PSY 307 – Statistics for the Behavioral Sciences Chapter 17 – Repeated Measures ANOVA (F-Test)
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Transcript of PSY 307 – Statistics for the Behavioral Sciences Chapter 17 – Repeated Measures ANOVA (F-Test)
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.