Abstract
The classical assumption in generalized linear measurement error models (GLMEMs) is that measurement errors (MEs) for covariates are distributed as a fully parametric distribution such as the multivariate normal distribution. This paper uses a centered Dirichlet process mixture model to relax the fully parametric distributional assumption of MEs, and develops a semiparametric Bayesian approach to simultaneously obtain Bayesian estimations of parameters and covariates subject to MEs by combining the stick-breaking prior and the Gibbs sampler together with the Metropolis–Hastings algorithm. Two Bayesian case-deletion diagnostics are proposed to identify influential observations in GLMEMs via the Kullback–Leibler divergence and Cook’s distance. Computationally feasible formulae for evaluating Bayesian case-deletion diagnostics are presented. Several simulation studies and a real example are used to illustrate our proposed methodologies.
Highlights
Generalized linear models (GLMs) are widely used to fit responses that do not satisfy the usual requirements of least-squares methods in biostatistics, epidemiology, and many other areas
Stefanski and Carroll (1985) developed a bias-adjusted estimator, a functional maximum likelihood estimator and an estimator exploiting the consequences of sufficiency for a logistic regression when covariates were subject to measurement errors (MEs); Stefanski and Carroll (1987) studied parameter estimation in GLM with canonical form when the explanatory vector was measured with an independent normal error; Buzas and Stefanski (1996) investigated instrumental variable estimation in generalized linear measurement error models (GLMEMs) with canonical link functions; Aitkin and Rocci (2002) presented an EM algorithm for maximum likelihood estimation in GLMs with continuous MEs in the explanatory variables; Battauz (2011) developed a Laplace-based estimator for GLMEMs; Battauz and Bellio (2011) proposed a structural analysis for GLMs when some explanatory variables were measured with error and the ME variance was a function of the true variables
All the above mentioned studies assume that the covariate MEs in GLMEMs are distributed as a fully parametric distribution such as the multivariate normal distribution
Summary
Generalized linear models (GLMs) are widely used to fit responses that do not satisfy the usual requirements of least-squares methods in biostatistics, epidemiology, and many other areas. Bayesian case deletion approaches to detect influential observations (or sets of observations) have been proposed for some statistical models such as linear regression models (Carlin and Polson 1991), GLMs (Jackson et al 2012) and generalized linear mixed models (Fong et al 2010) based on the Kullback–Leibler divergence (K– L divergence) and the conditional predictive ordinate.
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