BAYESIAN INFERENCE ON VARIANCE COMPONENTS IN GENERALIZED LINEAR MIXED MODELS: AN EVALUATION OF DIFFERENT METHODS

Generalized linear mixed models are now popular in the animal breeding and biostatistics literature as these models allow inference on fixed and random effects for the exponential family of data distributions. In animal breeding, particular attention is directed towards variances of the random effects. We investigate three methods for marginal inference on variance components in binary data, including (1) the conventional expectation-maximization (EM) type algorithm, (2) Laplace\u27s method, and (3) exact Gibbs sampling methods. A simulation study involving probit animal models was used to compare the modal estimates computed under the three methods. It was found that EM estimates were badly biased downwards in comparison to Laplacian estimates. An application of all methods within a repeated measures probit analysis of mastitis incidence in dairy cows suggests that Laplacian and Gibbs sampling posterior marginal modes are somewhat congruent in moderately sized data sets, although the tail of the posterior density was lighter for the Laplacian approximation

Statistical Analysis of Efficient Unbalanced Factorial Designs for Two-Color Microarray Experiments

Experimental designs that efficiently embed a fixed effects treatment structure within a random effects design structure typically require a mixed-model approach to data analyses. Although mixed model software tailored for the analysis of two-color microarray data is increasingly available, much of this software is generally not capable of correctly analyzing the elaborate incomplete block designs that are being increasingly proposed and used for factorial treatment structures. That is, optimized designs are generally unbalanced as it pertains to various treatment comparisons, with different specifications of experimental variability often required for different treatment factors. This paper uses a publicly available microarray dataset, as based upon an efficient experimental design, to demonstrate a proper mixed model analysis of a typical unbalanced factorial design characterized by incomplete blocks and hierarchical levels of variability

A general approach to mixed effects modeling of residual variances in generalized linear mixed models

We propose a general Bayesian approach to heteroskedastic error modeling for generalized linear mixed models (GLMM) in which linked functions of conditional means and residual variances are specified as separate linear combinations of fixed and random effects. We focus on the linear mixed model (LMM) analysis of birth weight (BW) and the cumulative probit mixed model (CPMM) analysis of calving ease (CE). The deviance information criterion (DIC) was demonstrated to be useful in correctly choosing between homoskedastic and heteroskedastic error GLMM for both traits when data was generated according to a mixed model specification for both location parameters and residual variances. Heteroskedastic error LMM and CPMM were fitted, respectively, to BW and CE data on 8847 Italian Piemontese first parity dams in which residual variances were modeled as functions of fixed calf sex and random herd effects. The posterior mean residual variance for male calves was over 40% greater than that for female calves for both traits. Also, the posterior means of the standard deviation of the herd-specific variance ratios (relative to a unitary baseline) were estimated to be 0.60 ± 0.09 for BW and 0.74 ± 0.14 for CE. For both traits, the heteroskedastic error LMM and CPMM were chosen over their homoskedastic error counterparts based on DIC values

Bayesian inference on genetic merit under uncertain paternity

A hierarchical animal model was developed for inference on genetic merit of livestock with uncertain paternity. Fully conditional posterior distributions for fixed and genetic effects, variance components, sire assignments and their probabilities are derived to facilitate a Bayesian inference strategy using MCMC methods. We compared this model to a model based on the Henderson average numerator relationship (ANRM) in a simulation study with 10 replicated datasets generated for each of two traits. Trait 1 had a medium heritability (h2) for each of direct and maternal genetic effects whereas Trait 2 had a high h2 attributable only to direct effects. The average posterior probabilities inferred on the true sire were between 1 and 10% larger than the corresponding priors (the inverse of the number of candidate sires in a mating pasture) for Trait 1 and between 4 and 13% larger than the corresponding priors for Trait 2. The predicted additive and maternal genetic effects were very similar using both models; however, model choice criteria (Pseudo Bayes Factor and Deviance Information Criterion) decisively favored the proposed hierarchical model over the ANRM model

HIERARCHICAL BAYESIAN METHODS TO MODEL HETEROGENEITY IN COW- AND HERD-LEVEL RELATIONSHIPS BETWEEN MILK PRODUCTION AND REPRODUCTION IN DAIRY COWS

Two of the most important broad classifications of phenotypes for successful dairy production are milk yield and fertility. The nature of the relationship between milk production and reproductive performance of dairy cows is uncertain due to conflicting results reported in many studies. A common deficiency in many such studies is an underappreciation of the dual dimension of the production-reproduction relationship, as defined by herd (random or u) level and cow (residual or e) level sources of (co)variation. Our overall hypothesis is that the e- and u- level relationships between milk production and reproduction in dairy cows are heterogeneous and depend upon various herd-related and management factors. Our objective is to develop hierarchical Bayesian extensions that capture heterogeneity in the relationships between traits by mixed effects modeling of u level and e level covariances between traits of interest. We specify a bivariate Bayesian model to jointly model two continuous traits and we apply a square-root free Cholesky decomposition to the variance-covariance matrices of the residuals (cow-level) and random effects (herd-level). As a result, the e- and u-level covariances among the traits are reparameterized into unconstrained and easily interpretable e- and u- regression parameters, respectively. These regression parameters specify the cow- and herd-level relationships, respectively, between the traits and can be easily modeled as functions of relevant fixed and random effects, thereby providing a mixed model extension of Pourahmadi’s method. We validate our method using a simulation study and apply it to data on 305-day milk yield and calving interval of Michigan dairy cows

Cumulative t-link threshold models for the genetic analysis of calving ease scores

In this study, a hierarchical threshold mixed model based on a cumulative t-link specification for the analysis of ordinal data or more, specifically, calving ease scores, was developed. The validation of this model and the Markov chain Monte Carlo (MCMC) algorithm was carried out on simulated data from normally and t4 (i.e. a t-distribution with four degrees of freedom) distributed populations using the deviance information criterion (DIC) and a pseudo Bayes factor (PBF) measure to validate recently proposed model choice criteria. The simulation study indicated that although inference on the degrees of freedom parameter is possible, MCMC mixing was problematic. Nevertheless, the DIC and PBF were validated to be satisfactory measures of model fit to data. A sire and maternal grandsire cumulative t-link model was applied to a calving ease dataset from 8847 Italian Piemontese first parity dams. The cumulative t-link model was shown to lead to posterior means of direct and maternal heritabilities (0.40 ± 0.06, 0.11 ± 0.04) and a direct maternal genetic correlation (-0.58 ± 0.15) that were not different from the corresponding posterior means of the heritabilities (0.42 ± 0.07, 0.14 ± 0.04) and the genetic correlation (-0.55 ± 0.14) inferred under the conventional cumulative probit link threshold model. Furthermore, the correlation (> 0.99) between posterior means of sire progeny merit from the two models suggested no meaningful rerankings. Nevertheless, the cumulative t-link model was decisively chosen as the better fitting model for this calving ease data using DIC and PBF

Sexual differentiation of the zebra finch song system: potential roles for sex chromosome genes

Abstract Background Recent evidence suggests that some sex differences in brain and behavior might result from direct genetic effects, and not solely the result of the organizational effects of steroid hormones. The present study examined the potential role for sex-biased gene expression during development of sexually dimorphic singing behavior and associated song nuclei in juvenile zebra finches. Results A microarray screen revealed more than 2400 putative genes (with a false discovery rate less than 0.05) exhibiting sex differences in the telencephalon of developing zebra finches. Increased expression in males was confirmed in 12 of 20 by qPCR using cDNA from the whole telencephalon; all of these appeared to be located on the Z sex chromosome. Six of the genes also showed increased expression in one or more of the song control nuclei of males at post-hatching day 25. Although the function of half of the genes is presently unknown, we have identified three as: 17-beta-hydroxysteroid dehydrogenase type IV, methylcrotonyl-CoA carboxylase, and sorting nexin 2. Conclusion The data suggest potential influences of these genes in song learning and/or masculinization of song system morphology, both of which are occurring at this developmental stage

Analysis of social interactions in group-housed animals using dyadic linear models

Understanding factors affecting social interactions among animals is important for applied animal behavior research. Thus, there is a need to elicit statistical models to analyze data collected from pairwise behavioral interactions. In this study, we propose treating social interaction data as dyadic observations and propose a statistical model for their analysis. We performed posterior predictive checks of the model through different validation strategies: stratified 5-fold random cross-validation, block-by-social-group cross-validation, and block-by-focal-animals validation. The proposed model was applied to a pig behavior dataset collected from 797 growing pigs freshly remixed into 59 social groups that resulted in 10,032 records of directional dyadic interactions. The response variable was the duration in seconds that each animal spent delivering attacks on another group mate. Generalized linear mixed models were fitted. Fixed effects included sex, individual weight, prior nursery mate experience, and prior littermate experience of the two pigs in the dyad. Random effects included aggression giver, aggression receiver, dyad, and social group. A Bayesian framework was utilized for parameter estimation and posterior predictive model checking. Prior nursery mate experience was the only significant fixed effect. In addition, a weak but significant correlation between the random giver effect and the random receiver effect was obtained when analyzing the attacking duration. The predictive performance of the model varied depending on the validation strategy, with substantially lower performance from the block-by-social-group strategy than other validation strategies. Collectively, this paper demonstrates a statistical model to analyze interactive animal behaviors, particularly dyadic interactions