Biology Reference
In-Depth Information
causal relationships may differ as a result of the selection variables. A key feature of
a selection diagram is that the selection variables explain all of the differences
between the probability distributions in the model and target. Letting
P
and
P
*be
the probability distributions in the model and target, respectively, this means that
any
P
* probability is equal to the corresponding
P
probability conditional on the set
of selection variables,
S
. This is particularly important when a probabilistic causal
relationship can be estimated in the model but not in the target (e.g., because an
experimental intervention was performed in the former but not in the latter).
Causal
effects
are one important type of probabilistic causal claim. The causal effect of
X
upon
Y
is the probability distribution
Y
conditional on an intervention on
X
(see
Pearl
2000
, p. 70). Pearl uses the “do-operator,” written as “
do
(
x
),” to indicate that
the value of the variable
X
has been set by an intervention rather than passively
observed, so that the causal effect of
X
upon
Y
would be written as
P
(
y
|
do
(
x
)).
7
Thus, if Fig.
10.2
is the correct selection diagram,
P
ð
, that
is, the causal effect of
U
on
C
in the target is equal to the corresponding causal
effect in the model conditional on the selection variable
S
. Since selection variables
are assumed to be unmeasured, it may not be possible to estimate
P
c
j
do
ð
u
ÞÞ¼
P
ð
c
j
do
ð
u
Þ
;
s
Þ
Þ
directly from data drawn from the model population. Consequently, extrapolating a
causal effect from the model to target requires some means of reducing
P
ð
c
j
do
ð
u
Þ
;
s
Þ
to a formula in which
do
(
u
) and
s
never occur in the same probability. In the
subsequent section, we will consider an example of how this can work. Pearl and
Bareinboim's selection diagrams also include dashed-double-headed arrows to
represent the presence of unmeasured common causes (as in Fig.
10.2
).
8
As a result
of the unmeasured common cause of
U
and
E
, the causal effect of
U
upon
C
cannot
be identified from observational data in the target (see Pearl
2000
, p. 94).
I now use selection diagrams to define extrapolation and integration. The
definition of extrapolation and direct extrapolation mostly parallel those given in
Pearl and Bareinboim (
2011
, p. 9) but diverge from them in one important respect
that I explain below. The definition of integration is original and useful for the
thinking about the Donohue and Levitt study.
ð
c
j
do
ð
u
Þ
;
s
Definition 10.1 (Extrapolation). Let
*the
target characterized by the probability distributions
P
and
P*
, respectively, and
let
D
be a selection diagram relating
be the model population and
Π
Π
and
*. Then a causal relation
Rcanbe
Π
Π
extrapolated from
to
* if and only if
R
(
*) is identifiable given the conjunc-
Π
Π
Π
tion
R
(
),
P
,
P
*, and
D
.
Π
Extrapolation is
direct
when
R
(
*). When extrapolation is
direct, no modification or adjustment to the causal relationship estimated in
the model is needed; it transfers as is to the target. As will be illustrated in the
) is the same as
R
(
Π
Π
7
Here I follow the convention of having lower-case letter represent particular values of the
variables represented by the corresponding upper-case letters.
8
DAGs with double-headed arrows representing unmeasured common causes are known as semi-
Markovian models.
Search WWH ::
Custom Search