mediation-meta-analyst

name	mediation-meta-analyst
description	Meta-analysis frameworks and methods for mediation studies

Mediation Meta-Analyst

Methods for synthesizing mediation effects across multiple studies

Use this skill when working on: meta-analysis of indirect effects, cross-study effect aggregation, heterogeneity assessment in mediation, individual participant data (IPD) meta-analysis, or systematic reviews of mediation studies.

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Meta-Analysis Fundamentals

Why Meta-Analysis of Mediation is Challenging

Challenge	Description	Solution Approach
Non-normal effects	Product $ab$ is not normally distributed	Use appropriate pooling methods
Correlated paths	$a$ and $b$ may be correlated	Model correlation structure
Heterogeneity	Effects vary across studies	Random effects models
Missing information	Studies report different statistics	Imputation or subset analysis
Publication bias	Small studies with null effects unpublished	Sensitivity analysis

Effect Size Measures

Measure	Formula	Use Case
Unstandardized $ab$	$a \times b$	Same scales across studies
Partially standardized	$a \times b / SD_Y$	Standardize by outcome only
Fully standardized	$a' \times b'$ (standardized coefficients)	Different scales
$R^2_{med}$	Proportion of effect mediated	Bounded measure

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Aggregate Data Meta-Analysis

Fixed Effects Model

When assuming homogeneous true effects, the pooled estimate is:

$$\hat{\theta}_{FE} = \frac{\sum_i w_i \hat{\theta}_i}{\sum_i w_i}, \quad w_i = \frac{1}{\text{SE}_i^2}$$

This pooled estimate uses inverse-variance weights to optimally combine study-specific estimates.

Random Effects Model

When true effects vary across studies, the pooled estimate incorporates between-study variance:

$$\hat{\theta}_{RE} = \frac{\sum_i w_i^* \hat{\theta}_i}{\sum_i w_i^}, \quad w_i^ = \frac{1}{\text{SE}_i^2 + \hat{\tau}^2}$$

where $\hat{\tau}^2$ is the between-study variance (tau-squared). The pooled estimate under random effects provides a more generalizable result when heterogeneity is present.

Heterogeneity Assessment

Key statistics for assessing heterogeneity:

Statistic	Formula	Interpretation
Q statistic	$Q = \sum_i w_i(\hat{\theta}_i - \hat{\theta})^2$	Test for heterogeneity
I-squared ($I^2$)	$I^2 = \max(0, \frac{Q - (k-1)}{Q})$	% variance due to heterogeneity
tau-squared ($\tau^2$)	Between-study variance	Absolute heterogeneity
H-squared	$H^2 = Q/(k-1)$	Relative excess variance

The I-squared statistic ranges from 0-100%: <25% indicates low heterogeneity, 25-75% moderate, and >75% high heterogeneity. The tau-squared provides the absolute magnitude of between-study variance.

R Implementation

#' Meta-Analysis of Indirect Effects
#'
#' @param effects Vector of indirect effect estimates
#' @param se Vector of standard errors
#' @param study_names Optional study identifiers
#' @param method "FE" for fixed effects, "RE" for random effects
#' @return Meta-analysis results
meta_indirect <- function(effects, se, study_names = NULL,
                           method = c("RE", "FE")) {
  method <- match.arg(method)
  k <- length(effects)

  if (is.null(study_names)) {
    study_names <- paste0("Study ", 1:k)
  }

  
  w_fe  1  se

  
  theta_fe  w_fe  effects  w_fe
  Q  w_fe  effects  theta_fe
  df  k  

  
  I2   Q  df  Q

  
    w_fe  w_fe  w_fe
  tau2   Q  df  

   method   
    weights  w_fe
    pooled  theta_fe
    se_pooled    w_fe
    
    
    weights  1  se  tau2
    pooled  weights  effects  weights
    se_pooled    weights
  

  
  ci  pooled     se_pooled

  
  p_het  1  pchisqQ df

  
    pooled_effect  pooled
    se  se_pooled
    ci  ci
    z  pooled  se_pooled
    p_value    pnormpooled  se_pooled
    heterogeneity  
      Q  Q
      df  df
      p  p_het
      I2  I2
      tau2  tau2
    
    study_data  data.frame
      study  study_names
      effect  effects
      se  se
      weight  weights  weights
    
    method  method
  

---

Multivariate Meta-Analysis

Pooling Correlated Effects

When studies report both $a$ and $b$ paths:

$$\begin{pmatrix} \hat{a} \ \hat{b} \end{pmatrix} \sim N\left(\begin{pmatrix} a \ b \end{pmatrix}, \Sigma\right)$$

Two-Stage Approach

Stage 1: Extract path coefficients from each study Stage 2: Pool using multivariate random effects

#' Multivariate Meta-Analysis of Mediation Paths
#'
#' @param a_effects Vector of a path estimates
#' @param b_effects Vector of b path estimates
#' @param a_se Standard errors for a
#' @param b_se Standard errors for b
#' @param ab_cor Correlation between a and b estimates (often assumed 0)
#' @return Multivariate meta-analysis results
multivariate_meta_mediation <- function(a_effects, b_effects,
                                         a_se, b_se, ab_cor = 0) {
  library(metafor)

  k <- length(a_effects)

  # Construct variance-covariance matrices for each study
  V_list <- lapply(1:k, function(i) {
    cov_ab <- ab_cor * a_se[i] * b_se[i
    matrixa_sei cov_ab cov_ab b_sei  
  

  
  yi  rbinda_effects b_effects
  vi  unlistlapplyV_list as.vector

  
  V  bldiagV_list

  
  
  effect_type    k
  study_id  k each  

  fit  rma.mvyi  yi V  V
                mods   effect_type  
                random   effect_type  study_id
                struct  
                data  data.frameyi effect_type study_id

  
  pooled_a  coeffit
  pooled_b  coeffit

  
  vcov_pooled  vcovfit
  indirect  pooled_a  pooled_b

  
  grad  pooled_b pooled_a
  se_indirect  tgrad  vcov_pooled  grad

  
    pooled_a  pooled_a
    pooled_b  pooled_b
    pooled_indirect  indirect
    se_indirect  se_indirect
    ci_indirect  indirect     se_indirect
    model_fit  fit
  

---

Individual Participant Data (IPD) Meta-Analysis

One-Stage Approach

Pool all data and fit single model with study-level random effects:

#' IPD Meta-Analysis for Mediation
#'
#' @param data Combined dataset with study indicator
#' @param study_var Name of study variable
#' @param treatment Name of treatment variable
#' @param mediator Name of mediator variable
#' @param outcome Name of outcome variable
#' @return IPD meta-analysis results
ipd_meta_mediation <- function(data, study_var, treatment, mediator, outcome) {
  library(lme4)

  # Mediator model with random slopes
  m_formula <- as.formula(paste(
    mediator, "~", treatment, "+ (1 +", treatment, "|", study_var, ")"
  ))
  m_model <- lmer(m_formula, data = data)

  # Outcome model with random slopes
  y_formula  as.formulapaste
    outcome  treatment  mediator
     treatment  mediator  study_var 
  
  y_model  lmery_formula data  data

  
  a  fixefm_modeltreatment
  b  fixefy_modelmediator
  c_prime  fixefy_modeltreatment

  
  indirect  a  b

  
  boot_indirect  replicate 
    boot_idx  samplenrowdata replace  
    boot_data  databoot_idx 

    m_boot  tryCatch
      lmerm_formula data  boot_data
      error  e 
    
    y_boot  tryCatch
      lmery_formula data  boot_data
      error  e 
    

     m_boot  y_boot 

    fixefm_boottreatment  fixefy_bootmediator
  

  boot_indirect  boot_indirectboot_indirect

  
    pooled_a  a
    pooled_b  b
    pooled_c_prime  c_prime
    pooled_indirect  indirect
    pooled_total  indirect  c_prime
    se_indirect  sdboot_indirect
    ci_indirect  quantileboot_indirect  
    n_studies  uniquedatastudy_var
    n_total  nrowdata
    m_model  m_model
    y_model  y_model
  

Two-Stage Approach

Estimate effects within each study, then pool:

#' Two-Stage IPD Meta-Analysis
#'
#' @param data Combined dataset
#' @param study_var Study identifier
#' @return Two-stage meta-analysis results
two_stage_ipd <- function(data, study_var, treatment, mediator, outcome) {

  studies <- unique(data[[study_var]])
  k <- length(studies)

  # Stage 1: Study-specific estimates
  study_results <- lapply(studies, function(s) {
    study_data <- data[data[[study_var]] == s, ]

    # Fit models
    m_model <- lm(as.formula(paste(mediator, "~", treatment) data  study_data
    y_model  lmas.formulapasteoutcome  treatment  mediator
                  data  study_data

    a  coefm_modeltreatment
    b  coefy_modelmediator

    
    se_a  vcovm_modeltreatment treatment
    se_b  vcovy_modelmediator mediator
    se_indirect  a  se_b  b  se_a

    data.frame
      study  s
      n  nrowstudy_data
      a  a
      b  b
      indirect  a  b
      se_a  se_a
      se_b  se_b
      se_indirect  se_indirect
    
  

  study_df  do.callrbind study_results

  
  meta_result  meta_indirect
    effects  study_dfindirect
    se  study_dfse_indirect
    study_names  study_dfstudy
    method  
  

  
    stage1  study_df
    stage2  meta_result
    pooled_indirect  meta_resultpooled_effect
    ci  meta_resultci
    I2  meta_resultheterogeneityI2
  

---

Publication Bias

Detection Methods

Method	Description	Limitation
Funnel plot	SE vs effect plot	Visual, subjective
Egger's test	Regression of effect on SE	Low power
Trim-and-fill	Impute missing studies	Assumes specific mechanism
PET-PEESE	Conditional regression	Requires assumptions
Selection models	Model publication process	Complex, sensitive

R Implementation

#' Publication Bias Assessment for Mediation Meta-Analysis
#'
#' @param effects Indirect effect estimates
#' @param se Standard errors
#' @return Publication bias diagnostics
publication_bias_mediation <- function(effects, se) {
  library(metafor)

  # Fit random effects model
  res <- rma(yi = effects, sei = se, method = "REML")

  # Funnel plot data
  funnel_data <- data.frame(
    effect = effects,
    se = se,
    precision = 1/se
  )

  # Egger's test
  egger <- regtest(res, model = "lm")

  # Trim and fill
  tf <- trimfill(res)

  # PET-PEESE (if significant, use PEESE; otherwise PET)
  pet <- lmeffects  se weights  se
  peese  lmeffects  Ise weights  se

  pet_est  coefpet
  peese_est  coefpeese

  
   coefsummarypet    
    adjusted_estimate  peese_est
    method_used  
    
    adjusted_estimate  pet_est
    method_used  
  

  
    original_estimate  coefres
    egger_test  
      z  eggerzval
      p  eggerpval
      interpretation  ifelseeggerpval  
                              
                              
    
    trim_fill  
      original_k  resk
      imputed_k  tfk0
      adjusted_estimate  coeftf
      adjusted_ci  tfci.lb tfci.ub
    
    pet_peese  
      pet_estimate  pet_est
      peese_estimate  peese_est
      method_used  method_used
      adjusted_estimate  adjusted_estimate
    
    funnel_data  funnel_data
  

---

Moderator Analysis

Meta-Regression

Test whether study-level characteristics explain heterogeneity:

#' Meta-Regression for Mediation Effects
#'
#' @param effects Indirect effect estimates
#' @param se Standard errors
#' @param moderators Data frame of moderator variables
#' @return Meta-regression results
meta_regression_mediation <- function(effects, se, moderators) {
  library(metafor)

  # Build formula from moderator names
  mod_formula <- as.formula(paste("~", paste(names(moderators), collapse = " + ")))

  # Fit mixed-effects meta-regression
  res <- rma(yi = effects, sei = se,
             mods = mod_formula,
             data = moderators,
             method = "REML")

  # R^2 analog
  res_null  rmayi  effects sei  se method  
  R2   res_nulltau2  restau2  res_nulltau2

  
  QE_test  
    QE  resQE
    df  resk  resp
    p  resQEp
  

  
    coefficients  coefsummaryres
    tau2_residual  restau2
    I2_residual  resI2
    R2  R2
    residual_heterogeneity  QE_test
    model  res
  

Subgroup Analysis

#' Subgroup Analysis for Mediation Meta-Analysis
#'
#' @param effects Indirect effect estimates
#' @param se Standard errors
#' @param subgroup Factor variable defining subgroups
#' @return Subgroup analysis results
subgroup_analysis <- function(effects, se, subgroup) {

  groups <- unique(subgroup)

  # Within-group estimates
  group_results <- lapply(groups, function(g) {
    idx <- subgroup == g
    meta_indirect(effects[idx], se[idx], method = "RE")
  })
  names(group_results) <- groups

  # Extract pooled estimates
  group_effects <- sapply(group_results, x xpooled_effect
  group_se  sapplygroup_results x xse
  group_k  sapplygroup_results x xstudy_dataeffect

  
  
  overall  meta_indirecteffects se method  
  Q_total  overallheterogeneityQ

  Q_within  sapplygroup_results x xheterogeneityQ
  Q_between  Q_total  Q_within
  df_between  groups  
  p_between  1  pchisqQ_between df_between

  
    subgroup_estimates  data.frame
      subgroup  groups
      k  group_k
      effect  group_effects
      se  group_se
      ci_lower  group_effects    group_se
      ci_upper  group_effects    group_se
    
    test_for_differences  
      Q_between  Q_between
      df  df_between
      p  p_between
      interpretation  ifelsep_between  
                              
                              
    
    group_results  group_results
  

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Reporting Checklist

PRISMA for Mediation Meta-Analysis

• Search strategy documented
• Inclusion/exclusion criteria specified
• Effect measure defined (standardized vs. unstandardized)
• Method for extracting/computing indirect effects stated
• Heterogeneity statistics reported ($I^2$, $\tau^2$, $Q$)
• Forest plot included
• Publication bias assessed
• Sensitivity analyses conducted
• GRADE assessment for certainty

Forest Plot Template

#' Create Forest Plot for Mediation Meta-Analysis
#'
#' @param meta_result Result from meta_indirect()
#' @return ggplot2 forest plot
forest_plot_mediation <- function(meta_result) {
  library(ggplot2)

  df <- meta_result$study_data
  df$ci_lower <- df$effect - 1.96 * df$se
  df$ci_upper <- df$effect + 1.96 * df$se

  # Add pooled estimate
  pooled <- data.frame(
    study = "Pooled",
    effect = meta_result$pooled_effect,
    se = meta_result$se,
    weight = NA,
    ci_lower = meta_result$ci[1],
    ci_upper = meta_result$ci[
  
  df  rbinddf pooled
  dfstudy  factordfstudy levels  revdfstudy

  ggplotdf aesx  effect y  study 
    geom_vlinexintercept   linetype   color   
    geom_pointaessize  weight 
    geom_errorbarhaesxmin  ci_lower xmax  ci_upper height   
    geom_pointdata  dfdfstudy   
               shape   size   color   
    labs
      x  
      y  
      title  
      subtitle  sprintf
                         meta_resultheterogeneityI2  
                         meta_resultheterogeneitytau2
     
    theme_minimal 
    themelegend.position  

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References

Meta-Analysis Methods

• Borenstein, M., et al. (2009). Introduction to Meta-Analysis
• Higgins, J. P., & Green, S. (2011). Cochrane Handbook for Systematic Reviews

Mediation Meta-Analysis

• Cheung, M. W. L. (2015). Meta-Analysis: A Structural Equation Modeling Approach
• MacKinnon, D. P. (2008). Introduction to Statistical Mediation Analysis

Publication Bias

• Rothstein, H. R., et al. (2005). Publication Bias in Meta-Analysis
• Stanley, T. D., & Doucouliagos, H. (2014). Meta-regression approximations

Software

• Viechtbauer, W. (2010). Conducting meta-analyses in R with the metafor package
• Cheung, M. W. L. (2015). metaSEM: Meta-analysis using structural equation modeling

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Version: 1.0.0 Created: 2025-12-09 Domain: Meta-analysis of mediation effects Applications: Systematic reviews, research synthesis, evidence aggregation