Mechanisms and Extrapolation in the Abortion-Crime Controversy - Mechanism and Causality in Biology and Economics - page 190

Fig. 10.4 A selection
diagram in which the causal
effect cannot be extrapolated

S 2

U E C

A S 1

Using the rules of probability, the second equation expands P ð c j do ð u Þ ; s Þ by

summing over A . The next equation eliminates the s from P

ð c j do ð u Þ ; a s Þ

, which is
justified by the Markov condition applied to the selection diagram. Now all that
needs to be done is to reduce P

;

to probabilities in which do ( u ) is absent,
which is easily done in this case as A is independent of U . 9 The final equation results
from a reapplication of the assumption that the selection variables account for all
differences between model and target. The right-hand side of the final equation,
then, consists of one probability, P

ð a j do ð u Þ ; s Þ

, that can be directly extrapolated
from model to target and another, P *( a ) that can be estimated using observational
data sampled from the target population. The right-hand side of the final equation is
an example of what Pearl and Bareinboim call a transport formula . A transport
formula specifies how a causal effect in the model can be adjusted so as to be
extrapolated to the target. In this example, then, P *( a ) is the only probability in the
transport formula that need be measured in the target.
However, whether extrapolation is possible depends on the selection diagram.
For example, consider the selection diagram resulting from adding a selection
variable pointing directly into E (as in Fig. 10.4 ). The presence of this selection
variable blocks the step from the second to third equations in the reasoning above,
because A does not d-separate S 2 from C . Indeed, P

ð c j do ð u Þ ; a Þ

cannot be extrapolated
from model to target given the selection diagram in Fig. 10.4 (see Appendix 2 for a
proof of this claim). Yet the harmful psychological effects represented by E would
plausibly be impacted by a variety of social, cultural, and economic factors that are
likely to vary from one place and time to another. As a result, it would be difficult to
justify the assumption that A or any other set of measured variables mediates all
selection variables relevant to E . In general, then, extrapolating a causal effect is
often very sensitive to difficult-to-justify assumptions about the absence of selec-
tion variables at crucial junctures in the selection diagram.
One way to deal with the problem of sensitivity to uncertain assumptions about
the selection diagram is to be less ambitious about what one wishes to extrapolate.
Causal effects are not the only type of causal claim that one might be interested in,
and extrapolating other sorts of claims may require less demanding assumptions
(see Steel 2008 , chapters 5 and 6). Examples include claims about positive or

ð c j do ð u ÞÞ

9 In the selection diagram in Fig. 10.3 , U is d-separated from A by the empty set, so P* ( a | do ( u ))
¼

P* ( a ) by rule 3 of Pearl's do-calculus.