Covariate-adjusted generalized linear models

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. Copyright 2009, Oxford University Press.

[1]  Hans-Georg Müller,et al.  Random Effect Threshold Models for Dose–Response Relationships with Repeated Measurements , 1997 .

[2]  Ben Armstrong,et al.  Measurement error in the generalised linear model , 1985 .

[3]  D. McLeish.  Dependent Central Limit Theorems and Invariance Principles , 1974 .

[4]  S Senn.  Covariance analysis in generalized linear measurement error models. , 1990, Statistics in medicine.

[5]  D. Ruppert,et al.  Measurement Error in Nonlinear Models , 1995 .

[6]  Raymond J. Carroll,et al.  Approximate Quasi-likelihood Estimation in Models with Surrogate Predictors , 1990 .

[7]  P. McCullagh.  Quasi-Likelihood Functions , 1983 .

[8]  L. Stefanski.  Correcting data for measurement error in generalized linear models , 1989 .

[9]  Stanley R. Johnson,et al.  Varying Coefficient Models , 1984 .

[10]  R J Carroll,et al.  Quasilikelihood estimation in measurement error models with correlated replicates. , 1996, Biometrics.

[11]  Damla Şentürk,et al.  Covariate-adjusted regression , 2005 .

[12]  George A Kaysen,et al.  Relationships among inflammation nutrition and physiologic mechanisms establishing albumin levels in hemodialysis patients. , 2002, Kidney international.

[13]  Damla Şentürk,et al.  Inference for covariate adjusted regression via varying coefficient models , 2006 .

[14]  Jianqing Fan,et al.  Efficient Estimation and Inferences for Varying-Coefficient Models , 2000 .

[15]  H. Muller,et al.  Inference for covariate adjusted regression via varying coefficient models , 2006, math/0607010.

[16]  T. Louis,et al.  Lipid Adjustment in the Analysis of Environmental Contaminants and Human Health Risks , 2005, Environmental health perspectives.

[17]  Raymond J. Carroll,et al.  Covariate Measurement Error in Logistic Regression , 1985 .

[18]  D. Hosmer,et al.  Applied Logistic Regression , 1991 .