Deriving covariance functions (CF) for Nordic Red dairy cattle yield evaluation model

Abstract

This study compares different covariance function (CF) structures regarding their fit to observed variance components and their effects on genomic estimated breeding values (GEBV) and validation outcomes. Parameter reduced CF have been utilized in the Nordic Red dairy cattle test-day model for production traits (9 traits, milk, protein, and fat for first three lactations) to reduce unknowns, smooth genetic variance curves, and decrease computational demand. For that model, applied CF were fitted separately to estimated variance-covariance matrices for genetic and non-hereditary effect, followed by a parameter reduction. Although the effect of applying parameter reduced CF on estimated breeding values has been studied, little is known about the effect on genomic prediction reliability. The base CF was derived similarly as that one used in the current Nordic test-day model, using a second-order Legendre polynomial combined with an exponential term (exp(-0.04d) on estimated variance components, and hereafter is named Legendre-Wilmink (LW) function (rank 36). The applied function was the same as used for estimating the original variance components. Two alternative CF were developed: a Wilmink function (rank 27) and a repeatability function (rank 9). CF dimensions were reduced by considering the largest eigenvalues explaining ≥99.5% of the variation. For LW, additional reductions considered 99.0, 98.0 or 97.0% of variance. Depending on function and variance retained, CF parameters for the genetic effect ranged from 5 to 20, and for non-hereditary effect from 6 to 25. The CF fit was assessed using Log Likelihood (LogL) values and daily heritability estimates, while the impact on GEBV was evaluated through linear regression (LR) method and correlations between yield deviations (YD) and GEBVs. Results indicated that considering even less variance (99.0 to 97.0%) for LW CF had only a minor effect on model fit (LogL), whereas applying the Wilmink or repeatability model caused more pronounced differences. Daily heritability estimates followed similar patterns: Wilmink produced slightly varying heritabilities across days compared to original values, while the repeatability model yielded a single heritability estimate for the entire lactation period. LR validation results, based on regression coefficients (b1 and coefficients of determination (R²), and correlations, showed that using the simpler functions when fitting CF is not the optimal choice for GEBV prediction accuracy. Validation results for milk yield showed that the repeatability model performed the weakest (b1 = 0.95, R² = 0.73), while the LW model CF that considered 99.5% of the variance achieved the best validation results (b1 = 0.98, R² = 0.78). Similar patterns were observed for protein and fat yields. Overall, parameter reduction within LW had negligible impact on reliability, but replacing LW with simpler CF structures compromised GEBV accuracy.