Constraint approaches to the estimation of relative risk

In medical and epidemiologic studies, relative risk is usually the parameter of interest. However, calculating relative risk using standard log-Binomial regression approach often encounters non-convergence. A modified Poisson regression, which uses robust variance, was proposed by Zou in 2004. Although the modified Poisson regression with sandwich variance estimator is valid for the estimation of relative risk, the predicted probability of the outcome may be greater than the natural boundary 1 for the unobserved but plausible covariate combinations. Moreover, the lower and upper bounds of confidence intervals for predicted probabilities could fall out of (0, 1). Chu and Cole, in 2010, proposed a Bayesian approach to overcome this issue. Posterior median was used to get the parameter estimation. However, the Bayesian approach may provide biased estimation, especially when the probability of outcome is high. In this article, we propose an alternative constraint optimization approach for estimating relative risk. Our approach can reach similar or better performance than Bayesian approach in terms of bias, root mean square error, coverage rate, and predictive probabilities. Simulation studies are conducted to demonstrate the usefulness of this approach. Our method is also illustrated by Prospective Registry Evaluating Myocardial Infarction: Event and Recovery data.