Bayesian Analysis for Categorical Data with Missing Traits Under a Multivariate Threshold Animal Model

Genetic variance and covariance components of the linear traits and the ordered categorical traits, that are usually observed as dichotomous or polychotomous outcomes, were simultaneously estimated in a multivariate threshold animal model with concepts of arbitrary underlying liability scales with Bayesian inference via Gibbs sampling algorithms. A multivariate threshold animal model in this study can be allowed in any combination of missing traits with assuming correlation among the traits considered. Gibbs sampling algorithms as a hierarchical Bayesian inference were used to get reliable point estimates to which marginal posterior means of parameters were assumed. Main point of this study is that the underlying values for the observations on the categorical traits sampled at previous round of iteration and the observations on the continuous traits can be considered to sample the underlying values for categorical data and continuous data with missing at current cycle (see appendix). This study also showed that the underlying variables for missing categorical data should be generated with taking into account for the correlated traits to satisfy the fully conditional posterior distributions of parameters although some of papers (Wang et al., 1997; VanTassell et al., 1998) presented that only the residual effects of missing traits were generated in same situation. In present study, Gibbs samplers for making the fully Bayesian inferences for unknown parameters of interests are played rolls with methodologies to enable the any combinations of the linear and categorical traits with missing observations. Moreover, two kinds of constraints to guarantee identifiability for the arbitrary underlying variables are shown with keeping the fully conditional posterior distributions of those parameters. Numerical example for a threshold animal model included the maternal and permanent environmental effects on a multiple ordered categorical trait as calving ease, a binary trait as non-return rate, and the other normally distributed trait, birth weight, is provided with simulation study.

다형질 Threshold 개체모형에서 Missing 기록을 포함한 이산형 자료에 대한 Bayesian 분석

한우의 근내지방도 또는 임신 여부 등과 같이 이산형 분포의 성질을 갖는 다수의 형질들에 대한 유전모수 및 종축의 유전능력을 평가하기 위한 방법으로써 Threshold 모형하에서 Bayesian 추론방법의 일종인 Gibbs sampling방법을 모의실험을 통하여 알아보았으며 기록이 누락된 다수의 형질을 포함하는 다형질 Threshold 개체모형에서의 종축평가 방법론을 제시하였다. 이산형 형질의 관측치에 대응하는 임의의 잠재변수는 기록을 갖고 있는 형질들에 대한 사전정보를 고려한 사후조건확률분포에서 Gibbs sampling을 할 때 모수에 근접하는 확률분포를 얻을 수 있었으며 이러한 이산형 기록들에 대한 육종가 추정치는 선형모형에서 보다 Threshold 모형에서의 추정치가 실제 모수에 더욱 근접하는 것을 알 수 있었다. 따라서 기록이 누락된 개체들에 대한 이산형 분포를 갖는 형질들에 대하여 선형분포를 갖는 형질들과 함께 동시 유전분석할 때 Threshod 모형이 일반 선형모형 보다 적합함을 알 수 있었다.