1. Analysis of Covariance
46-512: Statistics for Graduate Study in Psychology

2. Learning Outcomes What is an ANCOVA?
How does it relate to what we have done already?
When would we use it?
What are the issues & assumptions?
What are some limitations and alternatives?

3. Experiment: 3 instructional methods

4. First, let’s run it as an MRA…
Treat group 3 as control:
DC1 identifies Group 1
DC2 identifies Group 2
compute dc1=0.
compute dc2=0.
if (gpid=1) dc1=1.
if (gpid=2) dc2=1.
execute.
REGRESSION
/DESCRIPTIVES MEAN STDDEV CORR SIG N
/MISSING LISTWISE
/STATISTICS COEFF OUTS CI R ANOVA
/CRITERIA=PIN(.05) POUT(.10)
/NOORIGIN
/DEPENDENT y
/METHOD=ENTER dc1 dc2 .

5. Result…
R2 = / = .128

6. Which agrees with GLM

7. Back to MRA Enter our continuous variable (IQ)
Sans interaction term for the time being.
R2 = .527

8. What have we done? Analysis of Covariance What does it tell us?
In general, why would we use this technique? 1) 2) 3)

9. Examples of different applications
Elimination of systematic bias
The relationship between questionnaire responses and business performance, controlling for pre-existing differences in business performance.
Reduce Error Variance
In a random assignment experiment, looking at vigilance and using age as a covariate
Step-down Analysis
Studying the effects of an educational intervention on performance & self-esteem.

10. Effects & Extensions Types of Effects Extensions
Significance of Covariate(s)
Main Effects
Interactions among Factors
Interactions between factors and covariate(s) = bad news.
Extensions
Can have multiple covariates
Factorial Designs
Mixed Randomized by Repeated Designs
Within Subjects Designs

11. Back to our example (as one-way)

12. Run through GLM as ANCOVA
Why is GPID now significant?

13. Means and Adjusted Means
Adjusted Means calculated as…
For Group 1…

14. Parameter Estimates from SPSS
Compare to those from our MRA

15. Post-Hocs from SPSS

16. Bryant-Paulson Post Hoc
BPcrit = 3.55, Cell 3 is significantly higher than 1 & 2
Bryant-Paulson is an extension of Tukey’s Post-Hoc test, and more appropriate if X is random.

17. ANCOVA & Intact Groups Groups can still differ in unknown ways.
Question whether groups that are equivalent on the covariate ever exist – since ANCOVA adjusts for equivalence on the covariate.
Assumptions of linearity and homogeneity of regression slopes need to be satisfied.
Differential growth of subjects i.e., is difference due to treatment or differential growth?
Measurement error can produce spurious results.

18. Assumptions of ANCOVA
Larger sample sizes (because of the regression of the DV on the CV)
Absence of Multicollinearity and Singularity
Normality of sampling distributions (of the means)
Homogeneity of Variance
Linearity – of relationship between covariate and dependent variable
Homogeneity of regression
Reliability of covariates

19. Alternatives
In pre-post situations, using difference scores (assuming same metric)
Controversial and carries some risk
Incorporating pre-scores into a RM ANOVA design.
Residualize DV and run an ANOVA on the residualized scores.
Controversial, not a very popular approach
Blocking (rather than tackling!)
assigning/matching people based on pre-scores or creating appropriate IV categories of intact groups.
Utilizing the CV as a factor in the experiment, if it lends itself well to categorization.
This side-steps many issues, such as homogeneity of regression.
Johnson-Neyman technique
See Stevens (1999) for an alternative

20. Things to consider about covariates
Number
Reliability
Pre-screening
Multicollinearity
Loss of df

21. More complicated designs
More than one covariate
Factorial Designs
Repeated Measures Designs
For now, we will suspend discussion of more complicated designs, but revisit when we cover MANOVA and MANCOVA