Bello Nora M, Steibel Juan P, Tempelman Robert J
Department of Animal Science, Michigan State University, East Lansing, 48824-1225, USA.
Biom J. 2010 Jun;52(3):297-313. doi: 10.1002/bimj.200900182.
Bivariate mixed effects models are often used to jointly infer upon covariance matrices for both random effects (u) and residuals (e) between two different phenotypes in order to investigate the architecture of their relationship. However, these (co)variances themselves may additionally depend upon covariates as well as additional sets of exchangeable random effects that facilitate borrowing of strength across a large number of clusters. We propose a hierarchical Bayesian extension of the classical bivariate mixed effects model by embedding additional levels of mixed effects modeling of reparameterizations of u-level and e-level (co)variances between two traits. These parameters are based upon a recently popularized square-root-free Cholesky decomposition and are readily interpretable, each conveniently facilitating a generalized linear model characterization. Using Markov Chain Monte Carlo methods, we validate our model based on a simulation study and apply it to a joint analysis of milk yield and calving interval phenotypes in Michigan dairy cows. This analysis indicates that the e-level relationship between the two traits is highly heterogeneous across herds and depends upon systematic herd management factors.
双变量混合效应模型常用于联合推断两种不同表型之间随机效应（u）和残差（e）的协方差矩阵，以研究它们关系的结构。然而，这些（协）方差本身可能还取决于协变量以及促进跨大量聚类借用强度的额外可交换随机效应集。我们通过嵌入两个性状之间u水平和e水平（协）方差重新参数化的额外混合效应建模层次，提出了经典双变量混合效应模型的层次贝叶斯扩展。这些参数基于最近流行的无平方根Cholesky分解，易于解释，每个参数都方便地促进了广义线性模型表征。使用马尔可夫链蒙特卡罗方法，我们基于模拟研究验证了我们的模型，并将其应用于密歇根奶牛产奶量和产犊间隔表型的联合分析。该分析表明，这两个性状之间的e水平关系在不同牛群中高度异质，并且取决于系统的牛群管理因素。
