ReviewThree subjects measured in four conditions.

Find the sum of squares for condition differences, SStreatment

A. 84B. 152C. 252D. 336

Condition

Subject A B C D Ms

1 59 55 68 58 60

2 71 63 75 63 68

3 68 62 73 65 67Mi 66 60 72 62 65

ReviewThree subjects measured in four conditions.

Find the sum of squares for individual differences, SSsubject

A. 38B. 114C. 152D. 252

Condition

Subject A B C D Ms

1 59 55 68 58 60

2 71 63 75 63 68

3 68 62 73 65 67Mi 66 60 72 62 65

ReviewThree subjects measured in four conditions.

SStreatment = 252

SSsubject = 152

SStotal = 420

dftreatment = 3

dfresidual = 6

Calculate the F statistic for testing condition differences

A. 1.20B. 1.88C. 3.32D. 31.50

Condition

Subject A B C D Ms

1 59 55 68 58 60

2 71 63 75 63 68

3 68 62 73 65 67Mi 66 60 72 62 65

Factorial ANOVA

11/13

Multiple Independent Variables

• Simple (one-way) ANOVA tells whether groups differ– Compares levels of a single independent variable

• Sometimes we have multiple IVs– Factors– Subjects divided in multiple ways

• Training type & testing type

– Not always true independent variables• Undergrad major & sex

– Some or all can be within-subjects (gets more complicated)• Memory drug & stimulus type

• Dependent variable measured for all combinations of values• Factorial ANOVA

– How does each factor affect the outcome?– Extends ANOVA in same way regression extends correlation

Basic Approach

• Calculate sum of squares for each factor– Variability explained by that factor– Essentially by averaging all data for each level of that factor

• Separate hypothesis test for each factor– Convert SS to mean square– Divide by MSresidual to get F

TestingTraining Dominant Non-dominant MeanDominant [3,7,5,4,6] [14,15,11,13,12] 9Non-dominant [11,7,10,8,9] [10,12,13,11,9] 10Mean 7 12 9.5

Interactions

TestingTraining Dominant Non-dominantDominant M = 5 M = 13Non-dominant M = 9 M = 11Difference -4 +2

• Effect of one factor may depend on level of another– Pick any two levels of Factor A, find difference of means,

compare across levels of Factor B• Testable in same way as main effect of each factor

– SSinteraction, MSinteraction, F, p

• Can have higher-order interactions– Interaction between Factors A and B depends on

C• Partitioning variability– SStotal = SSA + SSB + SSC

+ SSA:B + SSA:C + SSB:C + SSA:B:C + SSresidual

Effect SS df MS F p

Delay 6000 1 6000 12.91 .0007

Injury 3160 2 1580 3.40 .041

Delay:Injury 1920 2 960 2.07 .136

Residual 25094 54 464.7

Brain Injury

Delay None Occipital MTL Mean

Short 78% 65% 73% 72%

Long 66% 53% 61% 52%

Difference 12% 12% 12%

Example: Memory and Brain Injury

37%

36%

Mean 72% 55% 59% 62%

Rule for an interaction:

• Pick any two levels of Factor A (A1, A2) and any two levels of Factor B (B1, B2)

• There’s an interaction if

• Equivalently:

Testing main effects and interactions:

Logic of Sum of Squares• Total sum of squares:

• Null hypothesis assumes all data are from same population– Expected value of is s2 for each raw score

– No matter how we break up SStotal, every individual square

has expected value s2

– SStreatment, SSinteraction, SSresidual are all sums of numbers

with expected value s2

• Every MS has expected value s2

– Average of many numbers that all have expected value s2

– E(MStreatment), E(MSinteraction), E(MSresidual) all equal s2,

according to H0

• If H0 false, then MStreatment and MSinteraction tend to be larger

– F is sensitive to such an increase

ReviewA factorial experiment compares men and women on their memory for different word types, with different distractor tasks.

Factors:• Sex (male, female)• Word type (noun, verb, adjective, preposition)• Second task (speech, manual, none)

How many groups of subjects are there?

A. 2B. 3C. 9D. 24

MenSpeec

h ManualNon

e WomenSpeec

h ManualNon

e

Noun Noun

Verb Verb

Adj. Adj.

Prep. Prep.

ReviewA factorial experiment compares people on their memory for different word types, with different distractor tasks.

Group Means:(ignoring sex)

Is there an interaction?

A. Yes, because adjectives and prepositions are differentially affected by the second task

B. Yes, because the difference between Speech and Manual is different for nouns than for verbs

C. No, because the difference between Manual and None is the same for all word types

D. Yes, because the overall averages for different word types are different

Speech Manual

None

Mean

Noun 15 13 17 15

Verb 10 11 15 12

Adj. 9 10 14 11

Prep. 8 9 13 10

ReviewA factorial experiment compares people on their memory for different word types, with different distractor tasks.

ANOVA table:

Find the Fs for the three effects

A. FWord type = 27.08, F2nd task = 30.33, FWord type:2nd task = 0.28

B. FWord type = 0.37, F2nd task = 0.28, FWord type:2nd task = 0.03

C. FWord type = 7.33, F2nd task = 8.53, FWord type:2nd task = 0.28

D. FWord type = 6.52, F2nd task = 3.37, FWord type:2nd task = 0.03

SS df MS F p

Word type 352 3117.3

37.33

.0003

2nd Task 273 2136.5

08.53

.0005

Interaction 27 6 4.500.28 .94

Residual 960 60 16