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.
