Bias‐reduced and separation‐proof conditional logistic regression with small or sparse data sets

Abstract

Conditional logistic regression is used for the analysis of binary outcomes when subjects are stratified into several subsets, e.g. matched pairs or blocks. Log odds ratio estimates are usually found by maximizing the conditional likelihood. This approach eliminates all strata-specific parameters by conditioning on the number of events within each stratum. However, in the analyses of both an animal experiment and a lung cancer case-control study, conditional maximum likelihood (CML) resulted in infinite odds ratio estimates and monotone likelihood. Estimation can be improved by using Cytel Inc.'s well-known LogXact software, which provides a median unbiased estimate and exact or mid-p confidence intervals. Here, we suggest and outline point and interval estimation based on maximization of a penalized conditional likelihood in the spirit of Firth's (Biometrika 1993; 80:27-38) bias correction method (CFL). We present comparative analyses of both studies, demonstrating some advantages of CFL over competitors. We report on a small-sample simulation study where CFL log odds ratio estimates were almost unbiased, whereas LogXact estimates showed some bias and CML estimates exhibited serious bias. Confidence intervals and tests based on the penalized conditional likelihood had close-to-nominal coverage rates and yielded highest power among all methods compared, respectively. Therefore, we propose CFL as an attractive solution to the stratified analysis of binary data, irrespective of the occurrence of monotone likelihood. A SAS program implementing CFL is available at: http://www.muw.ac.at/msi/biometrie/programs.

Soil Ingestion is Associated with Child Diarrhea in an Urban Slum of Nairobi, Kenya

“…Penalized maximum likelihood logistic regression was used because this method addresses the problem of small sample size/sparse-data bias that can overestimate ORs for small sample sizes when using ordinary multivariable logistic regression. 20,21 The following potential confounding variables were assessed for inclusion in the analysis: child's age, asset ownership index (to represent household income), number of household members, mother's educational attainment, drinking water contamination at time of visit, and whether the feces of all children under 5 years of age in the household are disposed of in a toilet or latrine. The drinking water contamination variable is a binary variable for whether any E. coli contamination was detected in a 100 mL sample of drinking water collected at the time of the household visit.…”

Separation Issues and Possible Solutions: Part I – Systematic Literature Review on Logistic Models ‐ Part II – Comparison of different methods for separation in logistic regression

“…The bias is removed by calculation of the posterior mode based on this prior. Heinze and Puhr (2010) proposed the modification of Firth method based on conditional likelihood:…”

The Impact of Predictor Variable(s) with Skewed Cell Probabilities on Wald Tests in Binary Logistic Regression

“…Refining these earlier findings, Albert and Anderson (1984) identified three types of data configurations that may affect estimation: complete separation, quasi-complete separation, and overlap. They mathematically proved that although overlap yields a finite and unique solution, MLEs do not exist for the other two data patterns, although it was left to future researchers to develop new techniques to overcome this obstacle (e.g., Barreto, Russo, Brasil, & Simon, 2014;Gordóvil-Merino, Guàrdia-Olmos, & Peró-Cebollero, 2012;Heinze & Puhr, 2010;Mîndrilã, 2010;Rousseeuw & Christmann, 2003).…”