Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 13

Covariates are characteristics of the participants in an experiment. These are characteristics outside of the actual treatment. If a researcher wants to compare means of multiple groups using the additional predictors, the covariates, then the ANCOVA is used. Examples of covariates could be love for puppies, softness of puppy fur.

Covariates can be included in an ANOVA for two reasons:

1. Reduce within-group error variance
   The unexplained variance is attributed to other variables, the covariates, which reduces the total error variance. This allows for a more sensitive test for the difference of group means.
2. Elimination of confounds
   By adding other variables, covariates, in the analysis, confounds are eliminated.

If there are covariates, the b-values represent the differences between the means of each group and the control adjusted for the covariate.

ASSUMPTIONS AND ISSUES WITH ANCOVA
There are two new assumptions for ANCOVA that are not present with ANOVA. These assumptions are independence of the covariate and treatment effect and homogeneity of regression slopes.

The ideal case is that the covariate is independent from the treatment effect. If the covariate is not independent from the treatment effect, then the covariate will reduce the experimental effect because it explains some of the variance that would otherwise be applicable to the experiment. The ANCOVA does not control for or balance out the differences caused by the covariate. The problem of covariates potentially explaining a bit of the data and wanting to filter these confounds is using randomizing participants to experimental groups or matching experimental groups on a covariate.

Another assumption of the ANCOVA is that the relationship between covariate and outcome variable holds true for all groups of participants and not only for a few groups of participants (e.g. for both males and females and not only males). This assumption can be checked by checking the regression line for all the covariates and all the conditions. The lines should be similar.

In order to test the assumption of homogeneity of regression slopes, the ANCOVA model should be customized on SPSS to look at the independent variable x the covariate interaction.

CALCULATING THE EFFECT SIZE
The partial eta squared is the effect size which takes the covariates into account. It uses the proportion of variance that a variable explains that is not explained by other variables in the analysis. It uses the following formula: