Estimating relative risks for longitudinal and clustered binary data

二元回归模型常用于观察到二响应变量和协变量向量的情况。经过适当的变换后，假设正响应的概率与协变量线性相关。转换，或链接函数，将概率连接到线性预测器。具有logit链接的逻辑回归特别受欢迎，因为回归参数可以解释为对数比值比（OR），并且OR的估计无论回顾性或前瞻性抽样设计都是有效的。关于在流行病学研究中使用OR作为暴露效应度量的适当性，文献中有许多讨论和兴趣。在病例对照研究中，由于病例组和对照组的抽样比例不同，逻辑模型更可取。然而，相对危险度（RR）比OR更易解释，尤其是在前瞻性队列研究中。当结果罕见时，逻辑模型的OR估计是RR的良好近似值，但当结果常见时，它可能会大大高估RR。Wacholder提出使用二项误差的对数-二项模型（相对风险回归）和log link函数来估计RR。对数二项模型的一个主要问题是计算不收敛。为了解决这个问题，人们提出了许多估计RR的替代方法。

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.