= Mediation Analysis =

'''Mediation analysis''' is a decomposition of causal effects based on a mediator variable.

<<TableOfContents>>

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== Description ==

In circumstances where a significant relationship has been found between an independent and dependent variable, but no causal mechanism has been identified, it is possible to propose hypotheses that a third '''mediator variable''' completes the causal model.

{{attachment:path.png}}

The effects pictured are:

 * '''direct effect''' of ''X'' on ''Y'' =  ''β,,1,,''
 * direct effect of ''X'' on ''M'' =  ''β,,2,,''
 * direct effect of ''M'' on ''Y'' =  ''β,,3,,''
 * '''indirect effect''' of ''X'' on ''Y'' = ''β,,2,,*β,,3,,''
 * '''total effect''' of ''X'' on ''Y'' = ''β,,1,, + (β,,2,,*β,,3,,)''

This can be estimated with the [[Statistics/SobelTest|Sobel test]], but best practice is to use a bootstrapping method.

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== Example ==

The [[Statistics/SobelTest|Sobel test]] approach, using the [[R/Multilevel|multilevel package]]:

{{{
library(multilevel)

sorel(data$X, data$M, data$Y)
}}}

This displays:
 * the component models, including the estimated coefficients and intercepts
 * the indirect effect
 * the pooled standard error
 * the calculated Z statistic

The bootstrapping approach, using the [[R/Mediation|mediation package]]:

{{{
XonY <- lm(Y ~ X, data = data)
XonM <- lm(M ~ X, data = data)
XMonY <- lm(Y ~ X + M, data = data)

library(mediation)

mediate(XonM, XMonY, treat='X', mediator='M', boot=TRUE, sims=500)
}}}

This displays:
 * the '''average causal mediation effects''' ('''ACME'''), the key measure of this bootstrapping method
   * if significantly different than 0, then there is a significant '''mediation effect'''
 * the '''average direct effects''' ('''ADE''')
 * the total effect
   * ''total effect = ACME + ADE''
 * the proportion of the total effect that was mediated
   * ''prop = ACME / total effect''



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