Generalized Linear Models for Small-Area Estimation

Abstract Bayesian methods have been used quite extensively in recent years for solving small-area estimation problems. Particularly effective in this regard has been the hierarchical or empirical Bayes approach, which is especially suitable for a systematic connection of local areas through models. However, the development to date has mainly concentrated on continuous-valued variates. Often the survey data are discrete or categorical, so that hierarchical or empirical Bayes techniques designed for continuous variates are inappropriate. This article considers hierarchical Bayes generalized linear models for a unified analysis of both discrete and continuous data. A general theorem is provided that ensures the propriety of posteriors under diffuse priors. This result is then extended to the case of spatial generalized linear models. The hierarchical Bayes procedure is implemented via Markov chain Monte Carlo integration techniques. Two examples (one featuring spatial correlation structure) are given to illu...

[1]  Kerrie Mengersen,et al.  [Bayesian Computation and Stochastic Systems]: Rejoinder , 1995 .

[2]  R. Fay,et al.  Estimates of Income for Small Places: An Application of James-Stein Procedures to Census Data , 1979 .

[3]  Scott L. Zeger,et al.  Generalized linear models with random e ects: a Gibbs sampling approach , 1991 .

[4]  Tapabrata Maiti,et al.  Hierarchical Bayes estimation of mortality rates for disease mapping , 1998 .

[5]  D. Clayton,et al.  Empirical Bayes estimates of age-standardized relative risks for use in disease mapping. , 1987, Biometrics.

[6]  Malay Ghosh,et al.  Bayesian Prediction in Linear Models: Applications to Small Area Estimation , 1991 .

[7]  Alan E. Gelfand,et al.  Model choice: A minimum posterior predictive loss approach , 1998, AISTATS.

[8]  R. Tsutakawa.  Mixed model for analyzing geographic variability in mortality rates. , 1988, Journal of the American Statistical Association.

[9]  Thomas J. Tomberlin,et al.  EMPIRICAL BAYES ESTIMATORS OF SMALL AREA PROPORTIONS IN MULTISTAGE DESIGNS , 1997 .

[10]  J. Besag,et al.  Bayesian Computation and Stochastic Systems , 1995 .

[11]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[12]  L Bernardinelli,et al.  Empirical Bayes versus fully Bayesian analysis of geographical variation in disease risk. , 1992, Statistics in medicine.

[13]  Walter R. Gilks,et al.  BUGS - Bayesian inference Using Gibbs Sampling Version 0.50 , 1995 .

[14]  Malay Ghosh,et al.  Robust Empirical Bayes Estimation of Means from Stratified Samples , 1987 .

[15]  J. Besag,et al.  Bayesian image restoration, with two applications in spatial statistics , 1991 .

[16]  Melvin R. Novick,et al.  Bayesian Full Rank Marginalization for Two-Way Contingency Tables , 1986 .

[17]  John A. Nelder,et al.  Generalized linear models. 2nd ed. , 1993 .

[18]  Adrian F. M. Smith,et al.  Bayesian Inference for Generalized Linear and Proportional Hazards Models Via Gibbs Sampling , 1993 .

[19]  Donald Malec,et al.  Bayesian Predictive Inference for Small Areas for Binary Variables in the National Health Interview Survey , 1993 .

[20]  Adrian F. M. Smith,et al.  Sampling-Based Approaches to Calculating Marginal Densities , 1990 .

[21]  N. Breslow,et al.  Approximate inference in generalized linear mixed models , 1993 .

[22]  D. Rubin,et al.  Inference from Iterative Simulation Using Multiple Sequences , 1992 .

[23]  P. McCullagh,et al.  Generalized Linear Models, 2nd Edn. , 1990 .

[24]  M. Ghosh,et al.  A Hierarchical Bayes Approach to Small Area Estimation with Auxiliary Information , 1992 .

[25]  L. Tierney,et al.  Accurate Approximations for Posterior Moments and Marginal Densities , 1986 .

[26]  R. Brooks,et al.  Approximate Likelihood Ratio Tests in the Analysis of Beta‐Binomial Data , 1984 .

[27]  B. Nandram,et al.  Bayesian Predictive Inference for a Finite Population Proportion: Two‐Stage Cluster Sampling , 1993 .

[28]  R. Tsutakawa,et al.  Empirical Bayes estimation of cancer mortality rates. , 1985, Statistics in medicine.

[29]  J. Albert.  Computational methods using a Bayesian hierarchical generalized linear model , 1988 .

[30]  O. William.  Journal Of The American Statistical Association V-28 , 1932 .

[31]  Thomas W.F. Stroud,et al.  Bayesian analysis of binary survey data , 1994 .

[32]  T. Stroud.  Hierarchical baxes predictive means and variances with application to sample survey inference , 1991 .

[33]  Bradley P. Carlin,et al.  Hierarchical Spatio-Temporal Mapping of Disease Rates , 1997 .

[34]  J. Rao,et al.  The estimation of the mean squared error of small-area estimators , 1990 .

[35]  K. Cowles,et al.  CODA: convergence diagnosis and output analysis software for Gibbs sampling output , 1995 .

[36]  R. Tsutakawa,et al.  Estimation of cancer mortality rates: a Bayesian analysis of small frequencies. , 1985, Biometrics.

[37]  Malay Ghosh,et al.  Small Area Estimation: An Appraisal , 1994 .

[38]  P. Lahiri.  Robust empirical bayes estimation in finite population sampling , 1986 .

[39]  G. Meeden,et al.  Empirical Bayes Estimation in Finite Population Sampling , 1986 .