📊 statistics

ShrinkageTrees: An R Package for Bayesian Tree Ensembles for Survival Analysis and Causal Inference

The paper introduces ShrinkageTrees, an R package that implements Bayesian additive regression tree models with advanced regularization strategies, including the novel Horseshoe Forest, to enable robust survival analysis and causal inference in high-dimensional settings.

Original authors: Tijn Jacobs

Published 2026-06-11

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Tijn Jacobs

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a doctor trying to predict how long a patient will live after a diagnosis, or perhaps trying to figure out if a new medicine actually works better than an old one. You have a lot of data: age, blood pressure, and maybe even thousands of gene readings. This is a messy, complicated puzzle, especially when some patients drop out of the study before the end (so you don't know exactly when they passed away) or when you only know an event happened "sometime between last month and this month."

This paper introduces a new tool called ShrinkageTrees. Think of it as a super-smart, highly disciplined team of detectives (a "tree ensemble") designed specifically to solve these medical puzzles.

Here is how it works, broken down into simple concepts:

1. The Team of Detectives (The Trees)

Imagine you don't ask one expert for an opinion; you ask a hundred. Each expert is a "decision tree."

How they work: Each tree asks simple yes/no questions like, "Is the patient older than 50?" or "Is Gene X high?" Based on the answers, the tree makes a small guess about the outcome.
The Team Effort: The final prediction is the sum of all these small guesses. Because there are so many trees, they can catch complex patterns that a single expert would miss.

2. The Problem: Too Many Clues, Too Few Suspects

In modern medicine, you often have thousands of clues (genes) but very few patients to study.

The Old Way: If you let the team of detectives run wild, they might get distracted by random noise. They might start making up rules based on coincidences (e.g., "Patients who like blue socks live longer") just to fit the data perfectly. This is called "overfitting." It looks great on paper but fails in the real world.
The Shrinkage Solution: The authors added a "disciplinarian" to the team. This is the Shrinkage part. It acts like a strict editor who says, "If a rule isn't strongly supported by the evidence, we shrink it down to zero."
- The Horseshoe Metaphor: The paper highlights a specific type of shrinkage called the "Horseshoe." Imagine a horseshoe magnet. It has a very strong pull in the middle (shrinking weak, useless clues to zero) but a wide, open curve at the ends (letting strong, important clues pass through untouched). This ensures the model ignores the noise but keeps the real signals loud and clear.

3. Handling Missing or Vague Clues (Censoring)

In survival studies, data is often incomplete.

Right-Censoring: A patient is still alive when the study ends. We know they survived at least that long, but not exactly when they will pass.
Interval-Censoring: We only know a patient passed away between their last check-up and their next one.
The Innovation: Most existing tools struggle with these vague timelines. ShrinkageTrees is built to handle these "fuzzy" timelines naturally, filling in the gaps mathematically without losing accuracy.

4. The "Two-Headed" Detective (Causal Inference)

Sometimes you want to know not just what will happen, but why it happened. Did the patient survive because of the drug, or just because they were younger?

The Old Way: Standard tools try to guess the outcome and the drug effect all at once, which can get confusing.
The New Way (The $\tau$ -learner): ShrinkageTrees splits the job into two specialized detectives:
1. The Prognostic Detective: Predicts how the patient would do regardless of the treatment (based on their age, genes, etc.).
2. The Treatment Detective: Predicts the extra effect the drug has.
  By separating these two, the model can tell you exactly how much the drug helped, even if the patient's underlying health was very different from others.

5. What the Paper Actually Proves

The authors tested this tool on two main things:

Real Data: They used a dataset of ovarian cancer patients with 1,000 gene readings. They showed that while older methods got "too confident" (fitting the noise), the new ShrinkageTrees method was more realistic and provided better uncertainty estimates.
Simulated Data: They created fake data where they knew the "truth." They tested the tool when the number of genes was huge (5,000 genes) but the number of patients was small. The ShrinkageTrees method (specifically the Horseshoe version) was the only one that stayed accurate and didn't get confused by the massive amount of data.

Summary

ShrinkageTrees is a new software package that helps researchers analyze survival data (time-to-event) and causal questions (did the treatment work?). It uses a team of decision trees but adds a special "shrinkage" filter to ignore useless data and focus only on what matters. It is the first tool of its kind to handle vague timelines (interval-censoring) and separate treatment effects from patient characteristics in high-dimensional settings (where you have more variables than patients).

The paper claims this tool is faster, more accurate in high-stakes scenarios, and provides a clearer picture of uncertainty than previous methods.

Technical Summary of "ShrinkageTrees: An R Package for Bayesian Tree Ensembles for Survival Analysis and Causal Inference"

Problem Statement
Survival analysis in high-dimensional settings ( $p \gg n$ ) faces three primary challenges: (1) the complex, unknown relationship between covariates and time-to-event outcomes requiring flexible non-parametric models; (2) the prevalence of imprecise observations, specifically right-censored and interval-censored data; and (3) the difficulty of estimating causal treatment effects under censoring and confounding, particularly when treatment effects vary across subgroups. Existing Bayesian Additive Regression Trees (BART) implementations fail to address all three simultaneously. Specifically, no R package currently offers Bayesian tree ensembles for interval-censored outcomes, the $\tau$ -learner architecture (which separates prognostic and treatment effects) has not been implemented for time-to-event endpoints, and standard BART regularization is often inadequate for high-dimensional genomic settings where overfitting is a risk.

Methodology
The paper introduces ShrinkageTrees, an R package implementing Bayesian tree ensembles within an Accelerated Failure Time (AFT) framework. The core model extends the standard BART formulation:
$\log T_i = \sum_{j=1}^m g(x_i; T_j, H_j) + \epsilon_i, \quad \epsilon_i \sim N(0, \sigma^2)$
where $T_i$ is the event time, $x_i$ are covariates, and the sum represents an ensemble of $m$ shallow decision trees. The package handles right- and interval-censored data via data augmentation in a Gibbs sampler.

The methodology distinguishes between two types of regularization, both of which are implemented and combinable:

Tree Structure Regularization: To encourage splitting on a subset of relevant variables, the package implements the Dirichlet splitting prior (DART). This places a Dirichlet prior on splitting probabilities, inducing structural sparsity.
Step Height Regularization: To shrink uninformative leaves toward zero while preserving strong signals, the package implements global–local shrinkage priors on the leaf values ( $H_j$ ). The paper highlights the Horseshoe Forest, which places a horseshoe prior on step heights. This utilizes a scale mixture of normals representation with a global shrinkage parameter ( $\tau$ ) and local scale parameters ( $\lambda_\ell$ ). The horseshoe prior allows for adaptive shrinkage, effectively handling the $p \gg n$ regime.

For causal inference, the package implements the $\tau$ -learner (Bayesian Causal Forests) adapted for survival outcomes. This decomposes the outcome into a prognostic function $\mu(x, \hat{e}(x))$ and a treatment effect function $\tau(x)$ , modeled by separate forests. This allows for distinct regularization and complexity control for the prognostic and causal components.

Key Contributions

First Implementation of Horseshoe Forests for Survival: The package provides the first implementation of the Horseshoe Forest, placing a horseshoe prior on step heights to enable high-dimensional survival analysis.
Interval-Censored Support: It is the first R package to provide Bayesian tree ensembles for interval-censored survival outcomes.
Causal Inference for Survival: It implements the $\tau$ -learner for time-to-event endpoints, allowing for the estimation of heterogeneous treatment effects (CATE) and average treatment effects (ATE) in survival contexts.
Flexible Regularization: It offers a modular framework where users can combine structural regularization (Dirichlet splitting) with step-height regularization (Horseshoe, Half-Cauchy, or Standard Normal).
Software Implementation: The package features an efficient C++ backend via Rcpp, supports multi-chain MCMC, and provides a full S3 workflow (fitting, prediction, convergence diagnostics via coda, and visualization).

Results
The paper validates the methodology through a semi-synthetic ovarian cancer dataset and a simulation study:

Ovarian Dataset: In a high-dimensional setting ( $n=357, p=1004$ ), standard BART achieved a near-perfect training C-index (0.989), indicating overfitting. In contrast, the Horseshoe Forest yielded a lower training C-index (0.727) but provided better-calibrated uncertainty. The causal analysis estimated a small, non-significant average treatment effect for carboplatin vs. cisplatin, with tight credible intervals.
Simulation Study: A simulation with known ground truth and interval-censored outcomes compared Standard BART, DART, and Horseshoe Forests across dimensions $p \in \{50, 500, 5000\}$ . The Horseshoe Forest consistently achieved the lowest Root Mean Squared Error (RMSE), the tightest credible intervals, and near-nominal 95% coverage rates across all dimensions. Standard BART produced overly wide intervals (over-coverage), while DART under-covered as dimensionality increased.
Computational Performance: Benchmarks show that while Horseshoe Forests are computationally more expensive than conjugate-prior models (due to reversible jump MCMC and sampling local scales), they scale linearly with sample size and are comparable in speed to existing BART implementations for conjugate models.

Significance and Claims
The paper claims that ShrinkageTrees addresses critical gaps in the existing Bayesian tree ensemble literature by unifying survival analysis, causal inference, and high-dimensional regularization. It positions the package as a tool that extends Bayesian tree ensembles to high-dimensional settings where standard BART fails to adapt. The authors emphasize that the package is the first to offer the $\tau$ -learner for survival endpoints and the first to implement Horseshoe Forests for this purpose. The work is presented as a practical implementation of the methodology developed in Jacobs et al. (2025), aimed at both practitioners new to Bayesian tree ensembles and experienced users requiring functionality not available in existing packages like BART, dbarts, or bcf. The authors modestly note limitations, such as the requirement for numeric covariates (necessitating dummy encoding for unordered categories) and the assumption of normally distributed errors in the AFT framework.