PSY 307 – Statistics for the Behavioral Sciences

1. PSY 307 – Statistics for the Behavioral Sciences Chapter 17 – Repeated Measures ANOVA (F-Test) Chapter 18 – Two Factors (Two-way ANOVA)

2. 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

3. 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.

4. 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.

5. Splitting Up the Variance:Repeated Measures ANOVA Total Variance SStotal Between Groups Within Groups SSbetween SSwithin Between Subjects Error SSsubject SSerror

6. Finding SSsubject SSsubject is calculated using row totals for each subject. SSerror is the amount remaining when SSsubject is subtracted from SSwithin.

7. Calculating SSerror and MSerror This is the same as for one-way ANOVA

8. 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.

9. 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. • h2is interpreted the same way as for one-way ANOVA. • Post-hoc t-tests need to be for paired samples, not independent groups.

10. 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.

11. Two Factors: Age and Sex Main effect for Age H0: myoung = mold H1: H0 is false Main effect for Sex H0: mmale = mfemale H1: H0 is false

12. What About Within the Cells? 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

13. 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.

14. 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.

15. Splitting Up the Variance: Two-Way ANOVA Total Variance SStotal Between Cells Within Cells SSbetween SSwithin Between Columns Between Rows Interaction SScolumns SSrows SSInteraction

16. 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

17. 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

18. 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

19. A Sample Two-Way ANOVA Table * Significant at the .05 level

20. 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

21. Calculating h2

22. 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.

23. 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.

24. 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.

25. 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.