Almost efficient estimation of relative risk regression

Relative risks (RRs) are often considered the preferred measures of association in prospective studies, especially when the binary outcome of interest is common. In particular, many researchers regard RRs to be more intuitively interpretable than odds ratios. Although RR regression is a special case of generalized linear models, specifically with a log link function for the binomial (or Bernoulli) outcome, the resulting log-binomial regression does not respect the natural parameter constraints. Because log-binomial regression does not ensure that predicted probabilities are mapped to the [0,1] range, maximum likelihood (ML) estimation is often subject to numerical instability that leads to convergence problems. To circumvent these problems, a number of alternative approaches for estimating RR regression parameters have been proposed. One approach that has been widely studied is the use of Poisson regression estimating equations. The estimating equations for Poisson regression yield consistent, albeit inefficient, estimators of the RR regression parameters. We consider the relative efficiency of the Poisson regression estimator and develop an alternative, almost efficient estimator for the RR regression parameters. The proposed method uses near-optimal weights based on a Maclaurin series (Taylor series expanded around zero) approximation to the true Bernoulli or binomial weight function. This yields an almost efficient estimator while avoiding convergence problems. We examine the asymptotic relative efficiency of the proposed estimator for an increase in the number of terms in the series. Using simulations, we demonstrate the potential for convergence problems with standard ML estimation of the log-binomial regression model and illustrate how this is overcome using the proposed estimator. We apply the proposed estimator to a study of predictors of pre-operative use of beta blockers among patients undergoing colorectal surgery after diagnosis of colon cancer.