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Modeling interval-censored, clustered cow udder quarter infection times through the shared gamma frailty model

Time to infection data are often simultaneously clustered and interval-censored. The time to infection is not known exactly; it is only known to have occurred within a certain interval. Moreover, observations often occur in clusters. Consequently, the independence assumption does not hold. Here we propose an extension of the shared gamma frailty model to handle the interval censoring and clustering simultaneously. We show that the frailties can be integrated out analytically, allowing maximization of the marginal likelihood to obtain parameter estimates. We apply our method to a longitudinal study with periodic follow-up on dairy cows to investigate the effect of parameters at the cow level (e.g., parity) and also parameters that can change within the cow (e.g., front or rear udder quarter) on time to infection. Dairy cows were assessed approximately monthly for the presence of a bacterial infection at the udder-quarter level, thus generating interval-censored data. Obviously, the four udder quarters are clustered within the cow. Based on simulations, we find that ignoring the interval-censored nature of the data can lead to biased parameter estimates.