Causality - Doing Data Science - page 286

Example: Z i is 1 if I smoked, 0 if I didn't (I am the unit). Y i 1 is 1 if I
got cancer and I smoked, and 0 if I smoked and didn't get cancer.
Similarly Y i 0 is 1 or 0, depending on whether I got cancer while not
smoking. The overall causal effect on me is the difference
Y i 1− Y i 0 . This is equal to 1 if I really got cancer because I smoked,
it's 0 if I got cancer (or didn't) independent of smoking, and it's -1 if I
avoided cancer by smoking. But I'll never know my actual value be‐
cause I only know one term out of the two.
On a population level, we do know how to infer that there are quite a
few “1"s among the population, but we will never be able to assign a
given individual that number .
This is sometimes called the fundamental problem of causal inference .
Visualizing Causality
We can represent the concepts of causal modeling using what is called
a causal graph .
Denote by W the set of all potential confounders. Note it's a big as‐
sumption that we can take account of all of them, and we will soon see
how unreasonable this seems to be in epidemiology research in the
next chapter.
In our example with Frank, we have singled out one thing as a potential
confounder—the woman he's interested in being beautiful—but if we
thought about it more we might come up with other confounders, such
as whether Frank is himself attractive, or whether he's desperate, both
of which affect how he writes to women as well as whether they re‐
spond positively to him.
Denote by A the treatment. In our case the treatment is Frank's using
the word “beautiful” in an introductory email. We usually assume this
to have a binary (0/1) status, so for a given woman Frank writes to,
we'd assign her a “1” if Frank uses the word “beautiful.” Just keep in
mind that if he says it's beautiful weather, we'd be measuring counting
that as a “1” even though we're thinking about him calling the woman
beautiful.
Denote by Y the binary (0/1) outcome. We'd have to make this well-
defined, so, for example, we can make sure Frank asks the women he
writes to for their phone number, and we could define a positive out‐
come, denoted by “1,” as Frank getting the number. We'd need to make
this as precise as possible, so, for example, we'd say it has to happen in