RECONSIDERING DEFINITIONS OF DIRECT AND
INDIRECT EFFECTS IN MEDIATION ANALYSIS
WITH A SOLUTION FOR A CONTINUOUS FACTOR
Ilya Novikov, Michal Benderly,
Laurence Freedman
Gertner Institute for Epidemiology and Health Policy Research, Tel
Hashomer, Israel
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Introduction
• Analysis of mediation is a part of causal
inferences in statistics
• During the last two decades the area was
substantially developed by Pearl, Robbins,
Greenland and others
• We present an implementation of the modified
Pearl’s approach for different situations
• The modification was proposed recently by
several authors but the implementation for
continuous factor is apparently new
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Mediation
Situation
- There are at least two factors (X, Z) affecting the outcome ( Y)
- There may be confounders ( C ) associated with X,Z, and Y
Causal Model
Factor X does not depend on Z and Y .
Mediator Z depends on X but not on Y
Outcome Y depends of X and Z
HOWEVER F(X|Z,Y,C)≠F(X|C), F(X,Z|Y,C) ≠F(X,Y|C)
Z
X
Y
C
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Effects – concept.
An effect is defined as the difference in the expectations
of the outcome in two situations.
Total effect of X on Y = effect on Y of changes in X
Direct effect of X on Y = effect on Y of changes in X while Z
remains unchanged
Indirect effect of X on Y =effect on Y of changes in Z, that
were induced by changes in X, while X remains unchanged
Z
X
Y
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Counterfactual approach. Binary factor
DATA: there are two data sets with X=0 and X=1 (red)
In order to estimate the direct and indirect effects we need two
unobservable data sets (gray)
Y(X=0,Z(0))
Y(X=0, Z(1))
Y(X=1,Z(0))
Y(X=1, Z(1))
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Effects – elaboration of the definitions.
What is the meaning of UNCHANGED? In comparison with what
situation?
Placing Z on causal path from X to Y provides the answer.
Direct effect: Z remains as it was at the initial values of X
Indirect effect: X remains at its new values
Y(X=0, Z(1))
Y(X=0,Z(0))
Direct
Y(X=1,Z(0))
Indirect
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Y(X=1, Z(1))
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Estimation of the effects. Linear model.
• For a linear model there is an exact solution. Let x,y,z are
continuous and all assumptions of linear regression are
fulfilled.
y=b0+b1*x+b2*z+e1
z=a0+a1*x + e2
e1,e2 – independent random errors, non-correlated with x
Then
Total effect = E(Y|X+1)-E(Y|X)= b1+b2*a1 (product formula)
Direct effect = E(Y|X+1,Z(X))-E(Y|X,Z(X))=b1
Indirect effect= E(Y|X+1,Z(X+1))-E(Y|X+1,Z(X))= b2*a1
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Problems with non-linear regression
• In non-linear situation (for example, for a binary outcome Y)
the product formula is not applicable.
• Various attempts to estimate indirect effect using coefficients
of non-linear regression were not commonly accepted
• The source of the problem is that the effect can not be
expressed using only regression coefficients but needs also
the distribution of the covariates
• The solution was found using a counterfactual approach
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Mediation formula
• Total effect=E(Y|X=1,Z(1))-E(Y|X=0,Z(0))
• Direct effect =E(Y|X=1,Z(0))-E(Y|X=0,Z(0))
• Indirect effect(Pearl)=E(Y|X=0,Z(1))-E(X=0,Z(0))
• Indirect effect (Modified)=
E(Y|X=1,Z(1))-E(Y|X=1,Z(0))
In general the total effect is not equal to the sum of the
direct and Pearl’s indirect effect
However the total effect is always equal to the sum of the
direct and modified indirect effect
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Estimation of the effects. Binary factor.
Since Y(X=1,Z(0)) does not exist, we estimate it using a
multiple imputation technique for missing values
• Z(0) for X=1 is estimated using regression Z(0) on C when
X=0
• Y(X=1,Z(0)) is estimated using regression Y on C,Z when X=1
Y(X=0,Z(0))
Direct
Y(X=1,Z(0))
Indirect
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Y(X=1, Z(1))
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Continuous factor. Definitions
• Data: For continuous X we have only one data set
• Requirement: in linear model it should lead to the exact
solution
• Definition (reduction to the binary situation): For each
object j, and “intervention” replaces x(j) by x’(j)=x(j)+1
Y(X,Z(X))
Direct
Y(X’,Z(X))
Indirect
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Y(X’,
Z(X’1))
Y(X’,Z(X’))
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Continuous factor. Estimation
• For each object “j” the potential outcome y’(j)=y(x+1,z(x)) is
estimated and imputed using the appropriate model y(x,z)
• For each object “j” the mediator z’(j)=z(x’) is estimated and
imputed using the appropriate model z(x)
• For each object “j” the potential outcome y’’(j)=y(x’,z(x’)) is
estimated and imputed using the appropriate model y(x,z)
Y(X,Z(X))
Direct
Y(X’,Z(X))
Indirect
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Y(X’, Z(X’))
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Realization. General logic
• We wrote a SAS macro that implements the general logic
described here and covers all 8 potential combinations of
binary or continuous types of variables X,Y,Z
• For binary X it assumes two data sets with X=0 and X=1
• For continuous X it uses one data set
• For imputation of Z the macro uses linear regression Z on X
and covariates C , and for binary Z it uses logistic regression
on X and C
• For imputation of Y the macro uses linear regression of Y on
X,Z and covariates C , and for binary Y it uses logistic
regression on X ,Z and C
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MACRO CALL (example)
%mediation8(data=all, ep=new_chf, mediator=totMIn,
predictor=CRPq, covbin=, covcont=age, RTF=y);
Thanks for your attention.
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