Intuitive Joint Priors for Bayesian Multilevel Models

• Applicant: Dr. Paul-Christian Bürkner
• Subject Area: Personality Psychology, Clinical and Medical Psychology, Methodology
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 500663361

Regression models are ubiquitous in the quantitative sciences making up a big part of all statistical analysis performed on data. In psychology, data often contains multilevel structure, for example, because of natural groupings of individuals or repeated measurement of the same individuals. Multilevel models (MLMs) are designed specifically to account for the nested structure in multilevel data and are a widely applied class of regression models in psychology and beyond. From a Bayesian perspective, the widespread success of MLMs can be explained by the fact that they impose joint priors over a set of parameters with shared hyper-parameters, rather than separate independent priors for each parameter. However, in almost all state-of-the-art approaches, different additive regression terms in MLMs, corresponding to different parameter sets, still receive mutually independent priors. As more and more terms are being added to the model while the number of observations remains constant, such models will overfit the data. This is highly problematic as it leads to unreliable or uninterpretable estimates, bad out-of-sample predictions, and inflated Type I error rates.The primary objective of the proposed project is to develop, evaluate, implement, and apply intuitive joint priors for Bayesian MLMs. Extending existing approaches, we propose to derive a prior on the coefficient of determination R2 (R-squared; also known as proportion of explained variance) and decompose it into individual variance components. Such R2 priors share several desirable properties with other shrinkage priors, which enable the estimation, interpretation, and selection of regression terms and prevent overfitting. In particular, we will substantially generalize the single-level R2-D2 prior of Zhang, Naughton, Bondell, & Reich (2020). The R2-D2 prior is an intuitive shrinkage prior based on R2 but so far only applicable for linear regression models without multilevel structure. We hypothesize that our proposed R2-D2-M2 prior framework (where M2 stands for Multilevel Models) will enable the reliable and interpretable estimation of much more complex Bayesian MLMs in psychology than was previously possible.The primary objective can be subdivided into four objectives realized in four corresponding work packages:(1) Joint priors for linear MLMs: Develop and evaluate the R2-D2-M2 prior for linear MLMs.(2) Joint priors for generalized linear MLMs: Develop and evaluate the R2-D2-M2 prior for various classes of generalized linear MLMs.(3) Benchmarks, application and refinement: Benchmark the R2-D2-M2 prior against state-of-the-art competitors. Apply the R2-D2-M2 prior in multiple research projects of collaborating psychological scientists and refine the methods based on the results and received feedback.(4) Clean, efficient implementation and provisioning as open source code: Implement the developed priors in modern and user-friendly open-source software.

• DFG Programme: Research Grants