Bayesian inference for the stereotype regression model: Application to a case-control study of prostate cancer

Abstract

The stereotype regression model for categorical outcomes, proposed by Anderson (J. Roy. Statist. Soc. B. 1984; 46:1-30) is nested between the baseline-category logits and adjacent category logits model with proportional odds structure. The stereotype model is more parsimonious than the ordinary baseline-category (or multinomial logistic) model due to a product representation of the log-odds-ratios in terms of a common parameter corresponding to each predictor and category-specific scores. The model could be used for both ordered and unordered outcomes. For ordered outcomes, the stereotype model allows more flexibility than the popular proportional odds model in capturing highly subjective ordinal scaling which does not result from categorization of a single latent variable, but are inherently multi-dimensional in nature. As pointed out by Greenland (Statist. Med. 1994; 13:1665-1677), an additional advantage of the stereotype model is that it provides unbiased and valid inference under outcome-stratified sampling as in case-control studies. In addition, for matched case-control studies, the stereotype model is amenable to classical conditional likelihood principle, whereas there is no reduction due to sufficiency under the proportional odds model. In spite of these attractive features, the model has been applied less, as there are issues with maximum likelihood estimation and likelihood-based testing approaches due to non-linearity and lack of identifiability of the parameters. We present comprehensive Bayesian inference and model comparison procedure for this class of models as an alternative to the classical frequentist approach. We illustrate our methodology by analyzing data from The Flint Men's Health Study, a case-control study of prostate cancer in African-American men aged 40-79 years. We use clinical staging of prostate cancer in terms of Tumors, Nodes and Metastasis as the categorical response of interest.

Figure 1

Posterior density estimates for covariate and category specific parameters of the stereotype model for unmatched FMHS data with numerical summaries as presented in Table 1. The results are based on 10,000 observations generated from the posterior distribution of each parameter. The solid line corresponds to the unordered model, whereas the dashed line corresponds to the ordered model.

Figure 2

Posterior density estimates for covariate and category specific parameters of the stereotype model for 1:3 matched FMHS data with numerical summaries as presented in Table 2. The results are based on 10,000 observations generated from the posterior distribution of each parameter. The solid line corresponds to the unordered model, whereas the dashed line corresponds to the ordered model.