階層概似函數方法於廣義線性混合模式之應用

Applications of the Hierarchical Likelihood Method in Generalized Linear Mixed Models

蔡秒玉(Miao-Yu Tsai) ； 李易儒(Yi-Ru Li)

《中國統計學報》 50卷4期 (2012/12) Pp. 199-219

https://doi.org/10.29973/JCSA.201212.0002

摘要

廣義線性混合模式（generalized linear mixed models; GLMMs）是一個相關性資料所常用的模式，若隨機效應參數較多，則此模式的邊際概似函數就必須以高維度積分的方式求得，因此參數估計則顯得相當複雜。Lee & Nelder（1996）提出hierarchical-likelihood的方法應用於結構複雜的模式，同時可避免困難的積分運算，但此方法必須配合使用牛頓演算法來估計固定效應及變異數成份，也因此在某些情況下會有參數估計值不收斂的問題，本文將使用牛頓演算法的修正式來進行改善。在模擬研究中，本文以hierarchical-likelihood以及peneralized quasilikelihood（PQL）的方法估計廣義線性混合模式中變異數成份的最大概似估計值以及有限制的最大概似估計值，從結果可以發現hierarchicallikelihood的方法比PQL來得穩定而且精準；在雙胞胎家庭近視資料實例中，本文亦使用hierarchical-likelihood的方法進行分析，從結果可知在調整了環境與視力相關變項後，顯示環境與遺傳因子皆與近視度數息息相關。

關鍵字

廣義線性混合模式 ； 階層概似函數 ； 牛頓演算法 ； peneralized quasi likelihood

並列摘要

Generalized linear mixed model (GLMM) is a common class of models for correlated data. However, the estimation of parameters in GLMMs is very complicated because the marginal likelihood often requires intractable high-dimensional integrations. Therefore, many current approaches rely on approximation methods for computation. The hierarchical likelihood (h-likelihood) method (Lee and Nelder 1996) can avoid such burdensome numerical integration and be applied to models with complex structures. However, it requires Newton's method to complete the parameter estimation. It is possible that Newton's method may lead to difficulties when implementing the h-likelihood method. In this article, we propose the modified Newton's algorithm to avert the tendency to fail to converge. In simulation studies, we use the h-likelihood and peneralized quasi likelihood (PQL) methods to obtain the maximum likelihood (ML) and restricted maximum likelihood (REML) estimates of variance components in GLMMs. The result shows that the performance of h-likelihood is more stable and accurate than that of PQL. Finally, we illustrate the h-likelihood method to a myopia twin study and the result reveals the importance of the genetic and environmental effects associated with myopia after adjusting for environmental and ocular covariates.

並列關鍵字

Generalized linear mixed models ； Hierarchical likelihood ； Newton-Raphson ； Peneralized quasi likelihood