# 臺灣博碩士論文加值系統

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 本文針對額外二項式變異量的模式建構與統計推論進行研究，主要以貝氏統計的觀點，區辨資料在未含影響因子或考慮影響因子之後的情況下是否存在額外二項式變異，並對其資料的模式、特性與分析過程加以探討。 在未含影響因子方面，本文以貝他 － 二項式模式來描述具有額外二項式變異的資料，此外，透過分析及模擬的結果可知，對於區辨資料是否存在額外二項式變異量，使用概似比檢定或貝氏因子皆為適宜有效的檢定方法，是以在模式的選擇上，可視情況配合運用。此外，在含有影響因子之額外二項式變異的資料架構中，目前在廣義線性模式有Williams 方法與隨機效應模式，除了簡述這兩個方法外，本文將以貝氏模式為主軸來對具有額外二項式變異之資料建構一個較適切的模式，並以貝氏因子做模式選擇的標準。 另外，由於額外二項式變異之模式為複雜的階層式模式，且貝氏因子亦牽涉到複雜的積分，時常會有計算上的困難，倘若這些積分無法以解析的方法求得真實解，將可藉由大樣本的估計方法、模擬的方式、或是兩者合併的方法求得積分的估計值。本文將採蒙地卡羅馬可夫鏈法（MCMC），以求得事後分配的樣本，透過此樣本便可對模式中所關心的事項作估計與推論。此外，將利用拉普拉斯法、Candidate’s estimate method、區域校正之拉普拉斯-梅托波里斯法以及最佳化區域校正之拉普拉斯—梅托波里斯法，來計算貝氏因子，並以做為模式選擇的依據。
 The focus of this paper is on inference of Bayesian models in extra-binomial variability among various groups of binomial trials. We discuss the modeling, its properties, and computational technique when dealing with data of extra-binomial variability. When there is no covariate involved, beta-binomial distribution may be considered due to its flexibility and conjugacy. When testing the existence of extra-binomial variability is of interest, the results based on simulation indicate that both the likelihood ratio test and Bayes factor are reliable and efficient methods. Furthermore, when there exist covariates and the logistic regression model is considered, the Williams method and random effect model are used to describe the extra-binomial variability. The Bayes factor is then used to test the existence of extra variability. However, to obtain the Bayes factor, one needs to carry out several integrations. In most situations, it is difficult to compute the integrals analytically. To overcome the difficulty in computation, there are several methods to be used, including MCMC method, Laplace approximations, Candidate’s estimate, Volume-corrected Laplace-Metropolis method, and Optimal volume-corrected Laplace-Metropolis method. We will demonstrate these approximations in two real applications.
 目錄 第一章、前言…………………………………………………………………1 第二章、額外二項式變異之模式與推論方法…………………………… 5 2.1 額外二項式變異之模式與其特性…………………………… 8 2.2 皮爾森卡方檢定（Pearson chi-square test）…………… 17 2.3 概似比檢定（Likelihood ratio test）…………………… 18 2.4 貝氏檢定（Bayesian hypothesis test）………………… 20 第三章、含解釋變數之額外二項式變異之模式與推論方法…………… 27 3.1 Williams 方法………………………………………………… 29 3.2 隨機效應模式（Random effect model）…………………… 37 3.3 貝氏模式（Bayesian model）……………………………… 40 第四章、實例與模擬的應用……………………………………………… 47 4.1 未含影響因子之額外二項式變異分析：以蜜蜂為例………47 4.2 含有影響因子之額外二項式變異分析：以末稍血管疾病手術為 例…………………………………………………………………52 4.3 模擬…………………………………………………………… 59 第五章、結論與討論……………………………………………………… 65 參考文獻…………………………………………………………………… 69 附錄一：拉普拉斯法（Laplace’s method）證明過程…………………71 附錄二：Candidate’s formula與參數的標準化程序………………… 76 附錄三：以區域校正之拉普拉斯－梅托波里斯法與最佳化區域校正之拉普 拉斯－梅托波里斯法估計積分值之S-PLUS程式……………… 78 附錄四：模擬之S-Plus 程式………………………………………………80
 Besag, J. (1989) A candidate’s formula: A curious result in Bayesian prediction. Biometrika, 76, 183.Collett, D. (1991) Modeling Binary Data, London: Chapman and Hall.Erkanli, A. (1994) Laplace Approximations for Posterior Expectations When the Mode Occurs at the Boundary of the Parameter Space, Journal of American Statistical Association, 89, 250-258.Gilks, W. R., Richardson, R., and Spiegelhalter, D. J. (1996) Markov Chain Monte Carlo in Practice, London: Chapman & Hall.Hsiao, C. K. (1997) Approximate Bayes Factors When a Mode Occurs on the Boundary, Journal of the American Statistical Association, 90, 656-663.Hsiao, C. K., Huang, S. Y., and Chang, C. W. (2001) Optimal Volume-Corrected Laplace-Metropolis Method for Computing Marginal Probability.Hsiao, C. K. and Kass, R. E. (2000）Bayesian Tests of Extra-binomial Variability.Jeffreys, H. (1961) Theory of Probability, 3rd edition, Oxford University Press.Jones, K. N. (1997) Analysis of pollinator foraging: tests for nonrandom behaviour, Functional Ecology, 11, 255-259.Kass, R. E. and Raftery, A. E. (1995) Bayes factors, Journal of American Statistical Association, 90, 773-795.Kass, R. E., Tierney, L., and Kadane, J. B. (1990) The Validity of Posterior Expansions Based on Laplace’s method, in Bayesian and Likelihood Methods in Statistics and Econometrics, eds. S. Geisser, J. S. Hodges, S. J. Press, and A. Zellner. New York: North-Holland.Kass, R. E., and Vaidyanathan, S. K. (1992) Approximate Bayes factors and orthogonal parameters, with application to testing equality of two binomial proportions, Journal of the Royal Statistical Society, Ser. B, 54, 129-144.Kass, R. E. and Wasserman, L. (1995) A Reference Bayesian Test for Nested Hypotheses and Its Relationship to the Schwarz Criterion, Journal of the American Statistical Association, 90, 928-934.Kass, R. E. and Wasserman, L. (1996) The Selection of Prior Distributions by Formal Rules’, Journal of the American Statistical Association, 91, 1343-1370.O’Hagan A (1994) Kedall’s Advanced Theory of Statistic, Volume 2B, Bayesian Inference. London: Edward Arnold.Risko, K. J. and Margolin, B. H. (1996) Some observations on detecting extra-binomial variability within the beta-binomial model, Statistics in Toxicology.Self, S. G. and Liang, K. Y. (1987) Asymptotic Properties of Maximum Likelihood Estimators and Likelihood Ratio Tests Under Nonstandard Conditions, Journal of the American Statistical Association, 82, 605-610.Spiegelhalter, D. J., Thomas, A., Best, H. G., and Gilks, W. R. (1995) BUGS: Bayesian Inference Using Gibbs Sampling, Version 0.50. MRC Biostatistics Unit, Cambridge.Tunis, S. R., Bass, E. B., and Steinberg, E. P. (1991) Adoption of angioplasty and associated trends in the surgical management of peripheral vascular disease: the Maryland experience, New England Journal of Medicine, 556-562.Williams, D. A. (1982) Extra-binomial Variation in Logistic Linear Model, Applied Statistics, 31, 144-148.
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