Approaches to Improving Causal
Inference from Mediation Analysis
David P. MacKinnon, Arizona State University
Pennsylvania State University
February 27, 2013
Background
Traditional Mediation Methods
Modern Causal Inference Methods
Example
Traditional and Causal Inference Methods
Summary
*Supported by National Institute on Drug Abuse and collaborations…
1
Mediating Variable
A variable that is intermediate in the causal process relating an
independent to a dependent variable.
Attitudes cause intentions which then cause
behavior (Azjen & Fishbein, 1980)
Prevention programs change norms which promote
healthy behavior (Judd & Kenny, 1981)
Increasing exercise skills increases self-efficacy
which increases physical activity (Bandura, 1977)
Exposure to an argument affects agreement with the
argument which affects behavior (McGuire, 1968)
2
Mediation is important
because …
Central questions in many fields are about
mediating processes
Important for basic research on mechanisms of
effects
Critical for applied research, especially
prevention and treatment
Many interesting statistical and mathematical
issues
3
Single Mediator Model
MEDIATOR
M
a
INDEPENDENT
VARIABLE
X
b
c’
DEPENDENT
VARIABLE
Y
4
Mediation Statements
• If norms become less tolerant about smoking then
smoking will decrease.
• If you increase positive parental communication then
there will be reduced symptoms among children of
divorce.
• If children are successful at school they will be less
anti-social.
• If unemployed persons can maintain their self-esteem
they will be more likely to be reemployed.
• If pregnant women know the risk of alcohol use for the
fetus then they will not drink alcohol during pregnancy.
5
S-O-R Mediator Model
a
Mental and
other
Processes
M
Stimulus
X
b
Response
c’
Y
6
Three-variable Models
Complex even though only one more variable is
added to a two-variable model.
Next complex model after randomized X to Y
relation.
Often includes hidden assumptions about how
variables are related.
Usually only one variable is randomized.
Special names for third-variable effects: mediator,
confounder, covariate, moderator.
7
Mediator Definitions
 A mediator is a variable in a chain whereby an
independent variable causes the mediator which
in turn causes the outcome variable (Sobel,
1990)
 The generative mechanism through which the
focal independent variable is able to influence
the dependent variable (Baron & Kenny, 1986)
 A variable that occurs in a causal pathway from
an independent variable to a dependent
variable. It causes variation in the dependent
variable and itself is caused to vary by the
8
independent variable (Last, 1988)
Applications
Two overlapping applications of mediation analysis:
(1) Mediation for Explanation
(2) Mediation by Design
9
Mediation for Explanation
• Observed relation and try to explain it.
• Elaboration method described by Lazarsfeld
and colleagues (1955; Hyman, 1955) where
third variables are included in an analysis to
see if/how the observed relation changes.
• Replication (Covariate)
• Explanation (Confounder)
• Intervening variable (Mediator)
• Specification (Moderator)
10
Mediation by Design
• Select mediating variables that are causally
related to an outcome variable.
• Intervention is designed to change these
mediators.
• If mediators are causally related to the
outcome, then an intervention that changes
the mediator will change the outcome.
• Common in applied research like prevention
and treatment.
11
Mediation in Intervention
Research Theory
• Mediation is important for intervention science.
Practical implications include reduced cost and more
effective interventions if the mediators of programs
are identified. Mediation analysis is an ideal way to
test theory.
• A theory based approach focuses on the processes
underlying interventions. Mediators play a primary
role. Action theory corresponds to how the program
will affect mediators. Conceptual Theory focuses on
how the mediators are related to the dependent
variables (Chen, 1990, Lipsey, 1993).
12
Intervention Mediation Model
MEDIATORS
Action
theory
M1 , M2 , M3 ,
…
Conceptual
Theory
INTEREVENTION
PROGRAM
OUTCOMES
X
Y
If the mediators selected are causally related to Y, then changing the
mediators will change Y.
13
Mediation Regression Equations
 For the single mediator model information
from some or all of three functional
equations are used.
 Now parametric and nonparametric
approaches.
 Regression equations are shown next. Can
also be more generally described as
Expectations.
14
Equation 1
MEDIATOR
M
INDEPENDENT
VARIABLE
X
c
DEPENDENT
VARIABLE
Y
1. The independent variable is related to the dependent variable:
15
Equation 2
MEDIATOR
M
a
INDEPENDENT
VARIABLE
DEPENDENT
VARIABLE
X
Y
2. The independent variable is related to the potential mediator:
16
Equation 3
MEDIATOR
M
a
INDEPENDENT
VARIABLE
X
b
c’
DEPENDENT
VARIABLE
Y
3. The mediator is related to the dependent variable controlling for
exposure to the independent variable:
17
Mediated Effect Measures
Indirect Effect = Mediated effect = ab = c-c’
Direct effect= c’ Total effect= ab+c’=c
18
Social Science Equations with Covariate C.
E[Y|X=x, C=c] = i1+ c X + c2 C
E[Y|X=x, M=m, C=c] = i2+ c’ X + b M + c3 C
E[M|X=x, C=c] = i3+ a X + a2 C
With XM interaction
E[Y|X=x, M=m, C=c] = i4+ c’ X + b M + h XM + c4 C
19
Identification Assumptions
1. No unmeasured X to Y confounders given
covariates.
2. No unmeasured M to Y confounders given
covariates.
3. No unmeasured X to M confounders given
covariates.
4. There is no effect of X that confounds the M to Y
relation.
VanderWeele & VanSteelandt (2009)
20
Assumptions
 Reliable and valid measures.
 Data are a random sample from the population of
interest.
 Coefficients, a, b, c’ reflect true causal relations
and the correct functional form.
 Mediation chain is correct. Temporal ordering is
correct: X before M before Y.
 No moderator effects. The relation from X to M
and from M to Y are homogeneous across
subgroups or other participant characteristics. 21
Significance Testing and Confidence
Limits
Product of coefficients estimation, ab, of the mediated effect
and standard error is the most general approach with best
statistical properties for the linear model given
assumptions. Best tests are the Joint Significance,
Distribution of the Product, and Bootstrap for confidence
limit estimation and significance testing again under
model assumptions.
For nonlinear models and/or violation of model
assumptions, the usual estimators are not necessarily
accurate. New developments based on potential outcome
approaches provide more accurate estimators (Robins &
Greenland, 1992; Pearl, 2001).
22
Causal Inference in Mediation
 Assumptions of true causal relations and selfcontained/comprehensive model. Blalock’s (1979)
presidential address states that ~50 variables are
involved in sociological phenomenon;Weinstein’s
comprehensive vs limited health psychology models.
How many variables are relevant in a research area?
 Problem with mediation analysis because M is not
randomly assigned but is self-selected (Holland, 1988;
Judd & Kenny, 1981; James, 1980; James & Brett,
1984; MacKinnon, 2008).
 Active research area. Frangakis & Rubin, (2002); Pearl
(2009), Geneletti (2007), Imai et al., 2010,
23
VanderWeele, 2010….
Modern Causal Inference in Mediation
Introduction of counterfactual/potential outcome
model illustrates problems with the causal
interpretation of results from mediation analysis.
Counterfactual is central to modern causal inference.
The counterfactual refers to conditions in which a
participant could serve, not just the condition that
they did serve in.
Problem with mediation analysis because M is not
randomly assigned but is self-selected.
This approach has led to several innovative methods.
24
Randomized Two Group Design
Unit (individual level) have observed data for one of
two conditions but not the other—the fundamental
problem of causal inference.
Randomization of a large number of persons solves the
fundamental problem of causal inference. The
average in each group can be compared and it is an
estimator of a causal effect. It is called an average
causal effect (ACE).
Randomization does not solve the problem for
mediation because some potential outcomes never
exist.
25
Problem with Mediation Analysis
For a participant in the treatment group, their mediator value if they
were in the control group is not observable because they are in the
treatment group.
That is, we will never get the value M for the control group if
instead they were in the treatment group. We will never get the
value of M for the treatment group, if instead they were in the
control group because they are in the treatment group.
One solution is the natural indirect effect is to use the value of M in
the control group as the counterfactual to compare the value of M
in the treatment group (Robins & Greenland, 1992; Pearl, 2001).
effect.
26
Analytical Causal Inference Methods
Natural Indirect Effects (Pearl 2001; Robins & Greenland,
1992, Imai et al., 2010; VanderWeele, 2010)
Inverse Probability Weighting (Coffman, 2011; Robins,
Hernán & Brumbeck, 2000)
Principal Stratification (Frangakis & Rubin, 2002; Jo, 2008)
Instrumental Variable Methods (Angrist et al., 1996;
Holland, 1988; MacKinnon, 2008; Sobel, 2007)
G-estimation (ten Have et al., 2007; Robins, 1989)
Decision Theory (Didelez et al., 2006, Geneletti, 2007)
Stochastic Processes (Steyer,2012; Mayer et al., 2012)
Focus on dealing with confounding.
Active area of research …
27
Which method is best?
 What are the best methods to find mediation when it
exists and not find it when it does not exist?
 How can we assess the accuracy of mediation methods?
(1) Mathematics
(2) Statistical simulations
(3) Use by Applied Researchers
(4) Applications to known mediation theory examples. This
is what is considered in this talk.
28
Memory Experiment
• It is well known in human memory research that making an
image increases word recall more than just repeating the
words over and over (Craik, 2002; Paivo, 1971;
Richardson, 1998)
• This is a situation where the mediation process is assumed
known– Manipulation to increase imagery (X) increases
imagery (M) which increases word recall (Y).
• Participants hear a list of 20 words with 10 seconds
between the words.
• Participants are randomly assigned to either make images
of the words or repeat the words.
• A way to validate mediation methods.
29
Rehearsal Manipulation
Subjects in the Primary Rehearsal Condition
P condition
As you hear each word, repeat it over and over until you hear the next
word.
Subjects in the Secondary Rehearsal Condition
S Condition
As you hear each word, make an image out of the word and the other
words that you hear. For example, if you heard the words camel and
woman, you would imagine a woman riding a camel.
30
Experiment Scoring
• I is answer to, “I made mental images of the
words.”
• R is answer to, “I repeated the words over
and over.”
• Scale 1-Not at all to 9-Absolutely
31
Experiment
Order of Words Read
Nice, Rock, Chair, Box, Calm, Think, Jump, Car,
Happy, Flag, Star, Study, Talk, Clock, Stick,
Caution, Rope, Send, Pole, Quick
Concrete
Rock, Chair, Box, Car, Flag, Star, Clock, Stick, Rope,
Pole
Not Concrete
Nice, Calm, Think, Jump, Happy, Study, Talk,
Caution, Send, Quick
32
Data are…
ID number
X primary or second rehearsal condition
M self-reported imagery score
R self-reported repetition score
Y total number of words recalled
Some data….
ID X M R Y
1 0 3 69
2 0 5 76
3 1 6 2 12
…
33
Memory Experiment Mediator
Model
Imagery
a
b
M
Rehearsal
Group
X
c’
Words
Recalled
Y
34
Memory Experiment: Imagery Mediator
Imagery
3.558 (.674)
M
Group
X
.614 (.218)
Recall
.569 (1.319)
Y
Mediated effect aˆbˆ = 2.185* (saˆbˆ =.880)
N=39, average M =6.38 (SD=2.93) Y =13.62 (SD=4.11)
35
Memory Experiment: Repetition Mediator
Repetition
-2.90* (.532)
M
Group
X
.050 (.300)
Recall
2.662+
(1.371)
Y
Mediated effect aˆbˆ = -.145 (saˆbˆ =.871)
36
Covariates
• Adjusted for Verbal GRE, Quantitative GRE, and Sex.
• Minimal change in coefficients.
Imagery Mediated effect = 2.177* (.944)
Repetition Mediated effect = -.360 (.762)
Evidence for Specificity of mediation because effect was
observed for imagery but not for repetition.
But there could be other confounders…
Note: Interaction of X and M was nonsignificant.
37
Two Mediator Model
 Straightforward extension of the single mediator
case to include both mediators in the same model.
 The product of coefficients methods is
straightforward for models with multiple
mediators but difference and causal step methods
can work, somewhat.
 Note the covariance between the two mediators,
estimated in the next figure, is not shown to make
the diagram easier to read.
38
Two Mediator Model
3.51**
(.673)
Imagery
M1
.702**
(.239)
1.32
(1.44)
INDEPENDENT
VARIABLE
X
Recall
Y
Repetition
-2.91**
(.532)
M2
.379
(.304)
39
Mediated Effect Measures
aˆ1bˆ1 = (3.505) (.702) = 2.46 for mediation through imagery and
aˆ bˆ
= (-2.91) (.379) = -1.101 for mediation through repetition. The total
mediated effect of aˆ1bˆ1 ( 2.46) plus aˆ bˆ (-1.101) equals 1.359 which is
equal to cˆ − cˆ' .
2 2
2 2
Theaˆ1bˆ1 mediated effect (s aˆ bˆ = .953 ) was statistically significant (z aˆ bˆ =
2.58 ) and the aˆ bˆ mediated effect (s aˆ bˆ = .912) was not (z aˆ bˆ = -1.21).
1 1
2 2
1 1
2 2
2 2
Imagery LCL=.593, UCL=4.04 and Repetition LCL=-2.89, UCL = .686
With covariates:
Mediated effect through imagery = 2.666* (1.016)
Mediated effect through repetition = -1.444 (.775)
40
Followup Experiments
Another question is, “What is the best next study or studies to
conduct after a statistical mediation analysis to test mediation
theory.”
1) Designs to address Consistency of the mediation relation.
2) Designs to address Specificity of the mediation relation.
MacKinnon, 2008; MacKinnon & Pirlott, 2010; 2012; related to
pattern matching (Shadish et al., 2002; Mark, 1986)
More detailed and different manipulations of
imagery with corresponding effects.
Different types of concrete words that differ in ease
of making images. Persons differ in imagery. 41
Replication Studies
• Conduct replications of the same experiment.
• Investigate the consistency of results across different
research participants.
• Conduct integrated data analysis, meta analysis.
• Analogous to Moderation of a mediated effect. The
moderator is research study. It is a nominal moderator.
• Data from eight different memory studies conducted
with different groups of participants.
42
Path Model for Mediation in Two
Studies.
c’study1
Study 1
Mediated
effect:
astd1bstd1
X
Study 2
Mediated
effect:
astd2bstd2
X
astudy1
astudy2
M
M
bstudy1
bstudy2
Y
Y
c’study2
43
Estimates of a, b, ab, and N
Obs
a
1
2
3
4
5
6
7
8
2.54500*
3.92420*
3.23813*
5.28333*
3.88333*
3.74561*
5.74825*
3.55833*
sa
0.84180
0.77710
0.50524
0.79539
0.76949
0.47772
0.60274
0.68940
b
sb
0.69580*
0.36685
0.65524*
0.49360
0.71480*
0.38900*
0.31689
0.61406 *
ab
0.15350
0.19522
0.12466
0.38804
0.20622
0.15988
0.35127
0.22613
1.77100*
1.43820
2.12175*
2.60785
2.77580*
1.45704*
1.82156
2.18503*
N
45
43
79
22
35
77
24
44
Ave 3.99
0.6824
0.5308
0.2256 2.022
44.4
*p<.05; complete cases
Average standardized effect size was .596.
Using average of a, sea, b, seb for distribution of the product, LCL=.338 and UCL=4.196
44
Causal Approaches
• Inverse Probability Weighting (Coffman, 2011; Robins et
al., 2000)
• Causal estimators and delta standard errors (Valeri and
VanderWeele, 2013)
• Plots of sensitivity to confounders. Also in traditional
mediation analysis but emphasized more in modern causal
inference approaches.
45
Causal Estimators
Natural Indirect Effect formulas (Pearl, 2001) and
standard errors derived via the multivariate delta
method. Analysis with the same three covariates.
Variable Indirect Effect (se)
Imagery 2.177
(1.023)
Repeat -.360
( .824)
LCL UCL
.171 4.18
-1.984 1.263
See Valeri & VanderWeele (2013).
46
Inverse Probability Weighting (IPW)
Method to adjust results for confounders.
Assumes no unmeasured confounding.
Weights observations as a way to deal with
confounding, missing data etc.
Here weights are used to adjust for confounding of
the M to Y relation; weights can differ for each
person.
Uses GENMOD to conduct weighted regression
analysis.
See Robins, Hernan, & Brumback (2000) and also
47
Coffman (2011).
Inverse Probability Weighting Results for
Imagery
from proc genmod
Standard 95% Confidence
Parameter Estimate Error
Limits
Z Pr > |Z|
Intercept 9.6814 1.3600 7.0158 12.3469 7.12 <.0001
exposure 1.7479 1.4291 -1.0530 4.5488 1.22 0.2213
mediator 0.5005 0.2434 0.0235 0.9775 2.06 0.0397
Weights ranged from .81 to 1.34
Imagery mediated effect = 1.90; LCL=.087 UCL=4.051
Repetition mediated effect = -.138; LCL = -1.664 UCL = 1.356
Coefficient was .5233 in unweighted analysis
48
Sensitivity Analysis for Confounding
Can be confounders of X to M and M to Y relation. Even
with randomization of X, there are possible confounders of
the M to Y relation.
Two types of confounding: (1) Measured and (2)
Unmeasured. If measures of confounders are available, you
can make statistical adjustments. What about confounders
that you have not measured?
Purpose of plots of confounder bias is to investigate how
much results would change for different effects of a
confounder. Methods refer to one confounder but it could
be viewed as a number of confounders with a single effect.
New area…
49
Directed Acyclic Graph
U
M
X
Y
50
Sensitivity Analysis for Confounding
How will results change with confounding of the M to Y
relation? X is randomized.
VanderWeele (2010), confounder effect on Y and difference in
proportions of the confounder between groups at level of M.
Imai et al. (2010), confounder effect as the correlation between
error terms.
Adaptation of Left Out Variables Error (LOVE; Mauro, 1990)
based on the correlation of a confounder with Y and the
correlation of a confounder with M (also in Imai et al. but as
r-squared).
Cox, et al. under review has examples and SAS and R
programs to make these plots.
51
Bias Correction for Memory Data
X was primary or secondary rehearsal, M was selfreported Imagery on a 1 to 9 scale and Y was the
number of words recalled.
γ -Corresponds to a binary confounder effect on the
number of words recalled. Perhaps a confounder like
knowing mnemonic techniques such as the pegword.
How large could that effect be? 6 words, 8 maximum.
δ-And how should the percentage of persons with versus
without this skill differ between groups at levels of M,
e.g., .05, .0, .2 maximum ? With randomization the
percentages should not be very different across groups.
(VanderWeele, 2010)
52
• Bias Plot Word Experiment
Mediated Effect Sensitivity to Confounding
VanderWeele(2010)
beta is relation of confounder U to Y
delta is difference proportion in U between groups
ab=2.185
Gamma
20
000
000
000
000
000
000
000
000
0000
0000
0000
0000
0000
00000
00000
00000
000000
000000
0000000
000000
00000000
00000000
000000000
000000000
0000000000
00000000000
000000000000
0000000000000
00000000000000
0000000000000000
0000000000000000000
00000000000000000000
00000000000000000000000
00000000000000000000000000
0000000000000000000000000000000
000000000000000000000000000000000000
000000000000000000000000000000000000000000
18
16
14
12
10
8
6
4
2
0
0.0
0.1
0.2
0.3
0.4
0.5
delta
rabnew
0
53
Imai et al., 2010
Based on Pearl’s natural and controlled effects describes
a method to conduct sensitivity analysis and has an R
program to make all the calculations. Possible to call R
from SAS but you need SAS 9.3.
The correlation between the error in the equation
predicting M and the error in equation predicting Y is
the measure of confounder bias.
An omitted variable that predicts both M and Y will lead
to a correlation between these residuals.
Program gives the correlation that makes aˆbˆ= δ(t)=0.
Dotted line is observed value of aˆbˆ ; ρ is the correlation
between residuals.
54
55
Left Out Variable Error Plots
This method is based on calculating how different
correlations for U with Y and U and M lead to different
values for the mediated effect.
The plot on the next page shows values where the
indirect effect will be zero for certain values for rMU and
rYU.
Adapted from Mauro (1980) based on early work on
consideration of confounding variables (James, 1980)
Still a work in progress...
56
Memory Data
57
Summary
 Mediation analysis is central to many disciplines because
it provides information about the causal mediating
mechanism between two variables.
 Mediation analysis is used to extract the most information
from a research study when measures of M are available
in addition to measures of X and Y.
 Programs of research address the many aspects of a
mediation relation with a variety of information, clinical
observations, qualitative data, experiments, and detailed
consideration of assumptions.
 Causal methods provide a general way to approach
mediation analysis for linear and nonlinear models
including sensitivity analysis.
58
Question
 How do we know that our statistical methods give the
correct answers?
 What other substantive mediation effects are considered
known? Cognitive Dissonance,…..
 When can we assume that there are no unmeasured
confounders?
 A man who carries a cat by the tail learns something
he can learn in no other way. Mark Twain
 A researcher who applies new statistical methods to real
data learns something she can learn in no other way. 59
Hypothesized Effects of Penn State
Mediation Analysis Presentation
Causal Challenges
of Mediation
Analysis
Traditional
Methods
Penn State
Entertainment
and Mediation
Analysis
Presentation
Causal Mediation
Analysis
Memory Example
For Validation
60
Thank You
61