Response to Marschner's letter to the editor.

We thank Dr Marschner for his insightful comments on our paper on relative risk (RR) regression (Fitzmaurice and others, 2014). In particular, his letter highlights an important distinction that may not have been transparent in our paper. Before discussing this issue, we must emphasize that the main motivation for developing the method proposed in our paper is the increasing use of Poisson regression estimating equations to circumvent convergence problems that commonly arise when fitting the log link binomial model. The estimating equations for Poisson regression yield consistent, but inefficient, estimators of the RR regression parameters. In our paper, we proposed using estimating equations with near optimal weights to yield almost efficient estimators while also avoiding convergence problems. In his letter, Dr Marschner emphasizes that the convergence problems that arise when fitting the log link binomial model are due to the algorithms or computation methods commonly used for finding the maximum likelihood estimator (MLE) rather than the maximum likelihood (ML) method of estimation per se. We are in complete agreement on this point. The computation method most widely used in standard commercial software for generalized linear models, e.g. SAS, Stata, and SPSS, relies on Newton– Raphson algorithms; convergence problems are common when fitting the log link binomial model using these algorithms despite the fact that the likelihood function has a unique finite maximum. The results of our simulation study reported in Table 2 highlight the magnitude of the problem for the Newton–Raphson algorithm implemented in PROC GENMOD in SAS; however, the simulation results cannot be extrapolated to other algorithms used for finding the MLE (including the more reliable algorithms discussed by Dr Marschner). For the most part, convergence failures of the fitting algorithms used in standard software have provided the impetus for the increasing use of less efficient estimators of the RR regression parameters (e.g. Poisson regression estimators) that avoid convergence problems. The quasi-likelihood method proposed in our paper and the reliable computation methods for finding the MLE discussed by Dr Marschner entail significant improvements to the fitting algorithms and yield efficient estimators of the RR regression parameters. In terms of asymptotic efficiency, we have demonstrated that our proposed method is a close competitor to the MLE. The additional simulation results presented by Dr Marschner indicate that the two methods are also very comparable in terms of finite sample bias. As a result, we find it difficult to advocate strongly for one method over the other; instead, we would recommend use of either approach for many applications of RR regression. In practice, the choice may depend in part on software preferences of the users