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Event history analysis with time-dependent covariates via landmarking supermodels and boosted trees

Oliver Lunding Sandqvist

TL;DR

Dynamic prediction in survival analysis with time-dependent covariates is challenging when covariates are high-dimensional. The authors propose a nonparametric estimator that marries landmarking supermodels with boosted trees to estimate the future conditional hazard $\lambda(t,s,\mathbf{W}(s))$, without Markov or joint-model assumptions, framed as a sieve M-estimator. They prove weak consistency and show the computation reduces to Poisson regression, enabling standard boosting implementations; simulations and a primary biliary cirrhosis data application demonstrate good predictive performance and personalized prognoses. The work provides a flexible, scalable approach for dynamic risk prediction with time-varying covariate information, with potential applications in medicine and insurance.

Abstract

We propose a nonparametric method for dynamic prediction in event history analysis with high-dimensional, time-dependent covariates. The approach estimates future conditional hazards by combining landmarking supermodels with gradient boosted trees. Unlike joint modeling or Cox landmarking models, the proposed estimator flexibly captures interactions and nonlinear effects without imposing restrictive parametric assumptions or requiring the covariate process to be Markovian. We formulate the approach as a sieve M-estimator and establish weak consistency. Computationally, the problem reduces to a Poisson regression, allowing implementation via standard gradient boosting software. A key theoretical advantage is that the method avoids the temporal inconsistencies that arise in landmark Cox models. Simulation studies demonstrate that the method performs well in a variety of settings, and its practical value is illustrated through an analysis of primary biliary cirrhosis data.

Event history analysis with time-dependent covariates via landmarking supermodels and boosted trees

TL;DR

Dynamic prediction in survival analysis with time-dependent covariates is challenging when covariates are high-dimensional. The authors propose a nonparametric estimator that marries landmarking supermodels with boosted trees to estimate the future conditional hazard , without Markov or joint-model assumptions, framed as a sieve M-estimator. They prove weak consistency and show the computation reduces to Poisson regression, enabling standard boosting implementations; simulations and a primary biliary cirrhosis data application demonstrate good predictive performance and personalized prognoses. The work provides a flexible, scalable approach for dynamic risk prediction with time-varying covariate information, with potential applications in medicine and insurance.

Abstract

We propose a nonparametric method for dynamic prediction in event history analysis with high-dimensional, time-dependent covariates. The approach estimates future conditional hazards by combining landmarking supermodels with gradient boosted trees. Unlike joint modeling or Cox landmarking models, the proposed estimator flexibly captures interactions and nonlinear effects without imposing restrictive parametric assumptions or requiring the covariate process to be Markovian. We formulate the approach as a sieve M-estimator and establish weak consistency. Computationally, the problem reduces to a Poisson regression, allowing implementation via standard gradient boosting software. A key theoretical advantage is that the method avoids the temporal inconsistencies that arise in landmark Cox models. Simulation studies demonstrate that the method performs well in a variety of settings, and its practical value is illustrated through an analysis of primary biliary cirrhosis data.
Paper Structure (10 sections, 2 theorems, 13 equations, 4 figures)

This paper contains 10 sections, 2 theorems, 13 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Proposed method
  3. Model specification
  4. Estimation
  5. Large sample properties
  6. Numerical study
  7. Setup
  8. Results
  9. Data application
  10. Discussion

Key Result

proposition 1

For the metric $d(F_1,F_2)=\left\lVert F_1-F_2\right\rVert_{\mu,1}$ induced by the $L^1(\mu)$ norm and with suitable choices of the hyperparameters $(m_k,d_{kj},\nu_{kj})$ and partitions $\mathcal{P}_k$, it holds that

Figures (4)

  • Figure 1: Simulation results. Root mean squared error for the estimated future conditional survival probabilities using the proposed boosted tree landmarking supermodel (black solid lines), the Cox landmarking supermodel (orange solid lines), the naive Cox estimator (orange dotted line), and the naive boosted tree estimator (black dotted line). The top row shows performance as a function of sample size $n$ (log scale) with $Q=10$ landmarks per subject; the bottom row shows performance as a function of $Q$ with sample size $n=100$. In the bottom row, the naive models appear as horizontal lines as they do not landmark the data. The columns correspond to Scenario 1 (Linear Markovian), Scenario 2 (Nonlinear Non-Markovian), and Scenario 3 (High-dimensional).
  • Figure 2: Partial dependence plot across bilirubin (left) and albumin (right) for the boosted tree estimator (line) and data points (ticks).
  • Figure 3: Marginal plot across bilirubin (left) and albumin (right) for the boosted tree estimator (line) and data points (ticks).
  • Figure 4: Survival curve for Subject A (left) and Subject B (right) for the boosted tree estimator.

Theorems & Definitions (6)

  • remark 1: At-risk process
  • remark 2: Relation to hazards
  • remark 3: Interval convention
  • proposition 1: Universal approximation theorem
  • theorem 1: Consistency
  • remark 4: Non-uniform landmark distribution