Schörgendorfer Angela, Branscum Adam J, Hanson Timothy E
IBM T.J. Watson Research Center, 1101 Kitchawan Road, Yorktown Heights, NY 10598, USA.
Biometrics. 2013 Jun;69(2):508-19. doi: 10.1111/biom.12007. Epub 2013 Mar 14.
Logistic regression is a popular tool for risk analysis in medical and population health science. With continuous response data, it is common to create a dichotomous outcome for logistic regression analysis by specifying a threshold for positivity. Fitting a linear regression to the nondichotomized response variable assuming a logistic sampling model for the data has been empirically shown to yield more efficient estimates of odds ratios than ordinary logistic regression of the dichotomized endpoint. We illustrate that risk inference is not robust to departures from the parametric logistic distribution. Moreover, the model assumption of proportional odds is generally not satisfied when the condition of a logistic distribution for the data is violated, leading to biased inference from a parametric logistic analysis. We develop novel Bayesian semiparametric methodology for testing goodness of fit of parametric logistic regression with continuous measurement data. The testing procedures hold for any cutoff threshold and our approach simultaneously provides the ability to perform semiparametric risk estimation. Bayes factors are calculated using the Savage-Dickey ratio for testing the null hypothesis of logistic regression versus a semiparametric generalization. We propose a fully Bayesian and a computationally efficient empirical Bayesian approach to testing, and we present methods for semiparametric estimation of risks, relative risks, and odds ratios when parametric logistic regression fails. Theoretical results establish the consistency of the empirical Bayes test. Results from simulated data show that the proposed approach provides accurate inference irrespective of whether parametric assumptions hold or not. Evaluation of risk factors for obesity shows that different inferences are derived from an analysis of a real data set when deviations from a logistic distribution are permissible in a flexible semiparametric framework.
逻辑回归是医学和人口健康科学中风险分析的常用工具。对于连续响应数据，通过指定阳性阈值来创建用于逻辑回归分析的二分结果是很常见的。经验表明，假设数据的逻辑抽样模型，对未二分的响应变量拟合线性回归，比分对二分终点进行普通逻辑回归能产生更有效的优势比估计。我们表明，风险推断对于偏离参数逻辑分布并不稳健。此外，当数据的逻辑分布条件被违反时，比例优势的模型假设通常不满足，从而导致参数逻辑分析的推断有偏差。我们开发了新颖的贝叶斯半参数方法，用于检验具有连续测量数据的参数逻辑回归的拟合优度。检验程序适用于任何截断阈值，并且我们的方法同时提供了进行半参数风险估计的能力。使用萨维奇 - 迪基比率计算贝叶斯因子，以检验逻辑回归的零假设与半参数推广。我们提出了一种完全贝叶斯和一种计算效率高的经验贝叶斯检验方法，并且我们提出了在参数逻辑回归失败时对半参数风险、相对风险和优势比进行估计的方法。理论结果确立了经验贝叶斯检验的一致性。模拟数据的结果表明，无论参数假设是否成立，所提出的方法都能提供准确的推断。对肥胖风险因素的评估表明，在灵活的半参数框架中允许偏离逻辑分布时，对真实数据集的分析会得出不同的推断。
