Classification using class-imbalanced data is biased in favor of the majority class. The bias is even larger for high-dimensional data, where the number of variables greatly exceeds the number of samples. The problem can be attenuated by undersampling or oversampling, which produce class-balanced data. Generally undersampling is helpful, while random oversampling is not. Synthetic Minority Oversampling TEchnique (SMOTE) is a very popular oversampling method that was proposed to improve random oversampling but its behavior on high-dimensional data has not been thoroughly investigated. In this paper we investigate the properties of SMOTE from a theoretical and empirical point of view, using simulated and real high-dimensional data.

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.

The goal of class prediction studies is to develop rules to accurately predict the class membership of new samples. The rules are derived using the values of the variables available for each subject: the main characteristic of high-dimensional data is that the number of variables greatly exceeds the number of samples. Frequently the classifiers are developed using class-imbalanced data, i.e., data sets where the number of samples in each class is not equal. Standard classification methods used on class-imbalanced data often produce classifiers that do not accurately predict the minority class; the prediction is biased towards the majority class. In this paper we investigate if the high-dimensionality poses additional challenges when dealing with class-imbalanced prediction. We evaluate the performance of six types of classifiers on class-imbalanced data, using simulated data and a publicly available data set from a breast cancer gene-expression microarray study. We also investigate the effectiveness of some strategies that are available to overcome the effect of class imbalance.

Background. Information on the course and outcome of early European Lyme neuroborreliosis is limited. Methods. The study comprised 77 patients (38 males, 39 females; median age, 58 years) diagnosed with painful meningoradiculitis (Bannwarth syndrome) who were followed up for 1 year at a single center. Results. Duration of neurological symptoms before diagnosis was 30 (interquartile range, 14–50) days. The most frequent symptoms/signs were radicular pain (100%), sleep disturbances (75.3%), erythema migrans (59.7%), headache (46.8%), fatigue (44.2%), malaise (39%), paresthesias (32.5%), peripheral facial palsy (PFP) (36.4%), meningeal signs (19.5%), and pareses (7.8%). Cerebrospinal fluid (CSF) analysis revealed lymphocytic/monocytic pleocytosis, elevated protein concentration, and intrathecal synthesis of borrelial immunoglobulin M and immunoglobulin G antibody in 100%, 81.1%, 63%, and 88.7% of patients, respectively. Borreliae (predominantly Borrelia garinii ) were isolated from CSF, skin, and blood in 15.6%, 40.6%, and 2.7% of patients, respectively. The outcome after 14-day treatment with ceftriaxone was favorable in 87.8% of patients. Control CSF examination at 3 months showed decreased leukocyte counts in all patients; however, 23.3% still had pleocytosis (>10 × 10 ⁶ cells/L). A model based on pretreatment data and the findings at the end of 14-day antibiotic treatment accurately predicted which patients would have an unfavorable outcome 6 or 12 months after treatment. Conclusions. Our patients had fewer pretreatment neurological complications (PFP, pareses) than reported for Bannwarth syndrome decades ago, probably as the result of earlier recognition and prompt antibiotic treatment. Unfavorable outcome was rare and was predicted by the continued presence of symptoms 14 days after commencement of treatment.

Background: In high-dimensional data (HDD) settings, the number of variables associated with each observation is very large. Prominent examples of HDD in biomedical research include omics data with a large number of variables such as many measurements across the genome, proteome, or metabolome, as well as electronic health records data that have large numbers of variables recorded for each patient. The statistical analysis of such data requires knowledge and experience, sometimes of complex methods adapted to the respective research questions.Methods: Advances in statistical methodology and machine learning methods offer new opportunities for innovative analyses of HDD, but at the same time require a deeper understanding of some fundamental statistical concepts. Topic group TG9 “High-dimensional data” of the STRATOS (STRengthening Analytical Thinking for Observational Studies) initiative provides guidance for the analysis of observational studies, addressing particular statistical challenges and opportunities for the analysis of studies involving HDD. In this overview, we discuss key aspects of HDD analysis to provide a gentle introduction for non-statisticians and for classically trained statisticians with little experience specific to HDD.Results: The paper is organized with respect to subtopics that are most relevant for the analysis of HDD, in particular initial data analysis, exploratory data analysis, multiple testing, and prediction. For each subtopic, main analytical goals in HDD settings are outlined. For each of these goals, basic explanations for some commonly used analysis methods are provided. Situations are identified where traditional statistical methods cannot, or should not, be used in the HDD setting, or where adequate analytic tools are still lacking. Many key references are provided.Conclusions: This review aims to provide a solid statistical foundation for researchers, including statisticians and non-statisticians, who are new to research with HDD or simply want to better evaluate and understand the results of HDD analyses.