A simple guide to Mediation
Lyytinen & Gaskin
Mediation and Multi-group Analyses
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Mediation
In an intervening variable model, variable X, is postulated to
exert an effect on an outcome variable, Y, through one or more
intervening variables called mediators (M)
mediational models advance an X M Y causal sequence, and seek to
illustrate the mechanisms through which X and Y are related.
(Mathieu & Taylor)
X
Y
M
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Mediation refers to causal relationships somewhat more complex
than the simple x predicts y relationship. When hypothesizing and
testing for mediation, a mediating variable, m, (click) plays some
role in the relationship between x and y. Thus we are hypothesizing
that the relationship between x and y is somewhat more complex than
a simple direct effect from IV to DV. More complex mediation can
also take place, for example, with two mediating variables, but
testing more complex mediation is not only beyond the scope of a
simple guide to mediation, there is also not currently an accepted
or agreed upon method for testing these types of mediation. So,
were going to keep it simple.
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Why Mediation?
Seeking a more accurate explanation of the causal effect the
antecedent (predictor) has on the DV (criterion , outcome) focus on
mechanisms that make causal chain possible
Missing variables in the causal chain
Intelligence Performance
Intelligence Work Effectiveness Performance
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So why even consider mediation? Why not just predict and test
direct effects? Often the reason is because we believe that some x
really does have an effect on some y, but we want to ensure that
there is not a more accurate explanation for this cause and effect
relationship. For example, lets say we want to predict what causes
people to either (Click) use an umbrella or not. We may say that
(click) rain causes this effect. However, this does not explain why
many people do not use an umbrella when it rains. Thus we can
explain more of the observed behavior in our sample if we include a
mediator called, (click) Desire to stay dry. Rain causes those who
wish to remain dry to use an umbrella when it rains, and those who
do not have this desire do not use an umbrella. So in this example
desire to stay dry mediates the effect rain has on using an
umbrella. Without the mediator, we can only explain the behavior of
those who use umbrellas in the rain; whereas, with the mediator, we
can also explain the behavior of those who do not use
umbrellas.
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Conditions for mediation
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(1) justify the causal order of variables including temporal
precedence;
(2) reasonably exclude the influence of outside factors;
(3) demonstrate acceptable construct validity of their
measures;
(4) articulate, a priori, the nature of the intervening effects
that they anticipate; and
(5) obtain a pattern of effects that are consistent with their
anticipated relationships while also disconfirming alternative
hypotheses through statistical tests.
Conditions for mediation
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Inferences of mediation are founded first and foremost in terms
of theory, research design, and the construct validity of measures
employed, and second in terms of statistical evidence of
relationships.
Mediation analysis requires:
1) inferences concerning mediational X MY relationships hinge on
the validity of the assertion that the relationships depicted
unfold in that sequence (Stone-Romero & Rosopa, 2004). As with
SEM, multiple qualitatively different models can be fit equally
well to the same covariance matrix. Using the exact same data, one
could as easily confirm a YMX mediational chain as one can an XMY
sequence (MacCallum, Wegener, Uchino, & Fabrigar, 1993).
Conditions for mediation
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2) experimental designs is to isolate and test, as best as
possible, XY relationships from competing sources of influence. In
mediational designs, however, this focus is extended to a three
phase XMY causal sequence requiring random assignments to both X
and M and related treatments
Because researchers may not be able to randomly assign
participants to conditions, the causal sequence of XMY is
vulnerable to any selection related threats to internal validity
(Cook & Campbell, 1979; Shadish et al., 2002). To the extent
that individuals status on a mediator or criterion variable may
alter their likelihood of experiencing a treatment, the implied
causal sequence may also be compromised. For example, consider a
typical: trainingself-efficacyperformance, mediational chain. If
participation in training is voluntary, and more efficacious people
are more likely to seek training, then the true sequence of events
may well be self-efficacytrainingperformance. If higher performing
employees develop greater self-efficacy (Bandura, 1986), then the
sequence could actually be performanceefficacytraining. If efficacy
and performance levels remain fairly stable over time, one could
easily misconstrue and find substantial support for the
trainingefficacyperformance sequence when the very reverse is
actually occurring. (Mathieu and Taylor 2006)
Conditions of mediation
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It is a hallmark of good theories that they articulate the how
and why variables are ordered in a particular way (e.g., Sutton
& Staw, 1995; Whetten, 1989). This is perhaps the only basis
for advancing a particular causal order in non-experimental studies
with simultaneous measurement of the antecedent, mediator, and
criterion variables (i.e., classic cross-sectional designs).
Implicitly, mediational designs advance a time-based model of
events whereby X occurs before M which in turn occurs before Y. It
is the temporal relationships of the underlying phenomena that are
at issue, not necessarily the timing of measurements
In other words, in mediation analyses, omitted variables
represent a significant threat to validity of the XM relationship
if they are related both to the antecedent and to the mediator, and
have a unique influence on the mediator. Likewise omitted variables
(and related paths) may lead to conclude falsely that no direct
effect XY exists, while in fact it holds in the population
Importance of theory Cause and effect
Performance
Self-efficacy
Training
Self-efficacy
Training
Performance
Training
Self-efficacy
Performance
Training
Performance
Self-efficacy
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Just as with other hypotheses, mediation suggests a cause and
effect relationship between variables. However, remember from
previous lessons that causation is not implied by correlation, and
statistically, the IVs, DVs, and mediating variables will likely be
correlated statistically. Thus it is critical to develop mediation
models based on theory first, then to test and support the
theoretical model through statistics. To illustrate this point,
consider the relationship shown. Self efficacy mediates the effect
training has on performance. In other words, training effects
performance through Self-efficacy. That makes sense.
Now consider this alternative model (click). Performance
mediates the effect Training has on Self-efficacy. Well, that also
makes sense, and it is supported by a statistical test. Now
consider these other models (click) (click), they also make sense
in certain contexts, and they are also supported by statistical
tests. So which model do we choose? Most likely, we have a certain
theory and model in mind prior to doing the statistical tests. Our
advice to you is to let this a priori theory guide your model
development, rather than first seeing what the statistics say, then
developing a model. See Mathieu and Taylor 2006 for a more in depth
discussion of this specific example.
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Types of Mediation
X
M
Y
Indirect Effect
M
X
Y
Partial Mediation
X
Y
M
Full Mediation
Significant Path
Insignificant Path
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Now, there are three main types of mediation. The first is
called indirect effect. Indirect effects predict no direct effect
from x to y while leaving the mediator out of the model. However, x
has a direct effect on the mediator, and the mediator has a direct
effect on y. Thus, x is said to have an indirect effect on y. This
hypothesis can only be supported if the direct effect of x on y is
insignificant, prior to testing for indirect effects. (Click) The
next type of mediation is called Partial Mediation. Partial
mediation predicts (click) significant direct (click) and indirect
effects from x to y. Thus the unmediated relationship is
significant, as well as the x to mediator and mediator to y
relationships. In order to avoid concluding that a partially
mediated effect is significant, when in fact, only the three direct
effects are individually significant, a significance test for
mediation must be performed. This will be explained more fully on
the next slide. (click) The third type of mediation is called Full
Mediation. Full mediation predicts that the direct effect of x on y
will be significant, only if the mediator is absent. (click) When
the mediator is present, this direct effect becomes (click)
insignificant (click), while the indirect effect is significant.
Lastly, if the x to mediator, and/or the mediator to y
relationships are insignificant, no mediation is taking place.
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More complex mediation structures
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X
M1
Y
Chain Model
M2
M3
X
M1
Y
M2
M3
Parallel Model
Hypothesizing Mediation
All types of mediation need to be explicitly and with good
theoretical reasons and logic hypothesized before testing them
Indirect Effect
You still need to assume and test that X has an indirect effect
on Y, though there is no effect in path XY
X has an indirect, positive effect on Y, through M.
Partial or Full
M partially/fully mediates the effect of X on Y.
The effect of X on Y is partially/fully mediated by M.
The effect of X on Y is partially/fully mediated by M1, M2,
& M3.
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Statistical evidence of relationships.
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Each type of mediation needs to be backed by appropriate
statistical analysis
Sometimes the analysis can be based on OLS, but in most cases it
needs to be backed by SEM based path analysis
There are four types of analyses to detect presence of mediation
relationships
Causal steps approach (Baron-Kenny 1986) (tests for significance
of different paths)
Difference in coefficients (evaluates the changes in
betas/coefficients and their significance when new paths are added
to the model)
Product of effect approach (tests for indirect effects a*b- this
always needs to be tested or evaluated using bootstrapping)
Sometimes evaluating differences in R squares
Statistical evidence of relationships
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Convergent validity is critical for mediation tests as this
forms the basis for reliability especially poor reliability of
mediator as to the extent that a mediator is measured with less
than perfect reliability, the MY relationship would likely be
underestimated, whereas the XY would likely be overestimated when
the antecedent and mediator are considered simultaneously (see
Baron & Kenny 1986)
Discriminant validity must be gauged in the context of the
larger nomological network within which the relationships being
considered are believed to reside. Discriminant validity does not
imply that measures of different constructs are uncorrelated the
issue is whether measures of different variables are so highly
correlated as to raise questions about whether they are assessing
different constructs. It is incumbent on researchers to demonstrate
that their measures of X, M, and Y evidence acceptable discriminant
validity before any mediational tests are justified.
Statistical evidence of relationships
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Statistical evidence of the relationships
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In simple partial mediation mx is the coefficient for X for
predicting M, and ym.x and yx.m are the coefficients predicting Y
from both M and X, respectively. Here yx.m is the direct effect of
X, whereas the product mx*ym quantifies the indirect effect of X on
Y through M. If all variables are observed then yx = yx.m + mx*ym
or mx*ym = yx - yx.m
Indirect effect is the amount by which two cases who differ by
one unit of X are expected to differ on Y through Xs effects on M,
which in turn affects Y
Direct effect part of the effect of X on Y that is independent
of the pathway through M
Similar logic can be applied to more complex situations
What would be the paths here?
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Statistical analysis
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The testing of the existence of the mediational effect depends
on the type of indirect effect
The lack of direct effect XY (yx is either zero or not
significant) is not a demonstration of the lack of mediated
effect
Therefore three different situations prevail (in this order)
The presence of a indirect effect (mx*ym is significant)
The presence of full mediation (yx is significant but yx.m is
not)
The presence of partial mediation (yx is significant and yx.m is
non zero and significant)
Testing for indirect effect
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Testing for full mediation
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Testing for partial mediation
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Observations of statistical analysis
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The key is to test for the presence of a significant indirect
effect just demonstrating the significant of paths yx, yx.m,mx.y,
and mx is not enough
One reason is that Type I testing of statistical significance of
paths is not based on inferences on indirect effects as products of
effects and their quantities
Can be done either using Sobel test (see e.g. www.quantpsy.org)
or bootstrapping
Sobel tests assumes normality of product terms and relatively
large sample sizes (>200)
Lacks power with small sample sizes or if the distribution is
not normal
Bootstrapping
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Bootstrapping (available in most statistical packages, or there
is additional code to accomplish it for most software packages)
Samples the distribution of the indirect effect by treating the
obtained sample of size n as a representation of the population as
a minitiature and then resampling randomly the sample with
replacement so that a sample size n is built by sampling cases from
the original sample by allowing any case once drawn to be thrown
back to be redrawn as the resample of size n is constructed
mx and ym and their product is estimated for each sample
recorded
The process is repeated for k times where k is large
(>1000)
Hence we have k estimates of the indirect effect and the
distribution functions as an empirical approximation of the
sampling distribution of the indirect effect when taking the sample
of size n from the original population
Specific upper and lower bound for confidence intervals are
established to find ith lowest and jst largest value in the ordered
rank of value estimates to reject the null hypothesis that the
indirect effect is zero with e.g. 95 level of confidence
Observations of statistical analysis
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In full and partial mediation bivariate XY (assessed via
correlation rYX or coefficient yx) must be nonzero in the
population if the effects of X on Y are mediated by M
Hence establishing a significant bivariate is conditional on
sample size
For example Assume that N=100 and sample correlations rXM=.30
and rMY =.30 and both would be significant at p
