Determining the prior mean in Bayesian logistic regression with sparse data: a nonarbitrary approach
针对稀疏数据下逻辑回归中最大似然估计的比值比偏倚问题，提出一种非任意、基于经验的方法来确定贝叶斯回归系数的先验均值，通过模拟和实际数据验证其能有效减小偏倚。
Abstract Logistic regression models lead to a bias of the odds ratio (OR) as estimated using the maximum-likelihood (ML) method, when there are few study participants at the binary outcome and factor levels. Although Bayesian methods are frequently applied to reduce this sparse data bias, the specification of the prior distribution for regression coefficients is the most controversial feature. We propose a nonarbitrary and empirical method to determine the prior mean for regression coefficients in Bayesian logistic regression analysis for sparse data. The proposed prior mean is calculated as the difference between the observed log OR and the quasi-expectation of log OR, and is interpreted as a shrinkage statistic of the ML estimate. Further, for easy and fast inference, the proposed method applies to Bayesian logistic regression with the normal prior and the log-F prior via data augmentation. Simulation results indicate that the OR bias based on the proposed method is consistently smaller than that based on the ML method. The OR bias estimated using the proposed method is generally smaller than that based on the mean prior of zero for the regression coefficient. We apply the proposed methods to 2 real data sets.