Tests of significance using regression models for ordered categorical data

Regression models of the type proposed by McCullagh (1980, Journal of the Royal Statistical Society, Series B 42, 109-142) are a general and powerful method of analyzing ordered categorical responses, assuming categorization of an (unknown) continuous response of a specified distribution type. Tests of significance with these models are generally based on likelihood-ratio statistics that have asymptotic chi 2 distributions; therefore, investigators with small data sets may be concerned with the small-sample behavior of these tests. In a Monte Carlo sampling study, significance tests based on the ordinal model are found to be powerful, but a modified test procedure (using an F distribution with a finite number of degrees of freedom for the denominator) is suggested such that the empirical significance level agrees more closely with the nominal significance level in small-sample situations. We also discuss the parallels between an ordinal regression model assuming underlying normality and conventional multiple regression. We illustrate the model with two data sets: one from a study investigating the relationship between phosphorus in soil and plant-available phosphorus in corn grown in that soil, and the other from a clinical trial comparing analgesic drugs.