The non-linear effect (determined by the penalised partial-likelihood approach) of milk-protein concentration on time to first insemination in Belgian dairy cows

Abstract

The time to first insemination in dairy cows depends partly on the energy balance of the cow. Because milk-protein concentration is related to the energy balance, we investigated whether milk-protein concentration predicted the hazard of being inseminated. The main objective of the paper is to demonstrate that the relationship between milk-protein concentration and the hazard of being inseminated was not linear and that this non-linear relationship was modelled adequately using cubic–splines. The semiparametric Cox model was used to introduce protein concentration into the model as a time-varying covariate and additionally herd was added to the model as a frailty term to adjust for the clustering of the cows within a herd. We extended the penalised partial-likelihood technique to fit the frailty model with cubic–splines for the effect of the protein concentration. The model was fitted for a large database consisting of 5114 multiparous cows from 181 different farms. Low milk-protein concentration (<2.7%) was associated with a negative energy balance and this probably led to the decreased hazard. On the other hand, high milk-protein concentration (>4.0%) was linked with low milk production and it was probably a farmer’s decision not to inseminate such cows, leading to the observed decreased hazard.