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.

Separation is encountered in regression models with a discrete outcome (such as logistic regression) where the covariates perfectly predict the outcome. It is most frequent under the same conditions that lead to small-sample and sparse-data bias, such as presence of a rare outcome, rare exposures, highly correlated covariates, or covariates with strong effects. In theory separation will produce infinite estimates for some coefficients. In practice however separation may be unnoticed or mishandled because of software limits in recognizing and handling the problem, and notifying the user. We discuss causes of separation in logistic regression and describe how common software packages deal with it. We then describe methods that remove separation, focusing on the same penalized-likelihood techniques used to address more general sparse-data problems. These methods improve accuracy, avoid software problems, and allow interpretation as Bayesian analyses with weakly informative priors. We discuss likelihood penalties and their relative advantages and disadvantages, including some that can be implemented easily with any software package. We illustrate ideas and methods using a case-control study of contraceptive practices and urinary tract infection.

Diabetes and chronic kidney disease (CKD) are a growing burden for health-care systems. The prevalence of diabetes has increased constantly during the last decade, although a slight flattening of end-stage renal disease as a result of diabetes has been observed recently in some European countries. In this study, we project the prevalence of CKD in patients with diabetes in European countries up to the year 2025. We analysed the population with diabetes and development of nephropathy in 12 European countries, which we computed from models published previously and on data from the annual reports of the European Renal Association (1998–2011). The prevalence of CKD stage 5 in patients with diabetes up to the year 2025 was projected by the Lee–Carter algorithm. Those for stage 3 and 4 were then estimated by applying the same ratios of CKD prevalences as estimated in the Austrian population with diabetic nephropathy. The estimated prevalence of CKD in patients with diabetes is expected to increase in all 12 countries up to the year 2025. For CKD stage 3, we estimate for Austria in 2025 a prevalence of 215 000 per million diabetic population (p.m.p.) (95% confidence interval 169 000, 275 000), for CKD4 18 600 p.m.p. (14 500, 23 700) and for CKD5 6900 p.m.p. (5400, 8900). The median prevalence in the considered countries is 132 900 p.m.p. (IQR: 118 500, 195 800), 11 500 (10 200, 16 900) and 4300 (3800, 6300) for CKD stages 3, 4 and 5, respectively. Altogether, these data predict in the years 2012–25 an annual increase of 3.2% in the prevalence of diabetic CKD stage 5. Due to the increase in prevalence of diabetes and CKD5, the costs of renal therapy are expected to rise. We believe that these data may help health-care policy makers to make informed decisions.

In this randomized, placebo-controlled crossover trial, patients with radicular symptoms experienced an average 32% reduction in pain after 4 weeks of treatment with duloxetine. The overall adverse event rate was similar between placebo and duloxetine treatments. Background: Among patients with chronic low back pain (CLBP), approximately 37% show signs of a neuropathic pain component (radicular pain). Treatment of this condition remains challenging. Therefore, the current study aimed to investigate the efficacy of duloxetine in the treatment of CLBP patients with neuropathic leg pain. Methods: The study was conducted as a prospective, randomized, placebo-controlled, double-blind crossover trial. CLBP with a visual analog scale (VAS) score greater than 5 and a neuropathic component that was assessed clinically and by the painDETECT questionnaire (score > 12) were required for inclusion. Patients were randomly assigned to either duloxetine or placebo for 4 weeks followed by a 2-week washout period before they crossed over to the alternate phase that lasted another 4 weeks. Duloxetine was titrated up to 120 mg/day. The primary outcome parameter was mean VAS score during the last week of treatment in each phase (VAS_week4). Results: Of 41 patients, 21 patients completed both treatment phases. In the intention-to-treat analysis (n = 25), VAS_week4 was significantly lower in the duloxetine phase compared with placebo (4.1 ± 2.9 vs. 6.0 ± 2.7; P = 0.001), corresponding to an average pain reduction of 32%. The painDETECT score at the end of each treatment phase was significantly lower in the duloxetine phase compared with placebo (17.7 ± 5.7 vs. 21.3 ± 3.6 points; P = 0.0023). Adverse events were distributed equally between the duloxetine (65%) and placebo phases (62%) (P = 0.5). Conclusion: In this crossover study, duloxetine proved to be superior to placebo for the treatment of CLBP with a neuropathic leg pain.

Background: For finite samples with binary outcomes penalized logistic regression such as ridge logistic regression has the potential of achieving smaller mean squared errors (MSE) of coefficients and predictions than maximum likelihood estimation. There is evidence, however, that ridge logistic regression can result in highly variable calibration slopes in small or sparse data situations.Methods: In this paper, we elaborate this issue further by performing a comprehensive simulation study, investigating the performance of ridge logistic regression in terms of coefficients and predictions and comparing it to Firth’s correction that has been shown to perform well in low-dimensional settings. In addition to tuned ridge regression where the penalty strength is estimated from the data by minimizing some measure of the out-of-sample prediction error or information criterion, we also considered ridge regression with pre-specified degree of shrinkage. We included ‘oracle’ models in the simulation study in which the complexity parameter was chosen based on the true event probabilities (prediction oracle) or regression coefficients (explanation oracle) to demonstrate the capability of ridge regression if truth was known.Results: Performance of ridge regression strongly depends on the choice of complexity parameter. As shown in our simulation and illustrated by a data example, values optimized in small or sparse datasets are negatively correlated with optimal values and suffer from substantial variability which translates into large MSE of coefficients and large variability of calibration slopes. In contrast, in our simulations pre-specifying the degree of shrinkage prior to fitting led to accurate coefficients and predictions even in non-ideal settings such as encountered in the context of rare outcomes or sparse predictors.Conclusions: Applying tuned ridge regression in small or sparse datasets is problematic as it results in unstable coefficients and predictions. In contrast, determining the degree of shrinkage according to some meaningful prior assumptions about true effects has the potential to reduce bias and stabilize the estimates.