A Structural Modelling Approach to Mediators, Moderators and Confounders a Counterfactual-Free Approach

Introduction Structural Modelling Mediation, moderation, confounding Concluding remarks

A structural modelling approach to mediators, 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, Belgium c Demography, Catholic University of Louvain, Belgium

January 23, 2013

Mouchart, Russo and Wunsch-Jan.2013 Structural Modelling 1 Introduction Structural Modelling Mediation, moderation, confounding Concluding remarks Outline

1 Introduction

2 Structural Modelling

3 Mediation, moderation, confounding

4 Concluding remarks

Mouchart, Russo and Wunsch-Jan.2013 Structural Modelling 2 Introduction Structural Modelling Mediation, moderation, confounding Concluding remarks Outline

1 Introduction

2 Structural Modelling

3 Mediation, moderation, confounding

4 Concluding remarks

Mouchart, Russo and Wunsch-Jan.2013 Structural Modelling 3 Introduction Structural Modelling Mediation, moderation, confounding Concluding remarks Introduction

Our previous papers: develop a structural modelling approach to causal analysis, i.e. establish causal relations by modelling structures.

This paper: present causal mediation analysis from a structural modelling point of view, i.e. determine the role of mediators and moderators in a causal structure.".

Mouchart, Russo and Wunsch-Jan.2013 Structural Modelling 4 Introduction Structural Modelling Mediation, moderation, confounding Concluding remarks Outline

1 Introduction

2 Structural Modelling

3 Mediation, moderation, confounding

4 Concluding remarks

Mouchart, Russo and Wunsch-Jan.2013 Structural Modelling 5 More speciﬁcally, let us consider X = (X1, ··· Xp). The joint distribution may be recursively decomposed as:

pX1,···Xp = pX1 pX2|X1 ··· pXp|X1,···Xp−1 (1)

Introduction Structural Modelling Mediation, moderation, confounding Concluding remarks Marginal-conditional decomposition

Explaining a multivariate, or complex, process means decomposing a complex mechanism in terms of an ordered sequence of simpler sub-mechanisms is most properly operated through a recursive decomposition of a multivariate distribution into a sequence of marginal and conditional distributions, each one representing a sub-mechanism of the global one.

Mouchart, Russo and Wunsch-Jan.2013 Structural Modelling 6 Introduction Structural Modelling Mediation, moderation, confounding Concluding remarks Marginal-conditional decomposition

More speciﬁcally, let us consider X = (X1, ··· Xp). The joint distribution may be recursively decomposed as:

pX1,···Xp = pX1 pX2|X1 ··· pXp|X1,···Xp−1 (1)

Mouchart, Russo and Wunsch-Jan.2013 Structural Modelling 7 Introduction Structural Modelling Mediation, moderation, confounding Concluding remarks The structural modelling approach

The structural modelling approach in a nutshell: the structurality of the statistical model is crucial, i.e. background knowledge invariance (or: stability) Causality is based on recursively decomposing a structural model into a sequence of sub-mechanisms: most systems of interest are of the multiple mechanisms type the mechanisms of interest are stochastic, represented by conditional distributions the effect of a cause is measured in terms of a variation of conditional distributions measuring the effect of a causing variable does not necessarily require the recourse to counterfactual concepts

Mouchart, Russo and Wunsch-Jan.2013 Structural Modelling 8 Introduction Structural Modelling Mediation, moderation, confounding Concluding remarks Outline

1 Introduction

2 Structural Modelling

3 Mediation, moderation, confounding

4 Concluding remarks

Mouchart, Russo and Wunsch-Jan.2013 Structural Modelling 9 Introduction Structural Modelling Mediation, moderation, confounding Concluding remarks Bridging the Structural Modelling Approach and Mediation Analysis

Basic Idea The classiﬁcation of variables into mediators, moderators or confounding variables refers to the “role - function” of a variable on the working of a mechanism or of a sub-mechanism.

We now examine the simplest case, namely the 3-variable one.

Mouchart, Russo and Wunsch-Jan.2013 Structural Modelling 10 Introduction Structural Modelling Mediation, moderation, confounding Concluding remarks Confounding-Mediating (1)

Let us consider a recursive decomposition of a 3-variate system:

pX,Z ,Y = pX · pZ|X · pY |X,Z (2) that may be represented by the directed acyclic graph (DAG) :

X H H HHa b H ? HH Hj- ZY c

Figure: A saturated 3-component completely recursive system

Mouchart, Russo and Wunsch-Jan.2013 Structural Modelling 11 Introduction Structural Modelling Mediation, moderation, confounding Concluding remarks Confounding-Mediating (2)

In this case: X is confounding the relation Z → Y Z is mediating the relation X → Y

Notice: If the labels on the arrows (i.e.a , b and c) stand for the coefﬁcients of the standardized regressions of Y on X, Z and of Z on X, Sewall Wright’s path analysis, in the 1920’s, leads to the “fundamental” relation a + bc = total effect of X on Y (3) which is the coefﬁcient the regression of Y on X, under a joint normality assumption, and therefore linear regression assumption. The structural modelling approach aims at enlarging this scope.

Mouchart, Russo and Wunsch-Jan.2013 Structural Modelling 12 Introduction Structural Modelling Mediation, moderation, confounding Concluding remarks Confounding-Mediating (3)

Consider the following “simplifying” hypotheses: (i) Y ⊥⊥X | Z i.e.

? ZY -

Figure: An unsaturated (1) 3-component completely recursive system

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

Mouchart, Russo and Wunsch-Jan.2013 Structural Modelling 13 Introduction Structural Modelling Mediation, moderation, confounding Concluding remarks Confounding-Mediating (4)

(ii) Y ⊥⊥Z | X i.e.

X HH H HH H ? HHj ZY

Figure: An unsaturated (2) 3-component completely recursive system

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

Mouchart, Russo and Wunsch-Jan.2013 Structural Modelling 14 Introduction Structural Modelling Mediation, moderation, confounding Concluding remarks Confounding-Mediating (5)

(iii) X⊥⊥Z i.e.

X HHj Y ¨¨* Z

Figure: An unsaturated (3) 3-component completely recursive system

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

Mouchart, Russo and Wunsch-Jan.2013 Structural Modelling 15 Introduction Structural Modelling Mediation, moderation, confounding Concluding remarks THUS : (i) The role of being “mediator” or “confounder” • depends not only on the recursive decomposition (i.e. on the identiﬁed 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 causing variable, say Z, may depend on the values of other causing variables, say X, and this: is a property of the conditional distribution pY |X,Z independently of the joint marginal distribution pX,Z is NOT representable in the DAG.

(iii) Moderation should be viewed in the framework of classifying different types of interaction.

Mouchart, Russo and Wunsch-Jan.2013 Structural Modelling 16 Introduction Structural Modelling Mediation, moderation, confounding Concluding remarks Outline

1 Introduction

2 Structural Modelling

3 Mediation, moderation, confounding

4 Concluding remarks

Mouchart, Russo and Wunsch-Jan.2013 Structural Modelling 17 Introduction Structural Modelling Mediation, moderation, confounding Concluding remarks Concluding remarks (1)

The structural modelling appproach, in short:

At the substantive level: decomposing a complex mechanism into an ordered sequence of sub-mechanisms, based on background knowledge and on invariance properties

At the statistical modelling level: recursive decomposition of a multivariate statistical model, often simpliﬁed by (tested) hypotheses (of, typically, conditional independences)

Mouchart, Russo and Wunsch-Jan.2013 Structural Modelling 18 Introduction Structural Modelling Mediation, moderation, confounding Concluding remarks Concluding remarks (2)

Implications for Mediation Analysis (MA)

MA should be based on a structural modelling rather than on empirical associations

When pY |X,Z represents the sub-mechanism of interest, MA involves 2 aspects:

analyzing and classifying the role -or function- of the explanatory variables, X and Z , and the properties of pY |X,Z viewed as a function of X and Z analyzing and classifying the role -or function- of the explanatory variables, X and Z , and the properties of pX,Z

Mouchart, Russo and Wunsch-Jan.2013 Structural Modelling 19 Introduction Structural Modelling Mediation, moderation, confounding Concluding remarks

Aknowledgement The authors thank Vincent Yserbyt (U.C.L.) for interesting comments

Mouchart, Russo and Wunsch-Jan.2013 Structural Modelling 20 Introduction Structural Modelling Mediation, moderation, confounding Concluding remarks 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 the sciences, Oxford University Press, 317-337.

MOUCHART M., F. RUSSOAND G.WUNSCH (2009), Structural modelling, exogeneity, and causality, Chap. 4 in Henriette Engelhardt, Hans-Peter Kohler, Alexia Prskawetz (eds), Causal Analysis in Population Studies: Concepts, Methods, Applications, Dordrecht: Springer, 59-82.

MOUCHART M., F. RUSSOAND G.WUNSCH (2010), Inferring Causal Relations by Modelling Structures, Statistica, LXX(4), 411-432.

Mouchart, Russo and Wunsch-Jan.2013 Structural Modelling 21 Introduction Structural Modelling Mediation, moderation, confounding Concluding remarks Selected bibliography

RUSSO F.(2009), Causality and Causal Modelling in the Social Sciences: Measuring Variations, Methodos Series Vol.5, Springer.

RUSSO F., G. WUNSCHAND M.MOUCHART (2011), Inferring Causality through Counterfactuals in Observational Studies: Some epistemological issues, Bulletin of Sociological Methodology/ Bulletin de Méthodologie Sociologique, 111, 43-64. DOI: 10.1177/0759106311408891.

Mouchart, Russo and Wunsch-Jan.2013 Structural Modelling 22 Introduction Structural Modelling Mediation, moderation, confounding Concluding remarks Selected bibliography

WUNSCH G. (1988), Causal Theory & Causal Modeling, Leuven University Press.

WUNSCH G. (2007), Confounding and control, Demographic Research, 16(4), 97-120. DOI: 10.4054/DemRes.2007.16.4

WUNSCH G.,M.MOUCHARTAND F. RUSSO (2012), Functions and mechanisms in structural-modelling explanation, submitted.

WUNSCH G., F. RUSSOAND M.MOUCHART (2010), Do we necessarily need longitudinal data to infer causal relations?, Bulletin of Sociological Methodology/ Bulletin de Méthodologie Sociologique, 106: 5-18, 2010. DOI: 10.1177/0759106309360114 On line version: http://bms.sagepub.com/cgi/content/abstract/106/1/5 Mouchart, Russo and Wunsch-Jan.2013 Structural Modelling 23 Introduction Structural Modelling Mediation, moderation, confounding Concluding remarks HAND WAVING

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Mouchart, Russo and Wunsch-Jan.2013 Structural Modelling 24