Procedures are discussed for the analysis of ordered categorical responses which are analogous to linear model procedures for a continuous response variable. The approach taken is based on the assumption that the observed categorical responses are determined by an underlying continuous variable, that satisfies a linear model. Previously, estimation procedures have been developed only for the case where the model for the assumed underlying variable is a fixed effects linear model;This approach is extended to the case where the linear model for the underlying continuous variable includes random as well as fixed effects. It will generally not be possible to evaluate the likelihood function. However, the EM algorithm is employed to develop procedures for computing maximum likelihood estimates for the fixed effects and predictions for the random effects. Estimation of variance ratios is also discussed. For the fixed effects case, conditions are given which ensure the asymptotic normality of the maximum likelihood estimator, and confidence intervals are derived for the probability for a response being in a particular category.