Bayesian Item Response Modeling in R with brms and Stan (1905.09501v3)

Abstract: Item Response Theory (IRT) is widely applied in the human sciences to model persons' responses on a set of items measuring one or more latent constructs. While several R packages have been developed that implement IRT models, they tend to be restricted to respective prespecified classes of models. Further, most implementations are frequentist while the availability of Bayesian methods remains comparably limited. We demonstrate how to use the R package brms together with the probabilistic programming language Stan to specify and fit a wide range of Bayesian IRT models using flexible and intuitive multilevel formula syntax. Further, item and person parameters can be related in both a linear or non-linear manner. Various distributions for categorical, ordinal, and continuous responses are supported. Users may even define their own custom response distribution for use in the presented framework. Common IRT model classes that can be specified natively in the presented framework include 1PL and 2PL logistic models optionally also containing guessing parameters, graded response and partial credit ordinal models, as well as drift diffusion models of response times coupled with binary decisions. Posterior distributions of item and person parameters can be conveniently extracted and post-processed. Model fit can be evaluated and compared using Bayes factors and efficient cross-validation procedures.

• The paper introduces a comprehensive Bayesian approach that overcomes limitations of traditional frequentist IRT models.
• It details the use of brms’ multilevel syntax to build complex models with hierarchical priors for person and item parameters.
• The work demonstrates practical applications across binary, ordinal, and response time data to improve psychometric assessments.

Overview of Bayesian Item Response Modeling in R with brms and Stan

The paper by Paul-Christian Bürkner presents an exhaustive framework for Bayesian Item Response Theory (IRT) modeling, utilizing the brms package within the R programming environment, supported by the probabilistic programming language Stan. IRT serves as a cornerstone in psychometrics for modeling responses to questionnaires or tests, often aimed at assessing latent constructs like abilities or attitudes across individuals.

Key Contributions

This paper makes notable contributions by addressing the limitations of traditional IRT models that are predominantly frequentist and confined to specific model classes. Bürkner champions a Bayesian approach, providing enhanced modeling flexibility and offering a unified framework for various IRT model types, including binary, ordinal, and response times data. The brms package becomes a pivotal tool that extends R's capabilities, supporting over 40 distributions that include standard IRT models, as well as more niche types like count data and response times.

Methodological Advancements

Bürkner's paper takes a deep dive into the syntax and functionality of brms, showcasing how its flexible multilevel syntax allows for the construction of complex models. This includes a comprehensive handling of person and item parameters which can exhibit dependencies both linearly and non-linearly. The inclusion of priors is crucial in Bayesian modeling, effectively utilizing the capacity for hierarchical prior distributions to enhance parameter estimation robustness—particularly useful in multidimensional models.

Detailed Examples

The author provides a range of hands-on applications demonstrating the efficacy and operationalization of brms. Examples span from binary models for verbal aggression datasets to graded response models for ordinal outcomes. Importantly, the application of drift diffusion models on response times illustrates the framework’s capacity to integrate cognitive process-oriented modeling, an area where Bayesian methods naturally excel due to their full probabilistic treatment.

Implications and Future Directions

For the psychometrics community and beyond, this framework lays the groundwork for more flexible and potentially more accurate measurement models. Its Bayesian foundation allows for the quantification of uncertainty in a comprehensive manner, opening pathways for more nuanced hypothesis testing and model evaluation. The flexible customization of response distributions and parameter structures can accommodate evolving research questions across various domains, not limited to psychology or education.

Future work could extend into refining computational efficiency, given the inherent time complexity of Bayesian analysis. Advances in the computational aspects of hierarchical Bayesian models could broaden brms application in fields where real-time or adaptive testing is pertinent, such as education technology platforms.

Conclusion

Bürkner’s work strongly advocates for the adoption of Bayesian statistics through brms in IRT and related modeling domains, representing a significant advance over pre-existing methodologies restricted by model specificity and frequentist paradigms. It is a compelling example of how computational and theoretical prowess can be synergistically wielded to tackle complex psychometric assessments. The paper serves as both a robust introduction and an advanced guide for practitioners keen to leverage R and Stan for state-of-the-art Bayesian modeling in the human sciences.