Product-limit survival functions with correlated survival times

A simple variance estimator for product-limit survival functions is demonstrated for survival times with nested errors. Such data arise whenever survival times are observed within clusters of related observations. Greenwood's formula, which assumes independent observations, is not appropriate in this situation. A robust variance estimator is developed using Taylor series linearized values and the between-cluster variance estimator commonly used in multi-stage sample surveys. A simulation study shows that the between-cluster variance estimator is approximately unbiased and yields confidence intervals that maintain the nominal level for several patterns of correlated survival times. The simulation study also shows that Greenwood's formula underestimates the variance when the survival times are positively correlated within a cluster and yields confidence intervals that are too narrow. Extension to life table methods is also discussed.