Choosing a confidence interval for P_med

pmed(method = ) offers five inference modes. This article states what each does and what the calibration study supports. The numbers come from the frozen calibration run of 2026-10-14 (36 cells, 2000 replications each; docs/reports/CALIBRATION-2026-10-14.md in the repository).

Decision guide

Situation Use Why
Quick look, no interval "plugin" Point estimate only
Gaussian M and Y, default "parametric_bootstrap" Fast; samples from N(theta, Sigma)
Normality questionable "nonparametric_bootstrap" Resamples rows and refits
Gaussian with covariates, want a deterministic interval "mbco" Likelihood-ratio inversion; never undercovers
Binary M or Y with covariates "model_bootstrap" Re-estimates residual scales on simulated data

MBCO

MBCO (model-based constrained optimization; Tofighi and Kelley, 2020) inverts a likelihood-ratio test for the product of the two paths. The result is deterministic.

  • Coverage. Every nominal cell with a well identified nonzero path passes (coverage .944 to .958 across N = 50, 250, 1000).
  • Conservative at the double null. At a = b = 0 coverage is about .997, and next to it (a = b = 0.05) about .994 at N = 50 and 250. Every failure in the study is over-coverage. The flag @diagnostics$pmed$near_double_null marks this region.
  • A calibrated cutoff was rejected. The conservativeness is not confined to the null test, and the candidate recalibrations failed (MBCO-NULL-T5-FEASIBILITY-2026-10-14.md).
  • Numerical status. Only verified endpoints set converged = TRUE. A disconnected confidence set or a failed profile returns missing bounds.

Model bootstrap

Opt-in, for binary designs. Coverage varies by design. Of the 15 cells, only Gaussian-Gaussian and Gaussian M with a 50% binary Y at N = 250 pass outright. A 5% binary outcome at small N is a fit-failure regime: at N = 250, 77% of replications return no interval, and coverage among the complete ones is .981. Do not quote 95% coverage for this method in general.

Percentile bootstrap

Percentile intervals from the parametric and nonparametric bootstraps carry Monte Carlo inflation at the stated number of draws. Use at least 1000 draws and report the seed.

Reporting checklist

  1. State the method and n_boot or the seed.
  2. For MBCO, report the interval status and whether near_double_null is set.
  3. For model_bootstrap, report the share of incomplete replications.
  4. Do not claim nominal coverage outside the calibrated design: single mediator, binary treatment, with or without one binary covariate.