1) Confounding occurs when a third variable is correlated with both the putative cause and outcome, making it difficult to isolate the impact of the cause on the outcome. 2) Controlling for confounding variables aims to evaluate the specific impact of a putative cause on an outcome, by keeping confounding variables constant. 3) From a structural modeling perspective, confounders can be seen as variables that influence the outcome through both direct and indirect paths, so accounting for all paths from confounders to outcomes is important.

1. CONFOUNDINGAND CONTROL IN A
MULTIVARIATE SYSTEM
AN ISSUE IN CAUSALATTRIBUTION
Federica RUSSO*, Michel MOUCHART** and Guillaume WUNSCH***
* Dipartimento di Studi Umanistici, Università degli Studi di Ferrara, Italy
**CORE and ISBA, University of Louvain, Belgium
***Demography, University of Louvain, Belgium

2. Confounding and Control (1)
• Given a correlation between two variables, X and Y, it is
possible that a third variable Z ‘confounds’ a putative
causal relation between X and Y insofar as it is correlated
with both X and Y.
• For example, one can observe a positive association between
income and health: the higher the income, the better the health.
• Though a higher income might indeed lead to better
health care, the association might also be due to the level
of education of the individuals concerned, as education is
correlated with both income and health.

3. Confounding and Control (2)
• In this elementary example, education ‘confounds’ the
relation between income and health; it is therefore
desirable to ‘control’ for education in order to isolate the
proper possible impact of income on health.
• Controlling for a confounding variable intends to define
and evaluate the specific impact of a putative cause on a
particular variable of interest, in a system with at least
three variables, of which one for the cause and another
one for the effect, the third variable being a possible
confounder of the cause and effect relation.

4. Confounding and Control (3)
• The distinction between a correlation due to a common
cause Z or a correlation due to a causal relation between
X and Y leads to the following definition of a confounder.
• A confounding variable or confounder is a variable that is
a common cause Z of both the putative cause X and its
outcome Y.
• In this sense, confounding requires a suitably defined
causal ordering of the variables.

5. Confounding and Control (4)
• In the framework of Directed Acyclic Graphs (DAGs),
Pearl (2000) has proposed a Back-Door Criterion in order
to determine the set Z of confounders to be controlled for.
• To control, in this context, means keeping Z constant, or
fixed, while examining the joint-variation between X and Y.

6. An ordered network of variables (1)
• We now reason on an ordered network of variables, and
we consider the simplest case, in the form of three
variables (X, Y, Z) where we specifically wish to evaluate
the impact of X on Y taking into account the action of Z.

7. An ordered network of variables (2a)
• Here, Z is a confounding
variable for the effect of X
on Y.
• Z intervenes in pY‫׀‬X,Z and
pX‫׀‬Z considered here as
two sub-mechanisms of
the mechanism (or DGP)
generating the data.
The saturated case : Z causes X and (Z,X) cause Y

8. An ordered network of variables (2b)
• In the present case, the direct effect
of X or of Z on Y is evaluated through
the parameters of the conditional
distribution Y‫׀‬X,Z. When interest
focuses on the effect of X on Y,
Z is a confounding variable that has
to be controlled for, in order to
determine the proper impact of X on
Y independently from Z.
• The status of confounder or not may
be affected by constraints or
simplifying assumptions imposed on
the saturated case, as we shall see.
The saturated case : Z causes X and (Z,X) cause Y

9. An ordered network of variables (3)
• In this case, the direct effect
of X or of Z is evaluated
through the conditional
distribution of Y‫׀‬Z, the latter
being a simplification of
pY‫׀‬X,Z given supplementary
condition Y ╨ X‫׀‬Z.
• The causal effect of X on Y is
null although X and Y are not
marginally independent.
• Z should be controlled for, in
order to examine the XY
relation independently from
the impact of Z.
A first unsaturated case: Y ╨ X ‫׀‬ Z

10. An ordered network of variables (4)
• In this case, the information
on X is sufficient for
predicting Y: adding
information on Z would not
improve the prediction on Y.
• Nevertheless, intervention
on Z would have an effect on
Y, mediated by the value of X
which should not be
controlled.
• Furthermore, Z should not
be controlled for, as
controlling for Z would freeze
a source of variation of X.
A second unsaturated case: Y ╨ Z ‫׀‬ X

11. An ordered network of variables (5)
• Once again, Z is not a
confounder anymore for the
relation between X and Y.
• Nevertheless, the effect of X
on Y is disturbed by the
impact of Z, because
variations in Y depend both
upon variations in X and
upon variations in Z.
• Z should therefore be
controlled for, in order to
detect more clearly the
specific impact of a variation
of X on Y (‘noise’ reduction).
A third unsaturated case: Z ╨ X

12. A structural modelling perspective (1)
• Mouchart, Russo and Wunsch (2010) identify three main
features of structural models:
• (i) a recursive decomposition of the joint distribution interpretable
as a sequence of sub-mechanisms, reflecting the causal ordering
of the variables;
• (ii) congruence with background knowledge;
• (iii) invariance or stability of the recursive decomposition across
changes of the environment.
• A structural model explains insofar as it represents the
mechanism of the DGP by a recursive decomposition
corresponding to the causal ordering of the variables.
This is an ideal goal, not always attainable in practice.

13. A structural modelling perspective (2)
• Consider again the
saturated case
• The DAG shows two
paths from Z to Y, a
direct path and an
indirect path mediated
through X.
• Issue of confounding
can be replaced by the
issue of incorporating all
relevant paths from the
confounder to the
outcome.

14. Conclusions
• From a structural modelling perspective, confounders can be
seen as variables that are at the origin of two or more different
paths leading to the outcome and may accordingly have both a
direct and an indirect effect on an outcome.
• Confounders do not appear per se in the system but are
subsumed within the larger framework of interrelations among
variables in the structural model.
• In a structural perspective, confounding is therefore a moot
issue because a structural model should incorporate the
multiple paths (e.g. direct and indirect) leading from the causes
to the outcome in the various sub-mechanisms, thus taking
possible confounders into account.

15. Selected readings
• Gaumé C. and Wunsch G. (2010), Self-rated Health in the Baltic Countries, 1994-1999,
European Journal of Population, 26(4): 435-457.
• Mouchart M. and Russo F. (2011), Causal explanation: recursive decompositions and
mechanisms, in P. McKay Illari, F. Russo, and J. Williamson (eds), Causality in the sciences,
Oxford University Press, Oxford, 317-337.
• Mouchart M. , F. Russo and G. Wunsch (2010), Inferring Causal Relations by Modelling
Structures, Statistica, LXX(4), 411-432.
• Pearl J. (2000; 2009), Causality, Cambridge University Press, Cambridge.
• Russo F.(2009), Causality and Causal Modelling in the Social Sciences: Measuring
Variations, Methodos Series Vol.5, Springer.
• Russo F., Mouchart M., Wunsch G. (2013). Confounding and control in a multivariate
system. An issue in causal attribution, Discussion Paper, ISBA, UcL (forthcoming).
• Wunsch G., Mouchart M. and Russo F. (in print), Functions and mechanisms in structural-
modelling explanations, Journal for General Philosophy of Science.