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############ Evaluating MEDIATION with Aroian Test and Regression ############
#
# Using the MacKinnon (2008) data from Chapter 3, page 56 (available on the R_SC web page).
# This script assumes you have downloaded and installed all available packages.
# Packages needed for this script will be noted as "library(xxxxx)" where xxxxx
# refers to the required package/library to be loaded.
# Make sure you have the R-Commander library loaded.
library(foreign)
library(Rcmdr)
# Use the 'foreign' library to load the "MediationExample.sav" data file.
med.ex <- read.spss("http://www.unt.edu/rss/class/Jon/R_SC/Module9/MediationExample.sav",
use.value.labels=TRUE, max.value.labels=Inf, to.data.frame=TRUE)
# From here on, simply use this script; submitted or pasted into the
# R Console.
attach(med.ex)
med.ex2 <- data.frame(X, M, Y)
detach(med.ex)
attach(med.ex2)
head(med.ex2)
cor(med.ex2)
# Equation (1): The mediator (M) predicted by the independent variable (X).
reg1 <- lm(M~X)
summary(reg1)
# Equation (2): The dependent variable (Y) predicted by the independent variable (X).
reg2 <- lm(Y~X)
summary(reg2)
# Equation (3): The dependent variable (Y) predicted by the mediator (M) and the independent variable (X).
reg3 <- lm(Y~M+X)
summary(reg3)
# AROIAN Z-TEST (1944/1947) similar to SOBEL (1982). Aroian is slightly more strict (conservative) than Sobel.
# Now take the following four values from the output of regression equation (1) and regression equestion (3). First
# we need a, which is the unstandardized coefficient of the IV (X) when predicting the mediator (M); which can be
# found in the output of regression equation (1). Here, a = 0.3386. Next we need Sa, which is the standard error
# of a; also found in the output of regression equation (1). Here, Sa = 0.1224. Next we need b, which is the
# unstandardized coefficient for the mediator (M) from the output of regression equation (3). Here, b = 0.4510. Next
# we need Sb, which is the standard error of b; also found in the output of regression equation (3). Here, Sb = 0.1460.
# Now, we can plug these values into the Aroian equation:
# a * b / sqrt(b^2 * Sa^2 + a^2 * Sb^2 + Sa^2 * Sb^2)
# Pluggin in the values from the output, we get the following:
Z.calc = .3386 * .4510 / sqrt(.4510^2 * .1224^2 + .3386^2 * .1460^2 +.1224^2 * .1460^2)
Z.calc
# Given that the calculated Z-value (2.003344) is greater than the critical Z-value for a two-tailed test at
# alpha = .05 (1.96); we can say that we have a significantly different from zero mediation effect.
# Going further, we can calculate the confidence interval for our effect. The estimated mediation effect is equal
# to a*b which for the current example is:
est.eff = .3386 * .4510
est.eff
# Then we can get the standard error of the mediation effect with the following equation:
# sqrt(a^2 * Sb^2 + b^2 * Sa^2)
# which for our example is:
std.err.eff = sqrt(.3386^2 * .1460^2 + .4510^2 * .1224^2)
std.err.eff
# So now, using our critical Z-value (+/- 1.96) and the calculated mediation effect (and standard error for it) we
# can construct our upper and lower 95% confidence limits.
# Upper.Limit = 0.1527086 + 1.96*.07410252
UL = est.eff + 1.96*std.err.eff
UL
# Lower.Limit = 0.1527086 - 1.96*.07410252
LL = est.eff - 1.96*std.err.eff
LL
# So, our confidence interval for our estimated mediation effect is (0.2979495, 0.00746766).
#### REFERENCES ####
# Aroian, L. A. (1944/1947). The probability function of the product of two normally distributed variables. Annals of
# Mathematical Statistics, 18, 265-271.
# Clark, M. J. (2004). Moderators and Mediators. RSS Matters, August, 2004.
# (http://www.unt.edu/benchmarks/archives/2004/august04/rss.htm).
# MacKinnon, D. P. (2008). Introduction to Statistical Mediation Analysis. New York: Lawrence Erlbaum Associates.
# Sobel, M. E. (1982). Asymptotic intervals for indirect effects in structural equations models. In S. Leinhart (Ed.),
# Sociological methodology 1982 (pp.290-312). San Francisco: Jossey-Bass.