The Kaplan–Meier Estimator as an Inverse-Probability-of-Censoring Weighted Average

Abstract

The Kaplan-Meier (product-limit) estimator of the survival function of randomly-censored time-to-event data is a central quantity in survival analysis. It is usually introduced as a nonparametric maximum likelihood estimator, or else as the output of an imputation scheme for censored observations such as redistribute-to-the-right or self-consistency. Following recent work by Robins and Rotnitzky, we show that the Kaplan-Meier estimator can also be represented as a weighted average of identically distributed terms, where the weights are related to the survival function of censoring times. We give two demonstrations of this representation; the first assumes a Kaplan-Meier form for the censoring time survival function, the second estimates the survival functions of failure and censoring times simultaneously and can be developed without prior introduction to the Kaplan-Meier estimator.

Nonparametric rank based estimation of bivariate densities given censored data conditional on marginal probabilities

“…whereπ n = 1 − n−1 j=1π j , see Satten and Datta (2001). In addition, it follows straightforward that similar to the standard empirical distribution function estimator we have for right-censored data that…”

“…As described in Section 1, and with a simple modification of the notation, the parameter estimates derived from the form of the likelihood at (3) corresponding to the marginal density f x were derived by Satten and Datta (2001) aŝ…”

“…Our derivation is based on extending the constrained maximum likelihood density based approach (Satten and Datta 2001;Zhou 2005) that is utilized in the univariate setting as an alternative strategy to the classical discrete nonparametric hazard function approach (Kaplan and Meier 1958). There are several methods for estimating a bivariate survival function across different censoring patterns that have been proposed in the literature based on extending the univariate hazard function approach or creating various decompositions (Akritas and Van Keilegom 2003;Gill et al 1995;Lin and Ying 1993;Prentice et al 2004;Wang and Wells 1997).…”

Estimation of Survival Distributions Under Right-Censoring When Sample Size is Random

“…Statistical properties of KM and NA have been widely studied (see e.g., Efron 1967;Breslow and Crowley 1974;Peterson 1977;Winter et al 1978;Chen et al 1982;Chang 1991;Klein 1991;Stute 1994a,b;Pawlitschko 1999;Satten and Datta 2001). It has been shown, among other things, that both estimators are biased.…”

On dealing with censored largest observations under weighted least squares

“…The IPCW approach is used in many studies in survival analysis [e.g. Robins and Finkelstein (2000), Satten and Datta (2001)]. The K−M weighting approach is also widely used in many studies such as Stute (1993Stute ( , 1994Stute ( , 1996, Stute and Wang (1994), Hu and Rao (2010), Khan and Shaw (2013).…”