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

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.