Discrete regression of outcomes with a discrete number of possible numeric values is addressed, as an alternative to multinomial and ordinal regression. The outcome values can represent categories but should be truly numeric, as for ordinal categories, and not nominal numeric codes for nonnumeric categories. The outcome values can also be actual numbers, as for pain level ratings. Formulations are provided for estimating correlation parameters for the directly specified exchangeable (EC), spatial autoregressive order 1 (AR1), and unstructured (UN) cases. Formulations are also provided for estimation for singleton univariate discrete outcomes as needed to generate initial estimates for parameter estimation of models for correlated sets of univariate discrete outcomes. Distributions for discrete outcomes are modeled using multinomial, ordinal, and censored Poisson probabilities. Non-constant dispersions are addressed using both extended and direct variance modeling. Discrete regression models allowing for covariance structures based on random effects/coefficients are considered as well.