Random slopes and the free random-effect covariance

The covariance is the quantity, not a nuisance

A random-intercept model, y ~ x + (1 | g), has a single variance component: how much the groups differ in their baseline. A random-slope model, y ~ x + (1 + x | g), has three: the intercept variance, the slope variance, and the correlation between them. That correlation is often the scientific question – do groups that start high also respond more steeply? – so it should be inferred, with its own uncertainty, not fixed at a point estimate.

tulpa() treats the whole random-effect covariance Sigma as the inferred object. When a term carries slopes it does not condition on a plug-in Sigma; it integrates over it.

Simulate a correlated random-slope data set

G   <- 60L                 # groups
npg <- 12L                 # observations per group
N   <- G * npg
grp <- rep(seq_len(G), each = npg)
x   <- rnorm(N)

# True Sigma: sd 0.7 (intercept), 0.5 (slope), correlation 0.4.
Sigma <- matrix(c(0.7^2,            0.4 * 0.7 * 0.5,
                  0.4 * 0.7 * 0.5,  0.5^2), 2)
u <- t(t(chol(Sigma)) %*% matrix(rnorm(2 * G), 2))   # G x 2 group effects
eta <- 0.2 + 0.5 * x + u[grp, 1] + u[grp, 2] * x
y   <- rpois(N, exp(eta))
d   <- data.frame(y = y, x = x, g = factor(grp))

Fit: the covariance is integrated, not plugged in

A (1 + x | g) term makes tulpa() route the Laplace path through the nested-Laplace integration over Sigma (tulpa_re_cov_nested()): a CCD grid in log-Cholesky coordinates, centred and rotated at the marginal-likelihood mode, with a weakly-informative PC + LKJ hyperprior. Each derived quantity – the standard deviations sigma_1, sigma_2 and the correlation rho_12 – is summarised after integration, as a weighted quantile of the joint posterior, so a skewed component is not collapsed to its mode.

fit <- tulpa(y ~ x + (1 + x | g), data = d, family = "poisson",
             mode = "laplace")
fit$posterior[, c("parameter", "median", "ci_lo", "ci_hi")]

The posterior medians track the truth (sigma_1 = 0.7, sigma_2 = 0.5, rho_12 = 0.4), each with a credible interval rather than a bare number.

Exact debias for small, low-count groups

The nested Laplace is fast and accurate when the per-group likelihood is close to Gaussian. For binary or low-count data in small groups the Laplace under-disperses Sigma – it pulls the variance components low. The exact counterpart, a Metropolis-within-Gibbs sampler with a conjugate inverse-Wishart draw for Sigma, corrects that bias. Ask for it with control$re_cov = "gibbs":

fit_gibbs <- tulpa(y ~ x + (1 + x | g), data = d, family = "poisson",
                   mode = "laplace",
                   control = list(re_cov = "gibbs",
                                  n_iter = 2000L, warmup = 1000L))

Both fits return the same accessors: fit$posterior holds the Sigma summary, and coef(fit) / summary(fit) report the fixed effects. The choice between them is the engine’s design in miniature – a cheap deterministic approximation, with an exact sampler available exactly where the approximation is biased.