Abstract
In disease mapping, the Bayesian approach is widely used for forming the prediction interval of relative risks. In this paper we propose a hierarchical-likelihood interval for disease mapping, which accounts for the inflation of standard error estimates caused by uncertainty in the estimation of the fixed parameters. Comparison is made with the Bayesian prediction intervals derived from penalized quasi-likelihood and fully Bayesian methods. Through simulation studies, we show that prediction intervals for random effects using hierarchical likelihood maintains the required level.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ainsworth LM, Dean CB (2006) Approximate inference for disease mapping. Comput Stat Data Anal 50: 2552–2570
Besag J, Higdon D (1999) Bayesian analysis of agricultural field experiments (with discussion). J R Stat Soc Series B 61: 691–746
Booth JG, Hobert JP (1998) Standard errors of prediction in generalized linear mixed models. J Am Stat Assoc 93: 262–272
Breslow NE, Clayton DG (1993) Approximate inference in generalized linear mixed models. J Am Stat Assoc 88: 9–25
Browne WJ (1998) Applying MCMC methods to multi-level models. Unpublished Ph.D. Dissertation, University of Bath
Browne WJ, Draper D (2006) A comparison of Bayesian and likelihood methods for fitting multilevel models (with discussion). Bayesian Anal 1: 473–550
Carlin BP, Gelfand AE (1991) Approaches for empirical Bayesian confidence intervals. J Am Stat Assoc 84: 717–726
Clayton D, Kaldor J (1987) Empirical Bayes estimates of age-standardized relative risks for use in disease mapping. Biometrics 43: 671–681
Cox DR, Reid N (1987) Parameter orthogonality and approximate conditional inference. J R Stat Soc Series B 49: 1–18
Datta GS, Rao JNK, Smith DD (2005) On measuring the variability of small area estimators under a basic area level models. Biometrika 92: 183–196
Dean CB, MacNab YC (2001) Modeling of rates over a hierarchical health administrative structure. Can J Stat 29: 405–419
Eberly LE, Carlin BP (2000) Identifiability and convergence issues for Markov chain Monte Carlo fitting of spatial models. Stat Med 19: 2279–2294
Ghosh M, Rao JNK (1994) Small area estimation: an appraisal. Stat Sci 9: 55–76
Ha I, Lee Y, MacKenzie G (2007) Model selection for multi-component frailty models. Stat Med 26: 4790–4807
Kass RE, Steffey D (1989) Approximate Bayesian inference in conditionally independent hierarchical models (parametric empirical Bayes models). J Am Stat Assoc 84: 717–726
Lee Y, Ha I (2010) Orthodox BLUP versus h-likelihood methods for inferences about random effects in Tweedie mixed models. Stat Comput 20: 295–303
Lee Y, Nelder JA (1996) Hierarchical generalized linear models (with discussion). J R Stat Soc Series B 58: 619–678
Lee Y, Nelder JA (2001) Hierarchical generalised linear models : a synthesis of generalised linear models, random effect models and structured dispersion. Biometrika 88: 987–1006
Lee Y, Nelder JA (2006) Fitting via alternative random effect models. Stat Comput 16: 69–75
Lee Y, Nelder JA, Noh M (2007) HL: problems and solutions. Stat Comput 17: 49–55
Lee Y, Nelder JA, Pawitan Y (2006) Generalized linear models with random effects: unified approach via h-likelihood. Chapman and Hall, New York
Lee Y, Lee W, Piepho H (2010) Inferences for random-effect models with singular precision matrix. Manuscript submitted for publication
Leroux BG (2000) Modelling spatial disease rates using maximum likelihood. Stat Med 18: 2321–2332
Leroux BG, Lin X, Breslow N (1999) Estimation of disease rates in small areas: a new mixed model for spatial dependence. In: Halloran ME, Berry D (eds) Statistical models in epidemiology, the environment and clinical trials. Springer, New York, pp 135–178
Lin X, Breslow NE (1996) Bias correction in generalized linear mixed models with multiple components of dispersion. J Am Stat Assoc 91: 1007–1016
MacNab YC, Dean CB (2000) Parametric bootstrap and penalized quasi–likelihood inference in conditional autoregressive models. Stat Med 19: 2421–2435
MacNab YC, Dean CB (2002) Spatio-temporal modelling of rates for the construction of disease maps. Stat Med 21: 347–358
MacNab YC, Farrell PJ, Gustafson P, Wen S (2004) Estimation in Bayesian disease mapping. Biometrics 60: 865–873
Noh M, Lee Y (2007) REML estimation for binary data in GLMMs. J Multivar Anal 98: 896–915
Rue H, Held L (2005) Gaussian Markov random fields: theory and applications, vol 104 of Monographs on Statistics and Applied Probability. Chapman & Hall, London
Tierney L, Kadane JB (1983) Accurate approximations for posterior moments and marginal densities. J Am Stat Assoc 81: 82–86
Tierney L, Kass RE, Kadane JB (1989) Fully exponential Laplace approximations to expectations and variances of nonpositive functions. J Am Stat Assoc 84: 710–716
Zhang Z (2007) A complete SAS program to iteratively run WinBUGS for Monte Carlo simulation, 2006; Retrieved April 19, 2007, from. http://www.psychstat.org/us/article.php/62.htm
Zeng D, Lin DY (2007) Maximum likelihood estimation in semiparameteric regression models with censored data (with discussion). J R Stat Soc Series B 69: 507–564
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lee, Y., Jang, M. & Lee, W. Prediction interval for disease mapping using hierarchical likelihood. Comput Stat 26, 159–179 (2011). https://doi.org/10.1007/s00180-010-0215-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00180-010-0215-3