Maximum Likelihood Estimation for Probit-Linear Mixed Models with Correlated Random Effects

Abstract

The probit-normal model for binary data (McCulloch, 1994, Journal of the American Statistical Association 89, 330-335) is extended to allow correlated random effects. To obtain maximum likelihood estimates, we use the EM algorithm with its h/I-step greatly simplified under the assumption of a probit link and its E-step made feasible by Gibbs sampling. Standard errors are calculated by inverting a Monte Carlo approximation of the information matrix rather than via the SEM algorithm. A method is also suggested that accounts for the Monte Carlo variation explicitly. As an illustration, we present a new analysis of the famous salamander mating data. Unlike previous analyses, we find it necessary to introduce different variance components for different species of animals. Finally, we consider models with correlated errors as well as correlated random effects.

QTL mapping in outbred half-sib families using Bayesian model selection

“…Bayesian mapping in half-sib families M Fang et al Sorensen et al, 1995). Then the conditional posterior of the thresholds also follows uniform distribution, fðt j y; w; j t jþ1 ; t jÀ1 Þ ¼ 1 min½minðy w ¼ j þ 1 j Þ; t jþ1 À max½maxðy w ¼ j j Þ; t jÀ1 ðB3Þ where min (y|w ¼ j þ 1) indicates the minimum value of the liabilities within observations in category j þ 1; similarly, max (max (y|w ¼ j)) denotes the maximum value of liabilities for observations in category j (Albert and Chib, 1993;Sorensen et al, 1995 where, y Ài denotes all elements except the ith (Chan and Kuk, 1997).…”

On the Multivariate Probit Model for Exchangeable Binary Data with Covariates

“…The likelihood given the observed data is now obtained by integrating out the latent variables, and its expression involves high-dimensional integrals which are not numerically tractable. To alleviate the computational difficulties of the likelihood approach, Chan and Kuk (1997) use the EM algorithm for estimation with non-exchangeable data. We propose instead the use of data augmentation as advocated by Tanner and Wong (1987) and Albert and Chib (1993).…”

“…Ochi and Prentice, 1984;Chan and Kuk, 1997;McCulloch, 1994), but they are computationally difficult due to the intractability of the expressions obtained by integrating out the latent variables (Chib and Greenberg, 1998). As an alternative, we propose the use of a Bayesian framework for estimation of the model parameters.…”

Copula analysis of mixture models

“…The theory is applied to a bivariate (temperature and humidity) climatological data set in Section 5.1, and compared with results obtained from the EM algorithm method in Section 5.2. Questions of identifiability have been studied by, e.g, Bock and Gibbons (1996), Chan and Kuk (1997), and Kuk and Chan (2001), with Kuk and Chan (2001) showing that when an identifiability problem exists, implementing the unconstrained EM algorithm is valid and that the loss of uniqueness of the estimates is usually not a major issue.…”