1. ANCOVA

2. What is Analysis of Covariance?
When you think of Ancova, you should think of sequential regression, because really that’s all it is
Covariates enter in step 1, grouping variable(s) in step 2
In this sense we want to assess how much variance is accounted for in the DV after controlling for (partialing out) the effects of one or more continuous predictors (covariates)
ANCOVA always has at least 2 predictors (i.e., 1 or more categorical/ grouping predictors, and 1 or more continuous covariates).
Covariate:
Want high r with DV; low with other covariates
It is statistically controlled in an adjusted DV
If covariate correlates with categorical predictor: heterogeneity of regression (violation of assumption)

3. Similarities with ANCOVA
Extends ANOVA: Assess significant group differences on 1 DV
While controlling for the effect of one or more covariate
Is Multiple Regression with 1+ continuous predictor (covariates) and 1+ dummy coded predictor
So incorporates both ANOVA and Multiple Regression
Allows for greater sensitivity than with ANOVA under most conditions
ANOVA assess is a univariate statistic used to compare differences in mean scores across groups.
Can have 1 way; 2 way; Factorial between group designs examines mean scores across levels of independent variable
Covariate reduces error

4. When to Use ANCOVA?
Experimental Design: Best use, but often difficult to implement
Manipulation of IV (2 group double blind drug trial)
Random Selection of Subjects
Random Assignment to Groups
Multivariate Analysis: Can use ANCOVA as a follow-up to a significant MANOVA
Conduct on each DV from a significant MANOVA with remaining DVs used as Covariates in each follow-up ANCOVA
Question to ask yourself however is, why are you interested in uni output if you are doing a multivariate analysis?
Quasi-Experimental Design (controversial use)
Intact groups (students taking stats courses across state)
May not be able to identify all covariates effecting outcome
1)Experimental Design
Can have pretest as covariate and post-test used as DV for two or more groups
Smoking Cessation—Tx and Control—smoking rate cigs day 1 = cov
Cig days2 = DV by condition
2) Non-Experimental: Quasi-experimental not causal—state limitations from unknown confounds
Design of 90s
more real life
confounding variables are problematic
problems with internal validity
MANOVA
Follow-up (post-hoc)
Same idea as doing the post-hoc Tukey test—trying to understand what is indicating a significant relationship.
Essentially this assesses the contributions of each DV : does this because it removes the variance of the the other DVs from the analysis—partials out the effects of multiple DV so we can see unique contribution
We will see that there will be a new linear combination on both sides of the equation when we move into multiple outcomes

5. ANCOVA Background Themes
Sample Data: Random selection & assignment
Measures: Group predictor, Continuous Covariate, Continuous DV
Methods: Inferential with experimental design & assumptions
Assumptions: Normality, Linearity, Homoscedasticity, and Homogeneity of Regression:
HoR: Slopes between covariate and DV are similar across groups
Indicates no interaction between IV and covariate
If slopes differ, covariate behaves differently depending on which group (i.e., heterogeneity of regression)
When slopes are similar, Y is adjusted similarly across groups

6. Testing for Homogeneity of Regression
First run the Ancova model with a Treatment X Covariate interaction term included
If the interaction is significant, assumption violated
Again, the interaction means the same thing it always has. Here we are talking about changes in the covariate/DV correlation depending on the levels of the grouping factor
If not sig, rerun without interaction term

7. ANCOVA Model Y = GMy +  + [Bi(Ci – Mij) + …] + E
Y is a continuous DV (adjusted score)
GMy is grand mean of DV
 is treatment effect
Bi is regression coefficient for ith covariate, Ci
M is the mean of ith covariate
E is error
ANCOVA is an ANOVA on Y scores in which the relationships between the covariates and the DV are partialed out of the DV.
An analysis looking for adjusted mean differences

8. Central Themes for ANCOVA
Variance: in DV
As usual, we are interested in accounting for the (adjusted) DV variance, here with categorical predictor(s)
Covariance: between DV & Covariate(s)
in ANCOVA we can examine the proportion of shared variance between the adjusted Y score and the covariate(s) and categorical predictor
Ratio: Between Groups/ Within Groups
Just as with ANOVA, in ANCOVA we are very interested in the ratio of between-groups variance over within-groups variance.

9. ANCOVA Macro-Assessment
F-test
The significance test in ANCOVA is the F-test as in ANOVA
If significant, 2 or more means statistically differ after controlling for the effect of 1+ covariates
Effect Size: ES= 2
2 = SSEFFECT / SSTOTAL, after adjusting for covariates
Partial 2 = SSEFFECT /(SSEFFECT + SSerror)

10. ANCOVA Micro-Assessment
Test of Means, Group Comparisons
Follow-up planned comparisons (e.g., FDR)
d-family effect size
While one might use adjusted means, if experimental design (i.e. no correlation b/t covariate and grouping variable) the difference should be pretty much the same as original means
However, current thinking is that the standardizer should come from the original metric, so run just the Anova and use the sqrt of the mean square error from that analysis
Graphs of (adjusted) means for each group also provide a qualitative examination of specific differences between groups.

11. Steps for ANCOVA Consider the following: All variables reliable?
Are means significantly different, i.e., high BG variance?
Do groups differ after controlling for covariate?
Are groups sufficiently homogeneous, i.e., low WG variance?
Low to no correlation between grouping variable & covariates?
Correlation between DV & covariates?
Can the design support causal inference (e.g., random assignment to manipulated IV, control confounds)?

12. ANCOVA Steps Descriptive Statistics Correlations
Means, standard deviations, skewness and kurtosis
Correlations
Across time for test-retest reliability
Across variables to assess appropriateness of ANCOVA
Test of Homogeneity of Regression
ANOVA (conduct as a comparison)
ANCOVA (ANOVA, controlling for covariates)
Conduct follow-up tests between groups

13. Ancova Example
This example will show the nature of Ancova as regression
Ancova is essentially performing a regression and, after seeing the difference in output between the stages in a hierarchical regression, we can see what Ancova is doing

14. Ancova Example
Consider this simple prepost setup from our mixed design notes
We want to control for differences at pre to see if there is a posttest difference b/t treatment and control groups
Pre Post
treatment 20 70
treatment 10 50
treatment 60 90
treatment 20 60
control
control
control
control

15. Ancova Example Dummy code grouping variable
Though technically it should probably already be coded as such anyway for convenience

16. Ancova Example
To get the compare the SS output from regression, we’ll go about it in hierarchical fashion
First enter the covariate as Block 1 of our regression, then run Block 2 with the grouping variable added

17. Ancova Example Ancova output from GLM/Univariate in SPSS
Regression output =

18. Ancova Example Compared to regression output with both predictors in
Note how partial eta-squared is just the squared partial correlations from regression
The coefficient for the dummy-coded variable is the difference between marginal means in the Ancova

19. Ancova
So again, with Ancova we are simply controlling for (taking out, partialing) the effects due to the covariate and seeing if there are differences in our groups
The only difference between it and regular MR is the language used to describe the results (mean differences vs. coefficients etc.) and some different options for analysis