Bayesian Multilevel Models with brms for Social Sciences (BMSS01)

Session 4 – 02:00:00 – Introduction to MCMC and brms

Break – 01:00:00

Session 6 – 02:00:00 – Bayesian Workflow: Approximate Inference, Priors, and Model Comparison

Having now fitted and diagnosed Bayesian linear regression models via full MCMC sampling, this session introduces a complementary workflow tool: fast approximate algorithms, such as Pathfinder variational inference, which fit a model in a fraction of the time that full MCMC requires. These are presented as a second tool to have available, useful for quickly catching coding errors or a badly specified model before committing to full sampling, and not as a replacement for MCMC. In brms and Stan, full MCMC sampling remains what participants will use the great majority of the time; approximate algorithms are a supplement to that practice, not an alternative introduction to it. The session then turns to prior specification, posterior predictive checking, and model comparison as further connected parts of the Bayesian workflow. We begin by examining default priors in brms using prior_summary, understanding that these are weakly informative priors designed to regularize without dominating the likelihood. We then cover how to specify custom priors using set_prior, including different priors for different parameters. Prior predictive checks are introduced: simulating data from the prior before seeing real data to ensure priors are sensible. Prior sensitivity analysis demonstrates how to assess whether posterior inference is robust to prior specification. The session then turns to model comparison, covering multiple approaches: WAIC and LOO cross-validation for comparing predictive accuracy, and Bayes factors for formal hypothesis comparison. We discuss when each method is appropriate, deliberately positioning Bayes factors as one tool among several rather than as the defining feature of Bayesian analysis. Practical workflows for comparing multiple models are demonstrated, closing out the non-multilevel portion of the course before Day 3 turns to multilevel models proper.

Session 7 – 02:00:00 – The Normal Random Effects Model and Partial Pooling

This session introduces the conceptual foundation of multilevel modelling: partial pooling as a principled middle ground between fitting each group separately (no pooling) and ignoring group structure entirely (complete pooling). We work through the normal random effects model, the natural multilevel model for continuous grouped data, using a dataset with many groups and unequal amounts of data per group, of the kind common in psychology and social science research, for example participants nested within labs or sites, or repeated observations nested within countries. We show how the group-level means are treated not as independent fixed unknowns but as draws from a shared population distribution, and how the resulting estimates for each group are pulled toward the overall mean by an amount that depends on the amount of data available for that group and the between-group variance relative to the within-group variance. The intraclass correlation is introduced as a measure of how much of the total variance is attributable to group membership. Throughout, the Bayesian workflow habits established on Day 2, fitting via full MCMC, checking diagnostics, and running posterior predictive checks, are applied to this new model structure, with fast approximate algorithms available as a secondary check when iterating on model specification.

Break – 01:00:00

Session 8 – 02:00:00 – Bayesian Linear Mixed Effects Models

This session covers Bayesian linear mixed effects models with varying intercepts and varying slopes, the workhorse model structure for grouped and repeated-measures data in psychology and social science research. The brms syntax for mixed models, using the lme4-style formula with random effects specified as (1 | group) for varying intercepts and (predictor | group) for varying slopes, is covered in detail, including the correlation structure between varying intercepts and slopes that this syntax implies. We compare brms output with lme4/lmer output directly, using the same datasets fit both ways, to build an intuition for where the two approaches agree and where they differ. A key advantage of Bayesian mixed models is demonstrated concretely: they often converge where lme4 produces singular fits or convergence warnings, particularly with complex random effects structures or limited numbers of groups, because weakly informative priors regularize the variance components in a way that flat-prior maximum likelihood estimation does not. MCMC diagnostics specific to multilevel models, including divergent transitions, are introduced.

Break – 01:00:00

Session 9 – 02:00:00 – Multilevel Models for Nested and Crossed Data

Session 7 – 02:00:00 – Group-Level Predictors and Cross-Level Interactions

This session covers models that include predictors measured at the group level as well as the individual level, for example, a classroom-level teaching method alongside student-level test scores, or a country-level policy variable alongside individual-level survey responses. We cover how to specify and interpret group-level predictors in brms, and how cross-level interactions, where the effect of an individual-level predictor depends on a group-level variable, are specified and interpreted. This is a common analytic need in social science research examining how context moderates individual-level effects, and the session works through several such examples in detail.

Break – 01:00:00

Session 8 – 02:00:00 – Bayesian Generalized Linear Mixed Models

This session extends Bayesian multilevel modelling to non-normal outcomes, combining the generalized linear model families from single-level modelling with the random effects structures covered on Day 3. We fit multilevel binary logistic regression models, multilevel ordinal logistic regression models for ordered categorical outcomes such as Likert-type survey responses, and multilevel Poisson and negative binomial models for count data. Throughout, we emphasize that the Bayesian workflow established earlier in the course, fit via MCMC, check diagnostics, check priors, check posterior predictive fit, compare models, remains the same regardless of the outcome distribution or the random effects structure. Real datasets from psychology and social science research illustrate these models in practice, chosen to reflect the kinds of outcomes participants are likely to encounter in their own work.

Break – 01:00:00

Session 9 – 02:00:00 – Evaluating Multilevel Models and Course Wrap-Up

This session covers approaches to evaluating multilevel models specifically: explained variance measures adapted to the multilevel setting, where a single R-squared value is less informative than it is in single-level regression, and posterior predictive checks adapted to grouped data, checking not only overall fit but fit at the group level. We close with a structured recap of the Bayesian workflow as it has been applied across the course, from the Bernoulli model on Day 1 through generalized multilevel models on Day 4, and discuss how participants can apply this workflow to multilevel models in their own research. Pointers to further learning, including Bayesian approaches to psychometric models and longitudinal growth curve models, are provided for those wishing to continue beyond the course.