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Conditional Aalen--Johansen estimation

Martin Bladt, Christian Furrer

TL;DR

The paper develops a nonparametric, covariate-adjusted framework for estimating conditional state-occupation probabilities in finite-state jump processes using a conditional Aalen–Johansen approach. By constructing kernel-based estimators for the conditional Nelson–Aalen rates and expressing the occupation probabilities as a product integral, it unifies kernel conditioning and landmark-type methods while handling right-censoring and both continuous and discrete covariates. It establishes strong uniform consistency and asymptotic normality under lax moment conditions, with perturbation to address division-by-zero and extensions to covariate atoms; it also provides practical plug-in covariance estimators for inference. The methodology generalizes classical survival analysis tools (e.g., conditional Kaplan–Meier) and applies to inhomogeneous semi-Markov and non-Markov settings, offering a flexible, nonparametric tool for duration and covariate-driven dynamics in multi-state models, with implementation in R and demonstrated via simulations and supplementary material.

Abstract

The conditional Aalen--Johansen estimator, a general-purpose non-parametric estimator of conditional state occupation probabilities, is introduced. The estimator is applicable for any finite-state jump process and supports conditioning on external as well as internal covariate information. The conditioning feature permits for a much more detailed analysis of the distributional characteristics of the process. The estimator reduces to the conditional Kaplan--Meier estimator in the special case of a survival model and also englobes other, more recent, landmark estimators when covariates are discrete. Strong uniform consistency and asymptotic normality are established under lax moment conditions on the multivariate counting process, allowing in particular for an unbounded number of transitions.

Conditional Aalen--Johansen estimation

TL;DR

The paper develops a nonparametric, covariate-adjusted framework for estimating conditional state-occupation probabilities in finite-state jump processes using a conditional Aalen–Johansen approach. By constructing kernel-based estimators for the conditional Nelson–Aalen rates and expressing the occupation probabilities as a product integral, it unifies kernel conditioning and landmark-type methods while handling right-censoring and both continuous and discrete covariates. It establishes strong uniform consistency and asymptotic normality under lax moment conditions, with perturbation to address division-by-zero and extensions to covariate atoms; it also provides practical plug-in covariance estimators for inference. The methodology generalizes classical survival analysis tools (e.g., conditional Kaplan–Meier) and applies to inhomogeneous semi-Markov and non-Markov settings, offering a flexible, nonparametric tool for duration and covariate-driven dynamics in multi-state models, with implementation in R and demonstrated via simulations and supplementary material.

Abstract

The conditional Aalen--Johansen estimator, a general-purpose non-parametric estimator of conditional state occupation probabilities, is introduced. The estimator is applicable for any finite-state jump process and supports conditioning on external as well as internal covariate information. The conditioning feature permits for a much more detailed analysis of the distributional characteristics of the process. The estimator reduces to the conditional Kaplan--Meier estimator in the special case of a survival model and also englobes other, more recent, landmark estimators when covariates are discrete. Strong uniform consistency and asymptotic normality are established under lax moment conditions on the multivariate counting process, allowing in particular for an unbounded number of transitions.
Paper Structure (15 sections, 12 theorems, 123 equations, 3 figures)

This paper contains 15 sections, 12 theorems, 123 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Conditional Aalen--Johansen estimator
  3. Setting
  4. Estimators
  5. Uniform consistency
  6. Results
  7. Inclusion of atoms
  8. Asymptotic normality
  9. Preliminaries
  10. Weak convergence
  11. Plug-in estimators for the covariances
  12. Proofs of consistency
  13. Consistency of density estimator
  14. Proofs of normality
  15. Bracketing central limit theorem

Key Result

Proposition 3.1

Suppose $\theta_x$ satisfies eq:right_endpoint. It then holds that

Figures (3)

  • Figure 1: Estimate of the state occupation probability $p_2$ (dashed) together with its true value (solid) for $n=1,000$ (left) and $n=5,000$ (right), respectively.
  • Figure 2: Estimates of the transition probability with $j=2$ (dashed) using the ordinary Aalen--Johansen estimator (magenta) and the conditional Aalen--Johansen estimator (black) together with the true value (solid) and for $n=1,000$ (left) and $n=5,000$ (right), respectively.
  • Figure 3: Estimates of the transition probability with $j=2$ (dashed) for $u=1$ (blue), $u=5$ (green), and without conditioning on duration (black) for $n=1,000$ (left) and $n=5,000$ (right), respectively. True values are in solid.

Theorems & Definitions (29)

  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Example
  • Example
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Proposition 3.1
  • Theorem 3.2: Strong uniform consistency
  • ...and 19 more