Semiparametric kernel logistic regression with longitudinal data

Abstract

Abstract Logistic regression is a well known binary classi cation method in the eld of sta-tistical learning. Mixed-e ect regression models are widely used for the analysis ofcorrelated data such as those found in longitudinal studies. We consider kernel exten-sions with semiparametric xed e ects and parametric random e ects for the logisticregression. The estimation is performed through the penalized likelihood method basedon kernel trick, and our focus is on the ecient computation and the e ective hyperpa-rameter selection. For the selection of optimal hyperparameters, cross-validation tech-niques are employed. Numerical results are then presented to indicate the performanceof the proposed procedure.Keywords: Generalized cross-validation function, kernel trick, logistic regression, lon-gitudinal data, mixed-e ects model, penalized likelihood. 1. Introduction Logistic regression (Amemiya, 1985; Agresti, 2002) is a popular method for binary classi -cation problems. The output of a logistic regression model can be interpreted as a posteriorestimate of the probability that an observation belongs to each of two disjoint classes. Theprobabilistic nature of the logistic regression model a ords many practical advantages, suchas the ability to accommodate unequal relative class frequencies in the training set or to ap-ply an appropriate loss matrix in making predictions that minimize the expected risk. As aresult, this model has been adopted in a diverse range of applications, including cancer classi- cation and analysis of DNA binding sites. For data that are clustered and/or longitudinal,mixed-e ect regression models are becoming increasingly popular (Hedeker and Gibbons,2006; Wu and Zhang, 2006). Mixed-e ects models constitute both xed and random e ects.In clustered data, subjects are clustered within an organization such as a hospital, school,clinic or rm. In longitudinal data where individuals are repeatedly assessed, measurements