1. Analysis of Covariance (ANCOVA)
Analysis of variance conducted after removing the relationship of some extraneous variable (covariate) with the dependent variable (Y)
What variability in DV can be explained by IV, AFTER removing variability explained by the covariate?

2. Reasons for Using ANCOVA
1. reduce ‘error’ (MSW) by removing effects of extraneous variable
2. adjust DV scores, what would they be in the absence of the covariate
(estimated through regression)

3. Ideally
small number of covariates
each correlated with the DV
uncorrelated with each other

4. Analysis of Covariance (ANCOVA)
Example – even after ‘randomly assigning’ participants to levels of the IV, some differences still exist before IV is introduced.
See example on next slide of ‘anxiety’ differences before any exposure to anxiety stimulus or any ‘treatment’ manipulation

6. Regression is used to adjust the DV scores
Adjusted scores reflect the removal of the variability in the DV explained
by the Covariate
what would Anxiety at time 3 be, when Anxiety at time 1 is held constant?
The regression residuals reflect variability not explained by the covariate
– it becomes the ‘error’ (MSW) term
and will be lower than the MSW would have been

7. Differences among the adjusted means
Differences among the adjusted means (MSB) can be evaluated relative to the unexplained variance that remains
after removal of the relationship of the DV with the covariate
What is IV – DV relationship after removing the Covariate – DV relationship

8. The adjusted scores of the DV are based on a regression equation
using the pooled regression coefficient
pooled across the separate regression
equations for each group in the design.
Thus, for each group the regression coefficient is the same, but the intercepts will vary (graph to follow)

9. Group means are adjusted to what they would be at mean of covariate1 1 1
Original mean 2 1 2 1
Adjusted mean 2
DV (Y) 1 2 3 3
Slopes the same, intercepts differ 2 2 3 3 3
Mean of Covariate 3
Group means are adjusted to what they would be at mean of covariate
Covariate

10. Assumptions for ANCOVA
Same as those for ANOVA, plus
Homogeneity of regression coefficients
since use ‘pooled’ estimate of regression coefficient
Linear relationship of DV with Covariate
since using linear regression to adjust scores on DV

11. What to report
Original Means and Standard Deviations
Adjusted Means
For multilevel variable use pairwise
comparisons option, not post hoc

12. ANOVA Without Covariate
Adjusted Means and SEs
Original Means and SDs
ANOVA Without Covariate
ANOVA With Covariate
MSW reduced due to covariate
MSB reduced due to covariate