Establish confidence intervals for daily milk yield measures by robust bootstrap

Prior to statistical analyses of daily milk yield data, outliers due to equipment malfunction or confirmed milk recording errors should be removed. However, those outliers that are caused by health status, body condition, stress, energy balance, or BST would be valid data. Hence, using the central limit theorem (CLT) to establish confidence intervals (CI) for yield measures could be misleading. The ”ordinary” bootstrap performs poorly in these situations. This study demonstrates the use of a robust bootstrap resampling algorithm to construct CI for daily milk yield. The double bootstrap algorithm advances the notion that CI can be constructed from a function of the sample and the mean whose distribution is independent of the mean, the sample, or any other unknown parameter using pivotal quantities. In the algorithm, the mean of the data is computed. Then the first stage bootstrap sample (F) of size n is obtained from the observed data, with replacement (WR). The difference between the mean of F and the mean of the observed data is divided by the SE of the mean (SEM) of F, is a pivotal quantity that provides a robust bootstrap-t distribution of the mean daily yield. Then, the second stage bootstrap sample (G) of size n is randomly drawn WR from F. The difference between the mean of this bootstrap sample and the mean of F is now divided by the SEM of G. The first and second stage bootstraps are repeated B and K times, respectively. The CI for the mean can be obtained from the percentiles of the bootstrap distributions. Daily milk records for 89 first lactation cows from a Michigan herd were used for demonstration with B=500 and K=500. The distribution was skewed to the right at peak and in late lactation. The 95% CI given by the CLT were widest. The ordinary bootstrap gave narrow CI while the bootstrap-t and double bootstrap methods gave relatively stable CIs. After computing 99% confidence intervals using this approach, data that do not fall within the limits of that interval could be removed prior to statistical analysis.