Firth's logistic regression has become a standard approach for the analysis of binary outcomes with small samples. Whereas it reduces the bias in maximum likelihood estimates of coefficients, bias towards one-half is introduced in the predicted probabilities. The stronger the imbalance of the outcome, the more severe is the bias in the predicted probabilities. We propose two simple modifications of Firth's logistic regression resulting in unbiased predicted probabilities. The first corrects the predicted probabilities by a post hoc adjustment of the intercept. The other is based on an alternative formulation of Firth's penalization as an iterative data augmentation procedure. Our suggested modification consists in introducing an indicator variable that distinguishes between original and pseudo-observations in the augmented data. In a comprehensive simulation study, these approaches are compared with other attempts to improve predictions based on Firth's penalization and to other published penalization strategies intended for routine use. For instance, we consider a recently suggested compromise between maximum likelihood and Firth's logistic regression. Simulation results are scrutinized with regard to prediction and effect estimation. We find that both our suggested methods do not only give unbiased predicted probabilities but also improve the accuracy conditional on explanatory variables compared with Firth's penalization. While one method results in effect estimates identical to those of Firth's penalization, the other introduces some bias, but this is compensated by a decrease in the mean squared error. Finally, all methods considered are illustrated and compared for a study on arterial closure devices in minimally invasive cardiac surgery. Copyright © 2017 John Wiley & Sons, Ltd.

Percutaneous groin cannulation using ACDs for establishing cardiopulmonary bypass in minimally invasive valve surgery significantly reduces groin complications, operation time and hospital stay. However, the remaining complications are mainly of vascular nature versus wound infection and lymph fistulae with cutdown.

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Background The performance of models for binary outcomes can be described by measures such as the concordance statistic (c-statistic, area under the curve), the discrimination slope, or the Brier score. At internal validation, data resampling techniques, e.g., cross-validation, are frequently employed to correct for optimism in these model performance criteria. Especially with small samples or rare events, leave-one-out cross-validation is a popular choice. Methods Using simulations and a real data example, we compared the effect of different resampling techniques on the estimation of c-statistics, discrimination slopes, and Brier scores for three estimators of logistic regression models, including the maximum likelihood and two maximum penalized likelihood estimators. Results Our simulation study confirms earlier studies reporting that leave-one-out cross-validated c-statistics can be strongly biased towards zero. In addition, our study reveals that this bias is even more pronounced for model estimators shrinking estimated probabilities towards the observed event fraction, such as ridge regression. Leave-one-out cross-validation also provided pessimistic estimates of the discrimination slope but nearly unbiased estimates of the Brier score. Conclusions We recommend to use leave-pair-out cross-validation, fivefold cross-validation with repetitions, the enhanced or the .632+ bootstrap to estimate c-statistics, and leave-pair-out or fivefold cross-validation to estimate discrimination slopes.