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

Kinds of ANOVAs Repeated Measures ANOVA Factorial ANOVA (Two-way) 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 Groups SSbetween Within Groups SSwithin Between Subjects SSsubject Error SSerror

Finding SSsubject Subject 24 48 XSubject A 3 6 B 4 8 C 2 10 Group Mean 5 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 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.

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. h2 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 10 Old 20 15 Main effect for Age H0: myoung = mold H1: H0 is false Main effect for Sex H0: mmale = mfemale H1: H0 is false

What About Within the Cells? Male Female Young 5 10 Old 20 15 Main effect for Age H0: myoung = mold H1: H0 is false Main effect for Sex H0: mmale = mfemale 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 interaction H1: 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 Cells Within Cells SSbetween SSwithin Between Columns Between Rows Interaction SScolumns SSrows SSInteraction

Calculating MScolumns and MSwithin dfcolumns = c-1, where c is the number of columns 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

Calculating MSrows and MSwithin dfrows = r-1, where r is the number of rows 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

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

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 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 h2 estimates the proportion of the total variance attributable to one of the two factors or the interaction. There is an h2 value for each main effect and one for the interaction. Cohen’s rule for interpreting h2 : .01 small effect .09 medium effect .25 large effect

Calculating h2

What is h2 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 (h2). 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.