Lyytinen & Gaskin Mediation and Multi-group Analyses
Slide 2
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 X Y Y M M 2 2
Slide 3
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 3 3
Slide 4
Conditions for mediation 4 4 (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.
Slide 5
Conditions for mediation 5 5 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 M Y 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 Y M X mediational chain as one can an X M
Y sequence (MacCallum, Wegener, Uchino, & Fabrigar, 1993).
Slide 6
Conditions for mediation 6 6 2) experimental designs is to
isolate and test, as best as possible, X Y relationships from
competing sources of influence. In mediational designs, however,
this focus is extended to a three phase X M Y 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 X M Y 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:
training self-efficacy performance, 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-efficacy training performance. If higher
performing employees develop greater self-efficacy (Bandura, 1986),
then the sequence could actually be performance efficacy training.
If efficacy and performance levels remain fairly stable over time,
one could easily misconstrue and find substantial support for the
training efficacy performance sequence when the very reverse is
actually occurring. (Mathieu and Taylor 2006)
Slide 7
Conditions of mediation 7 7 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 X M
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 X Y exists, while in fact it holds in the
population
Slide 8
Importance of theory Cause and effect Performance Self-efficacy
Training Self-efficacy Training Performance Training Self-efficacy
Performance Training Performance Self-efficacy 8 8
Slide 9
Types of Mediation X X M M Y Y Indirect Effect M M X X Y Y
Partial Mediation X X Y Y M M Full Mediation Significant Path
Insignificant Path 9 9
Slide 10
More complex mediation structures 10 X X M1M1 M1M1 Y Y Chain
Model M2M2 M2M2 M3M3 M3M3 X X M1M1 M1M1 Y Y M2M2 M2M2 M3M3 M3M3
Parallel Model
Slide 11
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 X Y 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 M 1, M 2, & M 3.
11
Slide 12
Statistical evidence of relationships. 12 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 1. Causal
steps approach (Baron-Kenny 1986) (tests for significance of
different paths) 2. Difference in coefficients (evaluates the
changes in betas/coefficients and their significance when new paths
are added to the model) 3. Product of effect approach (tests for
indirect effects a*b- this always needs to be tested or evaluated
using bootstrapping) 4. Sometimes evaluating differences in R
squares
Slide 13
Statistical evidence of relationships 13 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 M Y relationship would likely be underestimated,
whereas the X Y 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.
Slide 14
Statistical evidence of relationships 14
Slide 15
Statistical evidence of the relationships 15 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
Slide 16
What would be the paths here? 16
Slide 17
Statistical analysis 17 The testing of the existence of the
mediational effect depends on the type of indirect effect The lack
of direct effect X Y ( 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) 1. The presence of a
indirect effect ( mx * ym is significant) 2. The presence of full
mediation ( yx is significant but yx.m is not) 3. The presence of
partial mediation ( yx is significant and yx.m is non zero and
significant)
Slide 18
Testing for indirect effect 18
Slide 19
Testing for full mediation 19
Slide 20
Testing for partial mediation 20
Slide 21
Observations of statistical analysis 21 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 bootstrappingwww.quantpsy.org 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
Slide 22
Bootstrapping 22 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 i th lowest
and j st 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
Slide 23
Observations of statistical analysis 23 In full and partial
mediation bivariate X Y (assessed via correlation r YX 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 r XM =.30 and r MY =.30 and both
would be significant at p