Estimation & Prediction

Relationship between (co)variance components on the liability and observed scales for binary traits

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Abstract

The genomic evaluations for binary traits using threshold models often face challenges related to poor convergence and high computing time, particularly when data structure is complex. In such situations, an approximate analysis based on linear models can be used, since a linear model represents a first-order Taylor linearization of the threshold model. This linearization can be obtained by expanding the liability around the expectations of the random effects in the model. For example, under an animal model with direct () and permanent environmental () random effects, the binary phenotype can be approximated as: , where and are the cumulative distribution and probability density functions of a standard normal random variable evaluated at the population mean liability (). This approximation stablishes the connection between random effects on the liability and observed scales. A key implication of this framework is that the (co)variance components of the linearized model cannot be accurately estimated directly from the observed phenotypes, because binary variables have prevalence-dependent variances and covariances. Instead, the appropriate (co)variances must be derived from those on the liability scale using the linearization factors (). Without this transformation, fitting a linear model would yield biased variance and covariance estimates that do not correspond to the underlying liability model. Thus, accurately specifying the (co)variances of the linear approximation requires explicit formulas linking the two scales. Although the relationships between variances on the two scales is well known, the corresponding relationships for covariances, both within a trait and across traits have not been formally described. Our objective in this study was to derive formulas linking covariances on liability and observed scales under several common scenarios encountered in genomic evaluations of binary traits. Analogous to variances, covariances on the observed scale can be derived as the product of and the covariance on the liability scale for correlated random effects within a trait. For two binary traits, the covariance between random effects on the observed scale is given by multiplied by the covariance on the liability scale. We also present results for covariances between random effects for a binary and a continuous trait, as well as formulas for the residual covariances. These derivations provide the necessary (co)variance components for implementing approximate linear model analyses when threshold models are computationally prohibitive. They enable the construction of a broader range of genomic evaluation models for binary traits and facilitate efficient large scale genomic evaluations.

Keywords: 2026

How to Cite:

Bussiman, F., de Oliveira Padilha, D., Gianola, D., Hidalgo, J., Jarquin, D., Lourenco, D., Misztal, I. & Pérez, P., (2026) “Relationship between (co)variance components on the liability and observed scales for binary traits”, World Congress on Genetics Applied to Livestock Production Digital Archive 2026(1): 2287031. doi: https://doi.org/10.31274/wcgalp.24198

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