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ANCOVA

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ANCOVA

- 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)

- 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

- 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

- 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

- HoR: Slopes between covariate and DV are similar across groups

- 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

- 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

- 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.

- 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)

- 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.

- 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)?

- Descriptive Statistics
- 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

- 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

- 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

PrePost

treatment2070

treatment1050

treatment6090

treatment2060

treatment1050

control5020

control1010

control4030

control2050

control1010

- Dummy code grouping variable
- Though technically it should probably already be coded as such anyway for convenience

- 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

Ancova output from GLM/Univariate in SPSS

Regression output

4842.105-642.623 =4199.482

- 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

- 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