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



Outline

1 Introduction

2 Structural Modelling

3 Mediation, moderation, confounding

4 Concluding remarks




Outline

1 Introduction







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.".




Outline

1 Introduction







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)




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




Outline

1 Introduction







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.




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





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.





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





(ii) Y⊥⊥Z | X i.e.

X

Z Y

HHHHH

HHHj?


In this case:X is confounding the relation Z → YZ is NOT mediating the relation X → Y





(iii) X⊥⊥Z i.e.

X

Y

Z

HHj

��*


In this case:X is NOT confounding the relation Z → YZ is NOT mediating the relation X → Y




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.




Outline

1 Introduction







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)




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




AknowledgementThe authors thank Vincent Yserbyt (U.C.L.) for interestingcomments




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.





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.





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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