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Using latent representations to link disjoint longitudinal data for mixed-effects regression

Clemens Schächter, Maren Hackenberg, Michelle Pfaffenlehner, Félix B. Tambe-Ndonfack, Thorsten Schmidt, Astrid Pechmann, Janbernd Kirschner, Jan Hasenauer, Harald Binder

arXiv:2510.25531v2stat.MLcs.AIcs.LG

TL;DR

Rare-disease treatment-switch analyses face limited data and changing measurement instruments that complicate longitudinal modeling. The paper links item-level observations from different instruments in a shared latent representation for mixed-effects regression and statistical testing. In SMA, this approach aligned instruments to use a larger effective sample and detected treatment-switch effects that might otherwise have gone unidentified.

  • Problem

    Rare-disease studies have limited data, and changing measurement instruments can make longitudinal treatment-switch analysis difficult while conventional approaches discard item detail or restrict the observation window.

    Source: §1 · p. 3

  • Method

    Instrument-specific VAEs map item values into a shared low-dimensional representation linked over time by multivariate mixed-effects regression, with a bootstrap knockoff approach for statistical testing.

    Source: §1 · p. 4, §2.1 · p. 6, §2.3 · p. 9

  • Results

    In the SMA application, aligning instruments leveraged a larger effective sample than analyzing each instrument separately and enabled detection of treatment-switch effects that might not otherwise have been identified.

    Source: §4 · p. 19

  • Takeaways & Limitations

    Joint latent representations may offer relevant power gains for evaluating treatments in rare-disease registries where data are scarce.

    Source: §4 · p. 19

    The latent dimension must be chosen carefully: too few dimensions can limit modeled dynamics, while too many can increase overfitting.

    Source: §4 · p. 19

Abstract

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Many rare diseases offer limited established treatment options, leading patients to switch therapies when new medications emerge. To analyze the impact of such treatment switches within the low sample size limitations of rare disease trials, it is important to use all available data sources. This, however, is complicated when usage of measurement instruments change during the observation period, for example when instruments are adapted to specific age ranges. The resulting disjoint longitudinal data trajectories, complicate the application of traditional modeling approaches like mixed-effects regression. We tackle this by mapping observations of each instrument to a aligned low-dimensional temporal trajectory, enabling longitudinal modeling across instruments. Specifically, we employ a set of variational autoencoder architectures to embed item values into a shared latent space for each time point. Temporal disease dynamics and treatment switch effects are then captured through a mixed-effects regression model applied to latent representations. To enable statistical inference, we present a novel statistical testing approach that accounts for the joint parameter estimation of mixed-effects regression and variational autoencoders. The methodology is applied to quantify the impact of treatment switches for patients with spinal muscular atrophy. Here, our approach aligns motor performance items from different measurement instruments for mixed-effects regression and maps estimated effects back to the observed item level to quantify the treatment switch effect. Our approach allows for model selection as well as for assessing effects of treatment switching. The results highlight the potential of modeling in joint latent representations for addressing small data challenges.

1 Introduction

Rare-disease treatment-switch analyses must make use of limited data, but changing measurement instruments can leave longitudinal records disjoint. The paper proposes linking item-level measurements in a shared latent representation for mixed-effects modeling and statistical inference.

1 Introduction

  • Changing instruments can leave rare-disease longitudinal records disjoint, even though using all available data is important when treatment options and study populations are limited.In SMA, assessment instruments may change as patients age or their disease progresses.Source: §1 · p. 3
  • Analyzing only one instrument discards item-level information and can restrict follow-up, introducing bias and further reducing already limited statistical power.Different instruments also have distinct scales and within-patient correlations, making their results challenging to align.Source: §1 · pp. 3–4
  • The proposed approach embeds instrument-specific items into a shared low-dimensional representation and links time points with multivariate mixed-effects regression.A bootstrap knockoff approach is introduced for statistical testing that accounts for potential bias from fitting the mixed-effects model within the VAE.Source: §1 · p. 4
  • Existing linking approaches may require shared anchor items, while domain-adaptation methods often assume many observations and do not address short longitudinal records.The introduction identifies combining domain adaptation with longitudinal mixed-effects regression as a missing approach for treatment-switch inference.Source: §1 · pp. 4–5
  • The paper evaluates the approach using five motor-performance instruments in the SMArtCARE SMA registry, comparing it with per-instrument meta-analysis and testing recovery of artificial switches.It also illustrates model selection for the latent mixed-effects model.Source: §1 · p. 5

2 Methods

The methods combine instrument-specific variational autoencoders with multivariate mixed-effects regression to model longitudinal data in a shared latent representation. A bootstrap-knockoff procedure supports inference for the jointly estimated model.

2.1 Variational autoencoders

  • VAEs map high-dimensional observations into low-dimensional latent representations and reconstruct them in the original measurement space.The patient visits are treated as i.i.d. observations when network weights are optimized, yielding latent trajectories for subsequent mixed-effects regression.Source: §2.1 · pp. 6–7

2.2 Multivariate longitudinal mixed-effects regression

  • Multivariate mixed-effects regression represents each latent patient trajectory through fixed effects, patient-specific random effects, and residual errors.Fixed effects capture shared covariate influences; random effects allow individual deviations, with shared random-effect covariance across patients.Source: §2.2 · p. 8

2.3 Joining multiple measurement instruments in a latent representation

  • The approach uses instrument-specific encoders to map measurements into a joint latent trajectory, models it with mixed-effects regression, and decodes predictions to quantify treatment-switch effects per item.Encoder and decoder parameters and mixed-effects parameters are updated iteratively; treatment effects are assessed by comparing decoded trajectories with and without the switch.Source: §2.3 · pp. 9, 10, 12, Figure 1 · p. 10

2.4 Model selection

  • Because latent responses depend on jointly optimized VAE parameters, ordinary likelihood-ratio testing can suffer post-selection bias; bootstrap knockoffs provide inference for the joint estimator.The procedure constructs an empirical null by reinitializing and re-optimizing the VAEs across bootstrap samples; knockoff covariates act as negative controls.Source: §2.4 · pp. 12–13

3 Results

Across five SMA motor instruments, the latent model estimated positive post-switch improvements. Knockoff-calibrated tests and a covariance-aware meta-analysis also supported a nonzero treatment-switch effect.

3 Results

  • The analysis integrated five motor instruments for 522 patients, with a median of 17 observations per patient and 2.1 instruments used per visit on average.Instruments were used across different ages and motor-function stages.Source: §3 · p. 14

3.1 Quantifying the impact of treatment switches

  • Predicted improvement ranged from 1.8% to 5.4% of the maximal score across measurement instruments.Across ten random initializations, no instrument’s aggregated treatment effect changed from positive to negative.Source: §3.1 · pp. 14–15
  • Under an artificial annual switch adding +2 points, the approach recovered average effects of 0.78 to 1.97, with correct directionality and no overestimation.The perturbation ignored ceiling effects, within-observation structure, and cross-instrument alignment, so the authors expected the estimated magnitude to be understated.Source: §3.1 · p. 15
  • The latent model reduced ceiling effects relative to a standard data-level mixed model.Figure 2 contrasts observed scores (black), switched predictions (red), and hypothetical no-switch predictions (blue), with latent-model reconstructions above data-level reconstructions.Source: §3.1 · p. 15, Figure 2 · p. 16

3.2 Latent model selection

  • For d = 1, empirical knockoff null distributions closely followed theoretical chi-squared distributions; for d = 3, theoretical calibration was anti-conservative for fixed and random effects.Figure 3 compares those distributions across effect type and latent dimension.Source: §3.2 · p. 15, Figure 3 · p. 17
  • Treatment switches, ventilation, disease onset, and scoliosis exceeded knockoff thresholds, while sex and family history did not.In switch-free generated data, the treatment-switch likelihood ratio fell to 18.2, below the 23.6 threshold for d = 3 and r = 4.Source: §3.2 · pp. 16–17

3.3 Comparison with naive meta-analysis

  • A correlated-effects meta-analysis accounting for overlapping patients found a nonzero combined switch effect (global likelihood-ratio statistic 214.06, rd = 3; bootstrap-calibrated p < 10^-3).Its estimated magnitude was consistent with the latent mixed-effects model.Source: §3.3 · p. 18
  • At one-third of the original sample size, the latent approach still detected the switch (Λobs = 102.5), while data-level models often lacked stable fits; instrument-specific models also had smaller samples than the latent model.The latent model used 522 patients, compared with 308 for RULM, 318 for HINE-2, 239 for HFMSE, and 158 for CHOP-INTEND; ALSFRS-R had too few patients for the specified mixed model.Source: §3.3 · pp. 17–18

4 Discussion

The approach combines VAEs with latent longitudinal mixed-effects regression to align SMA measurements across instruments and assess treatment switches. The discussion reports detected switch effects, while noting that latent dimensionality constrains model complexity and requires careful selection.

4 Discussion

  • Encoder networks map items from different instruments to a unified latent representation, while knockoff variable bootstrap testing addresses potential fitting biases in inference.The unified representation handles systematic instrument changes during observation; the method also uses item-level details and aligns instruments with different scales.Source: §4 · pp. 18–19
  • The approach detected treatment switch effects in the SMA application that might not have been identified otherwise.The discussion describes this as a capability demonstrated in the application.Source: §4 · p. 19
  • Latent dimensionality must balance representing underlying dynamics against over-parameterization and overfitting, so its choice should reflect the dataset and ideally domain knowledge.The authors suggest estimating intrinsic dimension in a higher-dimensional space as one possible approach.Source: §4 · p. 19
  • The latent mixed-effects model should incorporate biomedical knowledge, with the proposed testing approach supporting model selection where needed.The authors note this may provide further insight into structure shared across measurement instruments.Source: §4 · p. 19
  • Hybrid deep-learning and statistical approaches may help model small, heterogeneous, high-dimensional, and multimodal datasets, an area the authors describe as promising for further research.The authors note that explicit statistical inference within deep-learning frameworks remains uncommon.Source: §4 · p. 19

5 Funding

The work was funded by the Deutsche Forschungsgemeinschaft through the SFB 1597 Small Data project.

5 Funding

  • Funding was provided by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), Project-ID 499552394, SFB 1597 Small Data.Source: §5 · p. 25

6 Data preprocessing and Hyperparameters

The analysis selected patients with treatment switches using eligibility and follow-up criteria, then prepared item-level motor-function data and specified the VAE and mixed-effects model. It also simulated an artificial treatment effect for evaluation.

6 Data preprocessing and Hyperparameters

  • The cohort included patients with treatment switches, excluding those with under six months of treatment before or after switching and retaining data only through the second switch for patients with more than two.Switches to medications with fewer than 10 patients were excluded, leaving two SMA medications; eligible patients needed at least four assessment visits, including two before and after the switch.Source: §6 · pp. 25–26
  • The resulting cohort comprised 522 patients, averaging 17.3 hospital visits and 2.1 of five motor-function tests per visit, with a 5.7-year median observation period.Median patient age was 6.7 years; test items were thermometer-encoded, and the decoder used an ordinal logistic distribution.Source: §6 · p. 26
  • Missing items within an assessment were zero-filled, while observations with more than one-quarter missing values were excluded.Zero-filling reflected clinical partners’ view that skipped items typically resulted from patient exhaustion.Source: §6 · p. 26
  • Covariates were standardized and generally interacted with age; SMN2 count, symptom-onset age, and years since switching were represented with specified category or medication-specific variables.The data-level mixed models included an intercept, whereas the latent model did not because latent variables were centered around zero.Source: §6 · p. 26
  • The encoder and decoder used dense neural networks with two hidden layers of 250 and 100 units and tanh activation.Source: §6 · p. 26
  • The VAE used a three-dimensional latent space chosen to represent three SMA aspects, and training alternated 100 VAE updates per epoch with mixed-model refitting for 20 epochs.The authors down-weighted the KL term by 0.5 and up-weighted alignment terms by 5; performance remained stable when these coefficients were halved or doubled.Source: §6 · p. 27
  • The artificial switch simulation added one motor-function item per test every six months after switching, selecting items below their maximum score and reallocating increments when possible.Source: §6 · p. 27

7 Tables

The tables cover the motor-function instruments and their included items, manuscript notation, and predicted item-level differences associated with treatment switches.

7 Tables

  • Table 1 presents the motor-function measurement instruments used in SMArtCARE and includes some items beyond those used to calculate official test scores.Those additional items were included to increase the amount of information.Source: Table 1 · p. 28
  • Table 2 lists the notation used throughout the manuscript.Source: Table 2 · p. 29
  • Table 3 reports mean predicted item-level differences one year after switching for five instruments, with a detected-switch block and an artificial +2-items-per-year switch block.Positive values indicate improved motor function; percentages express differences relative to each instrument’s maximum sum-score.Source: Table 3 · p. 30
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