Using the jackknife for estimation in log link Bernoulli regression models

Abstract

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.

Estimating the Relative Excess Risk Due to Interaction in Clustered-Data Settings

“…As such, it was important to estimate the RERI using relative risks rather than odds ratios. While there has been much literature dedicated to estimating adjusted relative risks for binary data, it has focused on the independentdata setting and not the context of estimating interaction effects (52,53,(61)(62)(63)(64)(65)(66)(67)(68)(69).…”

Relative Risk Regression for Binary Outcomes: Methods and Recommendations

“…) and the estimates may be biased (Lipsitz et al . ). Furthermore, as an approximate method of computing the MLE, the COPY method has been superseded by the reliable exact methods for computing the MLE discussed above.…”

“…; Lipsitz et al . ). Conditioning on the convergence of an unreliable computational method has the effect of distorting the sampling properties of the MLE and hence biasing comparisons with other estimation methods.…”

“…A drawback of a number of previous simulation studies of the log-binomial MLE is that they have used unreliable computational methods to obtain the MLE, for instance PROC GENMOD in SAS, which exhibits a high incidence of non-convergence (Yelland et al 2011;Fitzmaurice et al 2014;Lipsitz et al 2015). Conditioning on the convergence of an unreliable computational method has the effect of distorting the sampling properties of the MLE and hence biasing comparisons with other estimation methods.…”