Data aggregation can lead to biased inferences in Bayesian linear mixed models and Bayesian ANOVA: A simulation study

TL;DR

This study shows that aggregating data in Bayesian linear mixed models and ANOVA can bias results, especially when assumptions like sphericity are violated, and recommends analyzing non-aggregated data with full random effects.

Contribution

The paper demonstrates how data aggregation biases Bayesian inferences and advocates for modeling individual trial data with full random effects to improve accuracy.

Findings

Aggregated data analysis can bias Bayes factors under sphericity violations.

Bayesian ANOVA on aggregated data can lead to biased conclusions if assumptions are violated.

Modeling non-aggregated data with full random effects reduces bias.

Abstract

Bayesian linear mixed-effects models and Bayesian ANOVA are increasingly being used in the cognitive sciences to perform null hypothesis tests, where a null hypothesis that an effect is zero is compared with an alternative hypothesis that the effect exists and is different from zero. While software tools for Bayes factor null hypothesis tests are easily accessible, how to specify the data and the model correctly is often not clear. In Bayesian approaches, many authors use data aggregation at the by-subject level and estimate Bayes factors on aggregated data. Here, we use simulation-based calibration for model inference applied to several example experimental designs to demonstrate that, as with frequentist analysis, such null hypothesis tests on aggregated data can be problematic in Bayesian analysis. Specifically, when random slope variances differ (i.e., violated sphericity assumption), Bayes factors are too conservative for contrasts where the variance is small and they are too liberal for contrasts where the variance is large. Running Bayesian ANOVA on aggregated data can - if the sphericity assumption is violated - likewise lead to biased Bayes factor results. Moreover, Bayes factors for by-subject aggregated data are biased (too liberal) when random item slope variance is present but ignored in the analysis. These problems can be circumvented or reduced by running Bayesian linear mixed-effects models on non-aggregated data such as on individual trials, and by explicitly modeling the full random effects structure. Reproducible code is available from \url{https://osf.io/mjf47/}.