Transcript Document

Workshop on Statistical
Mediation and Moderation:
Statistical Mediation
Paul Jose
Victoria University of Wellington
27 March, 2008
SASP Conference
What do you want to know?
 Let’s briefly have each person state
what he or she would like to learn this
morning.
 Also, what is your level of statistical
knowledge/experience?
 Okay, let me tell you what I’m planning
to cover.
What am I doing today?
 I want to define mediation and
moderation
 How are they similar or different?
 Basic mediation and moderation
 Advanced mediation and moderation
 Questions and answers
Where does one start?
 I began to be interested in mediation and
moderation because I found that I was
increasingly using these approaches in
understanding “process” among variables.
 I found that there was little about these
techniques in traditional statistics textbooks—I
mostly obtained information through word-ofmouth.
 . . . and I was confused. I don’t like being
confused, so I did something about it. I educated
myself on these techniques. And now I can pass
on what I’ve learned. Let me list what I consider to
be the main sources of confusion.
Five major sources of confusion
1.
2.
3.
Moderation and mediation sound alike. It
makes it seem that they are very similar,
and or they derive from the same origin.
They are somewhat similar (cousins), but
they don’t come from the same place.
Second, statistics textbooks typically do not
do a very good job of explaining these two
approaches. Exception: Howell (2006).
Third, reports of moderation and mediation
in the research literature are not always
clear or accurately performed.
More confusion
4.
5.
Both are special cases of two separate
broad statistical approaches: mediation is a
special case of semi-partial correlations
(path modeling) and moderation is a special
case of statistical interactions (from
ANOVA). Both are included under GLM, but
this is not usually appreciated.
It’s not entirely clear what distinguishes a
moderating variable from a mediating
variable. Can one a priori define mediating
and moderating variables?
One last stumbling block
 Problem: there are no easily used statistics
programmes that compute mediation and
moderation. Can do analyses in SPSS and other
programmes that do regression, but there is no
graphing capability dedicated to either mediation or
moderation (except ModGraph and MedGraph).
 What we have here is a case of the users “getting
ahead” of the statisticians in the sense that
researchers frequently use mediation and moderation
but many statisticians aren’t even familiar with the
terms.
Background and history
 Most people’s awareness of this area comes from
this article:
Baron, Reuben M. & Kenny, David A.
(1986). The moderator-mediator variable distinction
in social psychological research: Conceptual,
strategic, and statistical considerations.
Journal of Personality and Social Psychology. Vol
51(6), pp. 1173-1182.
 Cited about 6,500 times by PsychInfo’s count. And
that’s just in Psychology.
 Most people are unclear about what they said and
how to perform the techniques.
Let’s get started:
Similarities and differences
Similarities:
 They both involve three variables;
 You can use regression to compute both;
 You wish to see how a third variable affects a basic relationship
(IV to DV).
Differences:
 You create a product term in moderation; not in mediation;
 You don’t have to centre anything in mediation;
 Moderation can be used on concurrent or longitudinal data, but
mediation is best used on longitudinal data.
 Graphing is critical for moderation; helpful for mediation.
How do you know if you have a
moderator or a mediator?
What’s the diff?

Moderators tend to be variables that are relatively
immune to change over time (personality trait,
gender, ethnic group, etc.).
 Mediators tend to be variables that change in
relation to other variables (anxiety, helpfulness,
honesty, mood).
 However, there is a class of variables (e.g., coping
efforts/strategies) that might be examined in both
ways. These two categories are not mutually
exclusive.
So let’s focus on mediation first
 Definition: “A mediating variable is one which
specifies how (or the mechanism by which) a
given effect occurs between an independent
variable (IV) and a dependent variable (DV).”
(Holmbeck, 1997, p. 599).
 The question you wish to answer is whether the
effect of the IV on the DV is at least partially
mediated by a third variable (MV).
 You can answer this question with two
regressions (and a correlation matrix).
 Let’s consider a specific example.
An example from my research
Stressor
intensity
Depression
Rumination
The theories
 Susan Nolen-Hoeksema believes that an individual
who ruminates more ends up more depressed. X =>
Y. Notice that it’s a causal statement.
 I don’t disagree with her, but I think that this simple
effect should be embedded within the stress and
coping context.
 We know that stress leads to depression. The
question I want to ask is whether at least part of the
effect of stress on depression occurs because certain
individuals ruminate about stressful events, and this
rumination leads to depression.
The basic relationship
Stressor
intensity
Depression
One must have a significant correlation between the IV
and DV (in fact among all 3 variables). The essential
question is whether by adding a third variable, one can at
least partially explain the basic relationship. Let’s look at
some real data.
The two steps
Step 1
Stressor
intensity
.45***
Depression
Step 2
Stressor
intensity
.45***
Depression
(.29**)
.51***
.46***
Rumination
(.32**)
Baron & Kenny’s 4 criteria
1.
2.
3.
4.
IV to MV must be significant
IV to DV must be significant
MV to DV must be significant (when
entered with the IV)
“The effect of the IV on the DV must be
less in the third equation than the second.
Perfect mediation holds if the IV has no
effect when the mediator is controlled.”


“must be less” is measured with the Sobel
formula (see following pages)
“Perfect mediation” occurs when the original
relationship goes to zero. This never happens
in psychology. I have a proposal for how to deal
with this issue, presented below.
What changed?
 Note that the beta weight from IV to DV
changed: from .45 to .29.
 What does that tell us?
 According to Baron and Kenny (1986), if one
obtains a significant drop in beta for this
relationship, then one has obtained significant
mediation.
 How can one test whether this is significant or
not? (It is not simply whether it goes from
significant to non-significant.) One needs to
compute the Sobel’s test:
z-value = a*b/SQRT(b2*sa2 + a2*sb2)
Who ‘ya gonna call?
 Many people have been using a web-site
by Preacher and Leonardelli, and it’s quite
useful for computing the Sobel’s statistic:
http://www.psych.ku.edu/preacher/sobel/s
obel.htm
 Let me show you how to use the site. It is
generally very helpful.
 I have invented my own programme to do
what P & L’s site does, and MORE. Let’s
check it out too.
Preparatory work

Before we run off to use these, please know
that you have to obtain some statistical
information first:
1.
2.
3.
Compute a correlation matrix of the 3 variables;
Perform a multiple regression of the IV on the
mediating variable; and
Perform a multiple regression of the IV and
mediator on the DV (simultaneous inclusion).
Correlation matrix
Correlations
Stress total
Stress total
Rumination total
depression total
Pearson Correlation
Sig . (2-tailed)
N
Pearson Correlation
Sig . (2-tailed)
N
Pearson Correlation
Sig . (2-tailed)
N
**. Correlation is sig nificant at the 0.01 level (2-tailed).
1
186
.508**
.000
186
.449**
.000
186
Rumination
total
.508**
.000
186
1
186
.464**
.000
186
depression
total
.449**
.000
186
.464**
.000
186
1
186
Results from the two regressions
1st regression (Stress on Rumination):
B
7.501 (unstand regression coefficient)
se
.938 (standard error)
2nd regression (Stress, Rumination on Depression):
You select the B and se for the mediating variable
here:
B
.069
se
.016
new beta for Stress
.288
new beta for Rumination
.317
(P & L web-site needs the first four values.)
Okay, go to the programmes
 It is necessary to have written down the
pertinent statistical output, or to have
printed off the relevant sections.
 Can do both programmes on the
internet.
 If you’re away from the internet you can
download the Excel macro of MedGraph
and run it whenever you want.
MedGraph output
Type of Mediation
Partial
Sobel z-value
3.795737 significance .000147
Standardized coefficient of Stress on Depression
Direct:
.288
Indirect:
.161
.449***
Independent Variable:
Stress
Outcome Variable:
Depression
(.288***)
.464***
.508***
(.317***)
Mediating Variable:
Rumination
Comparison of web-sites
 Preacher’s site has been around longer, it
allows variations on the Sobel formula, and
gives you an alternate way to compute the
Sobel’s t.
 My site results in a graphical presentation of
results, I think it’s harder to make mistakes
with my programme, and it has/will have
information about the type of mediation.
My criteria for type of mediation
 At present my programme stipulates:
 None: non-significant Sobel’s z-value
 Partial: significant Sobel’s and significant
basic relationship in the 2nd regression (IV to
DV)
 Full: significant Sobel’s and non-significant
basic relationship in the 2nd regression (IV to
DV)
 Dave Kenny argues against this (see his
web-site), and I tend to agree with him
now. My new approach is on the following
page.
What kind of mediation?
 None: non-significant Sobel’s z-value
 Partial: significant Sobel’s and ratio < .80.
(ratio is indirect/total; in this case it’s
.161/.449)
 Full: significant Sobel’s and ratio > .80
----------------------------------------------------- In the present case we have a significant
Sobel’s and ratio = .36. Thus, we have
partial mediation. Notice that I don’t use
the term “perfect mediation”. There is no
consensus on the partial/full mediation
issue.
Causal finding?
 Many researchers would be keen to argue
from this result that the experience of
stress leads to rumination, which in turn
partially leads to depressive symptoms,
i.e., a causal argument. Is this merited?
 Cole and Maxwell (2003) argue
strenuously that concurrent mediation
CANNOT support a causal statement.
They argue that few concurrent mediation
results actually turn out to hold up in
longitudinal data. What do they mean?
Shared and unique variance
Stress
Depressive
symptoms
Basic relationship is just a correlation between two variables.
Three variables: mediation
Direct effect
Stress
Depressive
symptoms
Indirect effect
Rumination
The green area indicates the degree of shared variance among
the three variables: that’s the size of the “indirect effect”. It is
hard to argue that these relationships are causal with these
data: they are the size of shared and unique variance.
Warnings!
 One must have all three correlations be significant before






launching this. K now suggests that 1st one may be optional.
Be sure that you do the regressions correctly, and that you are
taking the proper statistical information from the print-outs (B
vs. b).
Some people make causal arguments from these results. They
are shaky at best.
Types of specification error: 1) ordering of variables, 2)
variables with/without error, and 3) “third variable problem”
Longitudinal data are best.
Bootstrapping is best with small N samples.
Path models involving more than three variables is the general
case—don’t do a bunch of three-variable mediation analyses
when you can do one path model.
Specification error
 Major boogeyman in path model
analytic work: have you correctly
specified your model?
 Several issues here:
 Temporal
order of variables
 Variables measured with error
 Missing variable?
Why is your proposed model the
best?
Rumination
Stress intensity
Depressive
symptoms
There are exactly 6 combinations of any three variables—why is your
proposed model the best? Why not test all of them? I have, and in
the present case I find six instances of partial mediation. Which is
correct? They all tell us something useful about shared and unique variance.
Variables measured with error
 One can obtain biased estimates of the
indirect effect if the MV is measured with
significant error. (Same is true of the IV
and DV too, by the way.)
 Answer? Do mediation in a latent variable
path model in SEM. Possible but not easy.
Also, a lot of the times one doesn’t have a
sufficient N or multiple indicators of the
variables (3 indicators per variable).
Would look like this:
Latent variable path model
.30***
Stress intensity
(.20***)
.40***
Depression
.24***
Rumination
Indirect effect = .10; direct effect = .20; ratio = .33 (.36 in MR)
Missing variable?
 This is the old “third variable problem”, but
in this case we might wish to call it the
“fourth variable problem”.
 My student, Kirsty Weir, suggests that
anxiety/worry might “explain” the
relationship between rumination and
depression. Graph is on the following
page.
 One can never completely resolve this
question: include the likely candidates and
try to reject them.
The road from stress to depression
Note that the Rum to Dep path was removed because it was nonsignificant when we added the 4th variable (control). Is the 3-variable
mediation pattern wrong then?
Bootstrapping
 David MacKinnon and others have
argued that typical multiple
regression analysis is unbiased only
for large samples. (present case N =
575)
 They suggest:
 Large
sample: use MR
 Small sample: use bootstrapping
 What is bootstrapping?
Wave of the future
 Bootstrapping is a compilation of
regression results from many subsets of
the original dataset.
 The programme selects a subset of the
data (e.g., 50 from 100 participants), runs
the regression analysis, stores the result,
does it again and again up to a
predetermined number of times, and then
compiles the results of the repeated
analyses.
 Baron & Kenny didn’t mention this—
wasn’t used in 1986 very much at all. It is
performed now, but infrequently. It is the
wave of the future.
So how does one do this?
 If you toddle off to SPSS to do this, you
will be disappointed. Although it can
perform bootstrapping, it is not set up to
do mediation bootstrapping.
 Preacher and Hayes (see the Preacher
web-site on mediation) offers two different
macros: SAS and SPSS. Download it and
use it within SPSS. (not easy)
 Let’s look at the results of the SPSS
macro.
Macro output
Run MATRIX procedure:
DIRECT AND TOTAL EFFECTS
Coeff
s.e.
t
b(YX)
.3934 .0288 13.6685
b(MX)
1.0412 .0691 15.0779
b(YM.X) .1369 .0165 8.3200
b(YX.M) .2508 .0322 7.8002
Sig(two)
.0000
.0000
.0000
.0000
INDIRECT EFFECT AND SIGNIFICANCE USING NORMAL DISTRIBUTION
Value s.e. LL 95 CI UL 95 CI Z
Sig(two)
Sobel .1426 .0196 .1042 .1810
7.2723 .0000
BOOTSTRAP RESULTS FOR INDIRECT EFFECT
Mean s.e. LL 95 CI UL 95 CI LL 99 CI UL 99 CI
Effect .1434 .0239 .1001 .1939 .0879 .2113
SAMPLE SIZE
575
NUMBER OF BOOTSTRAP RESAMPLES
2000
It’s telling us that the indirect effect was significant—agrees with the
multiple regression result, but this is an unbiased estimate. (z = 3.80
before)
Mediation with longitudinal data
 . . . is very complicated but is very
illuminating.
 Much of structural equation modelling
(SEM) is devoted to trying to understand
mediational models.
 Path modelling with longitudinal data is
hard to do but will generate very
interesting and interpretable results.
 One should obtain the same variables at
different times of measurement to allow
residualisation.
Hierarchical multiple regression
Time 1
Time 2
Rum
2nd step
Dep
1st
step
Dep
This is N-H’s hypothesis: Rum1 should explain unique
variance in Dep2 after Dep1 is entered, i.e., explaining new
variance in the residual.
Back to Venn diagrams,
but with a difference
Dep2
Dep1
Stability coefficient: typically medium to large. The purple
area is the residual variance. It represents the change in
depression over this time period. The overlapping area refers
to the stability of depression over this time period.
Does Rum1 predict any of the residual?
Dep1
Dep2
Rum1
The red area is the amount of variance in Dep2 explained by
Rum1, i.e., the degree to which Rum1 explains change in
depression over time.
So what’s the answer?

Perform a hierarchical regression:
IV
DV
.72***
1) Dep1
Dep2
2) Rum1
.05ns
I found that N-H’s hypothesis was not
supported: Rum1 did not explain any of
the residual of Dep2 after Dep1 was
entered.
This is what it looks like
Dep1
Dep2
Rum1
Although Dep1 and Rum1 are significantly correlated, Rum1
doesn’t explain much new variance in Dep2 above and beyond
what Dep1 can do.
The other direction
IV
DV
.64***
1) Rum1
Rum2
.08*
2) Dep1
This result suggests that depression may
contribute to rumination over a 3-month
period of time, but not the other way
around.
It is recommended that you perform a path
analysis in SEM for this type of analysis:
allows for concurrent correlation (see next
page).
Two time points
Time 1
Time 2
Rum
Rum
Dep
Dep
SEM computes all of these relationships simultaneously,
allowing one to identify the unique relationships. Enact in
LISREL, EQS, AMOS, etc. What did I find?
Same basic results
Time 1
Rum
Time 2
.63***
Rum
.08*
.47**
Dep
.74***
.43**
Dep
But you get model fit indices, modification indices, and so
forth . . . I deleted the Rum1 to Dep2 path because it was nonsignificant.
Three time points and three variables
Time 1
Stress
Time 2
my
hypoth
Rum.
Stress
Stress
?
Rum.
Rum.
N-H
MR
Dep.
Time 3
Dep.
Dep.
SEM yielded this result
Stress
.74***
Stress
.59*** Stress
Rum.
.59***
.11*
Rum.
.61***
Rum.
.08*
Dep.
.72***
Dep.
.51***
Dep.
Powerful but hard to do
 Need to have three times of measurement
reasonably spread out so that stability
coefficients are not too high.
 Need to have good measures (small
measurement error) or do latent variable
longitudinal path modelling.
 This type of test of mediation is very
stringent because it occurs over time and
must be strong enough to exist against the
backdrop of the stability coefficients, i.e.,
these residualised effects explain change
in other variables.
Back to types of mediation
 Why do I think in terms of null, partial, and full
mediation?
 Because SEM-based path models yield those three
possible patterns.
 Sociological point: basic mediation (e.g., B&K) is
rooted in multiple regression where issues of model
specification are not salient. On the other hand, if you
learn SEM, then you will think in terms like I’ve
enunciated above. Confusions occur because of the
anachronisms in the field of mediation (harkening
back to MR rather than embracing path modelling).
Let’s bring mediation to a close
I’ve covered many powerful
techniques that derive from the
basic mediation paradigm.
 Remaining issues:

Logistic mediation
2. Mediation in other contexts: HLM
1.

Still much to learn and master, but
this is a good start.