A note on the normality assumption for Bayesian models of constraint in behavioral individual differences

TL;DR

This paper examines the normality assumption in Bayesian models of behavioral data, particularly response times, and finds that using a lognormal model does not alter inference, supporting the pragmatic use of simpler models.

Contribution

It demonstrates that assuming normality for response times in Bayesian models is acceptable despite their non-normal distribution, simplifying analysis of behavioral individual differences.

Findings

Using a shifted lognormal model does not change inference results.

Logarithmic effects are more interpretable but more complex to understand.

Normality assumption is pragmatically justified for response time data.

Abstract

To investigate the structure of individual differences in performance on behavioral tasks, Haaf and Rouder (2017) developed a class of hierarchical Bayesian mixed models with varying levels of constraint on the individual effects. The models are then compared via Bayes factors, telling us which model best predicts the observed data. One common criticism of their method is that the observed data are assumed to be drawn from a normal distribution. However, for most cognitive tasks, the primary measure of performance is a response time, the distribution of which is well known to not be normal. In this paper, I investigate the assumption of normality for two datasets in numerical cognition. Specifically, I show that using a shifted lognormal model for the response times does not change the overall pattern of inference. Further, since the model-estimated effects are now on a logarithmic scale, the interpretation of the modeling becomes more difficult, particularly because the estimated effect is now multiplicative rather than additive. As a result, I recommend that even though response times are not normally distributed in general, the simplification afforded by the Haaf and Rouder (2017) approach provides a pragmatic approach to modeling individual differences in behavioral tasks.