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Conditional GLMMs for reaction times in choice tasks

This study bridges cognitive diffusion models and Generalized Linear Mixed Models (GLMMs) by demonstrating that conditioning reaction time distributions on response alternatives allows for the use of Inverse Gaussian and Gamma distributions to both model reaction times and infer underlying cognitive processes.

Original authors: Mauricio Tejo, Cristian Meza, Fernando Marmolejo-Ramos

Published 2026-08-03

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Mauricio Tejo, Cristian Meza, Fernando Marmolejo-Ramos

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

The Brain's Secret Race Track

Imagine you are watching a race, but instead of runners on a track, the competitors are tiny sparks of thought zipping through your brain. Every time you see a sign, hear a sound, or feel a touch, your brain has to decide what to do. How long does that decision take? That split-second pause between seeing a stimulus and pressing a button is called a "reaction time." For over a century, scientists have been obsessed with these tiny delays because they are like a window into the mind's machinery. By measuring how fast or slow we react, researchers try to understand the invisible gears turning inside our heads.

To make sense of these reaction times, scientists usually use two different toolkits. The first toolkit is like a physics simulation: it imagines the brain as a ball rolling down a hill, gathering speed until it hits a wall. This "hitting time" is the moment you make a choice. The second toolkit is a statistical map called a Generalized Linear Mixed Model (GLMM). Think of this as a super-smart spreadsheet that can sort through messy data, separating out how much of your reaction time is due to the task itself versus how much is just because you are a unique individual with your own quirks. The big question has always been: Can we use the statistical map to reverse-engineer the physics simulation? Can we look at the messy numbers and say, "Aha! The brain was rolling at this specific speed"?

Connecting the Dots: From Spreadsheets to Brain Rolls

This paper by Mauricio Tejo, Cristian Meza, and Fernando Marmolejo-Ramos is like a translator trying to speak two different languages at once. The authors wanted to connect the "physics" of how the brain makes decisions with the "statistics" of how we measure those decisions. They focused on a specific type of mental race: a choice task where you have to pick between two options, like deciding if a face looks "sad" or "happy."

Here is the clever trick they used. Instead of looking at all the reaction times mixed together, they split the data into two piles: the times when people chose "sad" and the times when people chose "happy." They then applied a special kind of statistical model (a GLMM) to each pile separately. They discovered that if they used specific mathematical shapes for these piles—specifically the Inverse Gaussian and Gamma distributions—they could do something magical. These shapes aren't just random math tricks; they actually match the shapes you get when you simulate a ball rolling down a hill until it hits a wall.

By fitting these specific shapes to real human data, the authors could work backward. They took the numbers from the statistical model and used them to reconstruct the "hidden movie" of the brain's decision process. They could estimate things like the "drift rate" (how fast the information was gathering) and the "starting point" (where the mental race began). To prove their method worked, they ran computer simulations. They created fake data that followed these exact mathematical rules, ran their model on it, and successfully recovered the original "physics" they had put in. It was like building a house from a blueprint, then using the finished house to perfectly redraw the original blueprint.

The Real-World Test: Smiling Faces and Pen Tricks

To see if this worked in the real world, the authors applied their method to a famous experiment involving 116 people looking at faces. The faces ranged from "super sad" to "super happy," with many shades of gray in between. The participants had to quickly say if the face was sad or happy. Some of them even had to hold a pen in their teeth while doing it (a weird trick meant to force a smile or a frown), while others held no pen.

The results were fascinating. When the faces were very clear (a super sad face or a super happy face), the "drift rate" was high. This means the brain gathered evidence quickly and decisively, leading to fast reaction times. But when the faces were ambiguous (looking a bit like both), the drift rate slowed down. The brain had to roll the ball much longer before it hit the wall, resulting in slower, more variable reaction times.

Interestingly, the authors found that the "pen-in-teeth" trick didn't seem to change the fundamental speed of the brain's decision-making process. While the pen might have changed how people felt, it didn't seem to alter the core "physics" of how fast they could process the emotion. The statistical model showed that the brain's decision speed was driven almost entirely by how clear the face was, not by whether a pen was in someone's mouth.

What This Means (and What It Doesn't)

The authors are careful to say that this is a "moment-based approximation." Think of it like this: they are looking at the average speed and the average spread of the race to guess the track's layout. It's a very good guess, and their simulations show it works well, but it's not a perfect, pixel-by-pixel reconstruction of every single thought. They also noted that their model assumed the "non-decision" time (the time it takes to just see the face and move the finger, before the thinking even starts) was zero. In reality, that time exists, but their model treats the whole process as one smooth roll.

They also compared two different mathematical shapes (Gamma and Inverse Gaussian) and found that the Gamma distribution fit the real data slightly better than the Inverse Gaussian. This suggests that for these kinds of emotional face tasks, the Gamma shape might be the most accurate "blueprint" for the brain's decision process.

Ultimately, this paper doesn't claim to have solved the mystery of the human mind. Instead, it offers a powerful new bridge. It shows that we can use standard statistical tools, which are already built into many computer programs, to peek behind the curtain and see the hidden cognitive processes driving our choices. It turns a messy pile of reaction times into a clear story about how fast our brains roll toward a decision.

Technical Summary: Conditional GLMMs for Reaction Times in Choice Tasks

Problem Statement
Reaction time (RT) modeling in experimental psychology typically bifurcates into two distinct approaches: (1) quantitative distribution measurements that fit empirical data without clear cognitive interpretation (e.g., Ex-Gaussian), and (2) theoretical models derived from cognitive mechanisms (e.g., diffusion models) that often lack the hierarchical flexibility to handle complex experimental designs with multiple stimuli and individual variability. While Generalized Linear Mixed Models (GLMMs) offer a robust framework for handling fixed effects (stimuli) and random effects (individuals), they are rarely connected to the underlying cognitive processes (such as diffusion processes) that generate RTs. Furthermore, standard diffusion models often struggle to incorporate the complexity of modern choice tasks where responses are not strictly binary "correct/incorrect" but rather categorical alternatives (e.g., classifying facial expressions on a spectrum).

Methodology
The authors propose a unified framework that connects the first-hitting time of a simple diffusion model with Generalized Linear Mixed Models (GLMMs). The core methodology involves the following steps:

Conditional Modeling: Instead of modeling the overall RT distribution, the authors model the RT distribution conditioned on each specific response alternative ( $R_l$ ). This partitions the data into subsets corresponding to each response.
Theoretical Justification: The paper establishes that if a cognitive process is modeled as a diffusion process (specifically a Wiener process with drift), the first-hitting time to a barrier follows specific distributions. For a single barrier, this yields an Inverse Gaussian (IG) distribution; under modified assumptions, it yields a Gamma distribution. Both distributions belong to the Exponential Family, making them suitable for GLMMs.
GLMM Specification: A hierarchical GLMM is constructed where the dependent variable is the RT conditioned on a response.
- Fixed Effects: Represent the stimulus levels and conditions.
- Random Effects: Represent individual variability (random intercepts).
- Link Function: A log-link is used to connect the linear predictor to the mean RT.
- Variance Structure: The model utilizes the mean-variance relationship inherent to the chosen distribution family (IG or Gamma).
Moment-Based Reconstruction: A key theoretical contribution is the "reconstruction" of the underlying cognitive process parameters from the estimated GLMM parameters. By matching the marginal moments (mean and variance) of the GLMM estimates to the theoretical moments of the diffusion process, the authors derive estimates for the cognitive parameters:
- Drift Rate ( $\nu$ ): Represents the mean rate of information accumulation.
- Starting Point ( $a$ or shape parameter): Represents the initial processing state.
- Non-decision time ( $\theta$ ): In this specific implementation, $\theta$ is fixed at 0 to simplify the reconstruction, acknowledging this as a limitation.
Simulation and Validation: The authors validate the approach through simulations. They generate IG and Gamma distributed data, estimate GLMM parameters, and then reconstruct the diffusion process. Goodness-of-fit tests (PDF, CDF, Q-Q, P-P plots) confirm that the reconstructed process accurately recovers the original generating distributions.

Key Results
The methodology was applied to a real-world dataset from Marmolejo-Ramos et al. (2020), where participants classified facial expressions ranging from "sad" to "happy" under two conditions (holding a pen in teeth vs. no pen).

Model Fit: Both IG and Gamma GLMMs were fitted. The Gamma model demonstrated superior fit based on the Akaike Information Criterion (AIC) compared to the IG model, despite the IG model having a smaller dispersion parameter.
Cognitive Reconstruction: The reconstructed diffusion parameters revealed consistent patterns:
- Drift Rate: The estimated drift rate ( $\hat{\nu}$ ) increased with the emotional clarity of the stimulus in the direction of the response. For "sad" responses, drift was highest for clearly sad faces (stimuli 0–1) and decreased for happy faces. Conversely, for "happy" responses, drift was highest for clearly happy faces (stimuli 9–10).
- Ambiguity: Responses to ambiguous stimuli (e.g., stimulus 5) showed lower drift rates and higher variability, consistent with slower evidence accumulation.
- Pen Condition: The "pen-in-teeth" manipulation showed only weak evidence of modifying the drift rate and no evidence of shifting the starting point or boundary, suggesting the manipulation did not significantly alter the core cognitive accumulation process in this dataset.
Mixture Validation: The authors demonstrated that the overall RT distribution (the mixture of conditional distributions weighted by response frequencies) accurately reproduced the observed total RT distribution across the stimulus continuum.

Significance and Claims
The paper claims to provide a "relatively straightforward method" for bridging the gap between theoretical cognitive modeling and statistical data analysis. Its significance lies in two main areas:

Theoretical Grounding for GLMMs: It justifies the use of specific distributions (IG and Gamma) within GLMMs based on their derivation from diffusion processes, moving beyond purely empirical curve fitting.
Cognitive Inference from GLMMs: It allows researchers to infer parameters of the underlying cognitive process (drift rate, starting point) directly from the parameters of a fitted GLMM, provided the model is conditioned on response alternatives.

The authors maintain a modest tone regarding their claims. They explicitly note that the reconstruction is a "moment-based approximation" that is exact only when random effects are set to zero ( $C=0$ ). They acknowledge that the non-decision parameter ( $\theta$ ) was fixed at zero, which may introduce bias, and that the modeling of response probabilities themselves (rather than just RTs) was simplified. The paper positions this work as a foundation for future extensions, such as incorporating $\theta$ as a parameter, using Double Hierarchical GLMs (DHGLM), or applying Stochastic Approximation EM (SAEM) algorithms for estimation.

The Brain's Secret Race Track

Connecting the Dots: From Spreadsheets to Brain Rolls

The Real-World Test: Smiling Faces and Pen Tricks

What This Means (and What It Doesn't)

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