Markov chain Monte Carlo methods in biostatistics

Statistical Methods in Medical Research - Tập 5 Số 4 - Trang 339-355 - 1996
Andrew Gelman1, Donald B. Rubin2
1Department of Statistics, Columbia University, New York, NY 10027, USA
2Department of Statistics, Harvard University, Cambridge, Massachusetts USA

Tóm tắt

Appropriate models in biostatistics are often quite complicated. Such models are typically most easily fit using Bayesian methods, which can often be implemented using simulation techniques. Markov chain Monte Carlo (MCMC) methods are an important set of tools for such simulations. We give an overview and references of this rapidly emerging technology along with a relatively simple example. MCMC techniques can be viewed as extensions of iterative maximization techniques, but with random jumps rather than maximizations at each step. Special care is needed when implementing iterative maximization procedures rather than closed-form methods, and even more care is needed with iterative simulation procedures: it is substantially more difficult to monitor convergence to a distribution than to a point. The most reliable implementations of MCMC build upon results from simpler models fit using combinations of maximization algorithms and noniterative simulations, so that the user has a rough idea of the location and scale of the posterior distribution of the quantities of interest under the more complicated model. These concerns with implementation, however, should not deter the biostatistician from using MCMC methods, but rather help to ensure wise use of these powerful techniques.

Từ khóa


Tài liệu tham khảo

Rubin DB Using the SIR algorithm to simulate posterior distributions. In: Bernardo J ed. Bayesian statistics 3. Oxford: Oxford University Press, 1988: 395-402.

Spiegelhalter D., 1994, BUGS: Bayesian inference using Gibbs sampling, version 0.30

10.1080/01621459.1991.10475006

10.2307/2532317

10.2307/2986324

Longford N., 1993, Random coefficient models

Cowles MK, 1993, Bayesian Tobit modeling of longitudinal ordinal clinical trial compliance data

West M. Modelling with mixtures. In: Bernardo JM, Berger JO, Dawid AP, Smith AFM eds. Bayesian statistics 4. New York: Oxford University Press, 1992: 503-24.

10.1080/01621459.1992.10475258

10.1214/ss/1177010994

Baldi P., 1994, Proceedings of the National Academy of Sctence USA

Clayton D., Bernardinelli L. Bayesian methods for mapping disease risk. In: Elliott P, Cusick J, English D, Stern R eds. Geographical and environmental epidemiology: methods for small-area studies. Oxford: Oxford University Press, 1992: 205-20.

Gilks WR, 1993, Statistics in Medicine, 12, 1703, 10.1002/sim.4780121806

10.1007/978-94-015-7983-4_2

10.1201/9780429258411

Metropolis N., 1953, Physics, 21, 1087

10.1109/TPAMI.1984.4767596

10.1080/01621459.1987.10478458

10.1080/01621459.1990.10476213

10.1080/01621459.1990.10474968

Smith Afm, 1993, Journal of the Royal Statistical Society B, 55, 3, 10.1111/j.2517-6161.1993.tb01466.x

Besag J., 1993, Journal of the Royal Statistical Society B, 55, 25, 10.1111/j.2517-6161.1993.tb01467.x

Gilks WR, 1993, Journal of the Royal Statistical Society B, 55, 39, 10.1111/j.2517-6161.1993.tb01468.x

Gilks WR, 1996, Practical Markov Chain

10.1007/978-1-4684-0192-9

Carlin BP, 1996, Bayes and empirical Bayes methodsfor data analysis

10.1214/ss/1177011136

Gelman A., Roberts G., Gilks W. Efficient Metropolis jumping rules. In: Bernardo JM, Berger JO, Dawid AP, Smith AFM eds. Bayesian statistics 5. New York : Oxford University Press, 1996: 599-608.

Dempster AP, 1977, Journal of the Royal Statistical Society B, 30, 1, 10.1111/j.2517-6161.1977.tb01600.x

10.1080/01621459.1991.10475130

Laird NM, 1982, Journal of the Royal Statistical Society B, 44, 190, 10.1111/j.2517-6161.1982.tb01198.x

Breslow NE, 1993, Journal of the American Statistical Association, 88, 9, 10.1080/01621459.1993.10594284

10.1080/01621459.1965.10480829

10.2307/2289460

10.1080/01621459.1994.10476469

10.1093/biomet/57.1.97

Tierney L., 1995, Annals of Statistics

Gelman A., 1992, Computing Science and Statistics, 24, 433

Gelman A., Rubin DB A single series from the Gibbs sampler provides a false sense of security . In: Bernardo JM, Berger JO, Dawid AP, Smith AFM eds. Bayesian statistics 4. New York: Oxford University Press, 1996: 625-31.

10.1103/PhysRev.116.565

10.1016/0304-4076(95)01769-0

Gelman A. Inference and monitoring convergence. In: Gilks W, Richardson S, Spiegelhalter D eds. Practical Markov chain Monte Carlo. New York: Chapman & Hall, 1996: 131-43.

Besag J., 1986, Journal of the Royal Statistical Society B, 48, 259, 10.1111/j.2517-6161.1986.tb01412.x

Cowles MK, 1996, Journal of the American Statistical Association

10.1080/01621459.1996.10476708

Sinclair AJ, Proceedings of the Twentieth Annual Symposium on the Theory of Computing

Applegate D., 1990, Random polynomial time algorithms for sampling joint distributions

10.2307/2347565

10.2307/2986138

Hills SE, Smith Afm. Parameterization issues in Bayesian inference (with discussion). In: Bernardo JM, Berger JO, Dawid AP, Smith AFM eds. Bayesian statistics 4. New York: Oxford University Press, 1996: 227-46.

Geyer CJ, 1993, Annealing Markov chain Monte Carlo with applications to pedigree analysis. Technical report

10.1214/ss/1177010123

10.1002/sim.4780140805

10.1093/biomet/80.2.267