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Factorial Design. One Between-Subject Variable. One Within-Subject Variable. SS Total. SS between subjects. SS within subjects. Treatments by Groups. Groups. Subjects within groups. Treatments. Treatments by Subjects within groups. Differences Between Subjects.

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slide1

Factorial Design

One Between-Subject Variable

One Within-Subject Variable

SSTotal

SSbetween subjects

SSwithin subjects

Treatments by Groups

Groups

Subjects within groups

Treatments

Treatments by Subjects within groups

Differences Between Subjects

Differences Within Subjects

Groups – differences between groups of subjects

SS w/in Groups – differences between subjects w/in a group

Treatment – differences between subject’s scores across treatments

Treat x Groups – interaction between Treatments and Groups

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

slide3

Example

Speed (Repeated Measure)

Group

1

Group

2

=GT

=GM

slide4

Calculate MS

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)

slide5

Prepare Summary Table

7

What are the appropriate error terms?

(the denominators for the Fratios)

Interpolation?

8

repeated measures assumptions
Repeated Measures Assumptions

1)

normality

2)

homogeneity of variance

3)

compound symmetry

- constant variances on diagonal

- constant covariances off diagonal

A variance / covariance matrix for each group and overall

4)

T X Ss interactions are constant across groups

- test with Fmax

example
Example

No STRAT

Var/Covar Matrix

Speed

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

= a constant across all pairs of conditions

= .574

= .853

slide8

Simple Effects

Factorial Design

One B-S variable

One W-S variable

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

SSTotal = 67.44

SSTotal = 168.04

SSbs = 7.14

SSbs = 5.29

SSTreat = 54.69

SSTreat = 159.19

SSerror = 3.56

SSerror = 5.56

slide9

No STRAT

STRAT

+

=

67.44

168.04

235.48

SSTotal

SSTotal

(overall)

+

=

12.45

7.19

5.29

SSbs

SSbs

(overall)

+

159.19

=

213.88

SSTreat

54.69

SSTreat

+

SST X G

Why?

(overall)

=

SSerror

5.56

+

3.56

9.12

SST X S w/in group

(overall)

between subjects simple effects
Between-Subjects Simple Effects

We could do a separate analysis of each level

- unnecessary loss of df

SSgrp at 5

=

=

8.0

SSgrp at 15

=

=

4.5

SSgrp at 25

=

=

2.0

=

SSgrp at 35

=

8.0

MS all 1 df

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

Why?

MSerror

=