1. Education 793 Class Notes ANCOVA Presentation 11

2. 2 Review: Analysis of Variance ANOVA can be used to compare two or more means with one continuous dependent variables and two or more treatment levels:

3. 3 Review: Simple Linear Regression

4. 4 Simple Linear Regression Assumptions –independence –linearity –homogeneity We regress Y (dependent variable) on X (independent variable)  is the slope of the regression line

5. 5 ANCOVA—Extension of ANOVA ANCOVA is a method used to compare group means like ANOVA but we remove systematic individual differences from information collected on individuals before treatment (a covariate). The covariate can be nominal, ordinal or interval Including the covariate in the analysis reduces the within-group error term thus making it more powerful than ANOVA

6. 6 Design Requirements of ANCOVA There is one dependent variable with two or more levels. The levels of the dependent variables differ either quantitatively or qualitatively. A covariate is measured prior to the implementation of treatment and control conditions (this is often a pretest). A subject may appear in only one group.

7. 7 Assumptions for ANCOVA The subjects scores are independent The scores within each treatment (level) are normally distributed Homogeneity –equal variances across groups –equal variance at each level of the covariate

8. 8 Assumptions for ANCOVA When cell sizes are equal, ANCOVA is robust to violation of the homogeneity. Assume that the regression of the dependent variable (Y) on the covariate (X) is linear in each group. Assume that the regression slope of Y on X is the same in each group

9. 9 Visual Two possible scenarios with two groups, treatment and control and a measured covariate (pretest)

10. 10 Interpretation of Results We are interested in the Adjusted Group Means (after adjusting for the covariate) –After removing the systematic differences between subjects on the pretest, what are the differences attributable to treatment groups on the outcome?

11. 11 Example We want to look at the differences between male and female SAT Math scores for incoming 1998 UM freshmen, adjusting for their HSGPA’s.

12. 12 Output First the ANOVA TABLE Tests of Significance for SATM using UNIQUE sums of squares Source of Variation SS DF MS F Sig of F WITHIN+RESIDUAL 11609201.95 2404 4829.12 HSGPA 1171654.79 1 1171654.8 242.62.000 SEX 13727.09 1 13727.09 2.84.092 HSGPA BY SEX 68.71 1 68.71.01.905 What assumption does this test ?

13. 13 Output The adjusted means table Adjusted and Estimated Means Variable.. SATM SAT MATH SCORE CELL Obs. Mean Adj. Mean Est. Mean Raw Resid. Std. Resid. 1 680.985 680.985 680.985.000.000 2 649.068 649.068 649.068.000.000

14. 14 Further Details As with ANOVA, post-hoc comparisons can be done. This same procedure can be carried out with multiple regression techniques for next week.

15. 15 Next Week Multiple Regression –Chapter 18 p. 528-548