Using the jackknife for estimation in log link Bernoulli regression models

Bernoulli (or binomial) regression using a generalized linear model with a log link function, where the exponentiated regression parameters have interpretation as relative risks, is often more appropriate than logistic regression for prospective studies with common outcomes. In particular, many researchers regard relative risks to be more intuitively interpretable than odds ratios. However, for the log link, when the outcome is very prevalent, the likelihood may not have a unique maximum. To circumvent this problem, a 'COPY method' has been proposed, which is equivalent to creating for each subject an additional observation with the same covariates except the response variable has the outcome values interchanged (1's changed to 0's and 0's changed to 1's). The original response is given weight close to 1, while the new observation is given a positive weight close to 0; this approach always leads to convergence of the maximum likelihood algorithm, except for problems with convergence due to multicollinearity among covariates. Even though this method produces a unique maximum, when the outcome is very prevalent, and/or the sample size is relatively small, the COPY method can yield biased estimates. Here, we propose using the jackknife as a bias-reduction approach for the COPY method. The proposed method is motivated by a study of patients undergoing colorectal cancer surgery.