Estimating and
Testing Mediation
David A. Kenny
University of Connecticut
http://davidakenny.net/cm/MediationN.ppt
Overview
 Introduction (click to go there)
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The Four-Step Approach (click)
The Modern Approach (click)
Test the Indirect Effect (click)
Power (click)
Assumptions (click)
DataToText (click)
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Introduction
3
Interest in Mediation
• Mentions of “mediation” or
“mediator” in psychology abstracts:
– 1980: 36
– 1990: 122
– 2000: 339
– 2010: 1,198
4
Why the Interest in Mediation?
• Understand the mechanism
– theoretical concerns
– cost and efficiency concerns
• Find more proximal endpoints
• Understand why the intervention did not
work
– finding the missing link
– compensatory processes
5
The Beginning Model:
Basic Research
6
The Mediational Model
7
The Beginning Model:
Applied Research
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The Four Paths
•
•
•
•
X  Y: path c
X  M: path a
M  Y (controlling for X): path b
X  Y (controlling for M): path c′
(standardized or unstandardized)
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The Four-Step
Approach
10
In the 1980s Different Pairs of
Researchers Proposed a Series
of Steps to Test Mediation
• Judd & Kenny (1981)
• James & Brett (1984)
• Baron & Kenny (1986)
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The Four Steps
• Step 1: X  Y (test path c)
• Step 2: X  M (test path a)
• Step 3: M (and X)  Y (test path b)
• Step 4: X (and M)  Y (test path c′)
Note that Steps 3 and 4 use the same
regression equation.
12
Total and Partial Mediation
• Total Mediation
• Meet steps 1, 2, and 3 and find that c′ equal
zero.
• Partial Mediation
• Meet steps 1, 2, and 3 and find that c′ is
smaller in absolute value than c.
13
Example Dataset
• Morse et al.
– J. of Community Psychology, 1994
– treatment  housing contacts  days
of stable housing
– persons randomly assigned to treatment
groups.
– 109 people
(davidakenny.net/papers/twmr/example.txt)
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Variables in the Example
• Treatment
– 1 = treated (intensive case management)
– 0 = treatment as usual
• Housing Contacts: total number of contacts
per during the 9 months after the
intervention began
• Stable Housing
– days per month with adequate housing
(0 to 30)
– Averaged over 7 months from month 10
to month 16, after the intervention began
15
Morse et al. Example
 Step 1: X  Y
 c = 6.558, p = .009
 Step 2: X  M
 a = 5.502, p = .013
 Step 3: M (and X)  Y
 b = 0.466, p < .001
 Step 4: X (and M)  Y
 c′ = 3.992, p = .090
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When Can We Conclude
Complete Mediation?
When c′ is not statistically different from
zero, but c is? NOT REALLY
Ideally when a and b are substantial and c′ is
small, not just non-significant.
17
Standardized vs.
Unstandardized
For most everything in this presentation, the
variables can be standardized making the
coefficients Betas.
The key thing is to be consistent: If M is
standardized in Step 2, it is also
standardized in Step 3.
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Presenting Mediation
19
The Modern
Approach
20
Dissatisfaction with the
Steps Approach
Low power in the test of Step 1
No single measure and test of mediation
Work by MacKinnon, Hayes, Preacher and
others has led to the “Modern Approach.”
21
Decomposition of Effects
Total Effect = Direct Effect + Indirect Effect
c = c′ + ab
(This equality exactly holds for multiple regression,
but not necessarily for other estimation methods.)
The Indirect Effect or ab provides one number that
summarizes the amount of mediation.
Example:
6.558 = (5.502)(0.466)
And 100(2.56/6.56) = 39% of the total effect is
explained (ab/c or equivalently 1 - c′/c).
22
Two Sides of the Same Coin
Note that
ab = c - c′
That is, the indirect effect exactly equals
the amount of the reduction in the total
effect (c) after the mediator is introduced.
Some papers (e.g., Baron & Kenny)
emphasize one side and others emphasize
the other. Current work in mediation is
now focused on the “ab” side.
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Estimating the Total Effect (c)
The total effect or c can be inferred from
direct and indirect effect as c′ + ab.
We need not perform the Step 1 regression to
estimate c.
This can be useful in situations when c does
not exactly equally c′ + ab.
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Inconsistent Mediation
• ab and c′ have a different sign
• X as a “suppressor” variable
• Example: Stress and Mood with Coping as
a Mediator
• Consequences
– The Total Effect or c may not be significant
– Percent mediated greater than 100%
• Do we have mediation?
– Yes. There is an indirect effect (ab > 0).
– No. There is no effect that is “mediated.”
25
What to Present
Decomposition of Effects
Total Effect
Indirect Effect(s)
Direct Effects
Less of an emphasis on complete versus
partial mediation.
26
Test of the
Indirect Effect
27
Work on Determining How to
Test ab = 0
• The key piece of information in the modern
approach is the indirect effect.
• Many researchers developed approaches to
this problem.
28
Strategies to Test ab = 0
• Test a and b separately
• Sobel test
• Bootstrapping
• Monte Carlo Method
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Test a and b Separately
• Easy to do
• Works fairly well
• Does not provide a method for
a confidence interval for ab.
• Seem too much like the oldfashioned Four-Step Approach.
30
Sobel Test of Mediation
Compute the square root of a2sb2 +
b2sa2 which is denoted as sab
Note that sa and sb are the standard
errors of a and b, respectively; ta =
a/sa and tb = b/sb.
Divide ab by sab and treat that value as
a Z.
So if ab/sab greater than 1.96 in
absolute value, reject the null
hypothesis that the indirect effect is
zero.
31
Example
a = 5.502 and b = 0.466
sa = 2.182 and sb = 0.100
ab = 2.56; sab = 1.157
Sobel test Z is 2.218, p = .027
We conclude that the indirect effect
is statistically different from zero.
Website: http://www.people.ku.edu/~preacher/sobel/sobel.htm
32
Large values of “ab”
are more variable than
small values (i.e., 0).
The distribution of ab is highly skewed
which lowers the power of the test.
33
Bootstrapping
• “Nonparametric” way of computing a
sampling distribution.
• Re-sampling (with replacement)
• Many trials (computationally intensive)
• Current thinking is NOT to correct for
bias (i.e., the mean of the bootstrap
estimate differs slightly from the
estimate).
• Compute a confidence interval which is
asymmetric.
• Slight changes because empirically
derived.
34
Results of
Bootstrapping
95% Percentile Confidence Interval:
Lower Upper
.5325
5.0341
Note that the CI is asymmetric for an estimate of
2.598. Also values differ to sampling error.
(Done using the Hayes & Preacher macro from
http://www.afhayes.com/spss-sas-and-mplus-macros-and-code.html.)
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Monte Carlo Method
• Save a, b, sa (the standard error of a) and
sb from the mediation analysis.
• For some methods you have the
covariance of ab or sab to save.
• Use them and the assumption of
normality to generate estimates of a and b
and so ab.
• Use this distribution of ab’s to get a
confidence interval or a p value.
• Less computationally intensive than
bootstrapping and many trials are
feasible.
36
Results of the Monte
Carlo Method
95% Monte Carlo Confidence Interval (20,000 trials):
Lower Upper
0.551
5.158
Done using the Selig & Preacher web program at
http://www.quantpsy.org/medmc/medmc.htm; if
correlated use Tofighi’s Rmediation at
http://www.amp.gatech.edu/RMediation
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Power
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Power of the Test of c
If c′ is zero, then c equals ab.
If both a and b have a moderate effect size,
then c has a “smaller than small” effect size
(assuming c′ is zero).
Thus, it is very possible to have tests of
significance for a and b be statistically
significant, but c is not.
Note that for N = 100, if a and b have
moderate effect sizes (r = .3) and c′ = 0, the
power of the test of c is only .14 whereas
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the power for a is .87 and b is .83.
Relative Power of c and ab
Several authors, as early as Cox (1960), have
noted that the indirect effect has much
more power than the test of c even when
the two equal the same value as when c′ is
zero.
Kenny and Judd (2014) show that sometimes
the test of c need 75 times the number of
cases to have the same power as the test of
ab!
40
Relative Power of the Test of
c and c′ when b = 0
When b = 0, c = c′ and in this case c might be
statistically significant, yet c′ might not be
significant even though. Why?
There is multicollinearity between X and M
due to path a.
Note that as M becomes a more successful
mediator and path a gets larger,
multicollinearity becomes more of a issue
in the testing of c′ (and b).
41
Power and the Test of b
Let b = .3 (standardized). What happens to N need
to have 80% power as path a gets larger:
a
N
.1
86
.3
93
.5
112
(a is standardized; sample size needed for 80%
power to reject the null hypothesis that b = 0)
Conclusion: Power of the test of b declines as a
increases.
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Tool MedPowR
R based program to help in power analyses.
Can either give the power for a given value of N
and a, b, and c’ or give the N needed to achieve a
desired level of power.
Program can be downloaded at
http://davidakenny.net/progs/PowMed.R
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44
Power calculations have
begun...
Effect Size Power N
c
.390 .986 100
a
.300 .868 100
b
.300 .890 100
c'
.300 .890 100
ab
.090 .773 100
Alpha for all power
calculations set to .050.
Power calculations complete.
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Assumptions
46
Taking Assumptions
Seriously
• Older Mediation Analysis ignored the strong
assumptions required for the analysis.
• Current work, especially that within the Causal
Inference tradition, focuses much more on them.
• Researchers need to conduct “Sensitivity
Analyses” or “What If?”analyses.
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Assumptions:
Multiple Regression
• Linearity
– Is a problem in the example; housing contacts has a
quadratic effect on stable housing (p = .034).
• Normal Distribution of Errors
– Interval level of measurement of M and Y
• Equal Error Variance
• Independence
– No clustering
• X and M Do Not Interact to Cause Y
48
No XM Interaction:
Linear Mediation
• Called “Moderation” in Baron & Kenny
• Add XM (and possibly other interaction
terms, e.g., X2M) when explaining Y.
• Many contemporary analysts now see XM
interaction as part of mediation.
• Not significant for the example (p = .476)
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Causal Assumptions
• Perfect Reliability
– for M and X
• No Omitted Variables
– all common causes of M and Y, X and M,
and X and Y measured and controlled
• No Reverse Causal Effects
– M and Y not cause X
– Y may not cause M
(Guaranteed if X is manipulated.)
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Basic Mediational Causal Model
Note that U1 and U2 are
theoretical variables and not
“errors” from a regression
equation.
U1
1
M
a
b
U2
1
c'
X
Y
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Unreliability
• Usually safe to assume that X is
perfectly reliable.
• Measurement error in Y does not bias
unstandardized regression
coefficients.
• Measurement error in M is
problematic.
52
Unreliability in M
Error
1
U1
M
1
1
M
Latent
b
a
U2
1
c'
X
Y
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Effect of Unreliability in M
• b is attenuated (closer to zero)
• c′ is inflated (given consistent
mediation)
– more as a increases
– more as b increases
– Note that the bigger the indirect, the
greater the bias in c′.
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What to Do about Unreliability in M?
• Improve the reliability
• Adjust estimates using Structural Equation
Modeling
• Conduct Sensitivity Analyses assuming
different values of reliability.
55
Omitted Variables
U1
e
1
Omitted
Variable
M
f
a
b
U2
1
c'
X
Y
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Other Terms for an Omitted Variable
• Third variable.
• Confounder
– Term used in epidemiology
– Becoming increasingly popular
57
What is the Effect of Omitted
Variables?
• Usually, but not always, the sign of ef is the
same as b.
– Inflating the estimate of b
– Deflates the estimate of c′ (could produce
inconsistent mediation).
58
Effect of Vitamin A Supplements
in Northern Sumatra
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"Standard" Analysis
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Compliance should totally mediate this effect. Why is c' negative?
Results with an Omitted Variable
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Path c' fixed to zero. Omitted variables that causes M and Y.
What to Do about Omitted Variables?
• Do not omit them: Include them in the
analysis as covariates.
• If there is good reason to believe that c′ = 0,
they can be allowed for.
• Sometimes the omitted variable is “shared
method effects.” If an issue, measure M
and Y by different methods.
• Conduct sensitivity analyses.
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A Comforting Fact
• Unreliability in M deflates ab and inflates
c′.
• An omitted variable usually deflates ab
and inflates c′.
• As Fritz, Kenny, & MacKinnon have
shown these two biases can almost exactly
offset each other.
63
Reverse Causation
U1
1
M
a
b
g
U2
1
c'
X
Y
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Effect of Reverse Causation
• Typically, b and g have the same sign,
which likely makes the value of b inflated
and the value of c′ deflated.
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What to Do about Reverse
Causation?
• Longitudinal designs
• If c′ = 0, then the model can be estimated.
• Instrumental variable method.
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Timing of the Measurement of
the Mediator
• Mediator should be measured after X but
before Y.
• X might be measured at the same time as Y
(e.g., number of treatment sessions), but it
must be assumed that X has not changed
since when it affected Y.
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Controlling for Prior Values
• Obtain baseline measures of M and Y.
• Control for baseline M and Y in the
analysis.
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DataToText
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DataToText Project
Have the researcher tell DataToText
what is the research question.
DataToText performs the requisite
analyses.
DataToText gives the results from those
analyses:
computer output
a written description
70
DataToText Macros
• Macro developed to provide text,
tables, and figures of a simple
mediational analysis.
– SPSS version: MedText
http://davidakenny.net/dtt/mediate.htm
– R version: MedTextR
http://davidakenny.net/dtt/mediateR.htm
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Advantages of DataToText
• Does the analyses that should be done,
but often are not, e.g., tests for outliers
and nonlinearity.
• MedTextR issues up to 20 different
warnings.
• Produces a 3 page text describing the
results.
• Surprisingly “intelligent”
• Graphics
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Links
davidakenny.net/dtt/datatotext.htm
davidakenny.net/dtt/mediate.htm
davidakenny.net/dtt/mediateR.htm
davidakenny.net/dtt/MedTextR.pdf
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Topics Not Discussed
Use of SEM
simultaneous estimation
latent variables (reflective and formative)
instrumental variable estimation
Causal Inference Approach
Mediators or outcomes that are categorical or
counted
Clustering and multilevel mediation
Mediated moderation and moderated mediation
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Conclusion
• Mediational Analyses Are
Very Simple
• Mediational Analyses Are
Very Difficult
–Difficulties are more in
conceptualization and
measurement than in the
statistical analysis.
davidakenny.net/cm/mediationN.ppt
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