Retrospective-prospective symmetry in the likelihood and Bayesian analysis of case-control studies
本文证明病例对照研究中优势比的极大似然估计与逻辑回归结果一致，并揭示贝叶斯后验分布的前瞻与回顾模型等价性，为分层设计提供先验分布条件。
Prentice & Pyke (1979) established that the maximum likelihood estimate of an odds ratio in a case-control study is the same as would be found by fitting a logistic regression; in other words, for this specific target the incorrect prospective model is inferentially equivalent to the correct retrospective model. Similar results have been obtained for other models, and conditions have also been identified under which the corresponding Bayesian property holds, namely that the posterior distribution of the odds ratio is the same whether it is computed using the prospective or the retrospective likelihood. In this article we demonstrate how these results follow directly from certain parameter independence properties of the models and priors, and identify prior laws that support such reverse analysis, for both standard and stratified designs.