3 simple effects of Distraction

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3333

additivity of simple effects (ii)additivity of simple effects (ii)simple effects resimple effects re--partition the main effect and interaction partition the main effect and interaction variancevariance∑∑ simple effects of factor 2 = simple effects of factor 2 = ∑ main effect 2 + interaction∑ main effect 2 + interaction

in our examplein our example::SSSSconsumptionconsumption + + SSSSinteractioninteraction= = 3332.29 + + 1978.12= 5310.415310.41

SSSSconsumptionconsumption at D1 at D1 + + SSSSconsumptionconsumption at D2at D2= = 5208.33 + 102.08= 5310.415310.41

∑∑ dfdf for simple effects of factor 2 = for simple effects of factor 2 = ∑ ∑ dfdf main effect 2 + main effect 2 + dfdf interactioninteraction

intx

Consmain effect of Consumption + C x D interaction

2 simple effects of Consumption

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following up interactions following up interactions part 2part 2simple simple

comparisonscomparisons

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following up simple effects:following up simple effects:simple comparisons among simple comparisons among cell meanscell meansrecall the recall the significant simple effect of alcohol significant simple effect of alcohol consumptionconsumption for for distracted participantsdistracted participants: : –– for those who are distractedfor those who are distracted, there is at least 1 difference , there is at least 1 difference

in creativity between the 3 levels of alcohol consumption in creativity between the 3 levels of alcohol consumption (0 pints, 2 pints, 4 pints) (0 pints, 2 pints, 4 pints)

we need to follow up this simple effect using we need to follow up this simple effect using simple comparisonssimple comparisons (t tests or linear contrasts)(t tests or linear contrasts)–– the procedure is the procedure is identicalidentical to that used to follow up to that used to follow up

main effects...main effects...–– ...except now we are comparing ...except now we are comparing cell meanscell means, ,

rather than marginal meansrather than marginal means

only only significantsignificant simple effects should be followed upsimple effects should be followed up

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simple comparisons for consumptionsimple comparisons for consumptionin the distracted condition in the distracted condition –– SPSS output SPSS output

includes results for all pairwise comparisons

levels of consumption compared at

level 1 of distraction

factor (distracted)

1 = 0 pints2 = 2 pints3 = 4 pints

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simple comparisons for consumptionsimple comparisons for consumptionin the distracted condition in the distracted condition –– SPSS outputSPSS output

• SPSS output gives you the mean difference and the significancelevel (p value)

• but it does not tell you the calculated t-value

you will learn how to calculate this in Tutorial 3

significant difference or not? 0 vs 2 pints:0 vs 4 pints:2 vs 4 pints:

NONO

YES!YES!YES!YES!

3838

0 pints 2 pints 4 pints

Distracted 66.88 66.88 35.63

Contrast 1 2 -1 -1Contrast 2 0 1 -1

Consumption

simple comparisons for consumption simple comparisons for consumption in the distracted condition in the distracted condition ––

a quick look at linear contrastsa quick look at linear contrasts

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Distracted 66.88 66.88 35.63


Consumption

these are the cell means for distracted participants from our

data table earlier

a set of weights (aj) is used to define the contrasts:

contrast 1 compares 0 vs 2 & 4

contrast 2 compares 2 vs 4

contrasts are orthogonal:

∑ aj = 0 [sum of contrasts within a]

∑ ajbj = 0 [sum of products of each j set of contrasts]

number of contrasts = df for effect

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calculations for contrast 1calculations for contrast 1

96.3

883.04))1()1(2(

25.31222

=−+−+

=t


Distracted 66.88 66.88 35.63


Consumption

nMSaLt

errorj∑=

2

∑= jj XaL

abdferror −=

L = 2(66.88) – 1(66.88) – 1(35.63) = 31.25

tα=.05 (42) = 2.02 (unadjusted)

tα=.05 (42) = 2.33 (adjusted)(Bonferroni adjustment for 2 comparisons)

L = 2(66.88) – 1(66.88) – 1(35.63) = 31.25

nMSaLt

errorj∑=

2

∑= jj XaL

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contrast 1 contrast 1 –– what does it do?what does it do?

01020304050607080

0 2 4


Cre

ativ

ity

Distracted

contrast 1 compares (for distracted participants only) the mean creativity rating for participants who have had 0 pintswith the mean creativity rating for participants who have had 2 or 4 pints: t (42) = 3.96, p < .05 significant

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issues with follow-up comparisons

redundancy: explaining the same mean redundancy: explaining the same mean difference more than oncedifference more than once–– solution: solution: orthogonalorthogonal (independent) linear contrasts(independent) linear contrasts

increases in familyincreases in family--wise error ratewise error rate–– typetype--1 error rate is 1 error rate is αα for each test, this leads to higher for each test, this leads to higher

probability of committing a typeprobability of committing a type--1 error over all tests1 error over all tests

–– solution 1: use solution 1: use BonferroniBonferroni AdjustmentAdjustment for critical for critical tt(from (from BonferroniBonferroni t’t’--tables)tables)

–– solution 2: conduct contrasts defined solution 2: conduct contrasts defined a prioria priori, rather than , rather than exhaustive orthogonal set (i.e., do fewer contrasts)exhaustive orthogonal set (i.e., do fewer contrasts)

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mystery solved!main effect of alcohol consumption explained– creativity impaired at high levels of alcohol

(4 pints < 0 pints; and 4 < 2 pints)– creativity not impaired at low levels of alcohol

(0 pints = 2 pints) interaction effect explained– distraction reduces creativity at 4 pints of alcohol

(but not when 0 or 2 pints are consumed)

– alcohol affects creativity when distracted, but not in the control condition

– in the distraction condition:• creativity impaired at high levels of alcohol

(4 pints < 0 pints; and 4 < 2 pints)• creativity not impaired at low levels of alcohol

(0 pints = 2 pints)

see slides 17 + 18

see slides26 + 27

see slides 36 + 37

see slides 29 + 30

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steps for following up steps for following up main effectsmain effects

is main effect

significant?

NO

YES

NOYES

does the factor have >2 levels?

conduct follow-up tests / main effect comparisons

on marginal means (e.g., protected t-tests or

linear contrasts)

STOP

STOPSTOP

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steps for following up a steps for following up a 22--way interactionway interaction

is interaction significant?

NO

are tests for simple effects significant?YES

NOYES

does the factor have >2 levels?

NOYESconduct tests on cell means

for simple comparisons within level

of other factor

STOP

STOP

STOP

STOP

4646

summarymain effectsmain effects and and interactionsinteractions are are omnibus testsomnibus tests

Significant main effects Significant main effects for factors with more than two for factors with more than two levels are followed up with levels are followed up with main effect comparisonsmain effect comparisons–– Main effect comparisons Main effect comparisons test which marginal means of the factor

are different. Usually use sually use tt--teststests or or linear contrasts.linear contrasts.

interactionsinteractions are followed up with are followed up with simple effect testssimple effect testsSimple effect tests Simple effect tests for Factor A test whether the cell means for the levels of factor A are different, not overall but separately at each level of Factor B (e.g., A1B1 vs A2B1= simple effect of A at B1). We do a simple effect test of A for each level of factor B. Usually report F-test even if factor has only two levels.

Significant simple effects Significant simple effects for factors with more than two for factors with more than two levels are followed up with levels are followed up with simple comparisons simple comparisons –– Simple comparisons Simple comparisons show which cell means of the factor are

different. Usually use sually use tt--teststests or or linear contrasts linear contrasts focusing on focusing on theoretically relevant differences.theoretically relevant differences.

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readingsreadings

between-Ps ANOVA II: follow-up tests (this lecture)Field (3rd ed): Chapter 12 (pp 430-449 and pp 454-456)Field (2nd ed): Chapter 10 (pp 397-420 and pp 425-426)Howell (all eds): Chapter 13 (pp 413-445)

between-Ps ANOVA III: higher-order designs (next lecture)Field (3rd ed): review Chapter 12Field (2nd ed): review Chapter 10Howell (all eds): Chapter 13 (pp 446-453)

factorial betweenfactorial between--ps anova ii:ps anova iiuqwloui1/stats/3010 for post... ·...

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