Repeated Measures ANOVA Cal State Northridge

320Andrew Ainsworth PhD

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Major Topics

What are repeated-measures?An exampleAssumptionsAdvantages and disadvantagesEffect size

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Repeated Measures?

Between-subjects designs–different subjects serve in different

treatment levels– (what we already know)Repeated-measures (RM) designs–each subject receives all levels of at

least one independent variable– (what we’re learning today)

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Repeated Measures

All subjects get all treatments.All subjects receive all levels of the independent variable.Different n’s are unusual and cause problems (i.e. dropout or mortality).Treatments are usually carried out one after the other (in serial).

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Example: Counseling For PTSD

Foa, et al. (1991)–Provided supportive counseling (and other

therapies) to victims of rape–Do number of symptoms change with

time?• There’s no control group for comparison• Not a test of effectiveness of supportive

counseling

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Example: Counseling For PTSD

9 subjects measured before therapy, after therapy, and 3 months laterWe are ignoring Foa’s other treatment conditions.

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Example: Counseling For PTSD

Dependent variable = number of reported symptoms.Question: Do number of symptoms decrease over therapy and remain low?Data on next slide

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The DataPatient Pre Post Follow-Up Subject Mean

1 21 15 15 17.0002 24 15 8 15.6673 21 17 22 20.0004 26 20 15 20.3335 32 17 16 21.6676 27 20 17 21.3337 21 8 8 12.3338 25 19 15 19.6679 18 10 3 10.333

Mean 23.889 15.667 13.222 17.593SD 4.197 4.243 5.783 4.072

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Preliminary Observations

Notice that subjects differ from each other.–Between-subjects variabilityNotice that means decrease over time–Faster at first, and then more slowly–Within-subjects variability

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Partitioning Variability

This partitioning is reflected in the summary table.

SSbg

SSS SSError

SStotal

SSwg

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Sums of SquaresThe total variability can be partitioned into Between Groups (e.g. measures), Subjects and Error Variability

22 2

2 2

i GM j j GM i GM

i j i GM

Total BetweenGroups Subjects Error

T BG S Error

Y Y n Y Y g Y Y

Y Y g Y Y

SS SS SS SS

SS SS SS SS

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Deviation Sums of Squares

2

2 2 2

2 2 2

2 2 2

2 2 2

2 2

(___ ____) (___ ____) (___ ____)

(26 17.593) (32 17.593) (27 17.593)

(21 17.593) (25 17.593) (18 17.593)

(15 17.593) (15 17.593) (17 17.593)

(8 17.593) (15 17.593) (

Total i GMSS Y Y

23 17.593)1114.519

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Deviation Sums of Squares

2

2 2

2

[__*(____ ____) ] [__*(____ ____) ]

[__*(13.222 17.593) ] 562.074

BG j j GMSS n Y Y

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Deviation Sums of Squares

2

2 2

2 2

2 2

2 2

[__*(____ ____) ] [__*(____ ____) ]

[__*(____ ____) ] [__*(20.333 17.593) ]

[__*(21.667 17.593) ] [__*(21.333 17.593) ]

[__*(12.333 17.593) ] [__*(19.667 17.593) ]

[__*(10.333

Subject i GMSS g Y Y

217.593) ] 397.852

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Deviation Sums of Squares

2 2

2 2 2 2

2 2 2

2 2 2

2 2 2

(___ ____) (___ ____) (___ ____)

(26 23.889) (32 23.889) (27 23.889)

(21 23.889) (25 23.889) (18 23.889)

(15 15.667) (15 15.667) (17 15.667)

(

Error i j i GM

i j

SS Y Y g Y Y

Y Y

2 2 28 13.222) (15 13.222) (3 13.222)552.444552.444 397.852 154.592

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Computational ApproachPatient Pre Post Follow-Up Subject Sum

1 21 15 15 512 24 15 8 473 21 17 22 604 26 20 15 615 32 17 16 656 27 20 17 647 21 8 8 378 25 19 15 599 18 10 3 31

Sum 215 141 119 475SY² 9471

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Computational Sums of Squares

22

22

2 22

2

22

jbg

iSubject

j iError

Total

A TSSs as

S TSSa as

A S TSS Ys a as

TSS Yas

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Computational SS Example2 2 2 2

2 2 2 2 2 2 2 2 2

___ ___ 119 ___ _____ _______ __(__) __ __

_______ 8356.481 562.075

___ ___ ___ 61 65 64 37 59 31 8356.481__

_____ 8356.481 _______ 8356.481 397.852__

9471 8918.556 8754.333

bg

S

Error

SS

SS

SS

8356.481 154.5929471 8356.481 1114.519TSS

18

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Degrees of Freedom

1 3 1 2

( 1) 27 3 241 9 1 8

( 1)*( 1) 8*2 161 27 1 26

bg

ws

s

error

total

df g

df g n N gdf ndf n g

df N

From

B

G

AN

OVA

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Summary Table

Fcrit with 2 and 16 degrees of freedom is 3.63 we would reject h0

Source SS df MS FTime 562.074 2 281.037 29.087WS (from BG design)

552.444 24

Subject 397.852 8 49.731Error 154.593 16 9.662

Total 1114.519 26

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Plot of the Data

0

5

10

15

20

25

30

PreTest PostTest FollowUp

Repo

rted

Sym

ptom

s

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Interpretation

Note parallel with diagramNote subject differences not in error termNote MSerror is denominator for F on TimeNote SStime measures what we are interested in studying

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AssumptionsCorrelations between trials are all equal–Actually more than necessary, but close–Matrix shown below

Pre Post FollowupPre 1.00 .637 .434Post 1.00 .742Followup 1.00

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Assumptions

Previous matrix might look like we violated assumptions–Only 9 subjects–Minor violations are not too serious.Greenhouse and Geisser (1959) correction (among many)–Adjusts degrees of freedom

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Multiple Comparisons

With few means:– t test with Bonferroni corrections–Limit to important comparisonsWith more means:–Require specialized techniques

• Trend analysis

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Advantages of RM Designs

Eliminate subject differences from error term–Greater powerFewer subjects neededOften only way to address the problem–This example illustrates that case.

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Disadvantages of RM designs

Carry-over effects–Counter-balancingMay tip off subjects to what you are testingSome phenomenon cannot be tested in a repeated measure fashion (e.g. anything that requires tricking the participant)

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Effect SizeSimple extension of what we said for t test for related samples.Stick to pairs of means.OR–2 can be used for repeated measures

data as well–Some adjustments can make it more

meaningful