1. Mediation and Moderation: Modern Methods and Approaches
Alexander M. Schoemann
Department of Psychology
East Carolina University
Introduction and Basic Mediation

2. Let’s get to know each other
Alex Schoemann
PhD in Social and Quantitative Psychology from KU in 2011
Postdoctoral Researcher at the Center for Research Methods and Data Analysis at KU
Assistant Professor of Psychology at East Carolina University
Research focuses on latent variable modeling (SEM & MLM), missing data, and intergroup relations
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Overview: Page 1
Introductions
Who’s here?
Mediation vs. moderation
Software overview
Regression review
Correlation and covariance
Simple linear regression
Multiple linear regression
Categorical predictors
Other types of regression (e.g. logistic regression)
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Overview: Page 2
Basic mediation in regression
Fitting the mediation model
Testing indirect effects
Effect sizes
Pitfalls (suppressor variables, concurrent data)
Special topics in mediation
Latent variables
Multiple mediation
Logistic mediation
Mediation in multilevel modeling
Longitudinal mediation
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Overview: Page 3
Basic moderation in regression
Fitting the moderation model
Graphing the interaction
Simple slopes & regions of significance
Interpretation
Continuous vs. categorical moderator
Centering?
Special topics in moderation
Multiple moderators
Moderation in multilevel modeling
Latent variable moderation
Quadratic moderation & Polynomial relationships
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Overview: Page 4
Hybrid models of mediation and moderation
Conditional process models
Mediated moderation
Moderated mediation
Other hybrids
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Course Materials
All course materials should be posted on the Stats Camp website
This includes
All slides
Example data (mostly SPSS)
Example syntax (mostly SPSS)
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Software
This course will extensively use the PROCESS macro for SPSS and SAS (Hayes, 2014)
Available from:
SPSS users have the option of using either syntax or creating a drop down menu
We will use syntax in this course
SAS users must have SAS IML to use the macro
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Software
The nice thing about PROCESS is that the input and output for SPSS and SAS are almost the same.
For more complex analyses we will use other software
e.g. Mplus for SEM
If you have a question about other software packages (e.g. R, LISREL) I’ll try to answer it in the afternoon help sessions
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10. Why a course in mediation and moderation?
Both topics address complex research questions.
Not just does X predict Y but…
Why does X predict Y? (mediation)
When does X predict Y? (moderation)
Both topics address conditional effects
The relationship between two variables is influenced by a third variable
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11. Why not a course in mediation and moderation?
Despite the similarities mediation and moderation are two very different process.
They answer different questions
They make different assumptions
They use different analytic strategies
So why are we talking about them together?
They both start with M?
Baron and Kenny wrote an article about them in 1984?
Let’s define mediation and moderation before moving on
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12. Mediators and moderatorsX Y
mediator X Y
moderator
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Mediator vs. Moderator
Mediator: middle-person, letter carrier, delivery agent
X  M  Y
Key word: because
Moderator: “changer”
variable that alters the strength of another relationship (i.e. an interaction!)
Moderation occurs when the effect of X on Y depends on Z
Key word: depends
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Regression overview
Much of this class will focus on regression as the analysis of choice
SEM and MLM can be thought of as “fancy” version of regression
It is important that everyone is on the same page when it comes to regression basics
The next few slides will review correlation and regression before we get into mediation and moderation
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Regression overview
Say we have two variables, x and y. We might want to explore their characteristics with histograms and sample statistics.
Each variable’s marginal characteristics
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Regression overview
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Regression overview
We could also explore their joint characteristics with a scatter plot and measures of association.
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Regression overview
We could estimate a covariance or correlation to characterize the degree of linear relationship between x and y:
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Regression overview
Or we could do the same thing with regression using the intercept and slope.
Values of the intercept and slope are determined by the least squares criterion.
The values of the intercept and slope that minimize the squared distance between each point and the regression line.
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Regression overview
The simplest regression equation.
By the least square criterion the value of that minimizes the square residuals is the mean of y.
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Regression overview
We can explain more variance in y by including x as a predictor of y.
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Regression overview
We can include additional predictors of y.
With two predictors we are now searching for a plane of best fit.
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Regression overview
Interpreting regression coefficients
Intercept – The predicted score on y when all predictors are 0
Slope – The predicted increase in y for a 1 unit increase in x
With multiple predictors the slope is the predicted increase in y for a 1 unit increase in x holding all other predictors constant.
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Regression overview
Standardized vs. unstandardized slopes
The regression slopes we’ve discussed thus far are unstandardized slopes
Based on the original metric of the variable
SPSS call these “B”
We could also estimate standardized regression slopes
Convert y and all x’s to Z scores and fit the regression
SPSS calls these “Beta”
With one predictor the standardized slope = r
Be careful with these!
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Regression overview
Hypothesis tests for slopes
Almost every regression program will provide you with a test statistic (and p-value) for each regression slope
These are based on Wald statistics (either t or z)
Wald statistics is the parameter estimate divided by its standard error
Usually we test if the slope if different from zero
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Regression overview
Comparing models
For each model we get an estimate of R2
The proportion of variance in y explained by the x’s
We can compare the change in R2 across models (when adding predictors) to see if adding predictors improves the fit of the model.
When adding 1 predictor the test of the change in R2 is the same thing as testing for the significance of the slope.
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Regression overview
Example from Jose (2013)
File: Mediation_example.sav
Data from the International Wellbeing Survey
International sample of 364 adults between the ages of 17 and 79
Variables
ple = positive life events
grat = gratitude
shs = subjective happiness scale
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Regression overview
Thus far we have discussed predictors that are “continuous” variables
What if a predictor is a categorical variable?
The variable is coded into 1 or more “dummy” (or contrast) variables representing differences between groups
We always need k-1 dummy variables to represent k categories
Example: Sex would be coded into a dummy variable called Male with male = 1 and female = 0
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Regression overview
The slope of a dummy variable represents the mean difference on y between that group and all other groups.
With simple linear regression this is equivlent to doing a t-test
Example: experimental_mediation_example.sav
Does gratitude differ between treatment and control groups?
Treatment is a positive psychology based intervention for adolescents
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Regression overview
OLS regression assumes that the outcome variable (y) is measured on a “continuous” scale.
What if the outcome variable in regression is dichotomous?
Example: Has a participant been arrested?
Analyze with logistic regression!
Categorical DVs can make mediation a bit complicated. We’ll talk about this tomorrow.
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Three variable system
Predictor variable – X
Outcome variable – Y
Mediator variable – M
M (partially) explains the relationship between X and Y
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32. Pieces of the mediation puzzleX Y c X Y M c' a b
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33. How do we get estimates of a, b, and c?
We can gain estimates of a, b, c, and c’ through a series of separate regression models:
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Example: c path
PLE
Happiness
.485
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Example: a path
PLE
Happiness
Gratitude c'
1.752 b
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Example: b and c’ paths
PLE
Happiness
Gratitude
.269
1.752
.123
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37. What do the paths tell us about mediation?
When investigating mediation we are interested in if strength of the relationship between X and Y is reduced when M is included in the regression equation
Is the direct effect (c’) less than the total effect (c)?
Does c – c’ = 0?
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38. What do the paths tell us about mediation?
In our example c = .485 and c’ = .269
c – c’ = .485 – .269 = .216
c – c’ is also called the indirect effect
The indirect effect can also be estimated by multiplying a*b
a*b = 1.752*.123 = .216
a*b and c – c’ should provide the same value
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c – c’ = a*b
Start with the three regression equations for mediation
Plug in values for M and Y
Use algebra!
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Testing for mediation
When we test for mediation we test if the indirect effect is different from zero.
If the indirect effect equals zero there is no evidence of mediation
If the indirect effect does not equal zero there is evidence of mediation
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Testing for mediation
There have been many methods developed to assess the indirect effect:
Difference in coefficients (Baron & Kenny, 1984)
Sobel test
Distribution of the product method
Bootstrap confidence intervals
Monte Carlo confidence intervals
Profile likelihood confidence intervals …
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42. Difference in coefficients method
Fit regressions estimating a, b, c’, and c. Only continue if estimates of a, b, and c are significant.
Mediation has occurred if c’ is not significant (but c is)
THIS IS A SPECTACULARLY BAD IDEA!
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43. Difference in coefficients method
Cons
Not a test of the indirect effect
“The Difference Between “Significant” and “Not Significant” is not Itself Statistically Significant” –Gelman &Stern (2006)
Dependent on sample size
With larger samples c’ will always be significant…
c does not need to be significant to test for mediation
We’ll discuss this in more detail in a bit
The indirect effect may be significant even if a and/or b is not
This is rare but it can happen
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Sobel test
Compute a Wald statistic for the indirect effect
Results in a z statistic
We need estimates of the indirect effect (a*b) and its standard error
These estimates are based on the unstandardized regression slopes and their standard errors
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Sobel test
a and b are the unstandardized regression coefficients, sea and seb are the standard errors of a and b
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Sobel test
Who wants to do this by hand?
Built into the PROCESS macro
Kris Preacher’s webpage:
Other webpages:
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Preacher’s website
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Sobel test
Pros:
A test of the indirect effect
Easy to specify
We can use the standard error to compute a confidence interval for a*b (95% CI: )
Cons:
Assumes the distribution of a*b is normal (and CI are symmetric.
Even if a and b are normally distributed their product will not be normal!
Not recommended!
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Testing for mediation
There have been many methods developed to assess the indirect effect:
Difference in coefficients (Baron & Kenny, 1984)
Sobel test
Distribution of the product method
Bootstrap confidence intervals
Monte Carlo confidence intervals
Profile likelihood confidence intervals
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50. Detour: Hypothesis testing with CIs
When testing the significance of a*b with the remaining methods. we use a CI to test our null hypothesis.
H0: a*b = 0
If a*b is significant we say there is a less than 5% chance that a*b = 0 in the population
A 95% CI provides the same information
If 0 is not within the 95% CI: In 95% of samples of size n a*b ≠ 0. Significant mediation effect.
If 0 is within the 95% CI: In less than 95% of samples of size n a*b ≠ 0. Non-significant mediation effect.
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51. Distribution of the product method
If the distribution of a*b isn’t normal, can we use the (non-normal) distribution of a*b to generate a confidence interval?
Usually involves computing asymmetrical confidence limits (adjusting the z value used to compute the CI)
MacKinnon, Fritz, Williams, & Lockwood’s (2007) paper on PRODCLIN, a stand-alone program.
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52. Distribution of the product method
Software implementations:
PRODCLIN, a macro for SPSS or SAS (MacKinnon, Fritz, Williams, & Lockwood, 2007)
RMediation package for R (Tofighi & MacKinnon, 2011)
RMediation website:
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RMediation website
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54. Distribution of the product method
Pros:
A test of the indirect effect
Does not assume the distribution of a*b is normal
Cons:
Assumes the distribution of a and b are normal (this isn’t a big deal)
Requires specialized software
Can be difficult to implement
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Bootstrapping
Steps for bootstrapping
Draw a sample from the data of size n with replacement
Fit your model(s) to this data (e.g., estimate both a and b in two regressions)
Save the parameter estimates from Step 2
Repeat Steps s of times
The parameter estimates from Step 2 form a distribution for each parameter estimate
The 2.5th and 97.5th percentiles of the distribution form the 95% CI
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Bootstrapping
Orig Data
Bootstrap 1
Bootstrap 2
Bootstrap 3
Stanley
Andy
Phyllis
Jim
Dwight
Pam
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Bootstrapping
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Bootstrapping
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59. Bootstrapping details
How to do it?
Built into some SEM software (e.g., Mplus)
PROCESS macro for SPSS and SAS (see afhayes.com)
Other software (e.g., R, Stata) may require a bit of work
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Bootstrapping issues
Multilevel data (we’ll talk about this later)
Type of CI: percentile, Bias Corrected (BC), Bias Corrected, and Accelerated (BCa)
Studies indicate that BCa CIs perform best (if available)
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PROCESS
An SPSS/SAS macro that implements mediation and moderation (and all sorts of combinations of them) in a regression framework.
Currently there are 76 different models PROCESS will fit
A list of models can be found at:
SPSS and SAS syntax are very similar.
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PROCESS
To use a macro in SPSS or SAS we need to “install” it before use
This needs to be done once per session
Open the file processv211.sps or processv211.sas
Highlight and run the entire contents of the file
This creates a new command in SPSS or SAS syntax called “process”
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PROCESS
The code for our example would be:
SPSS
process vars = ple grat shs/ y = shs/ x = ple/ m = grat/ total = 1/ model = 4/ boot = 5000/ effsize = 1.
SAS
%process (data = example_mediation, vars = ple grat shs, y = shs, x = ple, m = grat, total = 1, model = 4, boot = 5000, effsize = 1);
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PROCESS
vars = ple grat shs The names of the variables used in the model. Any variables specified here but not in other parts of the syntax are control variables
y = shs The outcome variable name
x = ple The predictor variable name
m = grat The mediating variable(s) name. Multiple variable names means multiple mediators
total = 1 Print estimates of the total effect (1=yes, 0=no)
model = 4 The model we wish to fit (based on the PROCESS templates)
boot = 5000 Number of bootstrap resamples (larger is better…)
effsize = 1 Print estimates of effect size for the indirect effect (1=yes, 0=no)
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Monte Carlo CI
What if you don’t have the raw data?
What if bootstrapping isn’t possible?
Use a Monte Carlo CI!
Available at quantpsy.org
Or with the mc option in PROCESS
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Monte Carlo CI
Steps for Monte Carlo CI
Fit your models and save estimates of a, b, SEa and SEb
Assume that a and b are normally distributed with mean a (or b) and standard deviation SEa (and SEb)
Draw many (10000+) random values from the distribution of a and b. Compute a*b for each of these draws
Use random draws to construct a distribution of a*b
The 2.5th and 97.5th percentiles of the distribution form the 95% CI
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Monte Carlo CI
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Profile Likelihood CI
Based on Maximum Likelihood (ML) estimation
Finds values of a*b that increase the model log-likelihood by a set amount (e.g. 3.84)
Limited software availability.
OpenMX, others?
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69. Sobel vs. Bootstrapping vs. Monte Carlo CI
When both a and b are large and n is large, all three techniques are similar
When either a, b, or n is small, bootstrapping and Monte Carlo CIs are more accurate
When either a or b is not normally distributed, bootstrapping is most accurate
Take home message: When you can, bootstrap. When you can’t, use a Monte Carlo CI.
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Effect sizes
Confidence intervals can tell us if an indirect effect is significant and a range of possible values for an indirect effect.
They don’t tell us about the size of an indirect effect.
Many effect sizes have been proposed for indirect effects but most tend to be problematic (see Preacher & Kelly, 2011).
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Effect sizes
Effect sizes based on standardized coefficients
Partially standardized indirect effect
Standardize Y but not X
Still depends on the scale of X
Useful if X is dichotomous
Completely standardized indirect effect
Standardize X, M, and Y
Scale independent
SHOULD NOT BE USED WHEN X IS DICHOTOMOUS!!!
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Effect sizes
Effect sizes based on ratios
Ratio of the indirect effect to the total effect
Problematic when c is very small
Ratio of the indirect effect to the direct effect
Problematic when c’ is very small
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Effect sizes
Others…
R2 due to the indirect effect
Based on the assumption that X and Y are correlated
Preacher & Kelly’s Kappa squared
MAX(ab) is the largest possible value of the indirect effect given the data
Ratio of the indirect effect to the largest possible indirect effect
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Effect sizes
What to use?
None of the effect sizes are perfect, think about the context of your study
I would lean toward reporting a standardized estimate of the effect size and kappa squared
The ratio and R2 estimates make me a bit nervous, but there are situations where they perform well.
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Interpretation
When writing up the results from a mediation analyses it is easy to get stuck in the details of the analysis and miss the big picture
If there is a significant indirect effect, describe each path in the model and how they lead to each other.
To include a path diagram or not???
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Interpretation
PLE
Happiness
Gratitude
.269
1.752
.123
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Interpretation
“The results show that if someone experiences a high level of positive life events, then he or she is likely to report greater happiness. This relationship can be partially explained by detailing the involvement of gratitude. In essence, higher levels of positive life events led to higher levels of gratefulness, and, in turn, gratefulness led to higher levels of happiness.”
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78. Pitfalls (Alex’s soapbox time)
Does X have to predict Y to test for mediation?
Holdover belief from the days of Baron & Kenny
Your decision to test for an indirect effect should be based on theory, not on if a path is significant
It is possible that X does not predict Y (the total effect = 0) but there is a significant indirect effect (the indirect effect ≠ 0)
More on this when we talk about multiple mediators
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Pitfalls
“Full” vs. “Partial” mediation
Some people like the terms “full mediation” and “partial mediation”
Full mediation occurs when the direct effect (c’) is not significant
Partial mediation occurs when the direct effect (c’) is significant
I HATE these terms
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Pitfalls
“Full” vs. “Partial” mediation
The biggest problem with these terms is that “full mediation” is seen as more desirable than “partial mediation”
BUT full vs. partial mediation is a function of sample size!
With a large enough sample you will almost never have full mediation.
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Pitfalls
The issue of causation…
Mediation is a causal process
X causes changes in M and in turn M causes changes in Y
Most mediation models are cross-sectional
Like our example…
With 3 measured, cross-sectional variables, determining causal ordering is difficult/impossible
Is it X -> M -> Y or M -> Y -> X or…
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Pitfalls
The issue of causation…
How do we solve this issue?
Experimentally manipulate X?
Still a problem with M and Y
Use longitudinal data?
More on this later, but yes!
Other methods of causal modeling
See the field of causal mediation
Beyond the scope of this workshop
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83. That seems like enough for today…
To review:
Simple and multiple regression
Basic mediation involves 3 variables and 3 paths among them
X, M, Y
a, b, c’
When testing for mediation we test the indirect effect of a*b
Test this with bootstrapping!
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84. That seems like enough for today…
To review:
PROCESS makes estimating and testing mediation models straightforward with SPSS and SAS
Effect sizes for mediation can be tricky
Watch out for other pitfalls!
e.g. concurrent data
Any last questions?
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