One-Way Analysis of Covariance

1. One-Way Analysis of Covariance One-Way ANCOVA

2. ANCOVA • Allows you to compare mean differences in 1 or more groups with 2+ levels (just like a regular ANOVA), while removing variance from a 3rd variable • What does this mean?

3. ANCOVA

4. ANCOVA • Removing variance that is unrelated to the IV/intervention = removing error variance • Makes ANCOVA potentially a very powerful test (i.e. easier to find significant results than with ANOVA alone) by potentially reducing MSerror • Generally, the more strongly related are covariate and DV, and unrelated the covariate and IV, the more useful (statistically) the covariate will be in reducing MSerror

7. ANCOVA • Why would this be useful? • Any longitudinal research design needs to control for T1 differences in the DV • I.e. If assessing change in symptoms of social anxiety over time between 2 groups, we need to control for group differences in T1 social anxiety • Even if random assignment is used, use of a covariate is a good idea – Random assignment doesn’t guarantee group equality

8. ANCOVA • Why would this be useful? • Any DV’s with poor discriminant validity • I.e. SES and race are highly related – If we wanted to study the effects of SES, independent of race, on scholastic achievement we could use an ANCOVA using SES as the DV and race as a covariate

9. ANCOVA • Why would this be useful? • If you’re using 2+ DV’s (MANOVA) and want to isolate the effects of one of them • ANCOVA with the DV of interest and all other DV’s used as covariates • Note: In this case we’re specifically predicting that IV’s and covariates are related, it’s not ideal, but what can you do?

10. ANCOVA • However, ANCOVA should not be used as a substitute for good research design • If your groups are unequal on some 3rd variable, these differences are still a plausible rival hypothesis to your H1, with or without ANCOVA • Controlling ≠ Equalizing • Random assignment to groups still best way to ensure groups are equal on all variables

11. ANCOVA • Also, covariates change the meaning of your DV • I.e. We studying the effects of a tutoring intervention for student athletes – We find out our Tx group is younger than our control group – (Using age as a covariate)  (DV = class performance – age) • What does this new DV mean??? Effects of Tx over and above age (???)

12. ANCOVA • Also, covariates change the meaning of your DV • For this reason, DO NOT just add covariates thinking it will help you find sig. results • Adding a covariate highly correlated with a pre-existing covariate actually makes ANCOVA less powerful • df decreases slightly with each covariate • No increase in power since 2 covariates remove same variance due to high correlation

13. ANCOVA • Assumptions: • Normality • Homoscedasticity • Independence of Observations • Relationship between covariate and DV • Relationship between IV and covariate is linear • Relationship between IV and covariate is equal across levels of IV • AKA Homogeniety of Regression Slopes • I.e. an interaction between IV and CV

14. ANCOVA • Calculations • Don’t worry about them, in fact, you can skip pp. 577-585 in the text • Recall that in the one-way ANOVA we divided the total variance (SStotal) into variance attributable to our IV (SStreat) and not attributable to our IV (SSerror)

15. ANCOVA • In ANCOVA, we just divide the variance once more (for the covariate) • IV: Inferences are made re: its effects on the DV by systematically separating its variance from everything else • Covariate: Inferences are made by separating its variance from everything else, however this separated variance is not investigated in-and-of itself