## ABSTRACT

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, 12    , 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