An empirical saddlepoint approximation method for producing smooth survival and hazard functions under interval‐censoring

Abstract

We devise a new method to produce smooth estimates of baseline survival and hazard functions for incomplete data observed subject to interval‐censoring, that can in principle be viewed as being nonparametric. The key idea is to start from the nonparametric maximum likelihood estimate, and to then construct an empirical moment generating function for the underlying data generating mechanism, which is subsequently inverted via a saddlepoint approximation in order to obtain smooth distributional estimates. Unlike the typical spline‐based and other semiparametric methods that have thus far been devised for the same purpose, the proposed approach is unencumbered by the choice of tuning parameters. Simulation studies show that in terms of integrated squared error, the method is very close in performance to the parametric gold standard, and should generally be preferred over the well‐established spline‐based approach implemented in R package logspline. The methodology is illustrated on some publicly available real datasets, and its implications and limitations are discussed.

Improved nonparametric penalized maximum likelihood estimation for arbitrarily censored survival data

“…S1 also allowed us to compare the performance of our proposed algorithm to a common alternative semiparametric approach for estimating smoother survival functions based on B‐splines. It should be noted that the B‐spline method very commonly encountered nonconvergence issues in our simulations (also noted by others 15 ), whereas our method did not. This may introduce an unavoidable bias which could favor outperformance of the B‐spline method by systematically selecting for random simulation samples where B‐splines might perform better.…”

“…Further, S1 compares the performance of our proposed method to a commonly used B‐splines‐based method 11 implemented in the polspline R package 33 . Because nonconvergence is often an issue with this B‐spline routine, 15 we resampled until 100 simulations with convergence were obtained for each configuration. In Simulation 2 (S2), we tested the robustness of our method to an underlying disease process which is bimodal by performing 500 replicates under a single parameter configuration of (N, u, p, m).…”

“…KM and TB could be regarded as an NPMLE analogue of the histogram for censored data, so it is very natural to consider applying smoothing as a simple solution to excessive variability of the density estimate. A recent example in survival analysis used saddle point approximation to smooth survival and hazard function, 15 but has no convenient implementation in common statistical software.…”