---
kind: concept
slug: longitudinal-data-analysis
title: Longitudinal data analysis with tree ensembles
summary: >-
  Adapting random forests to repeated measurements of the same subjects, whose rows are correlated. Three routes: resample whole subjects, add mixed-effects random effects, or treat each subject's trajectory as one multivariate response. Trade-off: simple sampling fixes ignore time and random effects, while mixed-effects hybrids are richer but harder to fit and have little high-dimensional evidence.
tags: [longitudinal-data, repeated-measures, random-forest, mixed-models, survey]
papers: ["2208.04112"]
related: [random-forest, mixed-effects-model]
status: draft
updated_at: 2026-10-18
---

## Overview

Longitudinal (repeated-measures) data has several rows per subject: a patient measured every few months, or several samples taken at one visit. Rows from the same subject are correlated, and visits can be irregularly spaced or missing. Standard random forests assume independent rows, so two things break: observation-level bootstrap samples contain rows from nearly every subject, which makes the trees similar, and out-of-bag error looks better than it should [arxiv:2208.04112 §1, §2.1].

The invariant shared by every variant is that the method must account for the repeated rows, in one of three ways:

1. **Resample by subject.** Draw whole subjects into each tree, so the trees differ more and OOB rows are not twins of in-bag rows [arxiv:2208.04112 §2.2].
2. **Add random effects.** Keep a per-subject offset in a mixed-effects model and let a tree or forest model the fixed part [arxiv:2208.04112 §2.3].
3. **Model the trajectory as one response.** Treat a subject's outcomes as a vector or matrix and change the split criterion to compare whole trajectories [arxiv:2208.04112 §3].

The review that this page is compiled from covers all three. For multivariate responses, time-varying predictors also need the time-effect methods [arxiv:2208.04112 §3]. It runs no experiments of its own, and it reports that the packages have never been benchmarked against each other [arxiv:2208.04112 §5]. Use the notes below as a map, then go to the primary papers before implementing.

## Variants

### Clustered rows, no time effect

Subject-level resampling only. The model is a subject mean plus independent noise [arxiv:2208.04112 eq. 1], so the method corrects for clustering but uses neither time order nor random effects [arxiv:2208.04112 §2.2.2].

- **Averaging.** Collapse each subject's rows to one row and run a standard forest. This discards within-subject variation, hides unequal numbers of rows per subject, and cannot average categorical values that differ between visits. The cited study reports that it degrades more than a standard forest as the number of subjects shrinks [arxiv:2208.04112 §2.2.1].
- **Subject-level bootstrap (RF++).** For each tree, bootstrap subjects and keep all of their rows in-bag. A subject-level prediction is a majority vote over its rows, and the same vote gives a subject-level OOB error [arxiv:2208.04112 §2.2.2]. Card: [Hu and Szymczak's review](/papers/2208.04112).
- **Two-stage bootstrap (Adler et al.).** Resample subjects, then keep one randomly chosen row per selected subject. In the simulations the review cites, this beat a standard forest, averaging and RF++, but the review notes that results depend on the setting [arxiv:2208.04112 §2.2.2].

### Time effect, univariate outcome

**Historical random forest.** Predict each observation from its concurrent predictors plus a summary of the same subject's earlier observations. Time-varying predictors are summarised over a look-back window. The example summary counts past values below a threshold; the review also lists frequency and mean summaries. Only the count and frequency summaries are invariant to monotone transforms [arxiv:2208.04112 §2.3.1, eq. 2].

Each split searches the window and threshold along with the predictor. This is costly when there are many time-varying predictors, so the htree authors tune the cut-off on a random sub-sample [arxiv:2208.04112 §2.3.1]. Variable importance is available [arxiv:2208.04112 tab. 3].

**Mixed-effects hybrids, continuous outcome.** The model is a tree or forest for the fixed part plus a subject-level random effect, fitted by an alternating loop [arxiv:2208.04112 §2.3.2, eq. 3]:

- **MERT / MERF** (Hajjem et al.) use a tree or forest for the fixed part and assume residuals with a common variance. They resample observations rather than subjects, which the review says is justified only when the random effects capture all within-subject correlation [arxiv:2208.04112 §2.3.2].
- **RE-EM tree / REEMforest.** The tree only defines leaves. The leaf effects are then fitted in a linear mixed model, which allows a non-diagonal residual covariance such as autocorrelation [arxiv:2208.04112 §2.3.2]. Public R packages exist [arxiv:2208.04112 tab. 3].
- **Stochastic variants (SMERT, SMERF, SREEMtree, SREEMforest).** Add a subject-specific Gaussian process to capture smooth within-subject deviations [arxiv:2208.04112 §2.3.2].

Prediction differs by subject status. For a new subject, use only the fixed part. For a subject seen in training, add its estimated random effect [arxiv:2208.04112 §2.3.2].

**Mixed-effects hybrids, non-Gaussian outcome.** Two routes for binary outcomes:

- **GMERT / GMERF** (generalized linear mixed model with a tree or forest in the fixed part) use pseudo-likelihood iterations that need starting values. The GMET / GMERF variant starts from a plain GLM on all predictors, so it cannot be used when there are more predictors than observations [arxiv:2208.04112 §2.3.2].
- **BiMM tree / forest** (Bayesian mixed model for binary outcomes). The fixed part is a tree's leaf indicators, fitted by alternating a tree step with a Bayesian GLMM step until the posterior log-likelihood stabilises [arxiv:2208.04112 §2.3.3].

### Multivariate response: whole trajectories

These methods treat a subject's repeated outcomes as one object and change the impurity used to pick splits. They apply directly only to time-invariant predictors [arxiv:2208.04112 §3].

- **Distance-based impurity.** Segal's criterion uses a generalized sum of squares with a node covariance, estimated once in the parent and shared by both children. It requires the same observation times for all subjects [arxiv:2208.04112 §3.1.1, eq. 4]. With an independence covariance it reduces to the multivariate sum of squares used in several existing packages [arxiv:2208.04112 §3.1.1].
- **db-MRT.** Impurity is built from a distance matrix over pairs of observations. Responses may then differ in length or be irregularly timed, provided a distance is defined. Prediction inside a leaf needs care, because a plain average may not exist [arxiv:2208.04112 §3.1.1].
- **Likelihood-based impurity.** For several binary responses, Zhang's exponential-family model uses the maximised log-likelihood in each node. For multivariate normal responses, Abdolell et al. use a deviance that equals a Mahalanobis distance [arxiv:2208.04112 §3.1.2].
- **Spline coefficients.** Fit a spline to each subject's trajectory and regress the basis coefficients with a multivariate tree. This handles irregular times [arxiv:2208.04112 §3.2].

## When to use which

The review gives no benchmark ranking, so the choices below are drawn from the constraints each method's description and software support imply.

| Situation | Use | Why |
|---|---|---|
| Repeated outcome, no time-varying predictors, need a quick fix | Subject-level bootstrap (RF++) | Simple change to sampling; subject-level OOB error and variable importance are available [arxiv:2208.04112 §2.2.2 tab. 3] |
| Quick baseline on very few subjects | Compare against a subject-level method, not averaging | Averaging was reported to degrade faster than a standard forest as subjects shrink [arxiv:2208.04112 §2.2.1] |
| Time-varying predictors, time effect matters, variable importance needed | Historical random forest | Models history explicitly; importance is listed [arxiv:2208.04112 §2.3.1 tab. 3]. Cost grows with the number of time-varying predictors [arxiv:2208.04112 §2.3.1] |
| Continuous outcome, want explicit subject random effects | RE-EM tree / REEMforest | Public R packages; residual covariance can be non-diagonal [arxiv:2208.04112 §2.3.2 tab. 3] |
| Residuals clearly correlated over time (e.g. autocorrelated) | RE-EM family rather than MERT / MERF | MERT / MERF assume a common residual variance [arxiv:2208.04112 §2.3.2] |
| Binary outcome with random effects | BiMM forest (R code from paper supplement) or GMERT | Both pair a tree or forest with random effects for binary outcomes; BiMM uses a Bayesian GLMM step [arxiv:2208.04112 §2.3.2 §2.3.3 tab. 3] |
| Binary outcome, GLMM-style, but more predictors than observations | Not GMET / GMERF | Its GLM starting step cannot run when p exceeds n [arxiv:2208.04112 §2.3.2] |
| Irregular observation times, or a trajectory as the response | db-MRT or spline-coefficient tree | Neither needs common observation times [arxiv:2208.04112 §3.1.1 §3.2] |
| Same observation times for all subjects, time-invariant predictors | Segal-style criterion with a shared covariance | Covers AR(1), compound symmetry and sample covariance choices [arxiv:2208.04112 §3.1.1] |
| Omics-scale predictors | Check the evidence first | Few evaluations on high-dimensional data; see Contradictions [arxiv:2208.04112 §2.3.2 §5] |

## Implementation sketch

Language-agnostic pseudocode for the common core of the subject-level route. Only the sampling step is shared by all variants here; the mixed-effects loops are in the review and in the original papers [arxiv:2208.04112 §2.2.2 §2.3.2].

```pseudocode
# Per-tree training sample for clustered rows  [arxiv:2208.04112 §2.2.2]
def tree_sample(rows_by_subject, two_stage):
    subjects = list(rows_by_subject.keys())
    # Resample subjects with replacement, not rows  [arxiv:2208.04112 §2.2.2]
    drawn = sample_with_replacement(subjects, size=len(subjects))
    in_bag = []
    for subject in drawn:
        rows = rows_by_subject[subject]
        if two_stage:
            # Adler et al.: keep one random row per drawn subject
            in_bag.append(random_choice(rows))
        else:
            # RF++: keep every row of the drawn subject
            in_bag.extend(rows)
    # OOB is measured per subject, not per row  [arxiv:2208.04112 §2.2.2]
    out_of_bag = [
        row
        for subject in subjects if subject not in drawn
        for row in rows_by_subject[subject]
    ]
    return in_bag, out_of_bag
```

Where the variants differ:

- **Two-stage vs RF++:** the only change is how many rows per drawn subject enter the tree [arxiv:2208.04112 §2.2.2].
- **Historical RF:** each time-varying predictor is replaced by its windowed summary before the split search, so the window and threshold must be tuned [arxiv:2208.04112 §2.3.1]. The review does not state the candidate grid or sub-sample size.
- **Mixed-effects hybrids:** wrap the forest in an alternating loop that refits the random effects and checks a generalized log-likelihood. The review does not state the convergence tolerance [arxiv:2208.04112 §2.3.2].
- **Distance-based criteria:** the node impurity is a sum of pairwise or Mahalanobis terms. The review's text gives the split gain with a sign that would select the worst split if taken literally, and its formula is not normalised by node size. Check the original sources before coding it [arxiv:2208.04112 §3.1.1].

## Contradictions

- **Is MERT / MERF code available?** The software table lists an R program for MERT and MERF from the original author [arxiv:2208.04112 tab. 3]. The text says the MERT / MERF code is not publicly available [arxiv:2208.04112 §4, §5].
- **High-dimensional evidence.** The review reports that MERT / RE-EM based trees and forests work on high-dimensional data and beat both the linear mixed model and a standard forest, citing a 2021 study [arxiv:2208.04112 §2.3.2]. Its later summary says that, except for BiMM and REEMforest, the methods have not been evaluated on high-dimensional data [arxiv:2208.04112 §5].
- **Variable importance for historical RF.** The method section describes importance by deleting a variable and refitting [arxiv:2208.04112 §2.3.1]. The software table labels the historical RF importance as permutation importance [arxiv:2208.04112 tab. 3]. These are different procedures. Check which one the software computes.
- **Resampling unit for mixed-effects hybrids.** MERF resamples observations and relies on its random effects to absorb within-subject correlation [arxiv:2208.04112 §2.3.2]. The subject-level methods argue for resampling whole subjects to avoid that correlation [arxiv:2208.04112 §2.2.2].
- **Two-stage bootstrap.** Reported as the best option in simulations, with the caveat that results depend on the setting [arxiv:2208.04112 §2.2.2]. The review does not claim it generalises.

## Papers

- [Hu and Szymczak's review](/papers/2208.04112): narrative review of tree-ensemble methods for repeated measurements, with model equations, fitting loops and a software table. No experiments and no code of its own.