臺灣博碩士論文加值系統

摘 要
在對群集類別資料作分析時，找尋混合分配的參數估計是一個重要
的步驟。在Yang and Yu (1999)中曾利用最大概似(MLE)演算法，期望最
大概似(EM)演算法，分類最大概似(CML)演算法及模糊分類最大概似
(FCML)演算法估計多變量伯努力混合型的參數。這篇論文主要是將其
中EM, CML 及FCML 三種演算法推展至迴歸分析上以形容解釋變數對
反應變數的影響。此外，我們只著重於探討二元反應變數的資料，而這
也就是潛在類別類別羅吉斯迴歸模型的分析。接著利用電腦生成數值例
子並藉由推導出的演算法對潛在類別羅吉斯迴歸模型做參數估計，並討
論演算法之數值模擬結果的差異性。

Abstract
Mixtures of distributions are used to analyze the grouped categorical data. The
estimation of parameters is an important step for mixture distributions. According to
Yang and Yu (1999), they described maximum likelihood estimation (MLE) algorithm,
expection maximization (EM) algorithm, classification maximum likelihood (CML)
algorithm and fuzzy classification maximum likelihood (FCML) algorithm to estimate
the parameters of a mixture of multivariate Bernoulli distributions. In this paper, we
will extend EM, CML and FCML algorithms to regression analysis to describe the
effects of the explanatory variables on the response variable. This paper focus on
binary responses about the logistic regression analysis with a latent class model. We
then use the extend algorithms to estimate the parameters of the latent class logistic
regression model. The numerical comparisons are also made. Finally, we give
numerical results for these algorithms.

Contents
1. Introduction 1
2. A mixture model of distribution with latent class and fuzzy class 2
3. Logistic regression model 3
4. Latent class logistic regression analysis and its algorithm 4
4.1 A mixture model of logistic regression 5
4.2 Estimation 8
4.2.1 EM algorithm 9
4.2.2 CML algorithm 16
4.2.3 FCML algorithm 21
5. Numerical results 26
6. Conclusions 31
References 32

ReferenceAnderson, T. W. (1954), On estimation of parameters in latent structureanalysis. Psychometrika, 19, 1-10.Agresti, A. (1990), Categorical Data Analysis. New York: Wiley.DeSarbo, W. S., Wedel, M., and Bult, J. R. (1993), A latent class posi-tionregression model for heterogeneous count data. Journal of AppliedEconometrics, 8, 397-411.DeSarbo, W. S. and Wdel, M. (1995), A mixture likelihood approach forgeneralized linear models. Journal of Classification, 12, 21-55.Everitt, B. S. and Hand, D. J. (1981), Finite Mixture Distributions. NewYork: Chapman and Hall.Everitt,B.S. (1984), An Introduction to Latent Variable Models. NewYork: Chapman and Hall.Jobson, J. D. (1992), Applied Multivariate Data Analysis, Vol. 2. Cat-egoricaland Multivariate Methods. New York: Springer-Verlag.Lazarsfeld, P. F. (1950), The logical and mathematical foundation oflatent structure analysis. Psychometrika, 16, 151-166.McCullagh, P. and Nelder, J. A. (1989), Generalized Linear Models. NewYork : Chapman and Hall.McLachlan, G. J. and K. E. Basford (1988), Mixture Models: Inferenceand Aapplications to Clustering. New York: Marcel Dekker.Nelder, J. A. and Wedderburn, R. W. M. (1972), Generalized linearmodels. Journal of the Royal Statistical Society, Series A, 135, 370-384.Yang, M. S. (1993), On a class of fuzzy classification maximum likelihoodprocedures. Fuzzy Sets and Systems, 57, 365-375.Yang, M. S. and Yu, N. Y. (1999), On estimation of parameters in latentclass models using clustering algorithms. (submitted).Zadeh, L. A. (1965), Fuzzy sets. Inform. and control. 8, 338-353.33