Use of sequential estimation of regressions and effects on regressions to solve large multitrait test-day models

An alternative algorithm for the solution of random regression models for analysis of test-day yield was developed to allow use of those models with extremely large data sets such as the US database for dairy records. Equations were solved in two iterative steps: 1) estimation or update of regression coefficients based on test-day yields for a given lactation and 2) estimation of fixed and random effects on those coefficients. Solutions were shown to be theoretically equivalent to traditional solutions for this class of random regression models. In addition to the relative simplicity of the proposed method, it allows several other techniques to be applied in the second step: 1) a canonical transformation to simplify computations (uncorrelated regressions) that could make use of recent advances in solution algorithms that allow missing values, 2) a transformation to limit the number of regressions and to create variates with biological meanings such as lactation yield or persistency, 3) more complicated (co)variance structures than those usually considered in random regression models (e.g., additional random effects such as the interaction of herd and sire), and 4) accommodation of data from 305-d records when no test-day records are available. In a test computation with 176,495 test-day yields for milk, fat, and protein from 22,943 first-lactation Holstein cows, a canonical transformation was applied, and the biological variates of 305-d yield and persistency were estimated. After five rounds of iteration with a sequential solution scheme for the two-step algorithm, maximum relative differences from previous genetic solutions were <10% of corresponding genetic standard deviations; correlations of genetic regression solutions with solutions from traditional random regression were >0.98 for 305-d yield and >0.99 for persistency.