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 ($ I$ ) to model daily measurement error and an autoregressive matrix ($ E$ ) to account for biological change. $ E$ is defined mathematically as $ E_{ij} = r^{\left\vert i-j\right\vert}$ where $ i$ and $ j$ are test day DIM and $ 0 < r < 1$ . The $ r$ 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

\begin{xtabular}{lll}
MFP & 0.998 & 0.987 \\
SCS & 0.992 & 0.995 \\
\end{xtabular}
The matrix of correlations within traits ($ B$ ) was calculated as: $ B = b_1I + b_2E$ where the $ b_i$ are regression coefficients; separate functions were used to model the yield traits and SCS. Intercept terms ($ b_0$ ) were included in the calculation of $ B$ for first-parity SCS and later-parity MFP in order to guarantee the positive-definiteness of the resulting correlation matrices. In subsequent sections $ M$ and $ S$ denote the functions used to calculate correlations among MFP and SCS, respectively. Suppose that $ T$ is a $ 4\times4$ matrix of phenotypic correlations among traits partitioned as:

$\displaystyle T = \begin{pmat}[{..\vert.}]
t_{MM} & t_{MF} & t_{MP} & t_{MS} \...
... = \begin{pmat}[{\vert}]
T_{m} & T_{ms} \cr\-
T_{sm} & T_{ss} \cr
\end{pmat}$

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

$\displaystyle C = \begin{pmat}[{\vert}]
M \otimes T_m & M^{\frac{1}{2}} \cdot ...
...rac{1}{2}} \cdot M^{\frac{1}{2}} \otimes T_{sm} & S \otimes T_s \cr
\end{pmat}$

where $ C_{ik,jl}$ = correlation of trait $ i$ at DIM $ k$ with trait $ j$ at DIM $ l$ , $ \otimes$ is the Kronecker (direct) product operator, and $ \cdot$ denotes element-wise matrix multiplication.

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