Biology Reference
In-Depth Information
subsequent section, extrapolation need not be direct and can involve adjustments to
the causal relationship found in the model.
A few clarifications of Definition 10.1 are in order. First, for simplicity, Defini-
tion 10.1 is limited to the case in which there is only one model. The generalized
version of the definition allowing for multiple model populations
Π
n
would have corresponding probability distributions
P
1
through
P
n
, as well as
selection diagrams
D
1
through
D
n
(as each model could differ from the target its
own way). A further generalization of the definition would allow for distinct causal
relationships to be extrapolated from each model. The result of this generalization is
more properly regarded as a definition of
integration
rather than extrapolation,
since it involves the combination of a number of studies performed on several
populations.
Π
1
through
Definition 10.2 (Integration). Let
Π
n
be model populations
characterized, respectively, by the probability distributions
P
1
through
P
n
, and let
Π
through
Π
1
* be the target population, with probability distribution
P*
. Let
D
1
through
D
n
be
selection diagrams for the pairs
h
Π
1
,
*
i
through
h
Π
n
,
*
i
, respectively. Then
Π
Π
causal relations
R
1
(
Π
1
) through
R
n
(
Π
n
)
can be integrated to learn R
(
*) if and only
Π
if
R
(
*) is identifiable given conjunction of
R
1
(
Π
1
) through
R
n
(
Π
n
),
P
1
through
P
n
,
Π
P
*, and
D
1
through
D
n
.
This definition pertains to cases in which results from a number of studies of
disparate model populations are combined to infer a potentially new causal rela-
tionship in the target. For instance, Donohue and Levitt's scale-up model described
in Sect.
2
integrates a number of distinct results from studies performed in several
populations in order to form a rough estimate of a causal relationship not studied in
any of them, namely, the effect of legalized abortion in 1973 on crime in the 1990s
in the USA. Notice that Definition 10.1 is a special case of Definition 10.2 in which
there is only one model and the causal relationship to be inferred in the target is the
same one as that in the model.
A second clarification concerns the causal relation
R
. There are in fact several
types of causal relationships one might wish to extrapolate. Pearl and Bareinboim
(
2011
) focus on extrapolating causal effects. Steel (
2008
) is primarily concerned with
extrapolation of positive causal relevance, which is what is often the issue in cases
involving animal extrapolation, as when one wishes to know whether a chemical is a
human carcinogen. A causal effect is more informative than a claim about positive
causal relevance, and there are claims that fall between them in terms of specificity.
This is illustrated by Donohue and Levitt's extrapolation, from European studies, of
the claim that unwantedness doubles the chance of criminality later in life. In
addition, one might wish to extrapolate a claim about causal structure, for instance,
that socioeconomic status is a common cause of unwantedness and crime. The type of
claim at issue matters because more stringent background assumptions are typically
required for extrapolating more informative claims, a point which will be elaborated
more fully in the subsequent section.
The definition itself does not assume that the causal structures in the model
and target are represented by DAGs (with confounding arcs added) or that
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