General ANOVA/MANOVA and GLM - Covariates

The most common application of covariates is in between-groups designs, in cases when you have continuous variables that are likely to be correlated with the dependent variable of interest. Designs that include covariates can be analyzed in the GLM module.

For example, before beginning a special math training course, the IQ (Intelligence Quotient) of all participating students can be measured. Students are then randomly assigned to one of two courses, and their improvement in that course is measured as the major dependent variable of interest. In this study you might suspect that the large differences in intelligence among participating students contribute a lot of "random" variability to the dependent measure. In fact, this variability (due to differences in intelligence) may be so large that it will "mask" the differential effectiveness of the two training courses. In that case, you could specify the IQ measure as the covariate. If related to students' improvements in the math course, the covariate may significantly reduce the error variance.

Adjusted means

When the covariate is not only correlated with the dependent variable within each group in the design, but also correlated with the between-groups factors themselves, then you need to adjust the means before interpreting any effects. A covariate can be correlated with the between-groups factors if it is affected by them; for example, if in the above study you measured IQ after the math training courses, then it is conceivable that the covariate (IQ) could have been affected by the different courses. In those cases, you will see that the inclusion of the covariate not only affects the SS error term in the analysis but also SS effect terms for the between-groups factors. Also, the adjusted means (computed by GLM if requested via Covariate options on the GLM Results - Means tab) will be different from the raw observed means. When this happens, the inclusion of a covariate sometimes actually decreases the statistical significance of effects.

For more information, see:

See also, ANOVA/MANOVA Introductory Overview, General ANOVA/MANOVA and GLM - Notes, Methods for Analysis of Variance, General Linear Model (GLM), General Regression Models (GRM), Variance Components and Mixed Model ANOVA/ANCOVA, and Experimental Design (DOE); to analyze nonlinear models, see Generalized Linear/Nonlinear Model (GLZ).