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→Causal mediation analysis
* If both ''β''<sub>53</sub> in step 2 and ''β<sub>65</sub>'' in step 3 are significant, the moderator affects the relationship between the mediator and the dependent variable (path ''B'').
* If both ''β''<sub>53</sub> in step 2 and ''β<sub>65</sub>'' in step 3 are significant, the moderator affects the relationship between the mediator and the dependent variable (path ''B'').
* Either or both of the conditions above may be true.
* Either or both of the conditions above may be true.
==Causal mediation analysis==
===Fixing versus conditioning===
Mediation analysis quantifies the
extent to which a variable participates in the transmittance
of change from a cause to its effect. It is inherently a causal
notion, hence it cannot be defined in statistical terms. Traditionally,
however, the bulk of mediation analysis has been conducted
within the confines of linear regression, with statistical
terminology masking the causal character of the
relationships involved. This led to difficulties,
biases, and limitations that have been alleviated by
modern methods of causal analysis, based on causal diagrams
and counterfactual logic.
The source of these difficulties lies in defining mediation
in terms of changes induced by adding a third variables into
a regression equation. Such statistical changes are
epiphenomena which sometimes accompany mediation but,
in general, fail to capture the causal relationships that
mediation analysis aims to quantify.
The basic premise of the causal approach is that it is
not always appropriate to "control" for the mediator ''M''
when we seek to estimate the direct effect of ''X'' on ''Y''
(see the Figure above).
The classical rationale for "controlling" for ''M''"
is that, if we succeed in preventing ''M'' from changing, then
whatever changes we measure in Y are attributable solely
to variations in ''X'' and we are justified then in proclaiming the
effect observed as "direct effect of ''X'' on ''Y''." Unfortunately,
"controlling for ''M''" does not physically prevent ''M'' from changing;
it merely narrows the analyst's attention to cases
of equal ''M'' values. Moreover, the language of probability
theory does not possess the notation to express the idea
of "preventing ''M'' from changing" or "physically holding ''M'' constant".
The only operator probability provides is "Conditioning"
which is what we do when we "control" for ''M'',
or add ''M'' as a regressor in the equation for ''Y''.
The result is that, instead of physically holding ''M" constant
(say at ''M'' = ''m'') and comparing ''Y'' for units under ''X'' = 1' to those under
''X'' = 0, we allow ''M'' to vary but ignore all units except those in
which ''M'' achieves the value ''M'' = ''m''. These two operations are
fundamentally different, and yield different results,<ref>{{cite journal|last1=Robins|first1=J.M.|last2=Greenland|first2=S.|title=Identifiability and exchangeability for direct and indirect effects|journal=Epidemiology|date=1992|volume=3|issue=2|pages=143–155|doi=10.1097/00001648-199203000-00013|pmid=1576220}}</ref><ref name="pearl1994" >{{cite journal|last1=Pearl|first1=Judea|editor1-last=Lopez de Mantaras|editor1-first=R.|editor2-last=Poole|editor2-first=D.|title=A probabilistic calculus of actions|journal=Uncertainty in Artificial Intelligence 10|volume=1302|date=1994|pages=454–462|publisher=[[Morgan Kaufmann]]|location=San Mateo, CA|bibcode=2013arXiv1302.6835P|arxiv=1302.6835}}</ref> except in the case of no omitted variables.
To illustrate, assume that the error terms of ''M'' and ''Y''
are correlated. Under such conditions, the
structural coefficient ''B'' and ''A'' (between ''M'' and ''Y'' and between ''Y'' and ''X'')
can no longer be estimated by regressing ''Y'' on ''X'' and ''M''.
In fact, the regression slopes may both be nonzero
even when ''C'' is zero.<ref>{{cite journal|pmid=24885338|year=2014|last1=Pearl|first1=J|title=Interpretation and identification of causal mediation|journal=Psychological Methods|volume=19|issue=4|pages=459–81|doi=10.1037/a0036434|url=ftp://ftp.cs.ucla.edu/pub/stat_ser/r389-imai-etal-commentary-r421-reprint.pdf}}</ref> This has two
consequences. First, new strategies must be devised for
estimating the structural coefficients ''A,B'' and ''C''. Second,
the basic definitions of direct and indirect effects
must go beyond regression analysis, and should
invoke an operation that mimics "fixing ''M''",
rather than "conditioning on ''M''."
===Definitions===
Such an operator, denoted do(''M'' = ''m''), was defined in Pearl (1994)<ref name="pearl1994"/> and it operates by removing the equation of ''M'' and replacing it by a constant ''m''. For example, if the basic mediation model consists of the equations:
: <math> X=f(\varepsilon_1),~~M=g(X,\varepsilon_2),~~Y=h(X,M,\varepsilon_3) , </math>
then after applying the operator do(''M'' = ''m'') the model becomes:
: <math> X=f(\varepsilon_1),~~M=m,~~Y=h(X,m,\varepsilon_3) </math>
and after applying the operator do(''X'' = ''x'') the model becomes:
: <math>X=x, M=g(x, \varepsilon_2), Y=h(x,M,\varepsilon_3) </math>
where the functions ''f'' and ''g'', as well as the
distributions of the error terms ε<sub>1</sub> and ε<sub>3</sub> remain
unaltered. If we further rename the variables ''M'' and ''Y'' resulting from do(''X'' = ''x'')
as ''M''(''x'') and ''Y''(''x''), respectively, we obtain what
came to be known as "potential
outcomes"<ref>{{cite journal|last1=Rubin|first1=D.B.|title=Estimating causal effects of treatments in randomized and nonrandomized studies|journal=Journal of Educational Psychology|date=1974|volume=66|issue=5|pages=688–701|doi=10.1037/h0037350|url=https://semanticscholar.org/paper/545122e2990590524459ec9b59ccac6ce71e3b6a}}</ref> or "structural counterfactuals".<ref>{{cite journal|last1=Balke|first1=A.|last2=Pearl|first2=J.|editor1-last=Besnard|editor1-first=P.|editor2-last=Hanks|editor2-first=S.|title=Counterfactuals and Policy Analysis in Structural Models|journal=Uncertainty in Artificial Intelligence 11|volume=1302|date=1995|pages=11–18|publisher=[[Morgan Kaufmann]]|location=San Francisco, CA|bibcode=2013arXiv1302.4929B|arxiv=1302.4929}}</ref>
These new variables provide convenient notation
for defining direct and indirect effects. In particular,
four types of effects have been defined for the
transition from ''X'' = 0 to ''X'' = 1:
(a) Total effect –
: <math>TE = E [Y(1) - Y(0)] </math>
(b) Controlled direct effect -
: <math> CDE(m) = E [Y(1,m) - Y(0,m) ] </math>
(c) Natural direct effect -
: <math>NDE = E [Y(1,M(0)) - Y(0,M(0))] </math>
(d) Natural indirect effect
: <math> NIE = E [Y(0,M(1)) - Y(0,M(0))] </math>
Where ''E''[ ] stands for expectation taken over the error terms.
These effects have the following interpretations:
* ''TE'' measures the expected increase in the outcome ''Y'' as ''X'' changes from ''X=0'' to ''X'' =1'', while the mediator is allowed to track the change in ''X'' as dictated by the function ''M = g(X, ε<sub>2</sub>)''.
* CDE measures the expected increase in the outcome ''Y'' as ''X'' changes from ''X'' = 0 to ''X'' = 1, while the mediator is fixed at a pre-specified level ''M = m'' uniformly over the entire population
* ''NDE'' measures the expected increase in ''Y'' as ''X'' changes from ''X'' = 0 to ''X'' = 1, while setting the mediator variable to whatever value it ''would have obtained'' under ''X'' = 0, i.e., before the change.
* ''NIE'' measures the expected increase in ''Y'' when the ''X'' is held constant, at ''X'' = 1, and ''M'' changes to whatever value it would have attained (for each individual) under ''X'' = 1.
* The difference ''TE-NDE'' measures the extent to which mediation is ''necessary'' for explaining the effect, while the ''NIE'' measures the extent to which mediation is ''sufficient'' for sustaining it.
A controlled version of the indirect effect does not
exist because there is no way of disabling the
direct effect by fixing a variable to a constant.
According to these definitions the total effect can be decomposed as a sum
: <math>TE = NDE - NIE_r </math>
where ''NIE<sub>r</sub>'' stands for the reverse transition, from
''X'' = 1 to ''X'' = 0; it becomes additive in linear systems,
where reversal of transitions entails sign reversal.
The power of these definitions lies in their generality; they are applicable to models with arbitrary nonlinear interactions, arbitrary dependencies among the disturbances, and both continuous and categorical variables.
===The mediation formula===
[[File:Formulation of the indirect effect.png|thumb|Formulation of the indirect effect]]
In linear analysis, all effects are determined by sums
of products of structural coefficients, giving
: <math>
\begin{align}
TE & = C + AB \\
CDE(m) & = NDE = C, \text{ independent of } m\\
NIE & = AB.
\end{align}
</math>
Therefore, all effects are estimable whenever the model
is identified. In non-linear systems, more stringent
conditions are needed for estimating the
direct and indirect effects <ref name="Pearl-01"/><ref name="imai-etal-2010">{{cite journal|last1=Imai|first1=K.|last2=Keele|first2=L.|last3=Yamamoto|first3=T.|title=Identification, inference, and sensitivity analysis for causal mediation effects|journal=Statistical Science|date=2010|volume=25|issue=1|pages=51–71|doi=10.1214/10-sts321|arxiv=1011.1079|bibcode=2010arXiv1011.1079I}}</ref>
.<ref name="vanderweele-2009">{{cite journal|last1=VanderWeele|first1=T.J.|title=Marginal structural models for the estimation of direct and indirect effects|journal=Epidemiology|date=2009|volume=20|issue=1|pages=18–26|doi=10.1097/ede.0b013e31818f69ce|pmid=19234398}}</ref>
For example, if no confounding exists,
(i.e., ε<sub >1</sub>, ε<sub>2</sub>, and ε<sub>3</sub> are mutually independent) the
following formulas can be derived:<ref name="Pearl-01"/>
<!-- : <math> TE = \sum_m P(M=m) [E(Y\mid X=1, M=m) - E(Y\mid X=0, M=m) ]</math> -->
: <math>
\begin{align}
TE & = E(Y\mid X=1)- E(Y\mid X=0)\\
CDE(m) & = E(Y\mid X=1, M=m) - E(Y\mid X=0, M=m) \\
NDE & = \sum_m [E(Y|X=1, M=m) - E(Y\mid X=0, M=m) ] P(M=m\mid X=0) \\
NIE & = \sum_m [P(M=m\mid X=1) - P(M=m\mid X=0)] E(Y\mid X=0, M=m).
\end{align}
</math>
The last two equations are called ''Mediation Formulas'' <ref name="pearl-2009-r350">{{cite journal|last1=Pearl|first1=Judea|title=Causal inference in statistics: An overview|journal=Statistics Surveys|date=2009|volume=3|pages=96–146|url=http://ftp.cs.ucla.edu/pub/stat_ser/r350.pdf|doi=10.1214/09-ss057|doi-access=free}}</ref><ref name="vansteelandt-2012">{{cite journal|last1=Vansteelandt|first1=Stijn|last2=Bekaert|first2=Maarten|last3=Lange|first3=Theis|title=Imputation strategies for the estimation of natural direct and indirect effects|journal=Epidemiologic Methods|date=2012|volume=1|issue=1, Article 7|doi=10.1515/2161-962X.1014}}</ref><ref name="albert-2012">{{cite journal|last1=Albert|first1=Jeffrey|title=Distribution-Free Mediation Analysis for Nonlinear Models with Confounding|journal=Epidemiology|date=2012|volume=23|issue=6|pages=879–888|doi=10.1097/ede.0b013e31826c2bb9|pmid=23007042|pmc=3773310}}</ref>
and have become the target of estimation in many studies of mediation.<ref name="imai-etal-2010"/><ref name="vanderweele-2009"/><ref name="vansteelandt-2012"/><ref name="albert-2012"/> They give
distribution-free expressions for direct and indirect
effects and demonstrate that, despite the arbitrary nature of
the error distributions and the functions ''f'', ''g'', and ''h'',
mediated effects can nevertheless be estimated from data using
regression.
The analyses of ''moderated mediation''
and ''mediating moderators'' fall as special cases of the causal mediation
analysis, and the mediation formulas identify how various interactions coefficients contribute to the necessary and sufficient components of mediation.<ref name="vanderweele-2009"/><ref name="pearl-2009-r350"/>
[[File:Serial Mediation Model.png|thumb|A serial mediation model with two mediator variables.]]
===Example===
[[File:Parallel Mediation Model.pdf|thumb|A conceptual diagram that depicts a parallel mediation model with two mediator variables.]]
Assume the model takes the form
: <math>
\begin{align}
X & = \varepsilon_1 \\
M & = b_0 + b_1X + \varepsilon_2 \\
Y & = c_0 + c_1X + c_2M + c_3XM + \varepsilon_3
\end{align}
</math>
where the parameter <math>c_3</math> quantifies the degree to which ''M'' modifies the effect of ''X'' on ''Y''. Even when all parameters are estimated from data, it is still not obvious what combinations of parameters measure the direct and indirect effect of ''X'' on ''Y'', or, more practically, how to assess the fraction of the total effect <math>TE</math> that is ''explained'' by mediation and the fraction of <math>TE</math> that is ''owed'' to mediation. In linear analysis, the former fraction is captured by the product <math>b_1 c_2 / TE</math>, the latter by the difference <math>(TE - c_1)/TE</math>, and the two quantities coincide. In the presence of interaction, however, each fraction demands a separate analysis, as dictated by the Mediation Formula, which yields:
: <math>
\begin{align}
NDE & = c_1 + b_0 c_3 \\
NIE & = b_1 c_2 \\
TE & = c_1 + b_0 c_3 + b_1(c_2 + c_3) \\
& = NDE + NIE + b_1 c_3.
\end{align}
</math>
Thus, the fraction of output response for which mediation would be ''sufficient'' is
: <math> \frac{NIE}{TE} = \frac{b_1 c_2}{c_1 + b_0 c_3 + b_1 (c_2 + c_3)}, </math>
while the fraction for which mediation would be ''necessary'' is
: <math> 1- \frac{NDE}{TE} = \frac{b_1 (c_2 +c_3)}{c_1 + b_0c_3 + b_1 (c_2 + c_3)}. </math>
These fractions involve non-obvious combinations
of the model's parameters, and can be constructed
mechanically with the help of the Mediation Formula. Significantly, due to interaction, a direct effect can be sustained even when the parameter <math>c_1</math> vanishes and, moreover, a total effect can be sustained even when both the direct and indirect effects vanish. This illustrates that estimating parameters in isolation tells us little about the effect of mediation and, more generally, mediation and moderation are intertwined and cannot be assessed separately.
==Causal mediation analysis==
==Causal mediation analysis==