Bürkner 2021 - brms Specification Stan Code and Post-Processing

Summary

An IRT model in brms is the triple family + formula + prior. Group-level terms use the non-centered parameterization; the |ID| syntax correlates varying effects across formulas; non-linear formulas (nl = TRUE) express 2PL/3PL. Post-processing offers summaries, hypothesis, conditional_effects, posterior_predict, pp_check, loo, kfold, and bridge-sampling Bayes factors.

Overview

The package fits through Stan (adaptive HMC / NUTS). IRT-specific features are in the formula and prior arguments.

Main Content

Family. brmsfamily("bernoulli","logit"); each family has a primary parameter $\mu$ with a link and optional parameters (e.g. sigma with log link).

Formula patterns.

Goal	brms formula
1PL, pooled persons and items	y ~ 1 + (1 | item) + (1 | person)
1PL, non-hierarchical items	y ~ 0 + item + (1 | person)
Covariate varying over items	y ~ 1 + x + (1 + x | item) + (1 | person)
Correlated varying effects across parameters	(1 |i| item) and (1 |p| person) in every distributional formula
2PL (non-linear)	bf(y ~ exp(logalpha) * eta, eta ~ 1 + (1|i|item) + (1|id), logalpha ~ 1 + (1|i|item), nl = TRUE)
Non-linear transform shared	nlf(gamma ~ inv_logit(logitgamma))
Free thresholds per item (ordinal)	resp | thres(gr = item) ~ 1 + (1|id)
Category-specific effect	cs(Anger) with family = acat

Priors. set_prior("cauchy(0, 5)", class = "sd", group = "person") + set_prior("lkj(2)", class = "cor", group = "person"); dpar= and nlpar= target distributional/non-linear parameters; constant(1) fixes a parameter (e.g. ${\sigma }_{\theta }=1$); class “b” for non-hierarchical coefficients, coef= for item-specific priors.

Non-centered group-level effects in generated Stan code

brms declares standardized z_1 ~ normal(0,1) and sets r_1 = sd_1 * z_1 in transformed parameters, so the sampler explores independent standard normals (with Cholesky factors of correlation matrices when correlated). The linear predictor is built as mu = X * b plus a loop mu[n] += r_1_1[J_1[n]] * Z_1_1[n]; the likelihood uses bernoulli_logit_lpmf(Y | mu). The half-normal prior is written normal_lpdf(sd | 0, 3) - normal_lccdf(0 | 0, 3), where the constant makes bridge-sampling marginal likelihoods correct.

Estimation. Adaptive HMC with at least two critical tuning parameters set automatically; gradients by algorithmic differentiation. Convergence: $\hat{R}<1.05$ and effective sample size at least 400; divergences handled by raising adapt_delta (0.99 used for diffusion models).

Post-processing. summary, plot, coef/ranef (person and item parameters), hypothesis (posterior contrasts, e.g. SD difference), conditional_effects (probability scale), posterior_predict, pp_check, log_lik, loo (approximate LOO-CV), kfold, loo_compare, model_weights, pp_average, bridge_sampler, bayes_factor, post_prob.

Examples

Reading the generated 1PL Stan model

Setup. y ~ 0 + item + (1 | person) with normal(0,5) on item coefficients and normal(0,3) on the person SD. Solution. Stan block: parameters b, sd_1, z_1; transformed parameter r_1_1 = sd_1 * z_1; model block target += normal_lpdf(b|0,5); target += bernoulli_logit_lpmf(Y | X*b + r_1_1[J_1]). Interpretation. What you write in formula and prior maps one-to-one to blocks; vectorizing the likelihood is where brms earns its speed.

Connections

HMC and Stan in Practice · Posterior Predictive Checking

See Also

Bürkner 2021 - Worked Examples Binary Models VerbAgg