gms | German Medical Science

Introduction: Controlled, randomized, blinded trials are the gold standard in applied clinical sciences. However, blinding and randomization are not always possible or ethically feasible for quite a number of clinically relevant problems. Alternatively, there are observational studies that are prospective or retrospective in nature. In these cases, structural balanced is not guaranteed. For example, patients in the treatment arm could be older or more frequently diagnosed with diabetes compared to the control group. While presentation [1] was addressing the principle of adjusted relative risk estimation mainly, we now focus on numerical and computational issues for maximum likelihood estimation (MLE) in the framework of log-binomial regression described in [2] with respective implementation in R Cran [3].

Methods: For the case of retrospective studies, logistic regression is a standard approach for calculating adjusted odds ratios. For prospective studies with structural imbalances, the log-binomial regression model described in [2] allows for relative risk adjustment in the context of generalized linear models, theoretically. Compared to classical logistic regression, only one issue is modified: Instead of using the canonical link function of binomial distributions, the pure exponential function is used as response function considering link between the linear predictor and the binary outcome. This, however, may result in algorithmic MLE failure using standard software like IBM-SPSS or SAS [4]. Thus, we discuss two numerical sources of failed convergence and present new approaches for solutions: 1) Introduction of constrained maximum likelihood estimation (CMLE) for this particular modelling framework and 2) automatic calculation of a semioptimum and feasible starting vector for the nonlinear programming problem involved in CMLE.

Results: Two necessary improvements concerning CMLE in log-binomial regression established since the publication of Williamson et al. [4] are shown. Particularly, we address efficient specification of parameter constraints in order to ensure a range of zero to one for estimated probabilities. This holds true even for the case of very low or high event rates in the outcome variable. Furthermore, equations for constraints are transformed to end up with linear inequalities for ease of computation. A further aspect is the derivation of an automatically calculated feasible and semioptimal starting vector for this linear-constrained MLE problem. The resulting CMLE approach is implemented and solved efficiently by the BSW algorithm freely available [3].

Discussion: We show results of BSW applied to data analysis problems published and investigated with [3]. Moreover, ample simulations demonstrate the statistical behavior for a wide range of data scenarios. Numerical properties and convergence results can be derived from Kredler [5]. In particular, the local quadratic convergence property sticks out compared to EM-type approaches also presented for CMLE in log-binomial regression. Aspects of model specification have to be considered equally to other regression approaches.

Conclusion: Summarizing, we see that BSW solves the non-convergence problem for all the examples discussed in [4] and beyond. In this way, log-binomial regression with BSW is now a safe, globally and (locally) quadratic convergent as well as stable way for adjusted relative risk estimation in the general context of prospective epidemiological studies.

The authors declare that they have no competing interests.

The authors declare that an ethics committee vote is not required.