6.3 Modeling Correlations Among Test Day Yields

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.

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.

See