Schwarz’s criterion, also known as the Bayesian Information Criterion or BIC, is commonly used for model selection in logistic regression due to its simple intuitive formula. For tests of nested hypotheses in independent and identically distributed data as well as in Normal linear regression, previous results have motivated use of Schwarz’s criterion by its consistent approximation to the Bayes factor (BF), defined as the ratio of posterior to prior model odds. Furthermore, under construction of an intuitive unit-information prior for the parameters of interest to test for inclusion in the nested models, previous results have shown that Schwarz’s criterion approximates the BF to higher order in the neighborhood of the simpler nested model. This paper extends these results to univariate and multivariate logistic regression, providing approximations to the BF for arbitrary prior distributions and definitions of the unit-information prior corresponding to Schwarz’s approximation. Simulations show accuracies of the approximations for small samples sizes as well as comparisons to conclusions from frequentist testing. We present an application in prostate cancer, the motivating setting for our work, which illustrates the approximation for large data sets in a practical example.

中文翻译：

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单变量和多变量逻辑回归模型的贝叶斯因子的贝叶斯信息准则逼近

施瓦茨准则，也称为贝叶斯信息准则或 BIC，由于其简单直观的公式，常用于逻辑回归中的模型选择。对于独立和同分布数据以及正态线性回归中的嵌套假设的检验，先前的结果通过其与贝叶斯因子 (BF) 的一致近似来推动使用施瓦茨标准，贝叶斯因子 (BF) 定义为后验与先验模型的比率。此外，在为感兴趣的参数构建直观的单元信息先验以测试是否包含在嵌套模型中的情况下，先前的结果表明，Schwarz 的准则在更简单的嵌套模型附近将 BF 逼近到更高阶。本文将这些结果扩展到单变量和多变量逻辑回归，为任意先验分布和对应于施瓦茨近似的单位信息先验定义提供BF的近似值。模拟显示了小样本量的近似值的准确性以及与常客测试结论的比较。我们提出了一个在前列腺癌中的应用，这是我们工作的动机，它在一个实际示例中说明了大数据集的近似值。