ReviewMemory scores for subjects given three different study sequences:

Find the sum of squares between groups, SStreatment

A. 24B. 25C. 42D. 84

A B C

17 12 8

14 10 9

12 11 4

13

M = 14 M = 11 M = 7

ReviewMemory scores for subjects given three different study sequences:

Find the sum of squares within groups, SSresidual

A. 14B. 30C. 104D. 114

A B C

17 12 8

14 10 9

12 11 4

13

M = 14 M = 11 M = 7

ReviewMemory scores for subjects given three different study sequences:

SStreatment = 84SSresidual = 30dftreatment = 2 dfresidual = 7

Calculate F, for testing whether the group means are reliably different

A. 0.8B. 1.25C. 9.8D. 180

A B C

17 12 8

14 10 9

12 11 4

13

M = 14 M = 11 M = 7

Repeated-Measures ANOVA

11/11

Repeated-Measures Design

• Multiple measurements for each subject– Different stimulus types, conditions, times, etc.– All measurements are of the same variable, but in

different situations– Generalizes paired-samples design

• Is there an effect of the treatment?– Variation due to condition, time, stimulus, etc.– Do the means of the measurements vary?

• Same null hypothesis as simple ANOVA– m1 = m2 = … = mk

Repeated-Measures Data

• Individual differences – Variation from one subject to another– Affects all the scores of any given subject

Measurement

Subject 1 2 3 4

1 78 73 82 75

2 108 105 113 106

3 84 79 89 80

4 94 88 98 92

5 121 115 123 117

Accounting for Individual Differences

• Individual differences complicate hypothesis testing– Inflate variability of scores– Don’t affect random variability of treatment means

• Contribute to all measurements equally

• Basic idea– Subtract subject mean for each score– Do simple ANOVA on these differences (dfresidual changes)

Measurement Difference Score

Subject 1 2 3 4 Ms 1 2 3 4

1 78 73 82 75 77 1 -4 5 -2

2 108 105 113 106 108 0 -3 5 -2

3 84 79 89 80 83 1 -4 6 -3

4 94 88 98 92 93 1 -5 5 -1

5 121 115 123 117 119 2 -4 4 -2

Partitioning Variability• Break total variability into treatment, subjects, and residual error

• Total variability– Same as before:

• Variability due to treatment

– Same as before:

• Variability due to individual differences– Same idea as SStreatment

– Variability of subject means:• Residual variability

– Remaining variability– Can calculate directly, but not intuitive

Partitioning Variability

Measurement

Subject 1 2 3 4 Ms

1 78 73 82 75 77

2 108 105 113 106 108

3 84 79 89 80 83

4 94 88 98 92 93

5 121 115 123 117 119Mi 97 92 101 94 96

Repeated-Measures ANOVA• Does treatment explain significant portion of variability?

– Don't want to penalize for variability due to individual differences– Removing SSsubject reduces SSresidual and makes it a fair comparison

• Hypothesis test for repeated-measures ANOVA:

– Same as regular ANOVA, except we first remove SSsubject

– (SSsubject not meaningful with simple ANOVA because each subject is only in one group)

SStotal

Degrees of Freedom

dftotal = nk – 1

dftreatment = k – 1

dfresidual = nk – 1 – (k–1) – (n–1)= nk – n – k + 1 .

SStotal

dfsubject = n – 1

ReviewCoffee drinkers are given arithmetic tests on 3 different days: one after drinking coffee, one after decaf, and one with nothing.

Find the sum of squares for individual differences, SSsubject

A. 152B. 558C. 1674D. 2232

Condition

Subject C D N Ms

1 79 73 82 78

2 64 57 59 60

3 83 75 85 81

4 94 87 98 93Mi 80 73 81 78

ReviewCoffee drinkers are given arithmetic tests on 3 different days: one after drinking coffee, one after decaf, and one with nothing.

SStotal = 1860SStreatment = 152SSsubject = 1674SSresidual = 34

What would SSresidual be if these were 12 unrelated subjects?

A. 34B. 186C. 1708D. 1826

Condition

Subject C D N Ms

1 79 73 82 78

2 64 57 59 60

3 83 75 85 81

4 94 87 98 93Mi 80 73 81 78

ReviewCoffee drinkers are given arithmetic tests on 3 different days: one after drinking coffee, one after decaf, and one with nothing.

SStotal = 1860SStreatment = 152 dftreatment = 2SSsubject = 1674 dfsubject = 3SSresidual = 34 dfresidual = 6

Find the F statistic for testingdifferences across conditions

A. 0.14B. 13.41C. 98.47D. 111.88

Condition

Subject C D N Ms

1 79 73 82 78

2 64 57 59 60

3 83 75 85 81

4 94 87 98 93Mi 80 73 81 78