Factorial DesignOne Between-Subject Variable

One Within-Subject Variable

SSTotal

SSbetween subjects SSwithin subjects

Treatments by Groups

Treatments Treatments by Subjects within groups

Subjects within groups

Groups

Differences Between Subjects

Differences Within Subjects

Groups – differences between groups of subjectsSS w/in Groups – differences between subjects w/in a groupTreatment – differences between subject’s scores across treatmentsTreat x Groups – interaction between Treatments and Groups

Treats x Ss w/in Groups – interaction between Subjects and Treatments hold Groups factor constant

2)( GMySS ijtotal

2)( GMykSS ibsub

bsubtotalwsub SSSSSS

2)( GMynSS ijcells

2)( GMyknSS grpgrps

grpsbsubwgrps SSSSSS 2

· )( GMygnSS jtreat grpstreatcells SSSSSSSS grpstreat x

grptreat x /sub(wgrp)treat x SSSSSSSS treatsubw 2

·· )( GMyyy jiij

Sub # 5 15 25 35

1 7 7 4 3 5.25

2 8 8 5 6 6.75

3 9 7 4 3 5.75

4 8 6 3 3 5.00

Trc 32 28 16 15

8.0 7.0 4.0 3.75 5.69

1 10 5 2 1 4.5

2 10 6 3 2 5.25

3 9 5 4 2 5.00

4 11 6 3 2 5.5

Trc 40 22 12 7

10 5.5 3 1.75 5.06

Tc 72 50 28 22 172

9 6.25 3.5 2.75 5.38

rcy

y

rcy

=GT

=GMcy

Speed (Repeated Measure)

Example

Group

1

Group

2

Divide SS by appropriate df

SSbs by #Ss - 1

SSgrp by #Grps - 1

SSss w/in grps by (#Singrp-1) x (# of grps)

SSws by #Ss (# Treatments – 1)

SStreat by # Treatments - 1

SSTxG by (#grp – 1) (#Treats -1)

SSTxS w/in grpsby (#Treats -1) x (n-1) x (# of grps)

Calculate MS

Prepare Summary Table

Source SS Df MS F P

Btwn S 12.5 7

Grp 3.125 1 3.125 2 n.s.

Ss w/in Grp

9.375 6 1.563

Within S 223 24

Treat 194.5 3 64.833 127.89 < 0.01

TxG 19.375 3 6.458 12.74 < 0.01

TxS w/in grp

9.125 18 .507

Total 235.531

What are the appropriate error terms?

(the denominators for the Fratios)

Interpolation?

7

8

Repeated Measures Assumptionsnormality1)

2) homogeneity of variance

3) compound symmetry

- constant variances on diagonal

- constant covariances off diagonal

A variance / covariance matrix for each group and overall

1

))((

n

yyxxCov

1

N

N

yxxy

Cov

1

nn

TTTT

Cov

ji

ji

T X Ss interactions are constant across groups4)- test with Fmax

ExampleNo STRAT Var/Covar Matrix

5 15 25 35

5 0.66 0 0 0

15 0.66 0.66 1.0

25 0.66 1.0

35 2.25

Speed

The assumption of compound symmetry is usually replaced by the assumption of sphericity

2

2

2

1

2

12

21 NNyy

2

jTyiTy

4

66.

4

66.2

155 TT

4

25.2

4

66.2

355 TT

= .574

= .853

= a constant across all pairs of conditions

Simple Effects

One B-S variableOne W-S variable

Factorial Design

The W-S variable

- Separate One-Way ANOVAs (repeated measures)

∙ Error terms pooled = MS T X Ss w/in groups

∙ Or, use the MST X Ss for each separate analysis

No STRAT STRAT

SSTotal = 67.44

SSbs = 7.14

SSTreat = 54.69

SSerror = 5.56

SSTotal = 67.44 SSTotal = 168.04

SSbs = 5.29

SSTreat = 159.19

SSerror = 3.56

STRATNo STRAT

SSTotal

SSbs

SSTreat

SSerror

67.44 +

54.69

7.19

5.56

+

+

+

168.04 = 235.48

=

=

=

12.45

213.88

9.12

5.29

159.19

3.569df

3df 3df

9df

SSTotal

(overall)

SSbs

(overall)

SSTreat

SST X G

+

(overall)

SST X S w/in group

(overall)

Why?

Between-Subjects Simple Effects

We could do a separate analysis of each level

- unnecessary loss of df

SSgrp at 5

SSgrp at 15

SSgrp at 25

SSgrp at 35

=

=

=

=

=

=

=

=

8.0

4.5

2.0

8.0

MS all 1 df

SSerror term = SSw/cells = SSSs w/in grp + SS T X Ss w/in grps

MSerror =

grpsSw/in X T grpSw/in

cellsw/in SS

dfdf

Why?

77.MSerror 5.18SSerror 24df