- 學位論文
變異數成份在廣義線性混合模式下之貝氏統計推論
Bayesian Inference of Variance Components in Generalized Linear Mixed Models
摘要
在生物醫學統計中常接觸到具有相關性的資料，這些相關性可能是基因、環境、重複測量或是時間所造成的；在這一類型的研究中，我們通常不只對治療的效用有興趣，觀測值的變異以及觀測值之間的相關性亦可能是我們所關注的量，這些變異或相關形式都可以被稱為變異數成份(variance components)。 傳統的統計方法會使用有限制最大概似(restricted maximum likelihood, REML)估計法來估計變異數成份；貝氏統計分析則利用後驗分配來作推論。在廣義線性混合模式下，我們證明了在樣本數很大時，隨機效應之變異數成份的後驗眾數和REML估計值會近似相等(asymptotically equivalent)，同時也建立了後驗眾數和REML估計值的近似關係式，並提出了比較二種後驗眾數與REML估計值之間遠近的準則。我們利用此準則證明了以approximate Jeffreys’ prior為變異數成份的先驗分配，會比以approximate uniform shrinkage為變異數成份的先驗分配所得到的後驗眾數較接近REML估計值；而在有限樣本的模擬結果也有同樣的結果。最後，本文提出了連續型和離散型的兩個實例分析。
並列摘要
Longitudinal and correlated data are commonly modeled with generalized linear mixed models (GLMM) which contain both fixed and random effects. The source of the random effect may come from the genetic heredity, familial aggregation, or environmental heterogeneity. The inference of its variance component is usually difficult due to the dimension of its covariance matrix (more than one random effect) and the complexity of the likelihood function. In this research, we discuss the restricted maximum likelihood (REML) estimation of variance components, and focus mainly on the Bayesian approach with posterior distribution. We will demonstrate the specification of reference prior distribution on the random covariance matrix. We will consider the approximate uniform shrinkage prior and approximate Jeffreys’ prior. Both are formulated based on the approximated likelihood function. Under generalized linear mixed models, we show that the posterior mode under Jeffrey’s prior is asymptotically closer to the REML estimate than the mode under uniform shrinkage prior does. In fact, the relative distance converges to a positive integer for any square random matrix. We also conduct the formal Bayesian inference of the variance components using posterior samples obtained by Markov chain Monte Carlo method. Finally, we consider two real applications and simulation studies for the purpose of illustration.