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A stochastic-covariate failure model with an application to case-control analysis.

S M Berman1

  • 1Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, NY, New York 10012, USA. sberman@cims.nyu.edu

Mathematical Biosciences
|August 5, 2000
PubMed
Summary

This study introduces a method to estimate failure rates for rare events using stochastic processes. It establishes asymptotic independence between the marker process and failure time, enabling risk analysis in biomedical trials.

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Area of Science:

  • Statistics
  • Stochastic Processes
  • Biostatistics

Background:

  • Periodically stationary and ergodic stochastic processes are fundamental in modeling time-dependent phenomena.
  • Failure-time analysis often involves understanding the relationship between a marker process and the event occurrence.
  • Rare events in biomedical studies pose unique challenges for statistical modeling and risk assessment.

Purpose of the Study:

  • To investigate the limiting joint distribution and asymptotic independence of a stochastic process and its failure time.
  • To develop methods for estimating the unknown local failure-rate function from observed data.
  • To apply these methods to analyze rare events in case-control studies, specifically in biomedical trials.

Main Methods:

  • Utilized the theory of periodically stationary and ergodic stochastic processes.

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  • Assumed a local failure-rate function of the form up(x) where u approaches 0 for rare events.
  • Derived the limiting joint distribution of X(T(u)) and uT(u) and established their asymptotic independence.

    • Main Results:

    • Demonstrated that the marker process X(T(u)) and scaled failure time uT(u) are asymptotically independent as u approaches 0.
    • Provided explicit formulas for the marginal distributions of X(T(u)) and uT(u).
    • Developed an estimation procedure for the failure-rate function p(x) using independent copies of the limiting random variable Y.

    Conclusions:

    • The asymptotic independence simplifies the analysis of rare events in the presence of a marker process.
    • The proposed estimation method allows for risk assessment even when the failure rate is unknown.
    • The findings are directly applicable to biomedical trials, such as determining stroke risk based on anticoagulant levels.