An analysis of covariance (ANCOVA) procedure is used when the statistical model has both quantitative and qualitative predictors, and is based on the concepts of the General Linear Model (GLM). In ANCOVA, we combine the concepts we have learned so far in this course (applicable to categorical factors) with the principles of regression (applicable to continuous predictors, learned in STAT 501).

In this lesson, we will address the classic case of ANCOVA where the ANOVA model is extended to include the linear effect of a continuous variable, known as the covariate. In the next lesson, we will generalize the ANCOVA model to include the quadratic and cubic effects of the covariate as well.

Fun Facts: When SAS first came out they had only PROC ANOVA and PROC REGRESSION. Then people asked, "What about the case when want to do an ANOVA but have another continuous variable that you suspect will account for extraneous variability in the response?" In response, SAS came out with PROC GLM which is the general linear model. With PROC GLM you could place the continuous regression variable in the ANOVA model and it would run. Or, if you were running a regression, you could include a categorical variable in the regression model and it would also run. The GLM can handle both the regression and the categorical variables in the same model. Note, there is no PROC ANCOVA in SAS, but there is PROC MIXED. PROC GLM had problems when it came to random effects and was effectively replaced by PROC MIXED. The same sort of process was seen in Minitab and accounts for the multiple tabs under Stat > ANOVA and Stat > Regression, and eventually, Stat > General Linear Model (which works for random effects as well). So, we now have the capacity to include covariates and correctly work with random effects via SAS PROC MIXED or Minitab Stat > General Linear Model. But, enough history, let us get to this lesson.

A ‘classic’ ANOVA tests for differences in mean responses to categorical factor (treatment) levels. When there is heterogeneity in experimental units, sometimes restrictions on the randomization (blocking) can improve the accuracy of significance testing results. In some situations, however, the opportunity to construct blocks may not exist, but there may exist a continuous variable causing the heterogeneity. Such sources of extraneous variability are referred to as ‘covariates’, and historically have also been termed ‘nuisance’ or ‘concomitant’ variables.

Note that an ANCOVA model is formed by including a continuous covariate in an ANOVA model. As the continuous covariate enters the model as a regression variable, an ANCOVA requires a few additional steps that should be combined with the ANOVA procedure.