Multilevel structure in bqmm

bqmm uses lme4’s formula grammar, so random effects are written inline and nested or crossed structures come for free.

Random intercepts

library(bqmm)
bqmm(y ~ x + (1 | group), data, tau = 0.5)

Each group gets its own intercept deviation u_j ~ N(0, σ_u²). ranef() returns the posterior-median deviations; VarCorr() returns σ_u.

Random slopes

bqmm(y ~ x + (1 + x | group), data, tau = 0.5)            # diagonal
bqmm(y ~ x + (1 + x | group), data, tau = 0.5,
     cov = "unstructured")                                # correlated

With cov = "diagonal" (the default) the intercept and slope deviations are independent. With cov = "unstructured" they share an LKJ-correlated covariance and VarCorr() carries the correlation matrix:

fit <- bqmm(y ~ x + (1 + x | group), data, tau = 0.5, cov = "unstructured")
VarCorr(fit)
attr(VarCorr(fit), "correlation")

cov = "unstructured" currently supports a single grouping factor. Use the default diagonal covariance for multiple or crossed terms.

Nested and crossed grouping

bqmm(y ~ x + (1 | school/classroom), data, tau = 0.5)    # nested
bqmm(y ~ x + (1 | school) + (1 | neighbourhood), data, tau = 0.5)  # crossed

Both are parsed by lme4 and handled by the diagonal model — no special syntax is needed. The variance-component mapping (which random-effect column belongs to which (term, coefficient)) is built directly from lme4::mkReTrms() and is verified against lme4’s own design matrices in the package tests.

Practical notes

• Variance components and correlations need many groups to be estimable (≥ 20-30). With few groups the estimates shrink toward the prior; this is a feature of the data, not a defect.
• When random-effect SDs are small relative to the asymmetric-Laplace scale (which has SD ≈ 2.83·σ at τ = 0.5), the random structure is weakly identified.
• See vignette("bqmm-inference") for how the multilevel structure feeds into the fixed-effect variance correction.