Inge Riis Korsgaard

Bayesian mixed‐effects model analysis of a censored normal distribution with animal breeding applications

Adopting a Bayesian viewpoint, a method is presented for the analysis of censored records based on a Gaussian mixed effects model. The method uses the Gibbs sampler and data augmentation. Examples are presented using the motorette data of Schmee & Hahn (1979, Technometrics 21, 417–432) plus other simulated data sets that illustrate animal breeding applications.

A bivariate quantitative genetic model for a linear Gaussian trait and a survival trait

With the increasing use of survival models in animal breeding to address the genetic aspects of mainly longevity of livestock but also disease traits, the need for methods to infer genetic correlations and to do multivariate evaluations of survival traits and other types of traits has become increasingly important. In this study we derived and implemented a bivariate quantitative genetic model for a linear Gaussian and a survival trait that are genetically and environmentally correlated. For the survival trait, we considered the Weibull log-normal animal frailty model. A Bayesian approach using Gibbs sampling was adopted. Model parameters were inferred from their marginal posterior distributions. The required fully conditional posterior distributions were derived and issues on implementation are discussed. The twoWeibull baseline parameters were updated jointly using a Metropolis-Hastingstep. The remaining model parameters with non-normalized fully conditional distributions were updated univariately using adaptive rejection sampling. Simulation results showed that the estimated marginal posterior distributions covered well and placed high density to the true parameter values used in the simulation of data. In conclusion, the proposed method allows inferring additive genetic and environmental correlations, and doing multivariate genetic evaluation of a linear Gaussian trait and a survival trait.

A useful reparameterisation to obtain samples from conditional inverse Wishart distributions

A Bayesian joint analysis of normally distributed traits and binary traits, using the Gibbs sampler, requires the drawing of samples from a conditional inverse Wishart distribution. This is the fully conditional posterior distribution of the residual covariance matrix of the normally distributed traits and liabilities of the binary traits. Obtaining samples from the conditional inverse Wishart distribution is not straightforward. However, combining well-known matrix results and properties of the Wishart distribution, it is shown that this can be easily carried out by successively drawing from Wishart and normally distributed random variables.

Prediction error variance and expected response to selection, when selection is based on the best predictor – for Gaussian and threshold characters, traits following a Poisson mixed model and survival traits

In this paper, we consider selection based on the best predictor of animal additive genetic values in Gaussian linear mixed models, threshold models, Poisson mixed models, and log normal frailty models for survival data (including models with time-dependent covariates with associated fixed or random effects). In the different models, expressions are given (when these can be found – otherwise unbiased estimates are given) for prediction error variance, accuracy of selection and expected response to selection on the additive genetic scale and on the observed scale. The expressions given for non Gaussian traits are generalisations of the well-known formulas for Gaussian traits – and reflect, for Poisson mixed models and frailty models for survival data, the hierarchal structure of the models. In general the ratio of the additive genetic variance to the total variance in the Gaussian part of the model (heritability on the normally distributed level of the model) or a generalised version of heritability plays a central role in these formulas.

A bivariate quantitative genetic model for a threshold trait and a survival trait

Many of the functional traits considered in animal breeding can be analyzed as threshold traits or survival traits with examples including disease traits, conformation scores, calving difficulty and longevity. In this paper we derive and implement a bivariate quantitative genetic model for a threshold character and a survival trait that are genetically and environmentally correlated. For the survival trait, we considered the Weibull log-normal animal frailty model. A Bayesian approach using Gibbs sampling was adopted in which model parameters were augmented with unobserved liabilities associated with the threshold trait. The fully conditional posterior distributions associated with parameters of the threshold trait reduced to well known distributions. For the survival trait the two baseline Weibull parameters were updated jointly by a Metropolis-Hastings step. The remaining model parameters with non-normalized fully conditional distributions were updated univariately using adaptive rejection sampling. The Gibbs sampler was tested in a simulation study and illustrated in a joint analysis of calving difficulty and longevity of dairy cattle. The simulation study showed that the estimated marginal posterior distributions covered well and placed high density to the true values used in the simulation of data. The data analysis of calving difficulty and longevity showed that genetic variation exists for both traits. The additive genetic correlation was moderately favorable with marginal posterior mean equal to 0.37 and 95% central posterior credibility interval ranging between 0.11 and 0.61. Therefore, this study suggests that selection for improving one of the two traits will be beneficial for the other trait as well.

Analysis of rabbit doe longevity using a semiparametric log-Normal animal frailty model with time-dependent covariates

Data on doe longevity in a rabbit population were analysed using a semiparametric log-Normal animal frailty model. Longevity was defined as the time from the first positive pregnancy test to death or culling due to pathological problems. Does culled for other reasons had right censored records of longevity. The model included time dependent covariates associated with year by season, the interaction between physiological state and the number of young born alive, and between order of positive pregnancy test and physiological state. The model also included an additive genetic effect and a residual in log frailty. Properties of marginal posterior distributions of specific parameters were inferred from a full Bayesian analysis using Gibbs sampling. All of the fully conditional posterior distributions defining a Gibbs sampler were easy to sample from, either directly or using adaptive rejection sampling. The marginal posterior mean estimates of the additive genetic variance and of the residual variance in log frailty were 0.247 and 0.690.