Improving Dynamic Modeling of Intensive Longitudinal Data
Chair(s): Sofia Morelli (University of Tübingen, Germany), Kento Okuyama (University of Tübingen, Germany), Holger Brandt (University of Tübingen, Germany)
Intensive longitudinal data (ILD) provide a rich framework for capturing psychological, behavioral, and physiological processes over time. However, modeling ILD comes with unique challenges, including accounting for heterogeneity across and within subjects, evaluating model fit under complex dynamics, and handling unmeasured confounding and model uncertainty.
Driver introduces recent developments in the R package ctsem, focusing on efficiently modeling heterogeneity through non-linear filtering, refining model fit diagnostics beyond traditional SEM indices, and scaling to higher-dimensional systems via reduced random effects and backend optimizations. Morelli et al. propose a novel method to handle unmeasured confounding in latent mediation models, combining G-estimation and factor scores. Their two-stage method addresses measurement error and improves bias correction, offering an alternative to assumptions like sequential ignorability. Zierahn et al. investigate the limitations of Bayesian model fit indices in DSEM, particularly for small samples typical of ILD. They propose dynamic adaptations of fit indices to better detect localized misfit, such as omitted cross-loadings. Neduchal et al. introduce ensemble learning methods for detecting time-varying measurement model shifts in DSEM, using hierarchical and orthogonal stacking to dynamically recalibrate factor loadings and improve latent variable estimation. Finally, Okuyama et al. extend Bayesian model stacking to address uncertainty in structural and time-series components of DSEM. Their method mimics regime-switching behavior without the computational burden of full RSSS models, offering improved forecasting in complex systems.
Together, these contributions advance ILD modeling by improving robustness, fit diagnostics, and flexibility in capturing dynamic processes.
Presentations of the Symposium
Recent Developments in ctsem -- Heterogeneity Across and Within Subjects, Misfit, Dimensionality
Charles Driver
University of Zürich
The R package ctsem provides a unified framework for fitting both discrete and continuous time structural equation / state-space models. This talk reviews recent developments in the package across three broad areas: handling parameter heterogeneity, diagnosing model fit, and scaling to higher dimensional systems. For parameter heterogeneity across individuals in any of the system or measurement parameters, recent versions of ctsem use non-linear filtering by default, for much faster estimation than Bayesian approaches. There is now also some data driven approaches available -- automated individual and group level model selection routines, forward search of covariate moderators, and random forest approaches, and parameter heterogeneity across time. Regarding model fit, I will cover a few diagnostic tools designed to detect when dynamic system models need refinement—classical SEM indices are generally inadequate—and outline approaches for guiding targeted improvements. On the scalability front, I will discuss the combination of reduced dimension random effects coupled with the nonlinear filtering, and describe some experiments with a Julia backend.
Combining Factor Scores and G-estimation to Handle Unmeasured Confounding in Latent Mediation Analysis
Sofia Morelli, Roberto Faleh, Holger Brandt
University of Tübingen, Germany
Modelling mediation processes in longitudinal intervention studies provides a valuable framework for understanding underlying causal mechanisms. However, most standard mediation analyses rely on the often unrealistic assumption of no unmeasured confounding between the mediator and the outcome, called sequential ignorability.
By using G-estimation in place of standard estimation techniques such as maximum likelihood or least squares, this assumption can be relaxed and replaced with more plausible conditions, such as rank preservation or no essential heterogeneity. To extend this approach to latent constructs, we develop a factor score-based version using a two-stage method of moments to correct for bias introduced by measurement error.
We evaluate the performance of the proposed method through simulation studies, comparing it to standard structural equation modeling (SEM), and demonstrate its advantages in settings with unmeasured confounding.
Model fit evaluation in Dynamic Structural Equation Models
Kim Zierahn, Mihai Faculsan, Nikita Saprinsky, Holger Brandt
University of Tübingen, Germany
Intensive longitudinal data (ILD) has received increasing interest from researchers who want to model dynamic processes. In clinical psychology, ecological momentary assessments are used to examine participants measured at many measurement occasions to evaluate the underlying process of how a treatment may evolve. Recently, Dynamic Structural Equation Models (DSEM) were introduced that allow researchers to extract relevant information about these complex dynamic changes using latent variables. With this complexity – that is complex latent variable time series models in combination with measurement models that connect the latent and observed variables – the problem of potential misspecification increases dramatically. Model fit evaluation in Bayesian SEM, however, is still an understudied topic in general. While some research indicates that the respective Bayesian versions of fit indices only perform well in (very) large sample sizes, no comparable information is available for DSEM. We will present a simulation study that investigates the performance of Bayesian model fit indices for a reduced class of DSEM that can be used to evaluate model misfit such as omitted cross-loadings. Results indicate that standard fit indices (such as BRMSEA, BCFI) are not capable of identifying misfit in situations that reflect typical conditions in ILD (i.e. sample sizes below N=500). We propose a dynamic adaptation of these fit indices that based on a limited information approach can be used to locate if misfit occurs during the sequence of ILD collection.
Meta-Learning Approaches for improving misfit detection in Dynamic Structural Equation Models
Luca Neduchal, Kento Okuyama, Holger Brandt
University of Tübingen, Germany
Dynamic structural equation models (DSEM) are a popular framework for analyzing intensive longitudinal data. They combine time-series modeling with structural equation modeling. However, model evaluation remains difficult: traditional model fit indices become unreliable in complex settings and appropriate modification indices that identify measurement model misspecifications do not exist.
This article presents three ensemble methods -- linear stacking, hierarchical stacking and model orthogonalization --, which capture and account for time-varying factor loading shifts in the measurement model that would otherwise remain hidden. The methods stack time-specific likelihoods from an ensemble of base learner DSEM that solely differ with respect to their time-invariant factor loadings. By inflating and deflating base learner weights according to their local fits, the presented stacking approaches produce interpretable ensemble weights and better calibrated DSEM with respect to their latent variable estimates and time-specific likelihoods. The ensemble learners shed light on local misspecifications and efficiently adapt to shifts in the measurement model.
Results from a simulation study suggest that all three methods correctly identified the time point at which the shift occurred and adjusted their loading matrix in the following period. Hierarchical stacking and model orthogonalization made use of prior information from linear stacking and produced more likely predictions and less biased estimates. Further, manually accounting for shifting factor loadings through a weight-informed DSEM provided the best results among all learners.
This offers a practical view on detecting and accommodating misspecifications solely in the measurement model, leaving the structural dynamics untouched, until dedicated DSEM fit diagnostics become available.
Time-series forecasting in dynamic structural equation models under model uncertainty: Bayesian model stacking approach
Kento Okuyama, Luca Neduchal, Holger Brandt
University of Tübingen, Germany
Regime-switching state-space (RSSS) models are useful for forecasting critical states e.g., students' intention to dropout in math, abrupt mood switching in bipolar disorder, or even the 'hot hand' of the basketball player Vinnie Johnson.
Such models are confirmatory and complex; they are susceptible to misspecifications in many parts including measurement models (e.g., omitted cross-loading), structural/time-series models (e.g., wrong-order VAR model), Markov-switching models (e.g., omitted covariates), and distributional assumptions. An extension of RSSS to a large number of regimes that cover possible alternative specifications would be ideal but mostly not practical due to small class proportions and inflated computational costs.
To circumvent these issues, we propose a Bayesian model stacking approach as an alternative. Focusing on misspecifications in structural/time-series models, we demonstrate how candidate models (e.g., VAR(1)-VAR(3)) are weighted in a time- and covariate-dependent manner. Similar to RSSS models, our framework allows us to mimic regime-switches, but with a greater flexibility. Within- and between-person model weights are represented as posterior distributions, allowing for uncertainty quantification rather than relying on single point estimates. Utilizing a two-step approach, our method will have a computational advantage by estimating the model weights after fitting each individual model (parameters). We illustrate their utility through simulation studies and an empirical example using data on patients' development of their working alliance with their therapists.
