Bayesci Genelleştirilmiş Lineer Karma Modellerde Önsel Seçimleri ve Karşılaştırılması

Yıl 2017, Cilt: 21 Sayı: 3, 999 - 1010, 14.06.2017

Zeynep Öztürk Mehmet Ali Cengiz

Öz

Parametre tahmininde ve model seçiminde uzman görüşlerin modele katılmasını öngören Bayesci yaklaşımda en önemli nokta parametreler hakkında önsel bilgi seçimi ve kullanımıdır. Bu nedenle, bu çalışmada, önsel dağılımların seçimleri tanıtılmış, genelleştirilmiş lineer karma modeller için farklı önseller ile elde edilen modeller hem parametre tahmini hem de model uyumu anlamında karşılaştırılmıştır. Akciğer kanseri hastalarının yaşam kaliteleri ve aldıkları tedavinin çıktılarını ölçmek amacıyla yapılan bir çalışmaya ait gerçek bir veri kullanılmıştır. SAS 9.3 programında Markov Zinciri Monte Carlo algoritması ile elde edilen parametre tahminleri bulunmuş ve modeller kurulmuştur. Farklı önsel seçimleri ile Sapma Bilgi kriteri (DIC) ne göre en iyi model bilgilendirici önsel dağılım ile elde edilen model olmuştur.

Anahtar Kelimeler

Genelleştirilmiş lineer karma modeller, Bayesci yaklaşım; Önsel dağılımlar; Önsel seçimi

Kaynakça

• [1] Nelder, J. A and Wedderburn, R. W. M., 1972. Generalized Linear Models, Journal of the Royal Statistical Society. Series A (General) Vol. 135, No. 3 , pp. 370-384.
• [2] Zeger, S. L., Karim, M. R., 1991. Generalized Linear Models with Random Effects: A Gibbs Sampling Approach, Journal of the American Statistical Association, 86, 79–86.
• [3] Gamerman, D. Statistics and Computing (1997) 7: 57. doi:10.1023/A:1018509429360.
• [4] Booth, J. G., Hobert, J. P., 1999. Maximizing generalized linear mixed model likelihoods with an automated Monte Carlo EM algorithm. Journal of the Royal Statistical Society Series B (Statistical Methodology), 61, 265–285.
• [5] Natarajan, R., Kass, R. E., 2000. Reference Bayesian Methods for Generalized Linear Mixed Models, Journal of the American Statistical Association, 95, 227–237.
• [6] Zhao, Y., Staudenmayer, J., Coull, B.A., Wand, M.P., 2006. General design Bayesian generalized linear mixed models. Statistical Science, 21, 35–51.
• [7] Tsai, M. & Hsiao, C.K. Comput Stat (2008) 23: 587. doi:10.1007/s00180-007-0100-x.
• [8] Evangelou, E. A. 2009. Bayesian and Frequentist Methods for Approximate Inference in Generalized Linear Mixed Models. University of North Carolina at Chapel Hill. Doctor of Philosophy in the Department of Statistics and Operations Research (Statistics).
• [9] Bedrick, E. J., Christensen, R., Johnson, W., 1997. Bayesian binomial regression: Predicting survival at a trauma center. Amer. Statist. 51 211–218.
• [10] Natarajan, R., McCulloch, C.E., 1998. Gibbs sampling with diffuse proper priors: a valid approach to data-driven ınference? Journal of Computational and Graphical Statistics,7, 67/277.
• [11] Diggle, P. J., Tawn, J. A., Moyeed, R. A., 1998. Model-based Geostatistics, Journal of the Royal Statistical Society, Series C: Applied Statistics, 47, 299–326.
• [12] Berger, J. O., De Oliveira, V., Sans´o, B., 2001. Objective Bayesian Analysis of Spatially Correlated Data, Journal of the American Statistical Association, 96, 1361–1374.
• [13] Browne, W.J., Draper, D., 2006. A comparison of Bayesian and likelihood-based methods for fitting multilevel models, International Society for Bayesian Analysis, 1, Number 3, pp. 473-514.
• [14] Kass, R.E. , Natarajan, R., 2006. A Default Conjugate Prior for Variance Components in Generalised Linear Mixed Models (Comment on Article by Browne and Draper). Bayesian Analysis, 1(3), 535-542.
• [15] Gelman, A., 2006. Prior Distributions for Variance Parameters in Hierarchical Models, International Society for Bayesian Analysis, 1, Number 3, pp. 515-534.
• [16] Chen M-H, Ibrahim JG, Shao Q-M, Weiss RE, 2003. Prior Elicitation for Model Selection and Estimation in Generalized Linear Mixed Models. Journal of Statistical Planning and Inference; 111:57–76. 586.
• [17] Tiao, G. C. and Tan, W., 1965. Bayesian analysis of random effect models in the analysis of variance. i. Posterior distribution of variance components. Biometrika, 51: 37-53.
• [18] Box, G.,Tiao, G., 1973. Bayesian inference in statistical analysis, John Wiley&Sons.
• [19] Demidenko, E., 2004. Mixed models: Theory and Applications. In: Wiley Series in Probability and Statistics. Hoboken, New Jersey.
• [20] Congdon, P., 2003. Applied Bayesian Modelling. John Wiley Sons, England Jeffreys, A., 1961, The Theory of Probability, Cambridge University Press, Cambridge 2nd Edition. .
• [21] Antoni, K., Beirlant, J., 2006. Actuarial statistics with generalized linear mixed models, Insurance Mathematics ve Economics.
• [22] Jeffreys, H., 1961. Theory of probability Oxford univ Press, New York.
• [23] Mehmet A. Cengiz, M. A., Terzi, E., Şenel, T. ve Murat, N., 2012. Lojistik Regresyonda Parametre Tahmininde Bayesci Bir Yaklaşım, AKU J. Sci.12 (2012), 011302 ,(15-22).
• [24] Gelman, A., 2005. Analysis of variance: why it is more important than ever (with discussion). Annals of Statistics.
• [25] Daniels, M.J., 1999.A prior for the variance in hierarchical models. Canadian Journal of Statistics, 27:569/580.
• [26] Gelman, A., Jakulin, A., Pittau, M. G., Su¸S. 2008. A weakly informative default prior distribution for logistic and other regression models. Annals of Applied Statistics, 1360-1383.
• [27] Spiegelhalter, David J.; Best, Nicola G.; Carlin, Bradley P.; Van der Linde, Angelika (2002). Bayesian measures of model complexity and fit (with discussion)". Journal of the Royal Statistical Society, Series B. 64 (4): 583–639. doi:10.1111/1467-9868.00353. JSTOR 3088806. MR 1979380.
• [28] SAS Institute Inc. 2011. SAS/STAT 9.3 User’s Guide. Cary, NC, 7745 s.,USA.