Bias Correction for the Proportional Odds Logistic Regression Model with Application to a Study of Surgical Complications

The proportional odds logistic regression model is widely used for relating an ordinal outcome to a set of covariates. When the number of outcome categories is relatively large, the sample size is relatively small, and/or certain outcome categories are rare, maximum likelihood can yield biased estimates of the regression parameters. Firth (1993) and Kosmidis and Firth (2009) proposed a procedure to remove the leading term in the asymptotic bias of the maximum likelihood estimator. Their approach is most easily implemented for univariate outcomes. In this paper, we derive a bias correction that exploits the proportionality between Poisson and multinomial likelihoods for multinomial regression models. Specifically, we describe a bias correction for the proportional odds logistic regression model, based on the likelihood from a collection of independent Poisson random variables whose means are constrained to sum to 1, that is straightforward to implement. The proposed method is motivated by a study of predictors of post-operative complications in patients undergoing colon or rectal surgery (Gawande et al., 2007).