Equivalence of variance components between standard and recursive genetic models using LDL′ transformations

Recursive models are a category of structural equation models that propose a causal relationship between traits. These models are more parameterized than multiple trait models, and they require imposing restrictions on the parameter space to ensure statistical identification. Nevertheless, in certain situations, the likelihood of recursive models and multiple trait models are equivalent. Consequently, the estimates of variance components derived from the multiple trait mixed model can be converted into estimates under several recursive models through LDL′ or block-LDL′ transformations. The procedure was employed on a dataset comprising five traits (birth weight—BW, weight at 90 days—W90, weight at 210 days—W210, cold carcass weight—CCW and conformation—CON) from the Pirenaica beef cattle breed. These phenotypic records were unequally distributed among 149,029 individuals and had a high percentage of missing data. The pedigree used consisted of 343,753 individuals. A Bayesian approach involving a multiple-trait mixed model was applied using a Gibbs sampler. The variance components obtained at each iteration of the Gibbs sampler were subsequently used to estimate the variance components within three distinct recursive models. The LDL′ or block-LDL′ transformations applied to the variance component estimates achieved from a multiple trait mixed model enabled inference across multiple sets of recursive models, with the sole prerequisite of being likelihood equivalent. Furthermore, the aforementioned transformations simplify the handling of missing data when conducting inference within the realm of recursive models.