PSY 307 – Statistics for the Behavioral Sciences

Chapter 17 – Repeated Measures ANOVA (F-Test)

Kinds of ANOVAs

Repeated Measures ANOVA Used with 2 or more paired samples

Factorial ANOVA (Two-way) Use with 2 or more independent

variables Both IVs are independent samples

Mixed ANOVA A two-way ANOVA with both between-

subjects and repeated measures IVs

Repeated Measures ANOVA

Repeated Measures ANOVA – one-way ANOVA but the same subjects are measured in each group.

Estimates of variability are no longer inflated by random error due to individual differences. A more powerful F-test

A way to make the denominator smaller.

The Repeated Measures F-Ratio

The numerator contains the MS for between-groups, as for one-way ANOVA.

The denominator contains only the error variance (noise), not the entire within-group variance. Variance for individual subjects is

subtracted from the entire within-group variance, leaving the error variance.

Splitting Up the Variance:Repeated Measures ANOVA

Total Variance

SStotal

Between GroupsSSbetween

Within GroupsSSwithin

Between SubjectsSSsubject

ErrorSSerror

Finding SSsubject

Subject 0 24 48 XSubject

A 0 3 6 3

B 4 6 8 6

C 2 6 10 6

Group Mean

2 5 8 5 (Grand Mean)

SSsubject is calculated using row totals for each subject.

SSerror is the amount remaining when SSsubject is subtracted from SSwithin.

Calculating SSerror and MSerror

error

errorerror df

SSMS

error

between

MS

MSF

This is the same as for one-way ANOVA

Effect Size for Repeated Measures ANOVA

The effect size for repeated measures is called the partial squared curvilinear correlation.

It is called partial because the effects of individual differences have been removed.

errorbetween

betweenp SSSS

SS

2

Comparison to One-Way ANOVA

Other assumptions hold (e.g., normality, equal variance), but sphericity is an added assumption. Sphericity means data are uncorrelated.

Counterbalancing may be needed. is interpreted the same way as for one-

way ANOVA. Post-hoc t-tests need to be for paired

samples, not independent groups.

Two-Way ANOVA

Two-way (two factor) ANOVA – tests hypotheses about two independent variables (factors).

Three null hypotheses are tested: Main effect for first independent

variable Main effect for second independent

variable Test for an interaction between the two

variables.

Two Factors: Age and Sex

Male Female

Young 5 5 10

Old 10 10 20

15 15

Main effect for Age

Main effect for Sex

H0: male = female

H1: H0 is false

H0: young = old

H1: H0 is false

What About Within the Cells?

Male Female

Young 5 10 10

Old 10 5 20

15 15

Main effect for Age

Main effect for Sex

H0: male = female

H1: H0 is false

H0: young = old

H1: H0 is false

An interaction occurs when the pattern within the cells is different depending on the level of the IVs.

H0: There is no interactionH1: H0 is false

Interaction

Two factors interact if the effects of one factor on the dependent variable are not consistent for all levels of a second factor.

Interactions provide important information about the question at hand. Interactions must be discussed in your

interpretation of your results because they modify main effects.

Applet Demonstrating Two-Way ANOVA

http://www.ruf.rice.edu/~lane/stat_sim/two_way/index.html

Try this at home to understand what an interaction looks like when graphed and in a data table.

Splitting Up the Variance: Two-Way ANOVA

Total Variance

SStotal

Between CellsSSbetween

Within CellsSSwithin

Between ColumnsSScolumns

Between RowsSSrows

InteractionSSInteraction

Calculating MScolumns and MSwithin

columns

columnscolumns df

SSMS

within

columns

MS

MSF This is the same as for one-way

ANOVA but with different df:dfwithin = N-(c)(r), where c is columns and r is rows

dfcolumns = c-1, where c is the number of columns

Calculating MSrows and MSwithin

rows

rowsrows df

SSMS

within

rows

MS

MSF This is the same as for one-way

ANOVA but with different df:dfwithin = N-(c)(r), where c is columns and r is rows

dfrows = r-1, where r is the number of rows

Calculating MSint and MSwithin

int

intint df

SSMS

withinMS

MSF int This is the same as for one-way

ANOVA but with different df:dfwithin = N-(c)(r), where c is columns and r is rows

dfint = (c-1)(r-1), where c is the number of columns and r is the number of rows

A Sample Two-Way ANOVA Table

Source SS Df MS F

Column 72 2 72/2 = 36 36/5.33 = 6.75*

Row 192 1 192/1 = 192 192/5.33 = 36.02*

Interaction 56 2 56/2 = 28 28/5.33 = 5.25*

Within 32 6 32/6 = 5.33

Total 352 11

* Significant at the .05 level

Proportion of Explained Variance

estimates the proportion of the total variance attributable to one of the two factors or the interaction.

There is an value for each main effect and one for the interaction.

Cohen’s rule for interpreting : .01 small effect .09 medium effect .25 large effect

Calculating

withincolumn

column

SSSS

SScolumn

)(2p

withinrow

row

SSSS

SSrow

)(2p

withinSSSS

SS

int

int2p )ninteractio(

What is used for?

When a main effect is non-significant due to small sample size, the size of the effect can be examined.

Some journals require effect sizes to always be reported along with inferential tests.

Simple Effects

A simple effect is the comparison of groups for each level of one IV, at a single level of the other IV. Example: Young and old males. Example: Young males and females.

Calculation of the simple effect is the same as doing a one-way ANOVA on just one column or row.

Use t-tests to follow up.

Assumptions

Similar to those for one-way ANOVA: Normal distribution, equal variances

All cells should have equal sample sizes.

Use Cohen’s guidelines for effect size (2).

Use t-tests for multiple comparisons within cells.

Mixed ANOVAs

One variable is between subjects. One or more variables are within

subjects (paired or repeated measures).

A mixed ANOVA is performed using the Repeated Measures General Linear Model menu choice on SPSS.

Formulas are complex and beyond the scope of this course.