probitordinal

Probitordinal refers to what is commonly called the ordinal probit model, a regression framework for ordered categorical outcomes. It assumes an underlying latent continuous propensity y* that relates to predictors x via y* = xβ + ε with ε ~ N(0,1). The observed ordinal response Y takes values 1,...,K according to thresholds τ_0 = -∞ < τ_1 < ... < τ_{K-1} < τ_K = ∞, with Y = j if τ_{j-1} < y* ≤ τ_j.

Parameters are estimated by maximum likelihood. The probability of category j is P(Y=j|x) = Φ(τ_j − xβ) − Φ(τ_{j−1} − xβ), where Φ is the standard normal CDF. The coefficients β indicate the direction of influence on the latent propensities; larger xβ increases the probability of higher categories. Interpretation often focuses on marginal effects, which vary with x.

Assumptions include the normality of the error term and the proportional-odds-like structure that a single set of coefficients applies across thresholds (though relaxed forms exist, such as generalized or partial-proportional-odds models). The model requires ordered categories and independent observations. Mis-specification can bias estimates and predictions.

Applications span social sciences, economics, and political science for survey data and ratings. Software implementations exist in R (e.g., MASS::polr with method='probit'), Stata (oprobit), and Python (statsmodels' OrderedModel with probit link).