Biometrika Advance Access originally published online on April 7, 2009
Biometrika 2009 96(2):457-468; doi:10.1093/biomet/asp003
Article |
Jackknife estimation of mean squared error of small area predictors in nonlinear mixed models
Department of Mathematics and Statistics, Arizona State University, Tempe, Arizona 85287-1804, U.S.A. sharon.lohr{at}asu.edu
School of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S 5B6, Canada jrao{at}math.carleton.ca
Received for publication 1 October 2007. Revision received 1 August 2008.
Empirical Bayes predictors of small area parameters of interest are often obtained under a linear mixed model for continuous response data or a generalized linear mixed model for binary responses or count data. However, estimation of the unconditional mean squared error of prediction is complicated, particularly for a nonlinear mixed model. Jiang et al. (2002) proposed a jackknife method for estimating the unconditional mean squared error and showed that the resulting estimator is nearly unbiased. The leading term of this estimator does not depend on the area-specific responses in the nonlinear case, whereas the posterior variance of the small area parameter given the model parameters is area-specific. Rao (2003) proposed an alternative method that leads to a computationally simpler jackknife estimator with an area-specific leading term. We show that a modification of Rao's method leads to a nearly unbiased area-specific jackknife estimator, which is also nearly unbiased for the conditional mean squared error given the area-specific responses. We examine the relative performances of the jackknife estimators, conditionally as well as unconditionally, in a simulation study, and apply the proposed method to estimate small area mean squared errors in disease mapping problems.
Key Words: Area-specific • Beta-binomial model • Binary response • Disease mapping • Empirical Bayes • Generalized linear mixed model
References
-
Booth J. G., Hobert J. P. Standard errors of prediction in generalized linear mixed models. J. Am. Statist. Assoc. (1998) 93:262–72.[CrossRef][Web of Science]
Breslow N. E., Clayton D. G. Approximate inference in generalized linear mixed models. J. Am. Statist. Assoc. (1993) 88:9–25.[CrossRef][Web of Science]
Clayton D., Kaldor J. Empirical Bayes estimates of age-standardized relative risks for use in disease mapping. Biometrics (1987) 43:671–81.[CrossRef][Web of Science][Medline]
Datta G. S., Rao J. N. K., Smith D. D. On measuring the variability of small area estimators under a basic area level model. Biometrika (2005) 92:183–96.
Fay R. E., Herriot R. A. Estimates of income for small places: an application of James-Stein procedures to census data. J. Am. Statist. Assoc. (1979) 74:269–77.[CrossRef][Web of Science]
Fuller W. A. Prediction of true values for the measurement error model. Statistical Analysis of Measurement Error Models and Applications—Brown P. J., Fuller W. A., eds. (1990) 112. Providence, RI: American Mathematical Society. 41–57. Contemporary Mathematics.
Hoadley B. Asymptotic properties of maximum likelihood estimators for the independent not identically distributed case. Ann. Math. Statist. (1971) 42:1977–91.[CrossRef]
Householder A. S. The Numerical Treatment of a Single Nonlinear Equation (1970) New York: McGraw-Hill.
Jiang J., Lahiri P., Wan S.-M. A unified jackknife theory for empirical best prediction with M-estimation. Ann. Statist. (2002) 30:1782–810.[CrossRef]
Kleinman J. C. Proportions with extraneous variance: single and independent samples. J. Am. Statist. Assoc. (1973) 68:46–54.[CrossRef][Web of Science]
Lohr S. Discussion of papers on resampling methods in small area estimation. Proc. Survey Res. Meth. Sect. Am. Statist. Assoc. (2007) Alexandria, VA: American Statistical Association. 3265–7.
Longford N. On standard errors of model-based small area estimators. Survey Methodol. (2007) 33:69–79.
Quenouille M. H. Notes on bias in estimation. Biometrika (1956) 43:353–60.
Rao J. N. K. Small Area Estimation (2003) New York: Wiley.
R Development Core Team. R: A Language and Environment for Statistical Computing (2008) Vienna: R Foundation for Statistical Computing.
Smith D. M. Algorithm AS 189: Maximum likelihood estimation of the parameters of the beta binomial distribution. Appl. Statist. (1983) 32:196–204.[CrossRef]
Wakefield J. Disease mapping and spatial regression with count data. Biostatistics (2007) 8:158–83.
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