Let represent a response variable of interest that is to be subjected to treatments within a categorical explanatory variable. Suppose, further, that it is known (or at least suspected) that the response also depends linearly on continuous explanatory variables, 12 , also referred to as covariates . For = 1 2 and = 1 2 , denote the th observed response within the th treatment by , and denote the corresponding levels of the covariates by 1 2 . As with one-factor models, although desireable, treatment sample sizes, , need not be equal in size. Two equivalent models are presented. In the default setting, R uses treatment coding as for the one-factor case.
For this approach, the starting model has the form