A Tutorial on Bayesian Linear Mixed Models with Stan
The paper by Sorensen and Vasishth serves as a comprehensive tutorial aimed at researchers in psychology, linguistics, and cognitive science who are interested in fitting linear mixed models (LMMs) using the Bayesian framework. It leverages the probabilistic programming language, Stan, to model and analyze data from repeated measures designs, which are prevalent in these fields. The authors specifically choose Stan due to its scalability and flexibility in fitting complex models that are beyond the capabilities of other Bayesian modeling languages such as WinBUGS and JAGS.
The tutorial begins by introducing the concept of Bayesian linear mixed models, emphasizing the advantages they offer over traditional frequentist approaches, such as repeated measures ANOVA. These models allow researchers to account for multiple sources of variance within a single model without aggregating over subjects and items. Consequently, this facilitates the investigation of more nuanced effects and interactions in experimental data.
To elucidate the process of building Bayesian LMMs, the paper uses data from a self-paced reading paper as a running example, initially employing a simple two-condition design. The authors guide the reader through fitting increasingly complex models, starting with a fixed-effects model, moving to a varying intercepts model, and finally, a varying intercepts and slopes model. They meticulously detail the implementation of these models in Stan, providing code snippets and explanations of the statistical modeling concepts involved.
The tutorial further explores the interpretation and inference of Bayesian model outputs. The concept of credible intervals is introduced as a tool for making judgments about parameter estimates. Unlike frequentist confidence intervals, credible intervals express a range where the true parameter value is believed to reside with a certain probability, given the data and prior beliefs. In Bayesian modeling, this allows researchers to assess the probability of hypotheses directly.
While the two-condition example serves as a preparatory step, the tutorial extends these concepts to a more complex 2×2 factorial design. This section demonstrates how the Bayesian framework and Stan's matrix formulation can be generalized to accommodate factorial designs, making it suitable for a wide array of experimental setups. Such flexibility signifies the power of Stan in handling sophisticated models, enabling researchers to investigate intricate effects in their data.
The paper concludes by suggesting further readings for researchers interested in deepening their understanding of Bayesian modeling. A list of recommended textbooks is provided, catering to varying levels of technical expertise and backgrounds, thereby supporting the reader in advancing their knowledge of Bayesian statistics.
The implications of adopting Bayesian LMMs for research in psychology, linguistics, and cognitive science are substantial. Bayesian models offer richer interpretations of data by incorporating prior information and providing probabilistic statements regarding hypotheses. Moreover, they facilitate the fitting of models that are not tractable under frequentist methodologies, thereby potentially advancing theoretical and experimental investigations in these domains.
In summary, Sorensen and Vasishth's tutorial is a valuable contribution for researchers seeking to navigate the complexities of Bayesian linear mixed models using Stan. The step-by-step approach, combined with practical examples, serves as a solid foundation for integrating Bayesian methods into empirical research frameworks. This tutorial paves the way for the broader application of Bayesian techniques, potentially igniting new discussions and innovations in experimental research methodologies.