Multivariate estimation of genetic parameters – Quo vadis?

INTRODUCTION Estimation of genetic parameters is one of the basic tasks in quantitative genetics. As recording schemes become more sophisticated and breeding objectives more detailed, the number of traits of interest is increasing continually. This necessitates multivariate analyses considering more than just a few traits simultaneously. Fortunately, we are at a stage were advances in modelling, computational algorithms and the corresponding software for estimation, paired with modern day computer hardware are bringing large-scale analyses comprising numerous traits and records on tens of thousands of animals within the realms of reality. For example, Tyrisevä et al. (2011) recently presented a 25-trait analysis involving more than 100 000 sires. However, comparatively little attention has been paid to the problems of sampling variation inherent in multivariate analyses comprising multiple traits. It is well known that the eigenvalues of estimated covariance matrices are systematically over-dispersed (Lawley 1956) and that a large proportion of the sampling variances of genetic parameter estimates can be attributed to this excess variation. Moreover, the effects of this phenomenon increase dramatically with the number of traits. Hence, even multi-dimensional analyses based on relatively large data sets are likely to yield imprecise estimates. At the other end of the spectrum, we have numerous scenarios where the numbers of records are invariably limited. This includes records for new traits of interest or traits which are difficult or expensive to measure but which may have substantial impact on selection decisions in livestock improvement programmes. Typical examples are carcass characteristics of beef cattle. Similarly, evolutionary biologist concerned with quantitative genetics of natural populations are usually restricted to small samples. Hence, any avenue to ‘improve’ estimates, i.e. to obtain estimates which are on average closer to the population values, should be carefully considered. On the one hand, we have accumulated a substantial body of knowledge about genetic parameters for various traits. However, typically this is completely ignored. While the Bayesian paradigm directly provides the means to incorporate such prior information, analyses concerned with the estimation of covariance components more often than not assume flat or uninformative priors (Thompson et al. 2005). On the other hand, statistical techniques are available – often referred to as regularization methods – which substantially reduce sampling variance, albeit at the expense of introducing some bias, and thus yield ‘better’ estimates. Interest in regularized estimation for multivariate analyses dates back to the Seventies and earlier, stimulated in particular by the work of Stein (e.g. James and Stein 1961; Stein 1975). Recently, there has been a resurgence in attention with applications for estimation in very high-dimensional settings, in particular for genomic data (e.g. Warton 2008; Yap et al. 2009; Witten and Tibshirani 2009). This paper reviews the principles involved and examines the scope for adapting such techniques to estimation of genetic parameters for continuous traits in a mixed model framework. A penalized maximum likelihood scheme and suitable penalties are presented together with an application. *AGBU is a joint venture of NSW Department of Industry and Investment and the University of New England