Estimating survival curves with left-truncated and interval-censored data under monotone hazards.

We show that the nonparametric maximum likelihood estimator (NPMLE) of a survival function may severely underestimate the survival probabilities at very early times for left-truncated and interval-censored data. As an alternative, we propose to compute the (nonparametric) MLE under a nondecreasing hazard assumption, the monotone MLE, by a gradient projection algorithm when the assumption holds. The projection step is accomplished via an isotonic regression algorithm, the pool-adjacent-violators algorithm. This gradient projection algorithm is computationally efficient and converges globally. Monte Carlo simulations show superior performance of the monotone MLE over that of the NPMLE in terms of either bias or variance, even for large samples. The methodology is illustrated with the application to the Wisconsin Epidemiological Study of Diabetic Retinopathy data to estimate the probability of incidence of retinopathy.