Shared Uncertainty in Measurement ErrorProblems, With Application to Nevada TestSite Fallout DataYehua Li1, Annamaria Guolo2, F. Owen Hoffman3and Raymond J. Carroll4,∗1Department of Statistics, University of Georgia, Athens GA 30605, U.S.A.2Department of Statistics, University of Padova, Via Cesare Battisti, 241, 35121, Padova,Italy3SENES Oak Ridge, Center for Risk Analysis, 102 Donner Drive, Oak Ridge, TN 37830,U.S.A.4Department of Statistics, Texas A&M University, 3143 TAMU, Texas A&M University,College Station, TX 77843-3143, U.S.A.∗email: [email protected] radiation epidemiology, it is often necessary to use mathematical models in the absence ofdirect measurements of individual doses. When complex models are used as surrogates fordirect measurements to estimate individual doses that occurred almost 50 years ago, doseestimates will be associated with considerable error, this error being a mixture of (a) classicalmeasurement error due to individual data such as diet histories, and (b) Berkson measure-ment error associated with various aspects of the dosimetry system. In the Nevada Test SiteThyroid Disease Study, the Berkson measurement errors are correlated within strata. Thispaper concerns the development of statistical methods for inference about risk of radiationdose on thyroid disease, methods that account for the complex error structure inherence inthe problem. Bayesian methods using MCMC and Monte-Carlo EM-methods are described,with both sharing a key Metropolis-Hastings step. Regression calibration is also considered,but we show that regression calibration does not use the correlation structure of the Berksonerrors. Our methods are applied to the Nevada Test Site Study, where we find a strongdose-response relationship between dose and thyroiditis. We conclude that full considera-tion of mixtures of Berkson and classical uncertainties in reconstructed individual doses areimportant for quantifying the dose response and its credibility/confidence interval. Usingregression calibration and expectation values for individual doses can lead to a substantialunderestimation of the excess relative risk per gray and its 95% confid ence intervals.KEY WORDS: Bayes, Berkson error, Classic error, Correlation, Latent variables, Measure-ment error, MCMC, Monte-Carlo EM, Objective Bayes, Radiation epidemiology, Regressioncalibration, Thyroid disease.Short title: Shared Uncertainties in Measurement Error Models1 INTRODUCTIONThere are many studies relating radiation exp osure to disease. We focus here on data fromthe Nevada Test Site (NTS) Thyroid Disease Study. In the Nevada study, 2, 491 individualswho were exposed to radiation as children were examined for thyroid disease. The primaryradiation exposure to the thyroid glands of these children came from the ingestion of milkand vegetables contaminated with radioactive isotopes of iodine. The idea of the study wasto relate various thyroid disease outcomes to radiation exposure to the thyroid. The originalversion of this study was described by Stevens, et al. (1992), Kerber, et al. (1993) andSimon, et al. (1995). Recently, the dosimetry for the study was redone (Simon, et al., 2006),and the study results were reported (Lyon, et al., 2006).The estimation of individual radiation dose t hat occurred 50 years in th e past is well-known to be subject to large uncertainties, especially when mathematical models are em-ployed due to the absence of direct measurements of the concentrations of radioactivity infoods or within the thyroid gland of individuals. There are many references on this subject,a good introduction to which is given by Ron and Hoffman (1999) and the many statisticalpapers in that volume. Various statistical pap ers (Reeves, et al., 1998; Schafer, et al., 2001;Mallick, et al., 2002; Stram and Kopecky, 2003; Lubin, et al., 2004; Schafer and Gilbert,2006) describe measurement error properties and analysis in this context. What is typical inthese studies is to build a large dosimetry model that attempts to convert the known data,e.g., about the above-ground nuclear tests, to radiation actually absorbed into the thyroid.Dosimetry calculations for individual subjects were based on age at exposure, gender, res-idence history, whether as a child the individual was breast-fed, and a diet questionnairefilled out by the parent focusing on milk consumption and vegetables. The data were theninput into a complex model and for each individual, the point estimate of thyroid dose (thearithmetic mean of a lognormal distribution of dose estimates) and an associated error term(the geometric standard deviation) for the measurement error were reported.Generally, the authors engaged in dose reconstruction using mathematical models con-1clude that radiation doses are estimated with a combination of Berkson measurement errorand the classical type of measurement error. The type of model, pioneered by Reeves, et al.(1997), in the log-scale of dose says that true log-dose X is related to observed or calculatedlog-dose W by a latent intermediate L viaX = L + Ub; (1)W = L + Uc, (2)where Ubis the Berkson uncertainty and Ucis the classical uncertainty. If there is no classicaluncertainty, then W = L and we get the pure Berkson model X = W + Ub. If there is noBerkson uncertainty, then X = L and we get the classical additive error model W = X + Uc.The starting point of this paper is th e following. For reasons that we describe in moredetail in Section 2, the Berkson measurement errors produced as estimates by dose recon-struction models are now generally thought to be correlated across individuals. The originof correlated Berkson measurement errors for the NTS-Study are described in Section 7.2 ofMallick, et al. (2002). In a very different context, with a different dosimetry system, Stramand Kopecky (2003) also describe correlated Berkson errors.This paper is organized as follows. Section 2 describes the NTS-Study data in more detail,and gives a description of the modelling of mixtures of classical and correlated Berkson errors.The only paper of which we are aware that considers the analysis of data subject toa combination of classical measurement error and correlated Berkson errors is Mallick, etal. (2002). Their analysis, given in a brief concluding paragraph, was purely Bayesian, butit gave no details as to how to implement the MCMC-methods, sensitivity of the resultsto prior specification, etc.. Implementation is non-trivial, because the