Biometrika Advance Access originally published online on November 22, 2007
Biometrika 2007 94(4):905-919; doi:10.1093/biomet/asm063
Articles |
Using Hierarchical Likelihood for Missing Data Problems
Department of Preventive Medicine, University of Ulsan College of Medicine, Seoul 138-736, Korea ysch97{at}amc.seoul.kr
Department of Statistics, Seoul National University, Seoul 151-747, Korea youngjo{at}plaza.snu.ac.kr
London School of Hygiene and Tropical Medicine, London WC1E 7HT, U.K. mike.kenward{at}lshtm.ac.uk
Received for publication 1 June 2005. Revision received 1 March 2007.
Most statistical solutions to the problem of statistical inference with missing data involve integration or expectation. This can be done in many ways: directly or indirectly, analytically or numerically, deterministically or stochastically. Missing-data problems can be formulated in terms of latent random variables, so that hierarchical likelihood methods of Lee & Nelder (1996) can be applied to missing-value problems to provide one solution to the problem of integration of the likelihood. The resulting methods effectively use a Laplace approximation to the marginal likelihood with an additional adjustment to the measures of precision to accommodate the estimation of the fixed effects parameters. We first consider missing at random cases where problems are simpler to handle because the integration does not need to involve the missing-value mechanism and then consider missing not at random cases. We also study tobit regression and refit the missing not at random selection model to the antidepressant trial data analyzed in Diggle & Kenward (1994).
Key Words: Adjusted profile likelihood • Hierarchical likelihood • Marginal likelihood • Missing data • Restricted likelihood
References
-
Best N. G., Spiegelhalter D. J., Thomas A., Brayne C. E. G. Bayesian analysis of realistically complex models. J. R. Statist. Soc. A (1996) 159:323–42.[CrossRef]
Carlin B. P., Louis T. A. Bayesian and Empirical Bayesian Methods for Data Analysis (2000) London: Chapman and Hall.
Cox D. R., Reid N. Parameter orthogonality and approximate conditional inference (with Discussion). J. R. Statist. Soc. B (1987) 32:1–39.
Dempster A. P., Laird N. M., Rubin D. B. Maximum likelihood from incomplete data via the em algorithm (with Discussion). J. R. Statist. Soc. B (1977) 39:1–38.
Diebolt J., Ip E. H. S. Stochastic em: method and application. In: Markov Chain Monte Carlo in Practice—Gilks W. R., Richardson S., Spiegelhalter D. J., eds. (1996) London: Chapman and Hall. 259–68.
Diggle P., Kenward M. G. Informative drop-out in longitudinal analysis (with Discussion). Appl. Statist. (1994) 43:49–93.[CrossRef]
Heyting A., Essers J. G. A., Tolboom J. T. B. M. A practical application of the Patel-Kenward analysis of covariance to data from an anti-depressant trial with drop-outs. Statist. Appl. (1952) 2:259–307.
Horvitz D. G., Thompson D. J. A generalisation of sampling without replacement from a finite population. J. Am. Statist. Assoc. (1952) 47:663–85.[CrossRef][ISI]
Jansen I., Hens N., Molenberghs G., Aerts M., Verbeke G., Kenward M. G. The nature of sensitivity in missing not at random models. Comp. Statist. Data Anal. (2006) 50:830–58.[CrossRef]
Kenward M. G. Selection models for repeated measurements with nonrandom dropout: an illustration of sensitivity. Statist. Med. (1998) 17:2723–32.[CrossRef]
Kim D., Lee Y., Oh H. S. Hierarchical likelihood-based wavelet method for denoising signals with missing data. IEEE Sig. Proces. Lett. (2006) 13:361–4.[CrossRef]
Lee Y., Nelder. Hierarchical generalized linear models (with Discussion). J. R. Statist. Soc. B (1996) 58:619–78.
Lee Y., Nelder J. A. Hierarchical generalized linear models : A synthesis of generalised linear models, random-effect models and structured dispersion. Biometrika (2001) 88:987–1006.
Lee Y., Nelder J. A. Likelihood for random effect models (with Discussion). Statist. Oper. Res. Trans. (2005) 29:141–82.
Lee Y., Nelder J. A. Double hierarchical generalized linear models (with Discussion). Appl. Statist. (2006) 55:139–85.
Lee Y., Nelder J. A., Pawitan Y. Generalized Linear Models with Random Effects: Unified Analysis via H-likelihood (2006) New York: Chapman and Hall.
Little R. J. A., Rubin D. B. On jointly estimating parameters and missing data by maximizing the complete-data likelihood. Am. Statistician (1983) 37:218–20.[CrossRef]
Little R. J. A., Rubin D. B. Statistical Analysis with Missing Data (2002) 2nd ed. New York: John Wiley.
McCulloch C. E. Maximum likelihood algorithms for generalized linear mixed models. J. Am. Statist. Assoc. (1997) 92:162–70.[CrossRef][ISI]
Reilly M., Pepe M. S. A mean score method for missing and auxiliary covariate data in regression models. Biometrika (1995) 82:299–314.
Robins J. M., Rotnitzky A., Zhao L. P. Analysis of semiparametric regression models for repeated outcomes in the presence of missing data. J. Am. Statist. Assoc. (1995) 90:106–21.[CrossRef][ISI]
Rubin D. B. Inference and missing data. Biometrika (1976) 63:581–92.
Rubin D. B. Multiple Imputation for Nonresponse in Surveys (1987) Chichester: Wiley.
Rubin D. B. Multiple imputation after 18+ years. J. Am. Statist. Assoc. (1996) 91:473–89.[CrossRef][ISI]
Yates F. The analysis of replicated experiments when the field results are incomplete. Emp. J. Exp. Agric. (1933) 1:129–42.
Yun S., Lee Y. Robust estimation in mixed linear models with non-monotone missingness. Statist. Med. (2006) 25:3877–92.[CrossRef]
CiteULike 
Connotea 
Del.icio.us What's this?
| ||||||||||||||||||||||||||||||||||||||||||||||||||