# 6.3 Modeling Correlations Among Test Day Yields

Norman et al. NormanVRWS1999 used daily milk weights for Canadian cows and monthly test records of U.S. cows to estimate phenotypic correlations between test days within herd-year. They found that correlations between daily yields for a designated interval between test days generally were highest for mid-lactation and were lowest for early and late lactation. Seven possible sources of variation were examined and the resulting models for first- and later parities each included 3 sources of variation. The addition of more terms to the model generally improved fit, but gains often were very small. In addition to updating autoregressive parameters to account for changes in the shape of lactation curves over the last several years a simpler model for computing correlations matrices was developed.

Cole et al. ColeVRD2007 estimated correlations among test day yields using a simplified model that included an identity matrix ( ) to model daily measurement error and an autoregressive matrix ( ) to account for biological change. is defined mathematically as where and are test day DIM and . The for MFP and SCS were estimated separately for first- and later-parities (Table 6.2), and values for MFP were slightly larger than previous estimates [Norman, VanRaden, Wright, and SmithNorman et al.1999] due to the inclusion of the identity matrix. Parameters were not previously calculated for SCS separately from MFP.

Autoregressive parameters used in modeling correlations among test days & First & Later

The matrix of correlations within traits ( ) was calculated as: where the are regression coefficients; separate functions were used to model the yield traits and SCS. Intercept terms ( ) were included in the calculation of for first-parity SCS and later-parity MFP in order to guarantee the positive-definiteness of the resulting correlation matrices. In subsequent sections and denote the functions used to calculate correlations among MFP and SCS, respectively. Suppose that is a matrix of phenotypic correlations among traits partitioned as:

where is the phenotypic correlation of trait with trait . The complete correlation matrix ( ) can then be obtained as:

where = correlation of trait at DIM with trait at DIM , is the Kronecker (direct) product operator, and denotes element-wise matrix multiplication.