Moderated Mediation
Moderated Mediation
Moderated Mediation
Dr Brian K Cooper
Department of Management
Monash University
2015
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Mediation and Moderation
A mediator explains how or why an independent variable is related
to a dependent variable. Mediation is exemplified by the question
how did it work? The focus is on understanding the mechanism,
causal chain of events, or the underlying process.
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Mediation Analysis
As shown in Figure 1, in a basic mediation model an independent
variable (X) is hypothesized to influence a mediator (M) which, in
turn, influences the dependent variable (Y).
In more complex models there may be more than one mediator but
the principles of analysis remain largely the same.
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Figure 1: Basic Mediation Model
Controls
M = 0 + 1 X + e MEDIATOR
(M)
a b
INDEPENDENT DEPENDENT
VARIABLE VARIABLE
c
(X) (Y)
Y = 0 + 1 M + 2 X + e
Arrows indicate hypothesized effects
Indirect (mediated) effect of X on Y = a*b
Direct (unmediated) effect of X on Y = c 4
The Indirect Effect
As shown in Fig 1, to test mediation requires estimation of
coefficients in two regression equations:
1. Run a regression with the IV predicting the mediator. This will give
estimate a.
2. Run a regression with the IV and mediator predicting the DV. This
will give estimate b. Note the IV is controlled in the equation.
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Example of Mediation
Pollack et al. (2012) hypothesized that the relationship between
entrepreneurs' economic stress and withdrawal intentions was mediated
by depressed effect. In this example (see Figure 3):
Pollack, J., VanEpps, E. M., & Hayes, A. F. (2012). The moderating role of
social ties on entrepreneurs' depressed affect and withdrawal intentions in
response to economic stress. Journal of Organizational Behavior, 33, 789-810.
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Figure 2: Example Mediation Model
Depressed
Affect
Economic Withdrawal
Stress Intention
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Figure 3: Results of Mediation Model
DEPRESSED
AFFECT
ECONOMIC WITHDRAWAL
STRESS INTENTION
-.08 (-.09) R2 = .18
There are many statistical tests of the indirect effect (see Hayes &
Scharkow, 2013 for a review). Some of these tests (eg. the Sobel Z
test) assume the indirect effect is normally distributed.
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What is Bootstrapping?
Take a random sample of size N with replacement from the data (note each bootstrap
sample is thus the same size as the original sample).
Using the bootstrap sample calculate the desired estimates (eg. calculate ab).
We can then take the 2.5 and 97.5 percentiles of the empirical sampling distribution
to form a 95% confidence interval (CI) for the estimate. This is called a percentile
bootstrap.
The confidence interval gives a range of plausible values for the estimate. If the 95%
confidence interval does not contain zero at the selected level of confidence the
result is statistically significant (p < .05).
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PROCESS Macro
PROCESS is an easy to use add-on to SPSS or SAS for estimating
mediation, moderation, and moderated mediation models with
multiple regression (for continuous outcomes) or logistic
regression (for dichotomous outcomes).
Link: http://afhayes.com/introduction-to-mediation-moderation-
and-conditional-process-analysis.html
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Example of Bootstrapping
Refer back to the mediation analysis shown in Figure 3.
The a & b path coefficients are both statistically significant and in the
direction predicted. This provides some evidence in favor of
mediation.
Note the example in Figure 3 shows that the direct effect (-.08) is not
statistically significant at p > .05. Hence, we infer full mediation.
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Testing Interactions: Moderated
Regression
A moderator (Z) is a variable that affects the strength and/or direction of the
relationship between an independent variable (X) and a dependent variable (Y).
To test for moderation we first construct a new variable defined as the product of
scores on X and Z. This is called an interaction or product term.
We then include this interaction term as a predictor in a regression model along with
both X and Z as predictors. By including the product of X and Z in the equation we
allow the regression coefficient for X to vary as a linear function of Z.
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Basic Moderation Model
X Y
Controls
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Moderated Regression Equation
Y = B0 + B1 X + B2Z +B3 XZ+e
Z Y
XZ
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Example of Moderation
Age
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Steps in Moderated Regression
1. First Z-standardize or mean-centre X and Z. Although not
essential this step can assist in estimating and interpreting the
regression equation.
Typically, we plot the relationship between X and Y at high and low values
of Z.
Any plausible high and low values of the moderator can be plotted.
Aiken & West (1991) define high and low values as +/- 1 SD from the mean
for a continous moderator. Values for dichotomous moderators are simply
the two coded values of the moderator.
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Support for govt action
1
Low Neg emotions High Neg emotions
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Combining Mediation and
Moderation
Indirect effects can be moderated. This implies that the indirect or
mediated effect is itself contingent or conditional. This is also known
as moderated mediation or a conditional process.
Following Preacher et al. (20007) there are many ways in which the
magnitude of an indirect effect may be dependent upon one or more
moderators. We will consider the following cases:
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When the a Path is Moderated by W
(Process Model 7)
W
M
X Y
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When the b Path is Moderated by V
(Process Model 14)
V
M
X Y
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When the a Path is Moderated by W and the
b Path is Moderated by V (Model 21)
W
M V
X Y
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When the a and b Paths Are Both
Moderated by W (Process Model 58)
W
X Y
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Index of Moderated Mediation
Hayes (2015) has developed an Index of Moderation Mediation. This index
provides the most direct test for evidence of moderated mediation.
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Example: Path a Moderation
SELF-EFFICACY
DEPRESSED
AFFECT
ECONOMIC WITHDRAWAL
STRESS INTENTION
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Example 2: Path b Moderation
NEGATIVE
NEGATIVE EXPERIENCE
TONE
TEAM TEAM
DYSFUNCTION PERFORMANCE
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Model Inferences
Correct specification. Pay attention to including possible
confounders of the relationships between X and Y. Statistical
models are biased if not correctly specified. Control variables appear
in the PROCESS macro as covariates.
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PROCESS TIPS
Missing data. PROCESS requires complete
data. If missing data is an issue, it can be
imputed prior to using the macro. Mplus offers
further options for handling missing data.
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PROCESS TIPS
Multiple independent variables. If you have multiple
IVs, run PROCESS each time with one of the predictors
as the focal IV and the others as covariates. PROCESS
will control (partial out) the other IVs.
Multiple moderating variables. If desired PROCESS
can estimate three-way interactions (ie. two moderators
of an IV).
Multiple dependent variables. With multiple DVs,
simply run PROCESS for each DV in turn.
Categorical independent variables. The easiest
approach is to dummy code the categorical IV. If your IV
has k categories, construct k-1 dummy variables and
then run PROCESS using the approach outlined above.
This is similar to an ANOVA.
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PROCESS TIPS
Standardized coefficients. PROCESS coefficients are
unstandardized. Variables may be Z-scored prior to the
use of the macro to generate standardized (Beta)
coefficients. However, in PROCESS bootstrap
confidence intervals for the indirect effect should not be
interpreted as properly standardized. See Preacher &
Kelley (2011).
Dawson, J. (2013). Moderation in management research: What, why, when and how.
Journal of Business and Psychology. DOI: 10.1007/s10869-013-9308-7.
Edwards, J. R., & Lambert, L. S. (2007). Methods for integrating moderation and
mediation: A general analytical framework using moderated path analysis.
Psychological Methods, 12, 1-22.
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References
Hayes, A. F., & Scharkow, M. (2013). The relative trustworthiness of inferential tests of
the indirect effect in statistical mediation analysis: Does method really matter?
Psychological Science, 24, 1918-1927
Preacher, K. J., & Kelley, K. (2011). Effect size measures for mediation models:
Quantitative strategies for communicating indirect effects. Psychological Methods, 16,
93-115.
Preacher, K. J., Rucker, D. D., & Hayes, A. F. (2007). Assessing moderated mediation
hypotheses: Theory, methods, and prescriptions. Multivariate Behavioral Research,
42, 185-227.
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