Localized Logistic Regression for Categorical Influential Factors

Abstract

In localized logistic regression (cp. Loader, Local regression and likelihood, Springer, New York, 1999; Tutz and Binder, Statistics and Computing 15:155–166, 2005) at each target point where a prediction is required a logistic regression model is fitted locally. This is achieved by weighting the training observations in the log-likelihood based on their distances to the target observation. For interval-scaled influential factors these weights usually depend on Euclidean distances. This paper aims to combine localized logistic regression with dissimilarity measures more suitable for categorical data.

Categorical predictors are usually included into regression models by constructing design variables. Therefore, in principle distance measures can be defined based either on the original variables or on the design variables. In the first case matching coefficients, e.g., the simple or flexible matching coefficients, can be applied. In the second case Euclidean distances are suitable, too, since design variables can be considered interval-scaled.

Localized logistic regression with the proposed dissimilarity measures is applied to a SNP data set from the GENICA breast cancer study (cp. Justenhoven et al., Cancer Epidemiology Biomarkers and Prevention 13:2059–2064, 2004) in order to identify combinations of SNP variables that can be used to discriminate between cases and controls. By means of localized logistic regression one of the lowest error rates in combination with a maximal reduction of the number of predictors is achieved.