Efficiency of reduced logistic regression models

One feature of the usual polychotomous logistic regression model for categorical outcomes is that a covariate must be included in all the regression equations. If a covariate is not important in all of them, the procedure will estimate unnecessary parameters. More flexible approaches allow different subsets of covariates in different regressions. One alternative uses individualized regressions which express the polychotomous model as a series of dichotomous models. Another uses a model in which a reduced set of parameters is simultaneously estimated for all the regressions. Large‐sample efficiencies of these procedures were compared in a variety of circumstances in which there was a common baseline category for the outcome and the covariates were normally distributed. For a correctly specified model, the reduced estimates were over 100% efficient for nonzero slope parameters and up to 500% efficient when the baseline frequency and the effect of interest were small. The individualized estimates could have efficiencies less than 50% when the effect of interest was large, but were also up to 130% efficient when the baseline frequency was large and the effect of interest was small. Efficiency was usually enhanced by correlation among the covariates. For an underspecified reduced model, asymptotic bias in the reduced estimates was approximately proportional to the magnitude of the omitted parameter and to the reciprocal of the baseline frequency.