A diagnostic for Cox regression with discrete failure-time models

Changes in maximum likelihood parameter estimates due to deletion of individual observations are useful statistics, both for regression diagnostics and for computing robust estimates of covariance. For many likelihoods, including those in the exponential family, these delete-one statistics can be approximated analytically from a one-step Newton-Raphson iteration on the full maximum likelihood solution. But for general conditional likelihoods and the related Cox partial likelihood, the one-step method does not reduce to an analytic solution. For these likelihoods, an alternative analytic approximation that relies on an appropriately augmented design matrix has been proposed. In this paper, we extend the augmentation approach to explicitly deal with discrete failure-time models. In these models, an individual subject may contribute information at several time points, thereby appearing in multiple risk sets before eventually experiencing a failure or being censored. Our extension also allows the covariates to be time dependent. The new augmentation requires no additional computational resources while improving results.