The problem of estimating parameters and testing hypotheses pertaining to categorical data is well known in statistical analysis. Much of the literature on the subject specifies and fits linear models to multinomial data using methods such as weighted least squares. This article describes maximum-likelihood estimation and likelihood ratio tests for ordered categorical response variates with either discrete or continuous underlying probability distributions. Emphasis is on fitting and making inferences about parameters of mixture distributions, especially mixtures of normal distributions. Goodness-of-fit tests are given to check the adequacy of the fitted distributional models.