Analysis of Litter Size and Average Litter Weight in Pigs Using a Recursive Model

Abstract

An analysis of litter size and average piglet weight at birth in Landrace and Yorkshire using a standard two-trait mixed model (SMM) and a recursive mixed model (RMM) is presented. The RMM establishes a one-way link from litter size to average piglet weight. It is shown that there is a one-to-one correspondence between the parameters of SMM and RMM and that they generate equivalent likelihoods. As parameterized in this work, the RMM tests for the presence of a recursive relationship between additive genetic values, permanent environmental effects, and specific environmental effects of litter size, on average piglet weight. The equivalent standard mixed model tests whether or not the covariance matrices of the random effects have a diagonal structure. In Landrace, posterior predictive model checking supports a model without any form of recursion or, alternatively, a SMM with diagonal covariance matrices of the three random effects. In Yorkshire, the same criterion favors a model with recursion at the level of specific environmental effects only, or, in terms of the SMM, the association between traits is shown to be exclusively due to an environmental (negative) correlation. It is argued that the choice between a SMM or a RMM should be guided by the availability of software, by ease of interpretation, or by the need to test a particular theory or hypothesis that may best be formulated under one parameterization and not the other.M IXED linear models (Henderson 1984) are broadly used to predict breeding values and to estimate variance components for traits of interest in livestock and plant breeding and play an important role in evolutionary and theoretical quantitative genetics (Lande 1979;Cheverud 1984;Walsh 2003). In genetic improvement programs, the objective of selection includes typically several correlated traits. The classical approach for a multiple-trait analysis is to use models posing that the nature of the correlation between response variables (phenotypes) is due to linear associations between unobservables, such as additive genetic values or nongenetic sources, like permanent or temporary environmental effects.Structural equation models represent an extension of the standard linear model to account for links (feedback and/or recursiveness) involving either the phenotypes directly or latent variables; they are well established in econometrics and sociology (Goldberger 1972; Jöreskog 1973;Duncan 1975). These models were discussed in the early genetics literature by Wright (1921) but this work has not received much attention in quantitative genetics. Recently, Xiong et al. (2004) proposed the use of structural equation models for modeling and identifying genetic networks. In a quantitative genetics context, Gianola and Sorensen (2004) studied the consequences of the existence of simultaneous and recursive relationships between phenotypes on genetic parameters and presented statistical methods for inference. A recent application to study the relationship between somatic cell score and milk yiel...

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