Pool multiple imputation draws via Rubin's rules

Pools per-imputation block fits via Rubin's rules. When every draw also carries a per-coefficient skewness vector $gamma, the third cumulant is pooled by the law of total cumulants $$\kappa_3 = E[\kappa_3(X | k)] + 3\,\mathrm{Cov}(\mu_k, \sigma_k^2) + \kappa_3(\mu_k),$$ giving a pooled skewness $gamma alongside the usual $mean / $se. If any draw is missing $gamma, the third-cumulant path is skipped.

kappa3 is the third cumulant of the equal-weight mixture of the draws, so gamma is standardized by that mixture's variance, mean(se^2) + mean((beta - mean(beta))^2), not by V_total: Rubin's (1 + 1/K) inflation of the between-draw variance is a correction to the standard error for finite K, not a moment of the mixture.

Usage

rubins_pool(draws)

Arguments

draws

List of K draws. Each draw is a named list of submodel results, each with beta (numeric vector), se (numeric vector, same length as beta), and optionally gamma (numeric vector, same length as beta). Every draw of a submodel must carry the same number of coefficients. A submodel absent from some draws is pooled over the draws that carry it; one carried by a single draw is skipped with a warning.

Value

A named list of pooled submodel summaries (every submodel any draw carries), each with mean, se, V_within, V_between, V_total, K (number of draws that contributed), and when applicable gamma (pooled skewness) and kappa3 (pooled third cumulant).

Examples

# Three imputation draws of a two-coefficient submodel.
draws <- list(
  list(m = list(beta = c(1.0, 0.50), se = c(0.2, 0.10))),
  list(m = list(beta = c(1.2, 0.45), se = c(0.2, 0.10))),
  list(m = list(beta = c(0.9, 0.55), se = c(0.2, 0.10)))
)
pooled <- rubins_pool(draws)
pooled$m$mean
pooled$m$se