A structural modelling approach to mediators, moderators and
Transcript of A structural modelling approach to mediators, moderators and
IntroductionStructural Modelling
Mediation, moderation, confoundingConcluding remarks
A structural modelling approach tomediators, moderators and confounders
A counterfactual-free approach
MICHEL MOUCHART a , FEDERICA RUSSO b AND
GUILLAUME WUNSCH c
a Institute of Statistics, Biostatistics and Actuarial sciences(ISBA), Catholic University of Louvain, Belgium
b Center Leo Apostel, Vrije Universiteit Brussel, Belgiumc Demography, Catholic University of Louvain, Belgium
January 23, 2013
Mouchart, Russo and Wunsch-Jan.2013 Structural Modelling 1
IntroductionStructural Modelling
Mediation, moderation, confoundingConcluding remarks
Outline
1 Introduction
2 Structural Modelling
3 Mediation, moderation, confounding
4 Concluding remarks
Mouchart, Russo and Wunsch-Jan.2013 Structural Modelling 2
IntroductionStructural Modelling
Mediation, moderation, confoundingConcluding remarks
Outline
1 Introduction
2 Structural Modelling
3 Mediation, moderation, confounding
4 Concluding remarks
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IntroductionStructural Modelling
Mediation, moderation, confoundingConcluding remarks
Introduction
Our previous papers: develop a structural modelling approachto causal analysis, i.e. establish causal relations by modellingstructures.
This paper: present causal mediation analysis from a structuralmodelling point of view, i.e. determine the role of mediators andmoderators in a causal structure.".
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IntroductionStructural Modelling
Mediation, moderation, confoundingConcluding remarks
Outline
1 Introduction
2 Structural Modelling
3 Mediation, moderation, confounding
4 Concluding remarks
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IntroductionStructural Modelling
Mediation, moderation, confoundingConcluding remarks
Marginal-conditional decomposition
Explaining a multivariate, or complex, processmeans decomposing a complex mechanism in terms of anordered sequence of simpler sub-mechanismsis most properly operated through a recursivedecomposition of a multivariate distribution into asequence of marginal and conditional distributions, eachone representing a sub-mechanism of the global one.
More specifically, let us consider X = (X1, · · ·Xp). The jointdistribution may be recursively decomposed as:
pX1,···Xp = pX1 pX2|X1· · · pXp|X1,···Xp−1
(1)
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IntroductionStructural Modelling
Mediation, moderation, confoundingConcluding remarks
Marginal-conditional decomposition
Explaining a multivariate, or complex, processmeans decomposing a complex mechanism in terms of anordered sequence of simpler sub-mechanismsis most properly operated through a recursivedecomposition of a multivariate distribution into asequence of marginal and conditional distributions, eachone representing a sub-mechanism of the global one.
More specifically, let us consider X = (X1, · · ·Xp). The jointdistribution may be recursively decomposed as:
pX1,···Xp = pX1 pX2|X1· · · pXp|X1,···Xp−1
(1)
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IntroductionStructural Modelling
Mediation, moderation, confoundingConcluding remarks
The structural modelling approach
The structural modelling approach in a nutshell:the structurality of the statistical model is crucial, i.e.
background knowledgeinvariance (or: stability)
Causality is based on recursively decomposing a structuralmodel into a sequence of sub-mechanisms: most systemsof interest are of the multiple mechanisms typethe mechanisms of interest are stochastic, represented byconditional distributionsthe effect of a cause is measured in terms of a variation ofconditional distributionsmeasuring the effect of a causing variable does notnecessarily require the recourse to counterfactual concepts
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IntroductionStructural Modelling
Mediation, moderation, confoundingConcluding remarks
Outline
1 Introduction
2 Structural Modelling
3 Mediation, moderation, confounding
4 Concluding remarks
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IntroductionStructural Modelling
Mediation, moderation, confoundingConcluding remarks
Bridging the Structural Modelling Approach andMediation Analysis
Basic IdeaThe classification of variables into mediators, moderators orconfounding variables refers to the “role - function” of a variableon the working of a mechanism or of a sub-mechanism.
We now examine the simplest case, namely the 3-variable one.
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IntroductionStructural Modelling
Mediation, moderation, confoundingConcluding remarks
Confounding-Mediating (1)
Let us consider a recursive decomposition of a 3-variatesystem:
pX ,Z ,Y = pX · pZ |X · pY |X ,Z (2)
that may be represented by the directed acyclic graph (DAG) :
X
Z Y
HHHH
HHHHj
a
?b
-c
Figure: A saturated 3-component completely recursive system
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IntroductionStructural Modelling
Mediation, moderation, confoundingConcluding remarks
Confounding-Mediating (2)
In this case:X is confounding the relation Z → YZ is mediating the relation X → Y
Notice: If the labels on the arrows (i.e. a,b and c) stand for thecoefficients of the standardized regressions of Y on X ,Z and ofZ on X , Sewall Wright’s path analysis, in the 1920’s, leads tothe “fundamental” relation
a + bc = total effect of X on Y (3)
which is the coefficient the regression of Y on X , under a jointnormality assumption, and therefore linear regressionassumption.The structural modelling approach aims at enlarging this scope.
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IntroductionStructural Modelling
Mediation, moderation, confoundingConcluding remarks
Confounding-Mediating (3)
Consider the following “simplifying” hypotheses:(i) Y⊥⊥X | Z i.e.
X
Z Y?
-
Figure: An unsaturated (1) 3-component completely recursive system
In this case:X is NOT confounding the relation Z → YZ is mediating the relation X → Y
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Mediation, moderation, confoundingConcluding remarks
Confounding-Mediating (4)
(ii) Y⊥⊥Z | X i.e.
X
Z Y
HHHHH
HHHj?
Figure: An unsaturated (2) 3-component completely recursive system
In this case:X is confounding the relation Z → YZ is NOT mediating the relation X → Y
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IntroductionStructural Modelling
Mediation, moderation, confoundingConcluding remarks
Confounding-Mediating (5)
(iii) X⊥⊥Z i.e.
X
Y
Z
HHj
��*
Figure: An unsaturated (3) 3-component completely recursive system
In this case:X is NOT confounding the relation Z → YZ is NOT mediating the relation X → Y
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IntroductionStructural Modelling
Mediation, moderation, confoundingConcluding remarks
THUS :(i) The role of being “mediator” or “confounder”• depends not only on the recursive decomposition (i.e. on theidentified sub-mechanisms)•• BUT also on the possible presence of “simplifyng”assumptions (i.e. on the working of these sub-mechanisms)
(ii) Interaction means that the effect on, say, Y , of a causingvariable, say Z , may depend on the values of other causingvariables, say X , and this:is a property of the conditional distribution pY |X ,Z independentlyof the joint marginal distribution pX ,Z
is NOT representable in the DAG.
(iii) Moderation should be viewed in the framework ofclassifying different types of interaction.
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IntroductionStructural Modelling
Mediation, moderation, confoundingConcluding remarks
Outline
1 Introduction
2 Structural Modelling
3 Mediation, moderation, confounding
4 Concluding remarks
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IntroductionStructural Modelling
Mediation, moderation, confoundingConcluding remarks
Concluding remarks (1)
The structural modelling appproach, in short:
At the substantive level: decomposing a complex mechanisminto an ordered sequence of sub-mechanisms, based onbackground knowledge and on invariance properties
At the statistical modelling level: recursive decomposition of amultivariate statistical model, often simplified by (tested)hypotheses (of, typically, conditional independences)
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Concluding remarks (2)
Implications for Mediation Analysis (MA)
MA should be based on a structural modelling rather than onempirical associations
When pY |X ,Z represents the sub-mechanism of interest, MAinvolves 2 aspects:
analyzing and classifying the role -or function- of the explanatoryvariables, X and Z , and the properties of pY |X ,Z viewed as afunction of X and Z
analyzing and classifying the role -or function- of the explanatoryvariables, X and Z , and the properties of pX ,Z
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Mediation, moderation, confoundingConcluding remarks
AknowledgementThe authors thank Vincent Yserbyt (U.C.L.) for interestingcomments
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Selected bibliography
MOUCHART M. AND F. RUSSO (2011), Causal explanation:recursive decompositions and mechanisms, chap. 15 in P.McKay Illari, F. Russo, and J. Williamson (eds), Causality in thesciences, Oxford University Press, 317-337.
MOUCHART M., F. RUSSO AND G. WUNSCH (2009), Structuralmodelling, exogeneity, and causality, Chap. 4 in HenrietteEngelhardt, Hans-Peter Kohler, Alexia Prskawetz (eds), CausalAnalysis in Population Studies: Concepts, Methods,Applications, Dordrecht: Springer, 59-82.
MOUCHART M., F. RUSSO AND G. WUNSCH (2010), InferringCausal Relations by Modelling Structures, Statistica, LXX(4),411-432.
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IntroductionStructural Modelling
Mediation, moderation, confoundingConcluding remarks
Selected bibliography
RUSSO F.(2009), Causality and Causal Modelling in the SocialSciences: Measuring Variations, Methodos Series Vol.5,Springer.
RUSSO F., G. WUNSCH AND M. MOUCHART (2011), InferringCausality through Counterfactuals in Observational Studies:Some epistemological issues, Bulletin of SociologicalMethodology/ Bulletin de Méthodologie Sociologique, 111,43-64.DOI: 10.1177/0759106311408891.
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IntroductionStructural Modelling
Mediation, moderation, confoundingConcluding remarks
Selected bibliography
WUNSCH G. (1988), Causal Theory & Causal Modeling,Leuven University Press.
WUNSCH G. (2007), Confounding and control, DemographicResearch, 16(4), 97-120. DOI: 10.4054/DemRes.2007.16.4
WUNSCH G., M. MOUCHART AND F. RUSSO (2012), Functionsand mechanisms in structural-modelling explanation, submitted.
WUNSCH G., F. RUSSO AND M. MOUCHART (2010), Do wenecessarily need longitudinal data to infer causal relations?,Bulletin of Sociological Methodology/ Bulletin de MéthodologieSociologique, 106: 5-18, 2010.DOI: 10.1177/0759106309360114On line version:http://bms.sagepub.com/cgi/content/abstract/106/1/5
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HAND WAVING
� Further work in progress...
Thank you for your attention... and comments!
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