The model-based constrained optimization (MBCO) test of mediation tests \(H_0: a b = 0\). Because \(a b = 0\) iff \(a = 0\) or \(b = 0\), the constrained log-likelihood is a branch union — the better of the “drop the a-path” and “drop the b-path” fits:
\[ T = 2\left[\ell_{\text{full}} - \max(\ell_{a=0},\, \ell_{b=0})\right]. \]
That max() is non-linear, so the MBCO statistic of the
pooled estimate is not the pool of the
per-imputation MBCO statistics. You cannot pool first and test second.
Instead, missingmed keeps every per-imputation fit (exposed by
per_imputation_list()) and combines the
likelihood-ratio statistics with the
D4 rule (Chan & Meng, 2022; Grund, Lüdtke &
Robitzsch, 2021):
\[ d_S = \frac{\text{LRT(stacked data)}}{K}, \quad r_4 = \max\!\left(0, \tfrac{K+1}{k(K-1)}(\bar d - d_S)\right), \quad D_4 = \frac{d_S}{k(1 + r_4)} \sim F_{k,\nu}. \]
library(missingmed)
#>
#> Attaching package: 'missingmed'
#> The following object is masked from 'package:rmarkdown':
#>
#> run
set.seed(2026)
n <- 300
C <- rnorm(n)
X <- rbinom(n, 1, plogis(0.3 * C))
M <- 0.39 * X + 0.3 * C + rnorm(n)
Y <- 0.2 * X + 0.0 * M + 0.3 * C + rnorm(n) # interior null: b = 0
d <- data.frame(X = X, M = M, Y = Y, C = C)
d$M[sample(n, 45)] <- NA
d$Y[sample(n, 30)] <- NA
imp <- mice::mice(d, m = 20, method = "norm", printFlag = FALSE)
fit <- run(set_md_mediation(imp, Y ~ X + M + C, M ~ X + C,
treatment = "X", mediator = "M"))
infer(fit, type = "mbco")
#> D4 p r4 nu d_S
#> 0.1745614 0.6762864 0.1995802 454.9648684 0.2094004The per-imputation fits MBCO needs are available directly:
Asking for MBCO on the pooled result is an error by design: