Derivation and calculation of approximate reliabilities and daughter yield-deviations of a random regression test-day model for genetic evaluation of dairy cattle.

Test-day milk, fat, protein yield, and somatic cell score (SCS) were analyzed separately using data from the first 3 lactations and a random regression model. Data used in the model were from Austria, Germany, and Luxembourg and from Holstein, Red, and Jersey dairy cattle. For reliability approximation, a multiple-trait effective daughter contribution (MTEDC) method was developed under general multiple trait models, including random regression test-day models, by extending the single-trait daughter equivalents concept. The MTEDC was applied to the very large dairy population, with about 15.5 million animals. The calculation of reliabilities required less computer memory than the corresponding iteration program and a significantly lower computing time equivalent to 24 rounds of iteration. A formula for daughter-yield deviations was derived for bulls under multiple-trait models. Reliability associated with daughter-yield deviations was approximated using the MTEDC method. Both the daughter-yield deviation formula and associated reliability method were verified in a simulation study using the random regression test-day model. Correlations of lactation daughter-yield deviations with estimated breeding values calculated from a routine genetic evaluation were 0.996 for all bulls and 0.95 for young bulls having only daughters with short lactations.