Mediation

Klinkenberg

University of Amsterdam

15 nov 2022

Mediation

In statistics, a mediation model is one that seeks to identify and explain the mechanism or process that underlies an observed relationship between an independent variable and a dependent variable via the inclusion of a third hypothetical variable, known as a mediator variable (also a mediating variable, intermediary variable, or intervening variable).

Source: WIKIPEDIA

Example

Does the speed of recovery after sickness improve with the use of alternative medicine or is this effect mediated by a healthy lifestyle?

Photo by Pixabay from Pexels

Mediaton paths

Mediation Path

Simulate data

Create predictor variable

set.seed(1976)
## Set parameters for simulation
n     = 100
mu    = 10
sigma = 2
## Predictor
use.homeopathic.remedies = rnorm(n, mu, sigma)

Mediator

Create mediator

b0    = 2
b1    = 1.2
error = rnorm(n,0,.7)
healthy.lifestyle = b0 + b1*use.homeopathic.remedies + error

Specify model

Create outcome variable

b0    = 6
b1    = 1.2
b2    = 3
error = rnorm(n,0,1.4)
speed.of.healing = b0 + b1*use.homeopathic.remedies + b2*healthy.lifestyle + error

data <- data.frame(use.homeopathic.remedies, 
                   healthy.lifestyle, 
                   speed.of.healing)
data <- round(data, 4)

The data

Apply 3 models

m.1.out.pre     <- lm(speed.of.healing  ~ use.homeopathic.remedies)
m.2.med.pre     <- lm(healthy.lifestyle ~ use.homeopathic.remedies)
m.3.out.pre.med <- lm(speed.of.healing  ~ use.homeopathic.remedies + healthy.lifestyle)

Extract beta coëfficients

b.a        = m.2.med.pre$coefficients[2]
b.b        = m.3.out.pre.med$coefficients[3]
b.c        = m.1.out.pre$coefficients[2]
b.c.accent = m.3.out.pre.med$coefficients[2]

View beta coëfficients

b.a

use.homeopathic.remedies 
                1.210308

b.b

healthy.lifestyle 
         2.859761

View beta coëfficients

b.c

use.homeopathic.remedies 
                4.762102

b.c.accent

use.homeopathic.remedies 
                 1.30091

Visual

3D Visual

Interactive, give it a spin.

Calculate indirect effect

\(a \times b = b_a \times b_b\)

b.a*b.b

use.homeopathic.remedies 
                3.461192

b.c - b.c.accent

use.homeopathic.remedies 
                3.461192

Calculate indirect effect (partially standardized)

\(\frac{ab}{s_{Outcome}} = \frac{b_a b_b}{s_{Outcome}}\)

b.a*b.b/sd(speed.of.healing)

use.homeopathic.remedies 
               0.3833868

Calculate indirect effect (standardized)

\(\frac{ab}{s_{Outcome}} \times s_{Predictor} = \frac{b_a b_b}{s_{Outcome}} \times s_{Predictor}\)

b.a*b.b/sd(speed.of.healing)*sd(use.homeopathic.remedies)

use.homeopathic.remedies 
               0.6979258

Calculate \(P_M\)

\(\frac{ab}{c} = \frac{b_a b_b}{b_c}\)

b.a*b.b/b.c

use.homeopathic.remedies 
               0.7268202

Calculate \(R_M\)

\(\frac{ab}{c`} = \frac{b_a b_b}{b_{c`}}\)

b.a*b.b/b.c.accent

use.homeopathic.remedies 
                2.660593

Calculate \(R^2_M\)

\(R^2_{out,med} − (R^2_{out,pre \times med} − R^2_{out,pre})\)

m.4.out.med <- lm(speed.of.healing ~ healthy.lifestyle)
R2_out.med     = cor(m.4.out.med$fitted.values, speed.of.healing)^2
R2_out.pre.med = cor(m.3.out.pre.med$fitted.values, speed.of.healing)^2
R2_out.pre     = cor(m.1.out.pre$fitted.values, speed.of.healing)^2

R2_out.med - (R2_out.pre.med - R2_out.pre)

[1] 0.9161054

End

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