Source-linked AI summary
Random Hazard Forests
Hemant Ishwaran, Eileen M. Hsich, Udaya B. Kogalur, Donald K. K. Lee
TL;DR
Existing clinical risk methods often simplify irregular, repeatedly updated records into baseline or otherwise altered temporal representations, despite the need for prospective, continuously updated risk. RHF directly learns a continuous-time hazard map from predictable covariate processes using a nonparametric hazard likelihood and tree ensemble. Simulations and an ICU application show strong hazard accuracy and useful time-localized risk information, while pathwise survivor interpretation remains limited for internal covariates.
Problem
Clinical records contain irregular, asynchronous, evolving measurements, while many existing approaches simplify temporal structure before modeling continuously changing risk.
Method
RHF estimates a continuous-time hazard map with a tree ensemble and nonparametric hazard likelihood, routing predictable covariate paths without lookahead.
Results
RHF ranked first for hazard rMSE in challenging simulations and was among the best for discrimination, while the ICU case study revealed time-localized variable priorities.
Takeaways & Limitations
RHF’s flexibility is most valuable when risk depends on complex longitudinal states, and its pathwise analysis can expose how predictive variables change over follow-up.
Takeaways & Limitations
For internal covariates, pathwise survivor functionals generally lack their usual conditional-survival interpretation, so RHF targets the hazard map instead.
Abstract
from arXiv · showhide
Clinical data sources such as electronic health records and wearable sensors record patient status repeatedly over follow-up, often at irregular times and on different schedules for different measurements. These data create opportunities for continuously updated, individualized risk prediction. Existing approaches, however, often simplify the temporal structure before modeling it. We introduce Random Hazard Forests (RHF), a survival tree ensemble that learns how a patient's hazard changes in continuous time as new measurements become available. RHF formulates the estimation problem directly through a nonparametric hazard likelihood for predictable covariate processes. An efficient working model guides tree construction, after which flexible time-varying hazards are estimated for each terminal node. Given any predictable covariate path, each tree follows the path through its terminal nodes over time and assembles the corresponding node-level hazards into a trajectory. Averaging these trajectories across trees yields the RHF pathwise hazard estimate. Because routing at each time uses only the covariate state available immediately beforehand, RHF accommodates internal longitudinal covariates without lookahead. Simulations and an intensive care application show that RHF accurately estimates changing risk under irregular and asynchronous covariate updates.
1. Introduction.
RHF addresses continuously changing clinical risk by learning a continuous-time hazard from predictable, irregularly updated covariate histories. It combines hazard-likelihood estimation with tree ensembles to produce pathwise hazard trajectories without lookahead.
- Motivation: Modern clinical records contain irregular, asynchronous measurements and changing treatments, making baseline-only risk snapshots poorly suited to evolving patient status.ICU physiology, laboratory values, and organ-support therapies may change hour to hour, while trials, wearables, and electronic records collect repeated follow-up data.
- Contribution: RHF is a survival tree ensemble that learns hazard variation in continuous time from the current predictable covariate state.It follows evolving covariate histories rather than reducing each subject to a single baseline vector.
- Method: RHF maps time and covariate states to hazards, then evaluates that map along a supplied predictable covariate process to obtain a case-specific trajectory.Each tree routes the covariate path through terminal nodes over time, and the forest averages tree-specific trajectories.
- Method: RHF estimates the hazard map by minimizing empirical risk derived from a nonparametric hazard likelihood, guiding both tree splitting and terminal-node hazard estimation.This extends the random-survival-forest paradigm beyond baseline covariates.
- Design principle: Unlike landmarking, person-period grids, or feature summaries, RHF formulates the problem directly around a hazard likelihood for predictable covariate processes.Its routing convention requires the covariate state entering the hazard at time t to use information available immediately beforehand.
- Scope: For internal covariates, RHF targets the hazard map rather than a pathwise survivor functional because the latter generally lacks its usual conditional-survival interpretation.This scope boundary applies to longitudinal biomarkers or treatments tied to evolving health.
3. RHF tree construction.
RHF grows trees for evolving covariate histories using a likelihood-based split criterion, then estimates hazards within node–time cells. The resulting node hazards form a forest hazard map that can be evaluated along predictable covariate paths.
- RHF dynamically routes each subject through tree nodes using the current predictable covariate state, allowing one subject to contribute exposure and events to different nodes over time.
- Tree induction uses a working model with time-constant log hazards within current nodes, while time itself is excluded as a splitting variable.
- Candidate splits are selected by the largest empirical risk reduction under a nonparametric hazard likelihood, comparing child hazards with their pooled parent hazard.
- Terminal-node hazards are estimated for node–time cells as observed events divided by total at-risk exposure, with zero hazard when exposure is positive but events are absent.
- Averaging tree-level hazard maps yields the RHF forest hazard map, which produces a case-specific hazard trajectory when evaluated along a predictable covariate process.
- The evaluation grid controls temporal resolution, so finer grids provide greater event-time detail while coarser grids pool intervals for stability and computation.
4. RHF algorithm and implementation details.
RHF implements its continuous-time forest through start–stop records that encode predictable covariate states over intervals. Trees use finite exposure and event sums for splitting, and forest hazards are evaluated along supplied paths by routing each current state.
- Start–stop preprocessing represents each subject’s covariate, at-risk, and event processes as intervals with fixed routing states and event increments.
- Each tree uses a subject-level subsample, best-first search, sampled covariates, and admissible covariate cutpoints, selecting the split with the largest empirical risk reduction.
- At test time, a supplied predictable path is partitioned when its routing state changes, and each tree follows the selected terminal-node hazard on each interval.
- Because routing states are constant within intervals, exposure and event aggregates reduce to finite sums without numerical integration.
- Pointwise averaging of tree-specific trajectories equals evaluating the forest hazard map along the supplied predictable covariate path.
- RHF also accumulates node-level hazard increments to form a case-specific cumulative-hazard estimate over the specified path domain.
OOB case-specific hazard.
The OOB case-specific hazard averages predictions from trees that excluded the subject during training, providing an internally cross-validated trajectory.
- Each term comes from a tree fit without subject i, so the OOB ensemble provides an internally cross-validated case-specific estimate.
OOB empirical risk.
RHF evaluates OOB empirical risk with the same likelihood loss used for training, replacing each subject’s hazard by its OOB case-specific estimate. The framework also provides time-local variable-priority curves based on in-rule versus near-miss hazard exposure.
- OOB empirical risk uses the training likelihood loss while evaluating each subject only with trees for which that subject was out of bag.
- The aggregated OOB risk is a cross-validated analogue of empirical risk that can monitor convergence or compare forests with different tuning parameters.
- Time-local variable priority compares hazard exposure inside a terminal-node rule with exposure in a near-miss region created by removing one variable’s constraints.
- The variable-priority curve identifies time windows in which a coordinate has a significant local effect on the log integrated hazard.
5. Empirical evaluation.
The empirical evaluation uses synthetic data with time-varying covariates, varying observation resolution and comparing RHF with several competing methods.
- Synthetic event times arise from latent continuous-time covariate trajectories, while observed longitudinal measurements are sampled from those trajectories.
- The simulations vary the number of records per subject over R = 10,20,40, with larger R providing finer temporal resolution.
- RHF is compared against several competing methods in the benchmark experiments.
1. Smooth time variation.
The first simulation evaluates a smoothly varying hazard in which baseline covariates set overall risk and a smooth longitudinal signal has a baseline-dependent slope.
- The first simulation uses a smoothly varying hazard with baseline covariates determining overall risk level.
- A smooth scalar time-dependent signal contributes to risk, with its subject-specific slope determined by baseline covariates.
- This design favors Cox regression with time-dependent covariates because the signal is low dimensional, smooth, and represented by observed time-updated records.
2. Covariate-dependent zero-rate windows and latent longitudinal activation.
The second simulation tests whether methods recover a covariate-dependent zero-hazard interval and subsequent risk reactivation driven by an unobserved-coefficient longitudinal process.
- Baseline covariates define an extended interval during which the hazard is exactly zero.
- Outside that interval, one covariate region has modified hazard governed by a subject-specific longitudinal activation process.
- The latent coefficients of the activation process are not supplied to the learner.
- The design tests recovery of absent risk, returning risk, and dependence on the time-varying longitudinal state.
3. Localized longitudinal risk region.
The evaluation defines complementary discrimination and hazard-scale metrics, compares RHF with six competitors, and finds RHF strongest in the challenging time-varying simulations.
- 3. Localized longitudinal risk region.: The third simulation localizes elevated risk to a joint region of two longitudinal covariates that neither covariate identifies alone.
- Evaluation metrics: Because benchmark methods produce hazards, survival curves, relative-risk scores, or landmark risks, evaluation uses complementary metrics rather than one universal measure.
- Evaluation metrics: Cumulative AUC measures ranking of subjects failing by time t against those remaining event-free beyond t using a cumulative marker.
- Evaluation metrics: Incident AUC compares subjects failing at time t with those still at risk using an instantaneous marker.
- Evaluation metrics: Relative mean squared error evaluates hazard estimation on the numerical scale, with smaller rMSE indicating greater accuracy.
- Benchmark methods: RHF is compared with six competitors, including Cox-TDC, BoXHED, baseline RSF, neural hazard models, and landmark RSF.
- Results: RHF has the best overall performance in Simulation 2, combining the lowest hazard rMSE with the highest incident and near-highest cumulative discrimination.
- Results: In Simulation 3, RHF and BoXHED rank highest for cumulative and incident discrimination, while RHF has the lowest hazard rMSE.
6. Time-varying ICU risk in MIMIC-IV.
The MIMIC-IV analysis applies RHF to irregular, asynchronous ICU records to estimate and evaluate evolving cause-specific in-hospital death risk. RHF shows strong discrimination and time-localized variable priorities across ICU follow-up.
- Data and analysis: The cohort included 52,219 patients, 5,455 deaths, and 11,106,272 hourly start–stop records from first ICU stays.Follow-up ended at in-hospital death or discharge alive, with discharge treated as exit from the hospital-at-risk process.
- Data and analysis: RHF used temporally ordered interval records so measurements charted during an hour became available only in the following interval.This preserved prospective no-lookahead ordering while aligning irregular measurements and treatments to patient-specific hourly risk intervals.
- Performance evaluation: Cumulative AUC remained in the mid 0.9 range and incident AUC in the low 0.9 range across most follow-up in both samples.Incident AUC is the more local and demanding discrimination measure.
- Time-localized variable priority: Glasgow Coma Scale dominated priorities throughout follow-up, while lactate increased in importance over time and several laboratory markers became more important later.The analysis used window-specific contrasts of OOB integrated hazard exposure between forest rules and near-miss sets.
7. Discussion.
RHF directly models evolving hazard from predictable time-dependent covariates without reducing follow-up to baseline, landmark, or fixed discrete-time representations. Its flexibility is most useful for complex risk structures, while the current framework leaves calibration, observation-process modeling, and broader event settings open.
- Contribution: RHF estimates a time–covariate hazard map and produces path-specific hazard trajectories using covariate states available immediately beforehand.This accommodates internal longitudinal covariates without lookahead.
- When flexibility helps: In smooth, low-dimensional Simulation 1, Cox models with time-dependent covariates matched or outperformed RHF.This identifies a setting where RHF flexibility was not advantageous.
- When flexibility helps: In Simulations 2 and 3, RHF was among the best for discrimination and ranked first for hazard rMSE as risk depended on more complex longitudinal states.The discussion concludes that RHF flexibility becomes more valuable as risk complexity increases.
- Clinical interpretation: MIMIC-IV time-localized priorities revealed changing contributors to hazard that a single global ranking would miss.The case study links pathwise modeling with temporal interpretation of evolving physiology and treatment.
- Limitations and future work: RHF currently targets a single cause-specific event process, while calibration, observation-process modeling, competing risks, recurrent events, and multistate extensions remain open directions.Streaming implementations and richer local explanation tools are also identified as future needs for routine clinical use.
Funding.
The paper reports NIH support for its authors and provides implementation and data-access information for reproducibility.
- Funding: The authors were supported by NIH through NIGMS grant R35 GM139659 and NHLBI grant R01 HL164405.Support for Ishwaran and Kogalur included NIGMS and NHLBI funding, while Hsich and Lee were supported through NHLBI.
- Reproducibility: The randomForestRHF package is publicly available on CRAN, and MIMIC-IV data are accessible through PhysioNet after credentialing.The package includes simulation functions and the paper points to supplementary benchmark details for reproducibility.