Risk prediction models for discrete ordinal outcomes: calibration and the impact of the proportional odds assumption

TL;DR

This study evaluates calibration measures for ordinal outcome risk models, examines the impact of the proportional odds assumption on calibration and overfitting, and recommends multinomial logistic regression for better risk prediction accuracy.

Contribution

It provides a comprehensive comparison of calibration methods and assesses the effects of the proportional odds assumption on model performance and overfitting in ordinal risk prediction models.

Findings

Proportional odds models often have poor calibration when the true model is multinomial logistic.

Non-proportional odds models tend to overfit due to more parameters.

Multinomial logistic regression generally offers better risk estimates for ordinal outcomes.

Abstract

Calibration is a vital aspect of the performance of risk prediction models, but research in the context of ordinal outcomes is scarce. This study compared calibration measures for risk models predicting a discrete ordinal outcome, and investigated the impact of the proportional odds assumption on calibration and overfitting. We studied the multinomial, cumulative, adjacent category, continuation ratio, and stereotype logit/logistic models. To assess calibration, we investigated calibration intercepts and slopes, calibration plots, and the estimated calibration index. Using large sample simulations, we studied the performance of models for risk estimation under various conditions, assuming that the true model has either a multinomial logistic form or a cumulative logit proportional odds form. Small sample simulations were used to compare the tendency for overfitting between models. As a case study, we developed models to diagnose the degree of coronary artery disease (five categories) in symptomatic patients. When the true model was multinomial logistic, proportional odds models often yielded poor risk estimates, with calibration slopes deviating considerably from unity even on large model development datasets. The stereotype logistic model improved the calibration slope, but still provided biased risk estimates for individual patients. When the true model had a cumulative logit proportional odds form, multinomial logistic regression provided biased risk estimates, although these biases were modest. Non-proportional odds models require more parameters to be estimated from the data, and hence suffered more from overfitting. Despite larger sample size requirements, we generally recommend multinomial logistic regression for risk prediction modeling of discrete ordinal outcomes.