Assessing the Correlation Structure in Cow Udder Quarter Infection Times Through Extensions of the Correlated Frailty Model

Abstract

The association between infection times of the udder quarters of a dairy cow is essential information for the preventive control of udder infections in a dairy cow herd. Extensions of the correlated frailty model are proposed to investigate and compare different correlation structures among the four udder quarter infection times clustered within a cow. Such complex frailty models can be fitted with the SAEM-MCMC algorithm. It is demonstrated that substantial correlation exists between the udder quarter infection times, with the correlation within front and rear udder quarters being larger than between front and rear udder quarters. This signifies that an infected udder quarter is a risk factor for the other udder quarters, especially when the udder quarter is in the same region, i.e., front or rear.

Appendix

1.1 Expression of the Complete Log-Likelihood of the Correlated Frailty Model $\mathcal {M}_1$ with a Weibull Baseline Hazard

$$\begin{aligned} l_{N}(\mathbf{Y},\varvec{\Delta },\mathbf{b};\theta )= & {} \log (\lambda _0\gamma )\sum _{i=1}^{N} \sum _{j=1}^{4}\Delta _{ij}\\&+ \,(\gamma -1) \sum _{i=1}^{N} \sum _{j=1}^{4}\Delta _{ij}\log (Y_{ij}) \nonumber \\&+\,\sum _{i=1}^{N} \sum _{j=1}^{4} \; \; \Biggl ( \Delta _{ij}\Bigl (x_{ij}^t \beta + b_{ij}\Bigr ) -\lambda _0Y_{ij}^\gamma \exp (x_{ij}^t \beta + b_{ij}) \Biggr ) \nonumber \\&-\,\frac{N}{2} \log (\det (\Sigma )) -2N\log (2 \pi ) -\frac{1}{2}\sum _{i=1}^{N} b_{i}^t\Sigma ^{-1} b_{i}. \nonumber \end{aligned}$$

1.2 Expression of the Complete Log-Likelihood of the Correlated Frailty Model $\mathcal {M}_1$ with a Piecewise Constant Baseline Hazard

$$\begin{aligned} l_{N}(\mathbf{Y},\varvec{\Delta },\mathbf{b};\theta )= & {} \sum _{k=1}^K \log (\alpha _k) \sum _{i=1}^{N} \sum _{j=1}^{4}\Delta _{ij}\ \mathbbm {1}_{t_{k-1} \le Y_{ij}< t_k}\\&+\,\sum _{i=1}^{N} \sum _{j=1}^{4} \; \; \Delta _{ij}\ \Bigl (x_{ij}^t \beta + b_{ij}\Bigr ) \nonumber \\&-\, \sum _{i=1}^{N} \sum _{j=1}^{4} \sum _{k=1}^K \alpha _k \ (t_k - t_{k-1}) \ \exp \left( x_{ij}^t \beta + b_{ij}\right) \ \mathbbm {1}_{Y_{ij} \ge t_k} \nonumber \\&-\, \sum _{i=1}^{N} \sum _{j=1}^{4} \sum _{k=1}^K \alpha _k \ (Y_{ij} - t_{k-1}) \ \exp \left( x_{ij}^t \beta + b_{ij}\right) \ \mathbbm {1}_{t_{k-1} \le Y_{ij} < t_k} \nonumber \\&-\,\frac{N}{2} \log (\det (\Sigma )) -2N\log (2 \pi ) -\frac{1}{2}\sum _{i=1}^{N} b_{i}^t\Sigma ^{-1} b_{i}. \nonumber \end{aligned}$$