Moderator variable, Hayes, Process

1. Mediation
R22013
Prakriti Sinha

2. Mediation- An overview
• Mediator variable,
Intermediary variable,
Surrogate variable, and
Intermediate endpoint
• “lack of correlation does
not disprove causation”
and “correlation is
neither a necessary nor a
sufficient condition of
causality.” Bollen (1989)

3. Simple Mediation Model
Total Effect ->
(X → M1 → Y)
Direct Effect: c ′ estimates the difference between the
two group means holding M constant.
Indirect Effect: ab
a- quantifies how much two cases that differ by one unit
on X are estimated to differ on M
b- has an interpretation analogous to c ′ , except with M
as the antecedent. Two cases that differ by one unit on
M but that are equal on X are estimated to differ by b
units on Y.

4. Inference about the Indirect Effect of X on Y through M
Normal Theory Approach/ Sobel Test
• Assumes that the sampling distribution of the
indirect effect (ab) is normal.
• To conduct this test, you need an estimate of
the standard error of ab, which can be
calculated using standard errors of the
coefficients a and b.
• Once you have the standard error, you can
either perform a null hypothesis test to
determine whether the indirect effect is
different from zero or calculate a confidence
interval for the indirect effect.
LIMITATIONS- Firstly, it assumes a normal distribution of the sampling distribution of ab, which may not hold in
many empirical studies. Secondly, simulation studies have shown that this approach tends to have lower statistical
power and generates less accurate confidence intervals compared to alternative methods

5. Inference about the Indirect
Effect of X on Y through M
Bootstrap Method
• Treat the original sample as a representation
of the population.
• Bootstrapping generates multiple resamples by
randomly selecting observations from the original
data with replacement.
• By repeatedly calculating the statistic of interest
in each resample, an empirical distribution of the
statistic is constructed.
ADVANTAGES-
- First, it doesn't rely on the assumption of
normality,
- Second, it is particularly useful in small sample
sizes, where traditional methods may be less
reliable.
LIMITATIONS-
- It assumes that the original sample is
representative of the population, which may not
always be the case.
- The confidence intervals generated by
bootstrapping may vary slightly across runs

6. Baron and Kenny Approach
Causal Steps approach
Essential Criteria
• X is correlated to Y
• X is correlated to M
• M affects Y controlling for X.
The Direct Effect of X is compared to the Total Effect c. If c ′ is closer to zero than c and c ′ is not statistically significant,
then M is said to completely mediate X’s effect on Y.
But if c ′ is closer to zero than c but c ′ is statistically significant, then M partially mediates X’s effect on Y. Only part of
the effect of X on Y is carried through M.

7. Critiques the causal steps approach
• The first criticism is that the causal steps approach doesn't formally quantify the indirect effect.
Instead, it relies on a series of null hypothesis tests, which is unconventional in scientific research.
In contrast, other scientific inquiries rely on hypothesis tests or confidence intervals based on
direct quantifications.
• The second issue is that the causal steps approach requires multiple hypothesis tests, increasing
the likelihood of making errors and contravening the principle of using as few tests as needed.
• Third, it emphasizes qualitative claims about mediation, ignoring the principle of conveying
uncertainty. This neglects the advantages of quantifying indirect effects and constructing
confidence intervals.
• Lastly, the approach often demands a statistically significant total effect as a prerequisite for
testing indirect effects. This requirement is flawed because an indirect effect can exist even when
the total effect is statistically insignificant, leading to an under-analysis of data.

8. Problems in establishing causality using
mediation
• Epiphenomenal Association- M
may be correlated with some
other variable that X is actually
affecting, and if that other
variable causes Y rather than M,
one will go away with the
mistaken conclusion that X affects
Y indirectly through M when in
fact the other variable is the
mechanism variable through
which X exerts its effect indirectly.
• Confounding or Spurious
Association- if the association
between the variables can be
attributed to a third variable that
causally affects both.
In a mediation analysis, confounding and
epiphenomenal association due to C can be ruled out
by including C as a predictor in the models of M and Y

9. An example
IV- Economic Stress
DV- Withdraw from market
M- Emotion/ Feeling stressed
3 Control Variables
• Enterpreneur Self Effcacy
• Sex
• Tenure

10. Causal Order is hard to establish in mediation
• Cause must precede the Effect
• Experiments guarantee this precedence
• In mediation, there are several possibilities
(X → M → Y; X → Y → M; M → X → Y; M → Y → X; Y → X → M; Y → M → X).
Alternative ways of checking-
• Reversing causal pathways in a mediation model
• Piecing together an argument against competing causal orders
Only proper design that affords a clear causal interpretation for the direction of effects can solve this

11. Are the Effect size- Small, Medium or Large?
Partially Standardized Effect-
These effect sizes express the direct
and indirect effects relative to the
standard deviation of the outcome
variable Y, providing context by
comparing the effects to the
variability in Y.
Direct Effect ->
Indirect Effect ->
Total Effect ->

12. Are the Effect size- Small, Medium or Large?
Completely Standardized Effect-
Removing the scaling of X from the
partially standardized effect expresses
the direct and indirect effects in terms
of the difference in standard
deviations in Y between two cases
that differ by one standard deviation
in X. This yields the completely
standardized effect.
Direct Effect ->
Indirect Effect ->
Total Effect ->

13. Some (Problematic) Measures for Indirect Effects
Ratio of the Indirect Effect to the Total Effect.
PM is calculated as the ratio of the indirect effect (ab) to the total effect
(c). It quantifies the extent to which an independent variable X's effect
on the outcome Y operates indirectly through a mediator M. A higher PM
value suggests that a larger portion of the effect of X on Y is mediated
by M, while a lower PM value indicates a smaller indirect effect.
Proportion of Variance in Y Explained by the Indirect Effect.
It aims to quantify the proportion of variance in the outcome variable Y
that can be attributed to the indirect effect of the independent variable X
on Y through the mediator. M. R²med is calculated using squared correlations: r²MY (the
squared correlation between M and Y), r²XY (the squared correlation between X and Y), and R²Y.MX (the
squared multiple correlation estimating Y from both X and M).
Kappa-Squared.
To quantify the size of the indirect effect relative to its maximum
potential, κ² is calculated as the ratio of the indirect effect (ab) to its
maximum possible value in the given dataset (MAX(ab)). Unlike some
other effect size measures, κ² is bound between 0 and 1, making it a
true proportion.

14. Multiple X variables
Direct & Indirect Effect ->
Total Effect ->
When all k Xs are in the model simultaneously, the direct and
indirect effects of Xi are interpreted as the estimated difference
in Y between two cases differing by a unit on Xi but that are
equal on the other k − 1 Xi variables (or, rephrased, holding the
remaining k − 1 X variables constant, or controlling for those
variables). In other words, these represent the direct and
indirect effects of Xi on Y that are unique to Xi . As such, these
effects are interpreted just as they are when the remaining k −
1 X variables are conceptualized as statistical controls rather
than variables whose effects are substantively interesting

15. Multiple Y variables
A mediation model with k Y variables is displayed in the
form of a statistical diagram in Figure 4.6. A close
examination of this model shows that it is really just k
simple mediation models with a common X and M.
Because Yi is determined only by X and M, the direct
and indirect effects of X on Yi will be the same
regardless of whether they are estimated
simultaneously with the other k − 1 Y variables in the
model analytically or using k separate analyses, one for
each Y variable.

16. Parallel Multiple
Mediator Model
• The condition is that no mediator
causally influences another
• Specific Indirect effect- after controlling
for all other mediators in the model
Total Effect ->

17. Standard error Estimate
Pairwise Comparisons between
Specific Indirect Effects
• Whether one specific indirect effect is statistically different from another.
This comparison helps evaluate whether different mechanisms account
for varying portions of the effect of an independent variable (X) on a
dependent variable (Y).
1. Normal theory approach, involves calculating the difference between
specific indirect effects (aibi - ajbj) and dividing it by an estimate of its
standard error. This calculation yields a test statistic that can be
compared to the standard normal distribution to determine statistical
significance. Alternatively, a confidence interval for the difference can
be computed.
2. Bootstrapping, a resampling technique that approximates the
sampling distribution of the difference between specific indirect
effects. By repeatedly resampling the data and estimating the effects,
researchers can construct a bootstrap confidence interval. This interval
can then be used to assess whether the specific indirect effects are
statistically different.
• PROCESS, software can do both Use contrast=1 for the normal theory
approach or contrast=2 for the absolute value difference approach
(ignoring sign).

18. Serial Multiple Mediator
Model- 2 mediators
The assumption of no causal association
between two or more mediators is not
only relaxed, it is rejected outright a priori.
Total Effect ->
(X → M1 → Y; X → M2 → Y; and (X → M1 → M2 → Y)
c ′ - the estimated difference in Y between two cases that
differ by one unit on X but that are equal on all
mediators in the model.
Specific indirect effects - interpreted as the estimated
difference in Y between two cases that differ by one unit
on X through the causal sequence from X to mediator(s)
to Y.

19. Serial Multiple Mediator
Model – 3 mediators
The assumption of no causal association
between two or more mediators is not
only relaxed, it is rejected outright a priori.
Total Effect ->
(X → M1 → M2 → Y; X → M1 → M3 → Y; X → M2 →
M3 → Y) and (X → M1 → M2 → M3 → Y)
c ′ - the estimated difference in Y between two cases that
differ by one unit on X but that are equal on all
mediators in the model.
Specific indirect effects - interpreted as the estimated
difference in Y between two cases that differ by one unit
on X through the causal sequence from X to mediator(s)
to Y.

20. Models with Parallel and Serial Mediation Properties
Total Effect ->
Total Effect- > c’+ a1b1 + a2b2 + a3b3 + a1d21b2 + a1d31b3
Total Effect- > c’ + a1b1 + a2b2 + a3b3 + a4b4 + a1d41b4 +
a2d42b4 + a3d43b4

21. Complementarity and Competition among Mediators
• Specific Indirect Effects vs. Multiple Mediators: It's possible to find evidence of an
indirect effect of the independent variable (X) on the dependent variable (Y) through a
particular mediator (M1) when M1 is considered in isolation but not when it is included
in a model with multiple mediators (M2, M3, etc.). This can occur when the mediators
are correlated, as the paths from each mediator to the outcome are estimated while
controlling for all other mediators. Collinearity between mediators can lead to increased
sampling variance and more extensive confidence intervals, potentially making it harder
to detect specific indirect effects.
• Total Indirect Effect vs. Specific Indirect Effects: The total indirect effect quantifies how
X influences Y through all mediators together. Paradoxically, one might find situations
where specific indirect effects are individually significant, but the total indirect effect is
not statistically different from zero. This can happen when specific indirect effects have
opposite signs or when they are individually small but together contribute to a larger
total indirect effect.

22. • When X has more than 2 categories, various problematic approaches
in use are:-
• Discarding groups- Discard a group
• Combining subsets- Combined the two categories with viasuals
Mediation Analysis with a Multi-categorical
Antecedent
Test Only Test + Visual Elaborate Test + Visual
Test Only Test + Visual Elaborate Test + Visual

23. Mediation Analysis with a
Multi-categorical Antecedent
A multi-categorical antecedent variable
with g categories can be represented using
a dummy variable with g - 1 variations.
In mediation analysis with a multicategorical antecedent
variable X, there are g - 1 paths from X to the mediator M,
represented by the a coefficients. Each of these coefficients
captures a portion of the effect of X on M,
In the model of Y, there is only one regression coefficient (b) for
M, and it quantifies the effect of M on Y when controlling for X.
To assess whether M mediates the effect of X on Y, researchers
can examine the g - 1 relative indirect effects, defined as ajb,
where each captures a portion of the difference in Y between
groups resulting from X's influence on M. If at least one of
these relative indirect effects is significantly different from zero,
it suggests that M plays a mediating role in the relationship
between X and Y.

24. Thank You
Prakriti Sinha