Estimating relative risks for longitudinal and clustered binary data

二元回归模型常用于观察到一个二分响应变量和一个协变量向量的情况。假设在适当的变换后，肯定响应的概率与协变量线性相关。变换或链接函数将概率与线性预测器联系起来。Logistic回归特别受欢迎，因为回归参数可以解释为对数优势比(OR)，而且无论是回溯性还是前瞻性抽样设计，OR的估计都是有效的。在流行病学研究中，关于使用OR作为暴露效应的衡量标准的适当性，文献中有很多讨论和兴趣。在病例对照研究中，由于病例组和对照组的抽样比例不同，Logistic模型更可取。然而，相对风险(RR)比OR更容易解释，特别是在前瞻性队列研究中。当结果罕见时，Logistic模型的OR估计是RR的一个很好的近似值，但当结果常见时，它可能会大大高估RR。Wachold建议使用具有二项误差和对数连接函数的对数二项模型(相对风险回归)来估计RR。对数二项模型的一个主要问题是在计算中不能收敛。为了解决这个问题，人们提出了许多估算RR的替代方法。 到目前为止，对数二项模型仅适用于横断面研究的数据。近年来，纵向研究越来越多地应用于公共卫生、医学和社会科学。纵向研究收集很长一段时间的数据。在两个或更多不同的时间段内对每个变量进行测量。纵向研究可以将队列效应从治疗效应中分离出来，从而也允许研究人员测量变量随时间的变化。对于纵向数据，已经发展了两种主要的回归方法。一种是边际回归模型(MRM)，另一种是广义线性混合模型(GLIMMIX)。纵向二元反应数据的Logistic回归模型已被广泛讨论并在不同的背景下使用。使用对数二项模型估计相对风险的主要障碍是包含对数链接函数。 所提出的方法将提供一种解决方案，以估计纵向和集群二元反应的相对风险。这项技术有望在许多流行病学和临床研究中找到应用，以评估治疗和暴露的相对风险。

Binary regression model is often used in situation where a dichotomous response variable and a vector of covariates are observed. The probability of a positive response, after suitable transformation, is assume to be linearly related to the covariates. The transformation, or link function, connect the probability to the linear predictors. The logistic regression with logit link is particularly popular because the regression parameters can be interpreted as log odds ratios (OR) and because the estimates of OR remain valid regardless retrospective or prospective sampling designs. There have been much discussions and interests in the literature concerning the appropriateness of using OR as the measure of exposure effect in epidemiological research. In case-control studies, the logistic model are preferable because of different sampling fractions in case and control groups. However, relative risk (RR) is more interpretable than OR, especially in prospective cohort studies. When the outcome is rare, the OR estimate from the logistic model is a good approximation to the RR, but it may substantially over-estimate the RR when outcome is common. Wacholder proposed to use a log-binomial model (relative risk regression) with binomial error and log link function to estimate the RR. One major problem of the log-binomial model is failure of convergence in computation. To solve this problem, many alternative methods of estimating RR have been proposed. So far, the log-binomial model is only proposed to data from cross-sectional studies. In recent years, longitudinal studies are increasingly used in public health, medicine and social sciences. A longitudinal study collects data over long periods of time. Measurements are taken on each variable over two or more distinct time periods. The longitudinal studies can separate the cohort effect from the treatment effect and, thus, also allow the researchers to measure change in variables over time. Two predominant regression approaches have been developed for longitudinal data. One is the marginal regression model (MRM) and the other is the generalized linear mixed (GLIMMIX) model. Logistic regression models for longitudinal binary response data have been widely discussed and used in various context. The major obstacle of estimating relative risks using log-binomial model is the inclusion of log link function. The proposed method will provide a solution to estimating the relative risks for longitudinal and clustered binary responses. This technique is expected to find applications in many epideimiological and clinical studies to assess the relative risks of treatments and exposures.