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When a risk factor affects certain categories of a multinomial outcome but not others, outcome heterogeneity is said to be present. A standard epidemiologic approach for modeling risk factors of a categorical outcome typically entails fitting a polytomous logistic regression via maximum likelihood estimation. In this paper, we show that standard polytomous regression is ill equipped to detect outcome heterogeneity and will generally understate the degree to which such heterogeneity may be present. Specifically, nonsaturated polytomous regression will often a priori rule out the possibility of outcome heterogeneity from its parameter space. As a remedy, we propose to model each category of the outcome as a separate binary regression. For full efficiency, we propose to estimate the collection of regression parameters jointly using a constrained Bayesian approach that ensures that one remains within the multinomial model. The approach is straightforward to implement in standard software for Bayesian estimation.

A Multinomial Regression Approach to Model Outcome Heterogeneity