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Education 793 Class Notes

Education 793 Class Notes. ANCOVA Presentation 11. 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:. Review: Simple Linear Regression. Simple Linear Regression. Assumptions independence

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Education 793 Class Notes

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  1. Education 793 Class Notes ANCOVA Presentation 11

  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. Review: Simple Linear Regression

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

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

  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. 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. 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. 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. 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. Next Week • Multiple Regression • Chapter 18 p. 528-548

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