| Summary: | 碩士 === 國立臺北大學 === 統計學系 === 102 === Many recent studies often observe the response variables repeatedly to understand the influence of certain conditions longitudinally. The general linear model and generalized linear model for longitudinal data are used to make inference of this kind of data. Since the response variable is observed repeatedly, the model settings and estimations would need the multivariate distribution. Many continuous multivariate distributions have been proposed in the literatures. However, owing to the complexity of describing the association among the multivariate discrete random variables, it is lack of the well-known distribution. To estimate the parameter in the generalized linear model for longitudinal data, the generalized estimating equation (GEE) proposed by Liang and Zeger (1986) is a commonly used estimating method.
Based on the construction of the bivariate binomial distribution proposed by Biswas and Hwang (2002) and incorporating the first-order Markov chain to describe the association among the repeat measures, this paper proposes a new multivariate distribution for binary data. The basic properties of this distribution are proved and illustrated numerically. Also, the asymptotic properties of the maximum likelihood estimators (MLEs) of the parameters are discussed. Furthermore, the proposed multivariate distribution is used to model the generalized linear model for longitudinal data and the corresponding model parameters are obtained by the maximum likelihood estimation. Finally, the Monte Carlo simulations are conducted to evaluate the performance of MLE of parameters for the proposed multivariate distribution and the generalized linear model for longitudinal data. For the latter case, the performance is also compared with the current GEE approach.
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