1. Analysis of Covariance adjusting for potential confounds

2. Overall Purpose All ANVOVA designs have three things in common: All ANVOVA designs have three things in common: – at least one categorical IV – a quantitative outcome variable measured at least at the interval level, the DV – potential confounding variables

3. Overall Purpose ANCOVA is one strategy for handling potential confounding variables. ANCOVA is one strategy for handling potential confounding variables. Covariates are potential cofounding variables that are built into the design for the purpose of statistical adjustment. Covariates are potential cofounding variables that are built into the design for the purpose of statistical adjustment. ANCOVA can be used with all ANOVA designs. ANCOVA can be used with all ANOVA designs.

4. Overall Purpose ANCOVA can be used to control for pre- existing differences in groups. ANCOVA can be used to control for pre- existing differences in groups. ANCOVA can be used to statistically adjust for the influence of potential confounding variables on the dependent variable. ANCOVA can be used to statistically adjust for the influence of potential confounding variables on the dependent variable. ANCOVA can be used to increase the statistical power of an ANOVA design. ANCOVA can be used to increase the statistical power of an ANOVA design.

5. Overall Purpose The easiest way to understand ANCOVA is to think about the following design: The easiest way to understand ANCOVA is to think about the following design: We are attempting to compare the growth rates across time of two group, Rx and Control. We are attempting to compare the growth rates across time of two group, Rx and Control. We include the pretest as a covariate to control for non-equivalence of the groups at pre-test and to increase the power of the design. We include the pretest as a covariate to control for non-equivalence of the groups at pre-test and to increase the power of the design.

6. Overall Purpose Now we are actually comparing what are called adjusted means on the post test as our DV. Now we are actually comparing what are called adjusted means on the post test as our DV. The between-child variance that is due to the pretest is removed from the error variance in the design. The between-child variance that is due to the pretest is removed from the error variance in the design. The statistical power goes up quite a bit because the error variance is reduced. The statistical power goes up quite a bit because the error variance is reduced.

7. As a Regression Analysis We can run a regression analysis and use pretest as the IV and posttest as the DV. We can run a regression analysis and use pretest as the IV and posttest as the DV. Then we can add Group to the model and we will have a more sensitive test of the effect of Group or treatment condition. Then we can add Group to the model and we will have a more sensitive test of the effect of Group or treatment condition. The extent to which students vary around the grand mean of the postest, because of their pretest scores, has been removed. The extent to which students vary around the grand mean of the postest, because of their pretest scores, has been removed.

8. As A Regression Analysis This has the effect of simulating what would happen if both groups started out with the same pretest mean. This has the effect of simulating what would happen if both groups started out with the same pretest mean. However, you have to assume that you are adjusting the posttest scores along the same regression slope for each group. However, you have to assume that you are adjusting the posttest scores along the same regression slope for each group. This is called the assumption of Homogeneity of Regression. This is called the assumption of Homogeneity of Regression.

9. As A Regression Analysis Therefore, each ANCOVA design must meet the assumptions of the equivalent ANOVA design, and meet the assumption of Homogeneity of Regression. Therefore, each ANCOVA design must meet the assumptions of the equivalent ANOVA design, and meet the assumption of Homogeneity of Regression. You test for this by observing whether the regression slope between the covariates and the dependent variable is the same for each group. You test for this by observing whether the regression slope between the covariates and the dependent variable is the same for each group.

10. Interpretation It is important to examine both the adjusted and the unadjusted means and significance tests. It is important to examine both the adjusted and the unadjusted means and significance tests. This will help you understand the influence of the covariates. This will help you understand the influence of the covariates. Qualify interpretation of adjusted means. Qualify interpretation of adjusted means.

11. Interpretation No number of covariates is a substitute for randomization. No number of covariates is a substitute for randomization. You can never control for everything nor can you anticipate or measure all of the important covariates. You can never control for everything nor can you anticipate or measure all of the important covariates. Covariates are useful with randomization as they can really enhance power. Covariates are useful with randomization as they can really enhance power.