When discriminant analysis is used in practice for assessing the usefulness of diagnostic markers, the lack of control over covariates motivates the need for their adjustment in the analysis. This necessity for adjustment arises especially when the researcher's aim is classification based on a set of diagnostic markers and is not based on a set of covariates for which there exists known heterogeneity among the subjects with respect to the groups under consideration. The traditional covariance-adjusted approach is restrictive for such applications in that they assume linear covariates and a normal distribution for the the feature vector. Further, there is no available method for variable selection in using such covariance-adjusted models. In this paper, we generalize the traditional covariance-adjusted model to a general normal and logistic model, where these generalized models not only relax the distributional assumptions on the feature vector but also allow for nonlinear covariates. Exact and asymptotic tests are also derived for the problem of variable selection for these new models. The methodology is illustrated with both simulated data and an actual data set from a psychiatric study on using the Social Rhythm Metric for patients with anxiety disorders.