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Computational Growth Modeling: Assessing Longitudinal Change with Joint Models

This paper introduces and validates "computational growth modeling," a joint modeling approach that integrates hierarchical computational models with longitudinal data to more accurately estimate individual changes in behavioral processes over time compared to traditional two-stage methods, thereby offering improved insights into the etiology of psychopathology.

Original authors: Jeremy Haynes, Nathaniel Haines, Holly Sullivan-Toole, Thomas Olino

Published 2026-10-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jeremy Haynes, Nathaniel Haines, Holly Sullivan-Toole, Thomas Olino

Original paper licensed under CC BY 4.0 (https://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

Human behavior is rarely a single snapshot; it is a story that unfolds over time. To understand why people make the choices they do, psychologists often ask them to perform tasks that reveal how they learn from rewards and punishments. One such task, known as the Iowa Gambling Task, asks participants to pick cards from different decks. Some decks offer big wins but even bigger losses, while others offer smaller, steadier gains. By watching which decks people choose, researchers can see if they are learning to avoid the risky options. For years, scientists have used computer models to translate these choices into numbers, estimating things like how quickly a person updates their expectations after a win or a loss. These numbers, called parameters, act as a window into the mind's learning machinery. However, a persistent problem has been how to track these numbers as they change. When researchers want to see how learning evolves over months or years, they have typically relied on a two-step process: first, they calculate a person's learning numbers for a single session, and then they treat those numbers as perfect facts to see how they change later. This method assumes the first calculation was flawless, ignoring the natural fuzziness and uncertainty that comes with measuring human behavior.

A team of researchers led by Jeremy Haynes at Georgia Southern University has proposed a different way to look at this problem, one that treats the measurement and the change as a single, connected story. In a new study, they introduced a method they call computational growth modeling. Instead of separating the calculation of a person's learning style from the tracking of how that style changes over time, this new approach does both at once. By weaving the two together, the researchers can account for the uncertainty in the initial measurements, allowing the data from one session to inform the understanding of the next. To test if this idea works, the team first created a massive simulation. They generated thousands of fake datasets representing people playing a simple decision-making game over three sessions. In these simulations, they knew the exact truth about how the participants' learning rates were supposed to change. They then compared their new, integrated method against the traditional two-step approach. The results were clear: the new method was far better at recovering the true patterns of change. It correctly identified how fast people were learning and how their starting points varied, whereas the old method often missed the mark, particularly when trying to see how individual differences in starting ability related to the speed of improvement. The new approach successfully captured the "fog" of measurement error that the old method ignored, leading to a much sharper picture of how behavior evolves.

Encouraged by these simulations, the researchers applied their method to real-world data collected over three years from adults. These participants played the play-or-pass version of the Iowa Gambling Task five separate times. In this version, players can choose to take a risk and play a deck of cards or pass and skip the turn. Using their new integrated model, the team was able to trace how the participants' learning habits shifted across the five sessions. They found that as people gained experience, their sensitivity to immediate rewards and punishments actually decreased, suggesting they were becoming less reactive to single wins or losses and more focused on the long-term outcome. At the same time, their sensitivity to how often they won or lost increased, and they developed a stronger tendency to play regardless of the specific deck. This pattern of change suggested that participants were refining their strategy, moving away from impulsive reactions and toward a more calculated approach.

The power of this new method became even more apparent when the researchers added personal history into the mix. They examined whether a person's history of anxiety, depression, or substance use, as well as their sex, influenced how their learning changed over time. The model revealed distinct differences. For instance, men showed a higher baseline sensitivity to punishment than women, meaning they were initially more reactive to losses. More strikingly, the model showed that people with a history of depression started with a higher sensitivity to how often they won or lost, but their sensitivity to punishment dropped off much more sharply over time compared to those without such a history. This suggests that the way depression affects decision-making is not just a static trait but a dynamic process that changes as a person gains experience. The researchers noted that these findings about depression and learning rates were specific to the longer timeline of their study, a nuance that might have been missed with fewer data points.

This work does not claim to have solved the mystery of human decision-making, but it offers a more reliable tool for watching it unfold. The authors emphasize that their method is particularly useful for understanding how behavioral processes develop and how they might go awry in mental illness. By acknowledging that our measurements of the mind are never perfectly precise, and by building that uncertainty directly into the analysis of change, the computational growth modeling approach provides a clearer, more honest view of how we learn. It allows scientists to see not just where a person is on a single day, but how they are moving, and how that movement might differ based on their unique history and biology. As the researchers point out, this framework can be adapted to many different types of tasks and even animal studies, offering a versatile way to study the fluid nature of behavior across time.

Technical Summary: Computational Growth Modeling with Joint Models

Problem Statement
Longitudinal research in psychology frequently aims to characterize how behavioral processes change over time. A common methodological challenge arises when researchers attempt to analyze longitudinal data derived from computational models (e.g., reinforcement learning parameters). The standard practice involves a "two-stage" approach: first, fitting a computational model to behavioral data at each timepoint to extract person-level parameters; second, fitting a statistical growth model (e.g., Hierarchical Linear Models) to these extracted parameters.

The paper identifies a critical flaw in this two-stage approach: it treats the person-level computational parameters as "true scores" in the second stage, ignoring the measurement error and uncertainty inherent in their estimation from the first stage. This failure to propagate uncertainty from the behavioral model to the growth model can lead to biased inferences, reduced statistical power, and poor reliability in estimating longitudinal trajectories. While joint modeling has been used previously to improve test-retest reliability and cross-task associations, its application to assessing longitudinal change in computational parameters has been limited.

Methodology
The authors propose and implement "Computational Growth Modeling" (CGM), a framework that integrates computational behavioral models and longitudinal growth models into a single hierarchical Bayesian joint model.

Theoretical Framework:
- Person-Level Model: A computational model (e.g., Reinforcement Learning) characterizes trial-level behavior ( $Y$ ) based on task stimuli ( $C$ ) and person-specific parameters ( $\theta$ ).
- Group-Level Model: A longitudinal growth model characterizes how the person-level parameters ( $\theta$ ) change over time ( $T$ ) and vary across individuals. This is specified as a linear growth model where person-level intercepts ( $\beta_{0i}$ ) and slopes ( $\beta_{1i}$ ) are drawn from a multivariate normal distribution defined by group-level means ( $\gamma$ ), variances, and correlations.
- Integration: Unlike the two-stage approach, CGM estimates the computational parameters and the growth parameters simultaneously. This allows uncertainty in the trial-level parameter estimates to inform the estimation of the growth trajectories, and vice versa.
Simulation Study:
- Design: The authors simulated 900 datasets from a two-armed bandit task using a single-parameter Rescorla-Wagner reinforcement learning model. Data were generated for 50 "participants" across three sessions.
- Conditions: Simulations varied group-level slope effects (no change, moderate decrease, moderate increase) and intercept-slope correlations.
- Comparison: The CGM approach was compared against the traditional two-stage approach. Both were implemented using hierarchical Bayesian estimation in Stan.
- Metrics: Performance was evaluated based on parameter recovery (correlation between true and estimated parameters), mean absolute deviations (MAD) between true and estimated values, and 95% coverage probabilities (the frequency with which the 95% Highest Density Interval contained the true parameter).
Real Data Application:
- Task: The authors applied CGM to five waves of longitudinal data from adults performing the "play-or-pass" Iowa Gambling Task (IGT).
- Model: They utilized the Outcome Representation Learning (ORL) model, which includes four parameters: reward learning rate, punishment learning rate, win/loss frequency sensitivity, and response bias.
- Covariates: Two models were fitted: an unconditional growth model (baseline trajectories) and a conditional growth model including covariates for sex and lifetime history of anxiety, depression, and substance use disorder.

Key Results

Simulation Findings:
- Parameter Recovery: The CGM approach significantly outperformed the two-stage approach in recovering person-level growth parameters (intercepts and slopes). The two-stage approach showed poor recovery for intercepts (correlations $\le$ .51) compared to CGM (correlations $\ge$ .59).
- Bias and Uncertainty: The two-stage approach systematically underestimated the standard deviations of intercepts and slopes and failed to capture the true group-level intercepts and slopes in conditions with changing learning rates.
- Coverage: The two-stage approach exhibited poor 95% coverage probabilities for growth parameters (often far below the target 95%), indicating that the true parameters frequently fell outside the estimated confidence intervals. CGM maintained coverage probabilities much closer to the 95% target.
- Learning Rates: While both approaches performed well in estimating static learning rates at individual timepoints, CGM provided superior estimates of the change in these rates over time.
Real Data Findings:
- Unconditional Trends: Across five sessions over three years, reward and punishment learning rates decreased, while win/loss frequency sensitivity and response bias increased.
- Conditional Effects: The conditional model revealed meaningful individual differences. Male participants showed higher baseline punishment learning rates than females. Individuals with a history of depression exhibited higher baseline win/loss frequency sensitivity and a steeper decline in punishment learning rates over time compared to those without such a history.

Significance and Claims
The paper claims that Computational Growth Modeling offers a more robust and accurate method for assessing longitudinal change in behavioral processes compared to traditional two-stage approaches.

Methodological Advancement: By integrating computational and growth models, CGM properly accounts for measurement error and uncertainty, preventing the bias introduced when treating estimated parameters as fixed true scores.
Improved Inference: The simultaneous estimation framework yields better parameter recovery and more accurate confidence intervals (coverage), particularly for growth parameters (slopes and intercepts) which are often the primary focus of longitudinal research.
Flexibility: The framework is flexible enough to accommodate various computational models (from simple to complex) and can seamlessly integrate covariates (e.g., psychopathology history, sex) to test how individual differences moderate developmental trajectories.
Modest Scope: The authors note that while CGM shows promise, it is computationally intensive and requires bespoke programming (e.g., in Stan). They also acknowledge that hierarchical shrinkage, while improving reliability, compresses between-subject variability, which may be a consideration for researchers focused on individual organism behavior. The paper does not claim CGM is a universal solution but presents it as a superior alternative for specific contexts involving longitudinal computational modeling.

Technical Summary: Computational Growth Modeling with Joint Models

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