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On non-parametric maximum likelihood estimation of the bivariate survivor function.

R L Prentice1

  • 1Division of Public Health Sciences, Fred Hutchinson Cancer Research Center, 1100 Fairview Avenue North, MP-1002, P.O. Box 19024, Seattle, WA 98109-1024, USA. rprentic@fhcrc.org

Statistics in Medicine
|September 4, 1999
PubMed
Summary

This study introduces a non-parametric maximum likelihood estimator for bivariate survivor functions under independent censorship. The method maximizes likelihood by placing mass on specific grids, enabling unique estimation of parameters.

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Area of Science:

  • Statistics
  • Biostatistics
  • Survival Analysis

Background:

  • Estimating bivariate survivor functions is crucial for analyzing time-to-event data with multiple endpoints.
  • Independent censorship is a common challenge in survival data analysis.
  • Non-parametric methods offer flexibility when distributional assumptions are unknown.

Purpose of the Study:

  • To develop a non-parametric maximum likelihood estimator for the bivariate survivor function (F) under independent censorship.
  • To address the potential non-uniqueness of the estimator based on censoring patterns.
  • To establish a method for uniquely estimating parameters from a partially maximized likelihood.

Main Methods:

  • Maximizing the likelihood function by placing mass on specific grids (uncensored failure times, half lines, or upper right quadrant).

Related Experiment Videos

  • Accumulating mass along regions of flat likelihood to obtain a partially maximized likelihood.
  • Deriving score equations using Lagrange multipliers for parameter estimation.
  • Employing a modified Newton procedure for parameter calculation in simulations.

    • Main Results:

    • A non-parametric maximum likelihood estimator (&Fcirc;) for the bivariate survivor function was obtained.
    • The estimator's uniqueness depends on the configuration of censored data.
    • A method was developed to derive unique parameter estimates from a partially maximized likelihood function.
    • The modified Newton procedure was utilized for parameter estimation in simulation studies.

    Conclusions:

    • The proposed method provides a way to obtain a unique non-parametric maximum likelihood estimator for bivariate survivor functions.
    • Further development of non-parametric bivariate survivor function estimators is suggested.
    • The approach handles complexities arising from singly- and doubly-censored data.