Mediation

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

plot(data$use.homeopathic.remedies, data$speed.of.healing, col = 'red', xlab="alter", ylab="rocov")

fit.1 <- lm(speed.of.healing ~ use.homeopathic.remedies, data)

abline(fit.1, col = 'green')

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

Calculate \(\kappa^2\)

PROCESS version 2.16 no longer produces Preacher and Kelley’s kappa- squared as a measure of effect size for the indirect effect. This feature in earlier releases was disabled in version 2.16 in response to a recent article by Wen and Fan (2015, Psychological Methods) pointing out computational errors in its derivation.

\[\frac{ab}{max(ab)} = \frac{b_a b_b}{max(b_a b_b)}\]

What is \(max(ab)\)?