RMediation

Overview

RMediation computes confidence intervals for indirect effects in mediation analysis. It implements statistically rigorous methods that account for the non-normal distribution of the product of regression coefficients—addressing the well-known limitations of normal-theory approaches like the Sobel test.

The package supports:

• Single mediator models (X → M → Y)
• Sequential/serial mediators (X → M₁ → M₂ → Y)
• Parallel mediators with complex indirect effects
• Any product of normal random variables for custom applications

Try It Online (No Install)

The core confidence-interval methods run in the browser through the mediation-apps Shiny front end—no R installation required:

• Distribution-of-Product CI calculator — a live front end for medci()
• Monte Carlo / Asymptotic-Delta CI calculator — built on ci(type = "mc") and ci(type = "asymp")

Useful for quick checks, teaching, and for collaborators who do not use R.

Why RMediation?

Traditional methods (e.g., Sobel test) assume the indirect effect ab follows a normal distribution. This assumption is often violated because the product of two normal variables is not normally distributed—especially when effect sizes are small or sample sizes are modest.

RMediation provides methods that correctly handle this non-normality:

Method	Best For	Speed
DOP (Distribution of Product)	Two- and three-variable products; publication-quality results	Fast
Monte Carlo	Complex models; 4+ variable products	Moderate
MBCO (Model-Based Constrained Optimization)	Hypothesis testing; bootstrap inference	Slower

Installation

From CRAN (Stable)

install.packages("RMediation")

From GitHub (release 1.9.0, pre-release)

CRAN serves the stable 1.6.1. Release 1.9.0 is published on GitHub only, as a pre-release; it is not on CRAN.

# Using pak (recommended)
pak::pak("Data-Wise/[email protected]")

# Or using remotes
remotes::install_github("Data-Wise/rmediation", ref = "v1.9.0")

For the development branch, use @dev or ref = "dev" instead.

From r-universe (development binaries)

The development version is also published at the Data-Wise r-universe as a prebuilt binary (no compiler needed):

install.packages(
  "RMediation",
  repos = c("https://data-wise.r-universe.dev", "https://cloud.r-project.org")
)

Usage

Basic Example: Single Mediator

When you have coefficient estimates and standard errors (e.g., from a published study):

library(RMediation)

# Path coefficients: a = 0.5 (SE = 0.08), b = 0.6 (SE = 0.04)
# Compute 95% CI for the indirect effect ab
medci(mu.x = 0.5, mu.y = 0.6,
      se.x = 0.08, se.y = 0.04,
      rho = 0, type = "dop")

#> $`95% CI`
#> [1] 0.2010806 0.4053620
#> 
#> $Estimate
#> [1] 0.3
#> 
#> $SE
#> [1] 0.05209837

Sequential Mediators (3+ Variables)

For models with multiple sequential mediators (X → M₁ → M₂ → Y), use the ci() function with a formula:

# Four-path indirect effect: b1 * b2 * b3 * b4
ci(c(b1 = 1.0, b2 = 0.7, b3 = 0.6, b4 = 0.45),
   Sigma = c(0.05, 0, 0, 0,    # Covariance matrix (lower triangle)
             0.05, 0, 0,
             0.03, 0,
             0.03),
   quant = ~ b1 * b2 * b3 * b4,
   type = "MC",
   n.mc = 1e5)

With lavaan Models

RMediation integrates directly with lavaan SEM models:

library(lavaan)
library(RMediation)

# Define mediation model with labeled paths
model <- '
  m ~ a*x
  y ~ b*m + c*x

  # Define indirect effect
  ab := a*b
'

fit <- sem(model, data = mydata)

# Compute CI for indirect effect (auto-detects 'ab := a*b')
ci(fit, type = "dop")

Serial Mediation via medfit

For serial chains (X → M1 → M2 → Y), let medfit fit the model and extract the coefficients + covariance, then pass the result straight to ci(). medfit is an optional (Suggests) dependency.

library(RMediation)

# medfit extracts a SerialMediationData object from a lavaan or lm/glm fit...
mu <- medfit::extract_mediation(
  fit, treatment = "X", mediator = c("M1", "M2"), outcome = "Y"
)

# ...and RMediation computes the CI for the serial indirect effect a * d1 * b
ci(mu, level = 0.95, type = "MC")

See vignette("serial-mediation-with-medfit") for the full lavaan and lm/glm workflow.

Three-Variable Indirect Effects (ProductNormal3)

For sequential indirect effects of the form a1 * a2 * b (e.g., X → M1 → M2 → Y), use ProductNormal3 and its exact pprodnormal3() CDF:

library(RMediation)

# Path coefficients and their joint covariance
mu  <- c(a1 = 0.5, a2 = 0.4, b = 0.6)
Sigma <- diag(c(0.01, 0.01, 0.01))  # independent paths

# Exact 95% CI for the a1*a2*b sequential indirect effect
pn3 <- ProductNormal3(mu = mu, Sigma = Sigma)
ci(pn3, level = 0.95)

#>      lower      upper 
#> 0.04797581 0.21980280 

# Direct CDF query
pprodnormal3(q = 0, mean = mu, cov = Sigma)  # P(a1*a2*b <= 0)

Requires RMediation >= 1.7.0. Earlier versions return wrong values when the a1/a2 covariance is ill-conditioned — silently, and without a diagnostic. Strongly correlated path estimates are ordinary in mediation models, so this affects realistic input.

Using the S7 ProductNormal Class

For programmatic use, create ProductNormal objects directly:

# Define distribution of two correlated normal variables
pn <- ProductNormal(
  mu = c(0.5, 0.3),
  Sigma = matrix(c(0.01, 0.002,
                   0.002, 0.01), 2, 2)
)

# Compute confidence interval
ci(pn, level = 0.95, type = "dop")

# Get CDF and quantiles
cdf(pn, q = 0.15)
dist_quantile(pn, p = 0.975)

# View summary
summary(pn)

Hypothesis Testing with MBCO

mbco(h0, h1) compares a fitted restricted model (h0) with its full model (h1). OpenMx models support asymptotic and bootstrap tests; see ?mbco for clean-start product-null construction and bootstrap arguments.

For a product null (a*b = 0), use mbco_product_null() rather than one nonlinear constraint. A single constraint can land on the wrong branch, and OpenMx can silently drop an equality constraint whose Jacobian is zero at the starting values (OpenMx/OpenMx#433). When mbco() finds that h0 came back identical to h1 although its constraint was violated, it refits h0 and warns with class RMediation_constraint_not_enforced; the warning cannot tell this cause from a violation within the feasibility tolerance.

Two medfit SEMFit objects support asymptotic comparisons with medfit >= 0.8.0 (GitHub/r-universe), without OpenMx:

d <- as.data.frame(RMediation::memory_exp)
d$x <- as.numeric(d$x) - 1
syntax <- paste("repetition ~ a1*x", "imagery ~ a2*x",
                "recall ~ cp*x + b1*repetition + b2*imagery", sep = "\n")
full <- medfit::fit_sem(syntax, d)
null <- medfit::fit_sem(paste(syntax, "a1*b1 == 0", sep = "\n"), d)
RMediation::mbco(null, full) # statistic 0.0831, df 1, p = 0.7731

medfit fits each product-zero branch from clean starts and selects the smallest discrepancy. RMediation compares the fitted pair without refitting, retains medfit's compatibility checks, and refuses ordinary chi-square p-values for active tested bounds. Bootstrap types and OpenMx fitting controls are not supported for SEMFit pairs. The analyst remains responsible for substantive nesting and a valid reference distribution for other nonlinear restrictions.

Sensitivity to an Omitted Confounder

ml_omitted_confounder() shows how an unmeasured, standardized confounder of the mediator-outcome relation, described by its correlations with the treatment, mediator and outcome, would change the between- and within-cluster indirect effects of a two-level (1-1-1) model. See vignette("omitted-confounder-sensitivity").

R <- matrix(c(1, .3, .2, .3, 1, .4, .2, .4, 1), 3)
res <- ml_omitted_confounder(
  between = list(a = 0.3, b = 0.4, cor = R, sd = c(1, 1, 1)),
  r_za = c(0, 0.1)
)
res
plot(res, r_za = 0.1)

The default screen keeps only admissible (positive-definite) grid points; use screen = "none" to reproduce the published tables.

Key Functions

Function	Purpose
medci()	CI for product of two normal variables
ci()	CI for any function of parameters (formulas, lavaan, ProductNormal, ProductNormal3)
mbco()	Asymptotic and OpenMx bootstrap MBCO hypothesis tests
mbco_product_null()	MBCO test of a product null (a*b = 0, serial products) as an intersection-union test over single-path nulls, avoiding the wrong-branch failure of one nonlinear constraint
ml_omitted_confounder()	Sensitivity of two-level (1-1-1) indirect effects to an omitted confounder (Tofighi & Kelley, 2016)
ci_mediation_data(), ci_serial_mediation_data()	CI for a medfit MediationData or SerialMediationData object (also reached through ci())
pprodnormal()	CDF of product of two normals
qprodnormal()	Quantiles of product of two normals
dprodnormal()	Exact density of the product of two normals
pprodnormal3()	Exact CDF of product of three normals (a1*a2*b)
dprodnormal3()	Exact density of the product of three normals
pMC(), qMC()	Monte Carlo CDF and quantile for any function of coefficient estimates
ProductNormal()	S7 class for two-variable product distributions
ProductNormal3()	S7 class for three-variable sequential indirect effects

Learn More

• Quick Reference: which function answers which question, and the classed conditions it can signal.
• Cookbook: short recipes that run as written.
• Status and Roadmap: what is settled, the CRAN hold and its criteria, and what could come next.
• Technical Notes: why the integrator, the exact CDF, point masses, the densities and the MBCO checks behave as they do.

Part of the Mediationverse

RMediation is part of the mediationverse—a suite of R packages for comprehensive mediation analysis:

Package	Purpose
mediationverse	Meta-package: install and load all packages
medfit	Model fitting and coefficient extraction
probmed	Probabilistic effect size (P_med)
RMediation	Confidence intervals (DOP, Monte Carlo, MBCO)
medrobust	Sensitivity analysis
medsim	Simulation infrastructure

Install the entire ecosystem:

pak::pak("Data-Wise/mediationverse")
library(mediationverse)  # Loads all packages

Citation

If you use RMediation in your research, please cite:

Tofighi, D., & MacKinnon, D. P. (2011). RMediation: An R package for mediation
analysis confidence intervals. Behavior Research Methods, 43, 692-700.
https://doi.org/10.3758/s13428-011-0076-x

For Monte Carlo CIs in SEM:

Tofighi, D., & MacKinnon, D. P. (2016). Monte Carlo confidence intervals for
complex functions of indirect effects. Structural Equation Modeling: A
Multidisciplinary Journal, 23, 194-205.
https://doi.org/10.1080/10705511.2015.1057284

For sequential mediation and CI method comparisons:

Tofighi, D., & Kelley, K. (2020). Indirect effects in sequential mediation
models: Evaluating methods for hypothesis testing and confidence interval
formation. Multivariate Behavioral Research, 55(2), 188-210.
https://doi.org/10.1080/00273171.2019.1618545

For MBCO and bootstrap methods:

Tofighi, D., & Kelley, K. (2020). Improved inference in mediation analysis:
Introducing the model-based constrained optimization procedure.
Psychological Methods, 25(4), 496-515. https://doi.org/10.1037/met0000259

Tofighi, D. (2020). Bootstrap model-based constrained optimization tests of
indirect effects. Frontiers in Psychology, 10, 2989.
https://doi.org/10.3389/fpsyg.2019.02989

Getting Help

• Package documentation
• Report issues
• Mediationverse roadmap

License

GPL-3