Modeling the effect of a disease on culling: An illustration of the use of time-dependent covariates for survival analysis

This study demonstrated five different approaches, with and without time-dependent covariates, to determine the effect of disease on culling. It was also of interest to determine whether the time of the disease had an effect on subsequent culling (i.e., whether disease should be treated as a time-dependent covariate). To this purpose, five separate models were studied: Models 1 through 4 were Cox proportional hazards models, and Model 5 was a Weibull model. Model 1 treated disease as a binary, time-independent covariate. Model 2 treated disease as a time-dependent covariate, and one change of status was assumed to occur at the time of disease. Model 3 also assumed that one change in status occurred at the time of disease, but the effect of that change was assumed to be different depending on when the disease occurred. Models 4 (Cox) and 5 (Weibull) assumed an interaction between the occurrence of disease (time of disease) and the occurrence of culling (time of culling). As an illustration, the effect of mastitis on culling was studied for 2998 Holstein dairy cows in 10 herds. Parity and previous 305-d milk yield were also included as covariates; the data were stratified by herd. For all models, mastitis was a significant factor for culling. The significance tests for the estimates from Models 4 and 5 demonstrated that the hazard of culling differed for different stages of lactation, depending on when mastitis had occurred and when its effect on culling occurred; that is, time dependence exists between time of mastitis and time of culling.