Faculty Bibliography

Identifying regions with the highest and lowest mortality rates and producing the corresponding color-coded maps help epidemiologists identify promising areas for analytic etiological studies. Based on a two-stage Poisson-Gamma model with covariates, we use information on known risk factors, such as smoking prevalence, to adjust mortality rates and reveal residual variation in relative risks that may reflect previously masked etiological associations. In addition to covariate adjustment, we study rankings based on standardized mortality ratios (SMRs), empirical Bayes (EB) estimates, and a posterior percentile ranking (PPR) method and indicate circumstances that warrant the more complex procedures in order to obtain a high probability of correctly classifying the regions with the upper 100gamma% and lower 100gamma% of relative risks for gamma= 0.05, 0.1, and 0.2. We also give analytic approximations to the probabilities of correctly classifying regions in the upper 100gamma% of relative risks for these three ranking methods. Using data on mortality from heart disease, we found that adjustment for smoking prevalence has an important impact on which regions are classified as high and low risk. With such a common disease, all three ranking methods performed comparably. However, for diseases with smaller event counts, such as cancers, and wide variation in event counts among regions, EB and PPR methods outperform ranking based on SMRs

For the well-known Fay-Herriot small area model, standard variance component estimation methods frequently produce zero estimates of the strictly positive model variance. As a consequence, an empirical best linear unbiased predictor of a small area mean, commonly used in the small area estimation, could reduce to a simple regression estimator, which typically has an overshrinking problem. We propose an adjusted maximum likelihood estimator of the model variance that maximizes an adjusted likelihood defined as a product of the model variance and a standard likelihood (e.g., profile or residual likelihood) function. The adjustment factor was suggested earlier by Carl Morris in the context of approximating a hierarchical Bayes solution where the hyperparameters, including the model variance, are assumed to follow a prior distribution. Interestingly, the proposed adjustment does not affect the mean squared error property of the model variance estimator or the corresponding empirical best linear unbiased predictors of the small area means in a higher order asymptotic sense. However, as demonstrated in our simulation study, the proposed adjustment has a considerable advantage in the small sample inference, especially in estimating the shrinkage parameters and in constructing the parametric bootstrap prediction intervals of the small area means, which require the use of a strictly positive consistent model variance estimate

The quality of a production process is often judged by a quality assurance audit, which is essentially a structured system of sampling inspection plan. The defects of sampled products are assessed and compared with a quality standard, which is determined from a tradeoff among manufacturing costs, operating costs and customer needs. In this paper, we propose a new hierarchical Bayes quality measurement plan that assumes an implicit prior for the hyperparameters. The resulting posterior means and variances are obtained adaptively using a parametric bootstrap method. (C) Published in 2009 by John Wiley & Sons, Ltd. $$:

Empirical best linear unbiased prediction (EBLUP) method uses a linear mixed model in combining information from different sources of information. This method is particularly useful in small area problems. The variability of an EBLUP is traditionally measured by the mean squared prediction error (MSPE), and interval estimates are generally constructed using estimates of the MSPE. Such methods have shortcomings like under-coverage or over-coverage, excessive length and lack of interpretability. We propose a parametric bootstrap approach to estimate the entire distribution of a suitably centered and scaled EBLUP. The bootstrap histogram is highly accurate, and differs from the true EBLUP distribution by only O(d(3)n(-3/2)), where d is the number of parameters and n the number of observations. This result is used to obtain highly accurate prediction intervals. Simulation results demonstrate the superiority of this method over existing techniques of constructing prediction intervals in linear mixed models. $$: