Abstract
A fully Bayesian analysis using Gibbs sampling and data augmentation in a multivariate model of Gaussian, right censored, and grouped Gaussian traits is described. The grouped Gaussian traits are either ordered categorical traits (with more than two categories) or binary traits, where the grouping is determined via thresholds on the underlying Gaussian scale, the liability scale. Allowances are made for unequal models, unknown covariance matrices and missing data. Having outlined the theory, strategies for implementation are reviewed. These include joint sampling of location parameters; efficient sampling from the fully conditional posterior distribution of augmented data, a multivariate truncated normal distribution; and sampling from the conditional inverse Wishart distribution, the fully conditional posterior distribution of the residual covariance matrix. Finally, a simulated dataset was analysed to illustrate the methodology. This paper concentrates on a model where residuals associated with liabilities of the binary traits are assumed to be independent. A Bayesian analysis using Gibbs sampling is outlined for the model where this assumption is relaxed.
Highlights
In a series of problems, it has been demonstrated that using the Gibbs sampler in conjunction with data augmentation makes it possible to obtain samplingbased estimates of analytically intractable features of posterior distributions.Gibbs sampling [9,10] is a Markov chain simulation method for generating samples from a multivariate distribution, and has its roots in the MetropolisHastings algorithm [11,19]
Bayesian inference using Gibbs sampling in an ordered categorical threshold model was considered by [1,24,34]
In censored Gaussian and ordered categorical threshold models, Gibbs sampling in conjunction with data augmentation [25,26] leads to fully conditional posterior distributions which are easy to sample from
Summary
In a series of problems, it has been demonstrated that using the Gibbs sampler in conjunction with data augmentation makes it possible to obtain samplingbased estimates of analytically intractable features of posterior distributions. In censored Gaussian and ordered categorical threshold models, Gibbs sampling in conjunction with data augmentation [25,26] leads to fully conditional posterior distributions which are easy to sample from. This was demonstrated in Wei and Tanner [33] for the tobit model [27], and in right censored and interval censored regression models. The outline of the paper is the following: in Section 2, a fully Bayesian analysis of an arbitrary number of Gaussian, right censored Gaussian, ordered categorical and binary traits is presented for the particular case where all animals have observed values for all traits, i.e. no missing values.
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