Risk ratio estimation with the logistic model

The log-binomial model (the generalized linear model with binomial errors and log link) makes it possible to directly estimate the relative risk from cohort follow-up data, or the prevalence ratio from cross-sectional data, with adjustment for confounders. One of the problems with the use of this model is that the iterative estimation algorithm may fail to converge. Schouten et al recognized this problem, and proposed a clever solution to it. Their approach involves defining a dichotomous outcome variable (D) coded as D=1 for occurrence and D=0 for non-occurrence, and augmenting the original data by replicating the observations on subjects with the outcome (D=1) but with the outcome variable coded as D=0 in the second instance. (In the language of a case control study, each case is included both as a case and as a control). Schouten et al show that that a logistic regression model fitted to the expanded data set has the same parameters as the log-binomial model. They derive a consistent "information sandwich" estimator of the covariance matrix of the estimated coefficients that, with some data manipulation, can be obtained from the output of the logistic regression. The problem is that while a solution for the parameter vector can be obtained from nearly any set of data, it is not guaranteed to be admissible for the log-binomial model. We use Stata to demonstrate the method of Schouten et al, including the calculations required to obtain standard error estimates, and describe the frequency of inadmissible solutions in simulated data.