Multiplicative random regression model for heterogeneous variance adjustment in genetic evaluation for milk yield in Simmental

Abstract

A multiplicative random regression (M-RRM) test-day (TD) model was used to analyse daily milk yields from all available parities of German and Austrian Simmental dairy cattle. The method to account for heterogeneous variance (HV) was based on the multiplicative mixed model approach of Meuwissen. The variance model for the heterogeneity parameters included a fixed region x year x month x parity effect and a random herd x test-month effect with a within-herd first-order autocorrelation between test-months. Acceleration of variance model solutions after each multiplicative model cycle enabled fast convergence of adjustment factors and reduced total computing time significantly. Maximum Likelihood estimation of within-strata residual variances was enhanced by inclusion of approximated information on loss in degrees of freedom due to estimation of location parameters. This improved heterogeneity estimates for very small herds. The multiplicative model was compared with a model that assumed homogeneous variance. Re-estimated genetic variances, based on Mendelian sampling deviations, were homogeneous for the M-RRM TD model but heterogeneous for the homogeneous random regression TD model. Accounting for HV had large effect on cow ranking but moderate effect on bull ranking.

Across-country test-day model evaluations for Holstein, Nordic Red Cattle, and Jersey

“…The algorithm applied for solving the RRM evaluation model for EBV and the variance model for HV estimates was the same as explained in Lidauer et al (2008). Both models were solved by iterative methods.…”

“…Suitable residual variances for DNK and FIN were obtained by a calibration procedure during model development. Thus, the multiplicative model was solved and genetic variances were re-estimated from EBV by a full model sampling approach (Lidauer et al, 2008), to update the residual variances for DNK and FIN according to detected differences in genetic variance. The procedure was repeated until differences in across-country genetic variances were within ±1%.…”

Comparison of multiplicative heterogeneous variance adjustment models for genetic evaluations

“…4r 2 e þ n ij Àr ij 2 l is an overall mean; b 1j is the fixed effect of testyear-month j; b 2ip is the random effect of herd i 9 test-period p, where test-period p is either test-year k or test-year-month j, and is assumed to follow an AR (1) structure as described in Wade & Quaas (1993); e ij is the random residual of the variance model;ê ij has residuals for the mean model observations in stratum ij; n ij is the stratum size;r ij is an approximation of the loss in degrees of freedom due to estimation of fixed effects in the mean model (Lidauer et al 2008); andr 2 e is the residual variance applied in the mean model. The use of approximationr ij improves the estimation of s ij for strata with a low number of observations.…”

“…Robert-Grani e et al (1999) found an average autocorrelation of 0.69 for milk yield using the RGHM method, while Pena & Ibañez (2002) reported 0.90 using rMMM and 0.92 using RGHM, and Meuwissen et al (1996) 0.98 using MMM. In studies where an AR1-HTM structure was modelled for the variance model, Lidauer & M€ antysaari (2001) estimated an autocorrelation of 0.98 from Finnish Ayrshire test-day milk yield data, and Lidauer et al (2008) an autocorrelation of 0.9987 from Bavarian Fleckvieh test-day milk yield data. However, little has been reported in the literature about param-eter estimates for variance models applied to test-day milk yield data.…”

“…In production traits, HV across herds and time is increasing due to better environments and improved genetics (Wiggans & VanRaden 1991;Lidauer et al 2008). To improve the reliability of breeding value estimation, sophisticated test-day models have been developed and implemented in many national evaluation systems.…”