Biometrika Advance Access originally published online on April 21, 2009
Biometrika 2009 96(2):357-370; doi:10.1093/biomet/asp012
Article |
Covariate-adjusted generalized linear models
entürkDepartment of Statistics, Pennsylvania State University, University Park, Pennsylvania 16802, U.S.A. dsenturk{at}stat.psu.edu
Department of Statistics, University of California, Davis, One Shields Avenue, Davis, California 95616, U.S.A. mueller{at}wald.ucdavis.edu
Received for publication 1 April 2008. Revision received 1 October 2008.
We propose covariate adjustment methodology for a situation where one wishes to study the dependence of a generalized response on predictors while both predictors and response are distorted by an observable covariate. The distorting covariate is thought of as a size measurement that affects predictors in a multiplicative fashion. The generalized response is modelled by means of a random threshold, where the subject-specific thresholds are affected by a multiplicative factor that is a function of the distorting covariate. While the various factors are modelled as smooth unknown functions of the distorting covariate, the underlying relationship between response and covariates is assumed to be governed by a generalized linear model with a known link function. This model provides an extension of a covariate-adjusted regression approach to the case of a generalized linear model. We demonstrate that this contamination model leads to a semiparametric varying-coefficient model. Numerical implementation is straightforward by combining binning, quasilikelihood, and smoothing steps. The asymptotic distribution of the proposed estimators for the regression coefficients of the latent generalized linear model is derived by means of a martingale central limit theorem. Combining this result with consistent estimators for the asymptotic variance makes it then possible to obtain asymptotic inference for the targeted parameters. Both real and simulated data are used in illustrating the proposed methodology.
Key Words: Asymptotic inference • Confounding • Covariate-adjusted regression • Multiplicative effect • Quasilikelihood • Semiparametric modelling
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