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DAVID SONTAG: So today's lecture
is going to be about causality.

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Who's heard about
causality before?

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Raise your hand.

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What's the number one thing
that you hear about when

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thinking about causality?

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Yeah?

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AUDIENCE: Correlation
does not imply causation.

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DAVID SONTAG: Correlation
does not imply causation.

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Anything else come to mind?

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That's what came to my mind.

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Anything else come to mind?

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So up until now in
the semester, we've

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been talking about purely
predictive questions.

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And for purely
predictive questions,

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one could argue that
correlation is good enough.

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If we have some
signs in our data

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that are predictive of
some outcome of interest,

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we want to be able to
take advantage of that.

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Whether it's
upstream, downstream,

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the causal directionality is
irrelevant for that purpose.

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Although even that
isn't quite true, right,

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because Pete and I have been
hinting throughout the semester

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that there are times
when the data changes

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on you, for example, when you go
from one institution to another

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or when you have non-stationary.

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And in those situations,
having a deeper understanding

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about the data might
allow one to build

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an additional robustness to
that type of data set shift.

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But there are other
reasons as well why

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understanding something about
your underlying data generating

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processes can be
really important.

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It's because often,
the questions

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that we want to answer when
it comes to health care

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are not predictive questions,
their causal questions.

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And so what I'll do now is I'll
walk through a few examples

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of what I mean by this.

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Let's start out with what we saw
in Lecture 4 and in Problem Set

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2, where we looked
at the question

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of how we can do early
detection of type 2 diabetes.

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You used Truven
MarketScan's data

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set to build a
risk stratification

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algorithm for
detecting who is going

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to be newly diagnosed
with diabetes one

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to three years from now.

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And if you think about
how one might then

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try to deploy that
algorithm, you

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might, for example, try to
get patients into the clinic

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to get them diagnosed.

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But the next set of
questions are usually

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about the so what question.

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What are you going to do
based on that prediction?

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Once diagnosed, how
will you intervene?

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And at the end of the
day, the interesting goal

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is not one of how do
you find them early,

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but how do you prevent them
from developing diabetes?

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Or how do you prevent the
patient from developing

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complications of diabetes?

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And those are questions
about causality.

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Now, when we built
a predictive model

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and we introspected
at the weight,

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we might have noticed
some interesting things.

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For example, if you looked at
the highest negative weights,

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which I'm not sure if we did
as part of the assignment

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but is something that I did
as part of my research study,

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you see that gastric
bypass surgery has

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the biggest negative weight.

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Does that mean that if you give
an obese person gastric bypass

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surgery, that will prevent
them from developing type 2

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diabetes?

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That's an example of a causal
question which is raised

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by this predictive model.

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But just by looking
at the weight

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alone, as I'll
show you this week,

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you won't be able to
correctly infer that there

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is a causal relationship.

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And so part of what
we will be doing

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is coming up with a mathematical
language for thinking

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about how does one
answer, is there

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a causal relationship here?

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Here's a second example.

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Right before spring break
we had a series of lectures

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about diagnosis,
particularly diagnosis

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from imaging data of
a variety of kinds,

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whether it be
radiology or pathology.

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And often, questions
are of this sort.

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Here is a woman's breasts.

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She has breast cancer.

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Maybe you have an associated
pathology slide as well.

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And you want to know what is
the risk of this person dying

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in the next five years.

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So one can take a
deep learning model,

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learn to predict
what one observes.

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So in the patient in your
data set, you have the input

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and you have, let's
say, survival time.

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And you might use that
to predict something

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about how long it takes
from diagnosis to death.

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And based on those predictions,
you might take actions.

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For example, if you predict
that a patient is not risky,

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then you might
conclude that they

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don't need to get treatment.

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But that could be
really, really dangerous,

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and I'll just give
you one example

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of why that could be dangerous.

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These predictive models,
if you're learning them

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in this way, the outcome,
in this case let's

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say time to death, is
going to be affected

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by what's happened in between.

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So, for example,
this patient might

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have been receiving
treatment, and because

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of them receiving treatment in
between the time from diagnosis

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to death, it might have
prolonged their life.

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And so for this patient
in your data set,

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you might have observed that
they lived a very long time.

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But if you ignore what
happens in between

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and you simply learn to predict
y from X, X being the input,

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then a new patient comes
along and you predicted

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that new patient is going
to survive a long time,

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and it would be completely
the wrong conclusion

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to say that you don't need
to treat that patient.

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Because, in fact, the only
reason the patients like them

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in the training data
lived a long time

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is because they were treated.

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And so when it comes to this
field of machine learning

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and health care, we need
to think really carefully

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about these types of questions
because an error in the way

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that we formalize our
problem could kill people

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because of mistakes like this.

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Now, other questions
are ones about not how

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do we predict
outcomes but how do we

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guide treatment decisions.

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So, for example, as
data from pathology

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gets richer and
richer and richer,

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we might think that we
can now use computers

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to try to better predict
who is likely to benefit

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from a treatment than
humans could do alone.

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But the challenge
with using algorithms

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to do that is that people
respond differently

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to treatment, and the
data which is being

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used to guide treatment is
biased based on existing

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treatment guidelines.

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So, similarly, to the previous
question, we could ask,

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what would happen if we trained
to predict past treatment

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decisions?

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This would be the
most naive way to try

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to use data to guide
treatment decisions.

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So maybe you see David
gets treatment A,

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John gets treatment B,
Juana gets treatment A.

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And you might ask then,
OK, a new patient comes in,

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what should this new
patient be treated with?

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And if you've just
learned a model

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to predict from what you
know about the treatment

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that David is likely
to get, then the best

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that you could hope
to do is to do as well

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as existing clinical practice.

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So if we want to go beyond
current clinical practice,

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for example, to recognize
that there is heterogeneity

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in treatment response, then
we have to somehow change

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the question that we're asking.

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I'll give you one
last example, which

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is perhaps a more traditional
question of, does X cause y?

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For example, does
smoking cause lung cancer

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is a major question of
societal importance.

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Now, you might be familiar
with the traditional way

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of trying to answer questions
of this nature, which

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would be to do a randomized
controlled trial.

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Except this isn't
exactly the type

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of setting where you could do
randomized controlled trials.

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How would you feel if you were
a smoker and someone came up

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to you and said, you have
to stop smoking because I

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need to see what happens?

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Or how would you feel
if you were a non-smoker

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and someone came
up to you and said,

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you have to start smoking?

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That would be both not feasible
and completely unethical.

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And so if we want to
try to answer questions

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like this from data,
we need to start

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thinking about
how can we design,

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using observational
data, ways of answering

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questions like this.

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And the challenge
is that there's

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going to be bias in the data
because of who decides to smoke

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and who decides not to smoke.

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So, for example,
the most naive way

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you might try to
answer this question

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would be to look at the
conditional likelihood

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of getting lung
cancer among smokers

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and getting lung cancer
among non-smokers.

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But those numbers, as you'll
see in the next few slides,

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can be very misleading
because there

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might be confounding
factors, factors

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that would, for example, both
cause people to be a smoker

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and cause them to
receive lung cancer,

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which would differentiate
between these two numbers.

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And we'll have a
very concrete example

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of this in just a few minutes.

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So to properly answer
all of these questions,

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one needs to be thinking
in terms of causal graphs.

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So rather than the
traditional setup in machine

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learning where you just
have inputs and outputs,

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now we need to have triplets.

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Rather than having
inputs and outputs,

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we need to be thinking
of inputs, interventions,

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and outcomes or outputs.

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So we now need be having
three quantities in mind.

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And we have to start
thinking about, well,

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what is the causal relationship
between these three?

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So for those of you who have
taken more graduate level

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machine learning
classes, you might

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be familiar with ideas
such as Bayesian networks.

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And when I went to
undergrad and grad school

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and I studied machine
learning, for the longest time

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I thought causal
inference had to do

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with learning causal graphs.

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So this is what I thought
causal inference was about.

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You have data of the
following nature--

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1, 0, 0, 1, dot, dot, dot.

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So here, there are
four random variables.

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I'm showing the realizations
of those four binary variables

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one per row, and you have
a data set like this.

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And I thought
causal inference had

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to do with taking data like
this and trying to figure out,

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is the underlying
Bayesian network

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that created that data, is it
X1 goes to X2 goes to X3 to X4?

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Or I'll say, this is X1,
that's X2, x3, and X4.

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Or maybe the causal graph
is X1, to X2, to X3, to x4.

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And trying to distinguish
between these different causal

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graphs from observational
data is one type of question

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that one can ask.

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And the one thing you learn
in traditional machine

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learning treatments of
this is that sometimes you

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can't distinguish between
these causal graphs

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from the data you have.

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For example, suppose you just
had two random variables.

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Because any distribution could
be represented by probability

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of X1 times probability
of X2 given X1,

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according to just rule of
conditional probability,

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and similarly, any
distribution can be represented

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as the opposite, probability
of X2 times probability

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of X1 given X2, which would
look like this, the statement

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that one would make
is that if you just

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had data involving X1 and
X2, you couldn't distinguish

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between these two causal graphs,
X1 causes X2 or X2 causes X1.

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And usually another
treatment would say, OK,

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but if you have a third variable
and you have a V structure

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or something like X1 goes
to x2, X1 goes to X3,

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this you could distinguish from,
let's say, a chain structure.

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And then the final
answer to what

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is causal inference
from this philosophy

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would be something like, OK, if
you're in a setting like this

250
00:12:54,750 --> 00:12:57,150
and you can't distinguish
between X1 causes X2 or X2

251
00:12:57,150 --> 00:12:59,970
causes X1, then you
do some interventions,

252
00:12:59,970 --> 00:13:04,440
like you intervene on X1 and you
look to see what happens to X2,

253
00:13:04,440 --> 00:13:07,290
and that'll help you disentangle
these directions of causality.

254
00:13:09,612 --> 00:13:12,070
None of this is what we're
going to be talking about today.

255
00:13:14,950 --> 00:13:18,100
Today, we're going to be
talking about the simplest,

256
00:13:18,100 --> 00:13:20,500
simplest possible setting
you could imagine,

257
00:13:20,500 --> 00:13:22,030
that graph shown up there.

258
00:13:25,470 --> 00:13:29,130
You have three sets of
random variables, X,

259
00:13:29,130 --> 00:13:30,810
which is perhaps
a vector, so it's

260
00:13:30,810 --> 00:13:33,570
high dimensional, a
single random variable

261
00:13:33,570 --> 00:13:37,170
T, and a single
random variable Y.

262
00:13:37,170 --> 00:13:40,500
And we know the
causal graph here.

263
00:13:40,500 --> 00:13:42,090
We're going to
suppose that we know

264
00:13:42,090 --> 00:13:49,650
the directionality, that we
know that X might cause T

265
00:13:49,650 --> 00:13:54,660
and X and T might cause Y. And
the only thing we don't know

266
00:13:54,660 --> 00:13:58,180
is the strength of the edges.

267
00:13:58,180 --> 00:13:58,680
All right.

268
00:13:58,680 --> 00:14:00,730
And so now let's try to
think through this in context

269
00:14:00,730 --> 00:14:01,772
of the previous examples.

270
00:14:01,772 --> 00:14:02,780
Yeah, question?

271
00:14:02,780 --> 00:14:05,697
AUDIENCE: Just to make sure-- so
T does not affect X in any way?

272
00:14:05,697 --> 00:14:07,530
DAVID SONTAG: Correct,
that's the assumption

273
00:14:07,530 --> 00:14:10,150
we're going to make here.

274
00:14:10,150 --> 00:14:12,480
So let's try to
instantiate this.

275
00:14:12,480 --> 00:14:13,855
So we'll start
with this example.

276
00:14:17,300 --> 00:14:23,340
X might be what you know about
the patient at diagnosis.

277
00:14:23,340 --> 00:14:27,940
T, I'm going to assume for
the purposes of today's class,

278
00:14:27,940 --> 00:14:32,468
is a decision between two
different treatment plans.

279
00:14:32,468 --> 00:14:34,510
And I'm going to simplify
the state of the world.

280
00:14:34,510 --> 00:14:36,940
I'm going to say those
treatment plans only

281
00:14:36,940 --> 00:14:42,070
depend on what you know about
the patient at diagnosis.

282
00:14:42,070 --> 00:14:44,290
So at diagnosis, you
decide, I'm going

283
00:14:44,290 --> 00:14:46,780
to be giving them this
sequence of treatments

284
00:14:46,780 --> 00:14:49,750
at this three-month interval
or this other sequence

285
00:14:49,750 --> 00:14:52,000
of treatment at, maybe,
that four-month interval.

286
00:14:52,000 --> 00:14:53,680
And you make that decision
just based on diagnosis

287
00:14:53,680 --> 00:14:55,930
and you don't change it based
on anything you observe.

288
00:14:58,060 --> 00:15:03,140
Then the causal graph
of relevance there is,

289
00:15:03,140 --> 00:15:06,080
based on what you know about
the patient at diagnosis,

290
00:15:06,080 --> 00:15:07,910
which I'm going to
say X is a vector

291
00:15:07,910 --> 00:15:09,740
because maybe it's
based on images,

292
00:15:09,740 --> 00:15:11,240
your whole electronic
health record.

293
00:15:11,240 --> 00:15:14,180
There's a ton of data you have
on the patient at diagnosis.

294
00:15:14,180 --> 00:15:17,300
Based on that, you
make some decision

295
00:15:17,300 --> 00:15:19,310
about a treatment plan.

296
00:15:19,310 --> 00:15:23,390
I'm going to call that
T. T could be binary,

297
00:15:23,390 --> 00:15:27,290
a choice between two treatments,
it could be continuous,

298
00:15:27,290 --> 00:15:29,780
maybe you're deciding the
dosage of the treatment,

299
00:15:29,780 --> 00:15:33,380
or it could be
maybe even a vector.

300
00:15:33,380 --> 00:15:35,150
For today's lecture,
I'm going to suppose

301
00:15:35,150 --> 00:15:38,930
that T is just binary,
just involves two choices.

302
00:15:38,930 --> 00:15:41,120
But most of what
I'll tell you about

303
00:15:41,120 --> 00:15:45,440
will generalize to the setting
where T is non-binary as well.

304
00:15:45,440 --> 00:15:48,200
But critically,
I'm going to make

305
00:15:48,200 --> 00:15:49,880
the assumption for
today's lecture

306
00:15:49,880 --> 00:15:52,620
that you're not observing
new things in between.

307
00:15:52,620 --> 00:15:56,630
So, for example, in this
whole week's lecture,

308
00:15:56,630 --> 00:15:59,850
the following scenario
will not happen.

309
00:15:59,850 --> 00:16:05,520
Based on diagnosis, you make a
decision about treatment plan.

310
00:16:05,520 --> 00:16:08,220
Treatment plan starts,
you got new observations.

311
00:16:08,220 --> 00:16:09,720
Based on those new
observations, you

312
00:16:09,720 --> 00:16:11,428
realize that treatment
plan isn't working

313
00:16:11,428 --> 00:16:14,310
and change to another
treatment plan, and so on.

314
00:16:14,310 --> 00:16:17,440
So that scenario goes
by a different name,

315
00:16:17,440 --> 00:16:18,930
which is called
dynamic treatment

316
00:16:18,930 --> 00:16:22,710
regimes or off-policy
reinforcement learning,

317
00:16:22,710 --> 00:16:24,620
and that we'll learn
about next week.

318
00:16:24,620 --> 00:16:27,000
So for today's and
Thursday's lecture,

319
00:16:27,000 --> 00:16:28,830
we're going to suppose
you base on what

320
00:16:28,830 --> 00:16:31,650
you know about the patient at
this time, you make a decision,

321
00:16:31,650 --> 00:16:34,650
you execute the decision,
and you look at some outcome.

322
00:16:34,650 --> 00:16:38,250
So X causes T, not
the other way around.

323
00:16:38,250 --> 00:16:42,060
And that's pretty clear
because of our prior knowledge

324
00:16:42,060 --> 00:16:43,440
about this problem.

325
00:16:43,440 --> 00:16:46,350
It's not that the
treatment affects

326
00:16:46,350 --> 00:16:49,280
what their diagnosis was.

327
00:16:49,280 --> 00:16:52,760
And then there's the outcome
Y, and there, again, we

328
00:16:52,760 --> 00:16:55,190
suppose the outcome, what
happens to the patient, maybe

329
00:16:55,190 --> 00:16:59,810
survival time, for example, is
a function of what treatment

330
00:16:59,810 --> 00:17:03,740
they're getting and
aspects about that patient.

331
00:17:03,740 --> 00:17:05,119
So this is the causal graph.

332
00:17:05,119 --> 00:17:06,746
We know it.

333
00:17:06,746 --> 00:17:08,329
But we don't know,
does that treatment

334
00:17:08,329 --> 00:17:09,680
do anything to this patient?

335
00:17:09,680 --> 00:17:12,670
For whom does this
treatment help the most?

336
00:17:12,670 --> 00:17:14,420
And those are the types
of questions we're

337
00:17:14,420 --> 00:17:15,628
going to try to answer today.

338
00:17:18,185 --> 00:17:19,060
Is the setting clear?

339
00:17:31,390 --> 00:17:32,320
OK.

340
00:17:32,320 --> 00:17:36,030
Now, these questions
are not new questions.

341
00:17:36,030 --> 00:17:39,360
They've been studied
for decades in fields

342
00:17:39,360 --> 00:17:43,770
such as political science,
economics, statistics,

343
00:17:43,770 --> 00:17:45,967
biostatistics.

344
00:17:45,967 --> 00:17:48,300
And the reason why they're
studied in those other fields

345
00:17:48,300 --> 00:17:51,420
is because often you don't
have the ability to intervene,

346
00:17:51,420 --> 00:17:53,880
and one has to try to
answer these questions

347
00:17:53,880 --> 00:17:56,020
from observational data.

348
00:17:56,020 --> 00:18:01,140
For example, you might ask, what
will happen to the US economy

349
00:18:01,140 --> 00:18:05,880
if the Federal Reserve raises
US interest rates by 1%?

350
00:18:07,922 --> 00:18:10,130
When's the last time you
heard of the Federal Reserve

351
00:18:10,130 --> 00:18:13,100
doing a randomized
controlled trial?

352
00:18:13,100 --> 00:18:15,630
And even if they had done a
randomized controlled trial,

353
00:18:15,630 --> 00:18:18,130
for example, flipped a coin to
decide which way the interest

354
00:18:18,130 --> 00:18:21,377
rates would go, it wouldn't
be comparable had they

355
00:18:21,377 --> 00:18:23,960
done that experiment today to
if they had done that experiment

356
00:18:23,960 --> 00:18:26,210
two years from now because
the state of the world

357
00:18:26,210 --> 00:18:28,050
has changed in those years.

358
00:18:31,010 --> 00:18:33,620
Let's talk about
political science.

359
00:18:33,620 --> 00:18:38,990
I have close colleagues of mine
at NYU who look at Twitter,

360
00:18:38,990 --> 00:18:41,910
and they want to
ask questions like,

361
00:18:41,910 --> 00:18:44,210
how can we influence
elections, or how

362
00:18:44,210 --> 00:18:46,670
are elections influenced?

363
00:18:46,670 --> 00:18:54,060
So you might look at some
unnamed actors, possibly

364
00:18:54,060 --> 00:18:56,460
people supported by the
Russian government, who

365
00:18:56,460 --> 00:19:00,660
are posting to Twitter
or their social media.

366
00:19:00,660 --> 00:19:03,300
And you might ask the
question of, well,

367
00:19:03,300 --> 00:19:05,490
did that actually
influence the outcome

368
00:19:05,490 --> 00:19:08,378
of the previous
presidential election?

369
00:19:08,378 --> 00:19:09,920
Again, in that
scenario, it's one of,

370
00:19:09,920 --> 00:19:11,990
well, we have this
data, something

371
00:19:11,990 --> 00:19:15,110
happened in the
world, and we'd like

372
00:19:15,110 --> 00:19:17,420
to understand what was
the effect of that action,

373
00:19:17,420 --> 00:19:22,510
but we can't exactly go back
and replay to do something else.

374
00:19:22,510 --> 00:19:24,352
So these are fundamental
questions that

375
00:19:24,352 --> 00:19:26,560
appear all across the
sciences, and of course they're

376
00:19:26,560 --> 00:19:28,130
extremely relevant
in health care,

377
00:19:28,130 --> 00:19:30,130
but yet, we don't teach
them in our introduction

378
00:19:30,130 --> 00:19:32,050
to machine learning classes.

379
00:19:32,050 --> 00:19:34,930
We don't teach them in our
undergraduate computer science

380
00:19:34,930 --> 00:19:36,460
education.

381
00:19:36,460 --> 00:19:38,708
And I view this as a major
hole in our education,

382
00:19:38,708 --> 00:19:40,500
which is why we're
spending two weeks on it

383
00:19:40,500 --> 00:19:42,640
in this course, which
is still not enough.

384
00:19:46,070 --> 00:19:48,550
But what has changed
between these fields,

385
00:19:48,550 --> 00:19:51,480
and what is relevant
in health care?

386
00:19:51,480 --> 00:19:54,100
Well, the traditional way
in which these questions

387
00:19:54,100 --> 00:19:56,410
were asked in
statistics were ones

388
00:19:56,410 --> 00:19:59,975
where you took a huge
amount of domain knowledge

389
00:19:59,975 --> 00:20:02,350
to, first of all, make sure
you're setting up the problem

390
00:20:02,350 --> 00:20:04,850
correctly, and that's always
going to be important.

391
00:20:04,850 --> 00:20:08,680
But then to think through what
are all of the factors that

392
00:20:08,680 --> 00:20:11,580
could influence the
treatment decisions

393
00:20:11,580 --> 00:20:14,770
called the confounding factors.

394
00:20:14,770 --> 00:20:16,420
And the traditional
approach is one

395
00:20:16,420 --> 00:20:19,157
would write down 10,
20 different things,

396
00:20:19,157 --> 00:20:21,240
and make sure that you do
some analysis, including

397
00:20:21,240 --> 00:20:24,040
the analysis I'll show you about
in today and Thursday's lecture

398
00:20:24,040 --> 00:20:27,500
using those 10 or 20 variables.

399
00:20:27,500 --> 00:20:30,120
But where this field
is going is one of now

400
00:20:30,120 --> 00:20:31,440
having high dimensional data.

401
00:20:31,440 --> 00:20:34,292
So I talked about how you
might have imaging data for X,

402
00:20:34,292 --> 00:20:36,750
you might have the whole entire
patient's electronic health

403
00:20:36,750 --> 00:20:38,540
record data facts.

404
00:20:38,540 --> 00:20:41,310
And the traditional approaches
that the statistics community

405
00:20:41,310 --> 00:20:46,020
used to work on no longer
work in this high dimensional

406
00:20:46,020 --> 00:20:46,657
setting.

407
00:20:46,657 --> 00:20:48,990
And so, in fact, it's actually
a really interesting area

408
00:20:48,990 --> 00:20:51,030
for research, one that my
lab is starting to work on

409
00:20:51,030 --> 00:20:53,160
and many other labs, where
we could ask, how can we

410
00:20:53,160 --> 00:20:56,490
bring machine learning
algorithms that are designed

411
00:20:56,490 --> 00:20:58,230
to work with high
dimensional data

412
00:20:58,230 --> 00:21:01,380
to answer these types of
causal inference questions?

413
00:21:01,380 --> 00:21:04,860
And in today's lecture, you'll
see one example of reduction

414
00:21:04,860 --> 00:21:08,363
from causal inference
to machine learning,

415
00:21:08,363 --> 00:21:09,780
where we'll be
able to use machine

416
00:21:09,780 --> 00:21:13,110
learning to answer one of those
causal inference questions.

417
00:21:16,500 --> 00:21:19,640
So the first thing we need
is some language in order

418
00:21:19,640 --> 00:21:23,240
to formalize these notions.

419
00:21:23,240 --> 00:21:26,230
So I will work within what's
known as the Rubin-Neyman

420
00:21:26,230 --> 00:21:30,030
Causal Model, where
we talk about what

421
00:21:30,030 --> 00:21:31,930
are called potential outcomes.

422
00:21:31,930 --> 00:21:36,130
What would have happened under
this world or that world?

423
00:21:36,130 --> 00:21:38,850
We'll call Y 0,
and often it will

424
00:21:38,850 --> 00:21:42,160
be denoted as Y underscore
0, sometimes it'll

425
00:21:42,160 --> 00:21:47,290
be denoted as Y parentheses
0, and sometimes it'll

426
00:21:47,290 --> 00:21:59,990
be denoted as Y given
X comma do Y equals 0.

427
00:21:59,990 --> 00:22:05,330
And all three of these
notations are equivalent.

428
00:22:05,330 --> 00:22:08,290
So Y is 0 corresponds
to what would

429
00:22:08,290 --> 00:22:10,730
have happened to this
individual if you gave them

430
00:22:10,730 --> 00:22:12,800
treatment to 0.

431
00:22:12,800 --> 00:22:15,447
And Y1 is the potential
outcome of what

432
00:22:15,447 --> 00:22:17,780
would have happened to this
individual had you gave them

433
00:22:17,780 --> 00:22:19,010
treatment one.

434
00:22:19,010 --> 00:22:23,340
So you could think about Y1
as being giving the blue pill

435
00:22:23,340 --> 00:22:25,070
and Y0 as being
given the red pill.

436
00:22:28,400 --> 00:22:32,870
Now, once you can talk about
these states of the world,

437
00:22:32,870 --> 00:22:34,550
then one could start
to ask questions

438
00:22:34,550 --> 00:22:38,850
of what's better, the red
pill or the blue pill?

439
00:22:38,850 --> 00:22:41,000
And one can formalize
that notion mathematically

440
00:22:41,000 --> 00:22:43,640
in terms of what's called the
conditional average treatment

441
00:22:43,640 --> 00:22:45,800
effect, and this
also goes by the name

442
00:22:45,800 --> 00:22:48,970
of individual treatment effect.

443
00:22:48,970 --> 00:22:51,210
So it's going to take
as input Xi, which

444
00:22:51,210 --> 00:22:54,050
I'm going to denote as
the data that you had

445
00:22:54,050 --> 00:22:56,100
at baseline for the individual.

446
00:22:56,100 --> 00:23:00,600
It's the covariance, the
features for the individual.

447
00:23:00,600 --> 00:23:05,010
And one wants to know, well,
for this individual with what

448
00:23:05,010 --> 00:23:07,770
we know about them, what's the
difference between giving them

449
00:23:07,770 --> 00:23:11,430
treatment one or giving
them treatment zero?

450
00:23:11,430 --> 00:23:13,740
So mathematically, that
corresponds to a difference

451
00:23:13,740 --> 00:23:14,700
in expectations.

452
00:23:14,700 --> 00:23:20,340
It's a difference in
expectation of Y1 from Y0.

453
00:23:20,340 --> 00:23:22,860
Now, the reason why I'm
calling this an expectation

454
00:23:22,860 --> 00:23:26,340
is because I'm not going to
assume that Y1 and Y0 are

455
00:23:26,340 --> 00:23:31,780
deterministic
because maybe there's

456
00:23:31,780 --> 00:23:33,370
some bad luck component.

457
00:23:33,370 --> 00:23:36,820
Like, maybe a medication usually
works for this type of person,

458
00:23:36,820 --> 00:23:41,788
but with a flip of a coin,
sometimes it doesn't work.

459
00:23:41,788 --> 00:23:43,330
And so that's the
randomness that I'm

460
00:23:43,330 --> 00:23:47,980
referring to when I talk about
probability over Y1 given Xi.

461
00:23:47,980 --> 00:23:50,320
And so the CATE looks
at the difference

462
00:23:50,320 --> 00:23:51,910
in those two expectations.

463
00:23:51,910 --> 00:23:55,300
And then one can now talk about
what the average treatment

464
00:23:55,300 --> 00:23:59,570
effect is, which is the
difference between those two.

465
00:23:59,570 --> 00:24:04,480
So the average treatment effect
is now the expectation of--

466
00:24:04,480 --> 00:24:09,100
I'll say the expectation of
the CATE over the distribution

467
00:24:09,100 --> 00:24:14,980
of people, P of X.
Now, we're going

468
00:24:14,980 --> 00:24:17,900
to go through this in four
different ways in the next 10

469
00:24:17,900 --> 00:24:20,270
minutes, and then you're
going to go over it five more

470
00:24:20,270 --> 00:24:21,770
ways doing your
homework assignment,

471
00:24:21,770 --> 00:24:25,340
and you'll go over it two more
ways on Friday in recitation.

472
00:24:25,340 --> 00:24:27,387
So if you don't get it
just yet, stay with me,

473
00:24:27,387 --> 00:24:28,970
you'll get it by the
end of this week.

474
00:24:33,300 --> 00:24:38,870
Now, in the data that you
observe for an individual,

475
00:24:38,870 --> 00:24:41,700
all you see is what happened
under one of the interventions.

476
00:24:41,700 --> 00:24:45,650
So, for example, if the i'th
individual in your data set

477
00:24:45,650 --> 00:24:49,670
received treatment Ti equals
1, then what you observe,

478
00:24:49,670 --> 00:24:53,702
Yi is the potential outcome Y1.

479
00:24:53,702 --> 00:24:55,910
On the other hand, if the
individual in your data set

480
00:24:55,910 --> 00:24:58,880
received treatment
Ti equals 0, then

481
00:24:58,880 --> 00:25:00,770
what you observed
for that individual

482
00:25:00,770 --> 00:25:04,130
is the potential outcome Y0.

483
00:25:04,130 --> 00:25:08,750
So that's the observed
factual outcome.

484
00:25:08,750 --> 00:25:11,930
But one could also talk
about the counterfactual

485
00:25:11,930 --> 00:25:14,450
of what would have
happened to this person had

486
00:25:14,450 --> 00:25:17,190
the opposite treatment
been done for them.

487
00:25:17,190 --> 00:25:22,460
Notice that I just swapped each
Ti for 1 minus Ti, and so on.

488
00:25:22,460 --> 00:25:26,380
Now, the key challenge in the
field is that in your data set,

489
00:25:26,380 --> 00:25:29,470
you only observe the
factual outcomes.

490
00:25:29,470 --> 00:25:33,370
And when you want to reason
about the counterfactual,

491
00:25:33,370 --> 00:25:36,970
that's where you have to impute
this unobserved counterfactual

492
00:25:36,970 --> 00:25:38,720
outcome.

493
00:25:38,720 --> 00:25:40,640
And that is known as
the fundamental problem

494
00:25:40,640 --> 00:25:42,110
of causal inference,
that we only

495
00:25:42,110 --> 00:25:44,820
observe one of the two outcomes
for any individual in the data

496
00:25:44,820 --> 00:25:46,340
set.

497
00:25:46,340 --> 00:25:49,070
So let's look at a
very simple example.

498
00:25:49,070 --> 00:25:50,630
Here, individuals
are characterized

499
00:25:50,630 --> 00:25:54,400
by just one feature, their age.

500
00:25:54,400 --> 00:25:58,240
And these two curves
that I'm showing you

501
00:25:58,240 --> 00:26:00,010
are the potential
outcomes of what

502
00:26:00,010 --> 00:26:03,070
would happen to this
individual's blood pressure

503
00:26:03,070 --> 00:26:04,783
if you gave them
treatment zero, which

504
00:26:04,783 --> 00:26:06,700
is the blue curve, versus
treatment one, which

505
00:26:06,700 --> 00:26:08,990
is the red curve.

506
00:26:08,990 --> 00:26:09,490
All right.

507
00:26:09,490 --> 00:26:12,000
So let's dig in a
little bit deeper.

508
00:26:12,000 --> 00:26:13,830
For the blue curve,
we see people

509
00:26:13,830 --> 00:26:22,220
who received the control, what
I'm calling treatment zero,

510
00:26:22,220 --> 00:26:26,380
their blood pressure
was pretty low

511
00:26:26,380 --> 00:26:28,890
for the individuals who
were low and for individuals

512
00:26:28,890 --> 00:26:30,930
whose age is high.

513
00:26:30,930 --> 00:26:35,250
But for middle age individuals,
their blood pressure

514
00:26:35,250 --> 00:26:40,050
on receiving treatment zero
is in the higher range.

515
00:26:40,050 --> 00:26:42,230
On the other hand,
for individuals

516
00:26:42,230 --> 00:26:44,760
who receive treatment
one, it's the red curve.

517
00:26:44,760 --> 00:26:47,940
So young people have much
higher, let's say, blood

518
00:26:47,940 --> 00:26:53,790
pressure under treatment one,
and, similarly, much older

519
00:26:53,790 --> 00:26:55,965
people.

520
00:26:55,965 --> 00:26:57,340
So then one could
ask, well, what

521
00:26:57,340 --> 00:26:59,757
about the difference between
these two potential outcomes?

522
00:26:59,757 --> 00:27:02,850
That is to say the CATE, the
Conditional Average Treatment

523
00:27:02,850 --> 00:27:06,810
Effect, is simply looking at the
distance between the blue curve

524
00:27:06,810 --> 00:27:08,910
and the red curve
for that individual.

525
00:27:08,910 --> 00:27:11,310
So for someone with
a specific age,

526
00:27:11,310 --> 00:27:14,640
let's say a young person
or a very old person,

527
00:27:14,640 --> 00:27:17,400
there's a very big difference
between giving treatment

528
00:27:17,400 --> 00:27:19,980
zero or giving treatment one.

529
00:27:19,980 --> 00:27:21,658
Whereas for a
middle aged person,

530
00:27:21,658 --> 00:27:22,950
there's very little difference.

531
00:27:22,950 --> 00:27:30,090
So, for example, if treatment
one was significantly cheaper

532
00:27:30,090 --> 00:27:32,890
than treatment zero,
then you might say,

533
00:27:32,890 --> 00:27:34,500
we'll give treatment one.

534
00:27:34,500 --> 00:27:37,020
Even though it's not quite
as good as treatment zero,

535
00:27:37,020 --> 00:27:39,660
but it's so much cheaper and
the difference between them

536
00:27:39,660 --> 00:27:43,187
is so small, we'll
give the other one.

537
00:27:43,187 --> 00:27:45,270
But in order to make that
type of policy decision,

538
00:27:45,270 --> 00:27:46,560
one, of course,
has to understand

539
00:27:46,560 --> 00:27:47,700
that conditional
average treatment

540
00:27:47,700 --> 00:27:49,700
effect for that individual,
and that's something

541
00:27:49,700 --> 00:27:53,252
that we're going to want
to predict using data.

542
00:27:53,252 --> 00:27:54,710
Now, we don't always
get the luxury

543
00:27:54,710 --> 00:27:57,440
of having personalized
treatment recommendations.

544
00:27:57,440 --> 00:28:01,230
Sometimes we have
to give a policy.

545
00:28:01,230 --> 00:28:02,538
Like, for example--

546
00:28:02,538 --> 00:28:04,080
I took this example
out of my slides,

547
00:28:04,080 --> 00:28:06,018
but I'll give it to you anyway.

548
00:28:06,018 --> 00:28:07,560
The federal government
might come out

549
00:28:07,560 --> 00:28:12,885
with a guideline saying that
all men over the age of 50--

550
00:28:12,885 --> 00:28:14,010
I'm making up that number--

551
00:28:14,010 --> 00:28:19,320
need to get annual
prostate cancer screening.

552
00:28:19,320 --> 00:28:24,550
That's an example of a
very broad policy decision.

553
00:28:24,550 --> 00:28:28,450
You might ask, well, what is
the effect of that policy now

554
00:28:28,450 --> 00:28:31,940
applied over the
full population on,

555
00:28:31,940 --> 00:28:35,410
let's say, decreasing deaths
due to prostate cancer?

556
00:28:35,410 --> 00:28:37,990
And that would be
an example of asking

557
00:28:37,990 --> 00:28:40,655
about the average
treatment effect.

558
00:28:40,655 --> 00:28:42,280
So if you were to
average the red line,

559
00:28:42,280 --> 00:28:43,750
if you were to
average the blue line,

560
00:28:43,750 --> 00:28:45,460
you get those two dotted
lines I show there.

561
00:28:45,460 --> 00:28:46,750
And if you look at the
difference between them,

562
00:28:46,750 --> 00:28:48,250
that is the average
treatment effect

563
00:28:48,250 --> 00:28:50,350
between giving the
red intervention

564
00:28:50,350 --> 00:28:52,400
or giving the blue intervention.

565
00:28:52,400 --> 00:28:56,830
And if the average human
effect is very positive,

566
00:28:56,830 --> 00:29:00,370
you might say that, on
average, this intervention

567
00:29:00,370 --> 00:29:01,767
is a good intervention.

568
00:29:01,767 --> 00:29:03,850
If it's very negative, you
might say the opposite.

569
00:29:06,670 --> 00:29:08,980
Now, the challenge about
doing causal inference

570
00:29:08,980 --> 00:29:11,330
from observational data
is that, of course,

571
00:29:11,330 --> 00:29:14,620
we don't observe those
red and those blue curves,

572
00:29:14,620 --> 00:29:18,788
rather what we observe are
data points that might be

573
00:29:18,788 --> 00:29:20,080
distributed all over the place.

574
00:29:20,080 --> 00:29:23,120
Like, for example,
in this example,

575
00:29:23,120 --> 00:29:26,537
the blue treatment happens
to be given in the data more

576
00:29:26,537 --> 00:29:28,120
to young people, and
the red treatment

577
00:29:28,120 --> 00:29:31,540
happens to be given in the
data more to older people.

578
00:29:31,540 --> 00:29:33,530
And that can happen for
a variety of reasons.

579
00:29:33,530 --> 00:29:37,030
It can happen due to
access to medication.

580
00:29:37,030 --> 00:29:39,520
It can happen for
socioeconomic reasons.

581
00:29:39,520 --> 00:29:43,990
It could happen because existing
treatment guidelines say

582
00:29:43,990 --> 00:29:46,710
that old people should
receive treatment one

583
00:29:46,710 --> 00:29:49,330
and young people should
receive treatment zero.

584
00:29:49,330 --> 00:29:52,600
These are all reasons why
in your data who receives

585
00:29:52,600 --> 00:29:56,240
what treatment could
be biased in some way.

586
00:29:56,240 --> 00:30:00,025
And that's exactly what this
edge from X to T is modeling.

587
00:30:02,960 --> 00:30:04,380
But for each of
those people, you

588
00:30:04,380 --> 00:30:06,660
might want to know, well, what
would have happened if they

589
00:30:06,660 --> 00:30:07,952
had gotten the other treatment?

590
00:30:07,952 --> 00:30:10,230
And that's asking about
the counterfactual.

591
00:30:10,230 --> 00:30:13,980
So these dotted circles
are the counterfactuals

592
00:30:13,980 --> 00:30:17,300
for each of those observations.

593
00:30:17,300 --> 00:30:19,740
And by the way, you'll notice
that those dots are not

594
00:30:19,740 --> 00:30:21,990
on the curves, and the reason
they're not on the curve

595
00:30:21,990 --> 00:30:23,407
is because I'm
trying to point out

596
00:30:23,407 --> 00:30:25,930
that there could be some
stochasticity in the outcome.

597
00:30:25,930 --> 00:30:30,210
So the dotted lines are the
expected potential outcomes

598
00:30:30,210 --> 00:30:32,245
and the circles are the
realizations of them.

599
00:30:34,920 --> 00:30:35,490
All right.

600
00:30:35,490 --> 00:30:40,460
Everyone take out a calculator
or your computer or your phone,

601
00:30:40,460 --> 00:30:41,550
and I'll take out mine.

602
00:30:45,470 --> 00:30:49,090
This is not an opportunity to go
on Facebook, just to be clear.

603
00:30:49,090 --> 00:30:50,862
All you want is a calculator.

604
00:30:54,972 --> 00:30:56,930
My phone doesn't-- oh,
OK, it has a calculator.

605
00:30:56,930 --> 00:30:58,500
Good.

606
00:30:58,500 --> 00:31:00,450
All right.

607
00:31:00,450 --> 00:31:02,624
So we're going to do
a little exercise.

608
00:31:05,410 --> 00:31:08,530
Here's a data set on
the left-hand side.

609
00:31:08,530 --> 00:31:10,540
Each row is an individual.

610
00:31:10,540 --> 00:31:13,720
We're observing the
individual's age, gender,

611
00:31:13,720 --> 00:31:15,340
whether they exercise
regularly, which

612
00:31:15,340 --> 00:31:17,800
I'll say is a one or a zero,
and what treatment they got,

613
00:31:17,800 --> 00:31:21,750
which is A or B. On
the far right-hand side

614
00:31:21,750 --> 00:31:28,200
are their observed sugar
glucose sugar levels, let's say,

615
00:31:28,200 --> 00:31:29,340
at the end of the year.

616
00:31:33,010 --> 00:31:37,960
Now, what we'd like to
have, it looks like this.

617
00:31:37,960 --> 00:31:42,460
So we'd like to know what would
have happened to this person's

618
00:31:42,460 --> 00:31:45,700
sugar levels had they
received medication A

619
00:31:45,700 --> 00:31:47,830
or had they received
medication B.

620
00:31:47,830 --> 00:31:52,630
But if you look at
the previous slide,

621
00:31:52,630 --> 00:31:56,560
we observed for each individual
that they got either A or B.

622
00:31:56,560 --> 00:31:58,480
And so we're only
going to know one

623
00:31:58,480 --> 00:32:00,920
of these columns
for each individual.

624
00:32:00,920 --> 00:32:03,430
So the first row, for
example, this individual

625
00:32:03,430 --> 00:32:05,980
received treatment
A, and so you'll

626
00:32:05,980 --> 00:32:11,650
see that I've taken
the observed sugar

627
00:32:11,650 --> 00:32:14,730
level for that individual,
and since they received

628
00:32:14,730 --> 00:32:17,860
treatment A, that
observed level represents

629
00:32:17,860 --> 00:32:21,550
the potential outcome Ya, or Y0.

630
00:32:21,550 --> 00:32:27,370
And that's why I have a 6,
which is bolded under Y0.

631
00:32:27,370 --> 00:32:30,370
And we don't know what
would have happened

632
00:32:30,370 --> 00:32:32,200
to that individual
had they received

633
00:32:32,200 --> 00:32:36,580
treatment B. So in this
case, some magical creature

634
00:32:36,580 --> 00:32:38,762
came to me and told me
their sugar levels would

635
00:32:38,762 --> 00:32:40,720
have been 5.5, but we
don't actually know that.

636
00:32:40,720 --> 00:32:42,070
It wasn't in the data.

637
00:32:42,070 --> 00:32:43,737
Let's look at the
next line just to make

638
00:32:43,737 --> 00:32:45,230
sure we get what I'm saying.

639
00:32:45,230 --> 00:32:47,050
So the second
individual actually

640
00:32:47,050 --> 00:32:53,450
received treatment B. They're
observed sugar level is 6.5.

641
00:32:53,450 --> 00:32:55,790
OK.

642
00:32:55,790 --> 00:32:58,160
Let's do a little survey.

643
00:32:58,160 --> 00:33:00,950
That 6.5 number, should
it be in this column?

644
00:33:00,950 --> 00:33:01,698
Raise your hand.

645
00:33:01,698 --> 00:33:02,990
Or should it be in this column?

646
00:33:02,990 --> 00:33:04,740
Raise your hand.

647
00:33:04,740 --> 00:33:05,240
All right.

648
00:33:05,240 --> 00:33:08,050
About half of you
got that right.

649
00:33:08,050 --> 00:33:11,040
Indeed, it goes to
the second column.

650
00:33:11,040 --> 00:33:14,080
And again, what we would like
to know is the counterfactual.

651
00:33:14,080 --> 00:33:16,260
What would have been
their sugar levels

652
00:33:16,260 --> 00:33:18,267
had they received medication A?

653
00:33:18,267 --> 00:33:20,100
Which we don't actually
observe in our data,

654
00:33:20,100 --> 00:33:22,440
but I'm going to
hypothesize is--

655
00:33:22,440 --> 00:33:25,562
suppose that someone
told me it was 7, then

656
00:33:25,562 --> 00:33:27,270
you would see that
value filled in there.

657
00:33:27,270 --> 00:33:30,610
That's the unobserved
counterfactual.

658
00:33:30,610 --> 00:33:31,110
All right.

659
00:33:31,110 --> 00:33:33,900
First of all, is
the setup clear?

660
00:33:33,900 --> 00:33:34,470
All right.

661
00:33:34,470 --> 00:33:37,990
Now here's when you
use your calculators.

662
00:33:37,990 --> 00:33:40,720
So we're going to
now demonstrate

663
00:33:40,720 --> 00:33:43,420
the difference between
a naive estimator

664
00:33:43,420 --> 00:33:47,590
of your average treatment effect
and the true average treatment

665
00:33:47,590 --> 00:33:48,850
effect.

666
00:33:48,850 --> 00:33:51,190
So what I want you
to do right now

667
00:33:51,190 --> 00:33:59,270
is to compute, first,
what is the average sugar

668
00:33:59,270 --> 00:34:07,270
level of the individuals who
got medication B. So for that,

669
00:34:07,270 --> 00:34:11,440
we're only going to
be using the red ones.

670
00:34:11,440 --> 00:34:17,050
So this is conditioning
on receiving medication B.

671
00:34:17,050 --> 00:34:24,340
And so this is equivalent
to going back to this one

672
00:34:24,340 --> 00:34:27,130
and saying, we're only going to
take the rows where individuals

673
00:34:27,130 --> 00:34:29,350
receive medication
B, and we're going

674
00:34:29,350 --> 00:34:34,110
to average their
observed sugar levels.

675
00:34:34,110 --> 00:34:36,530
And everyone should do that.

676
00:34:36,530 --> 00:34:37,530
What's the first number?

677
00:34:42,600 --> 00:35:02,370
6.5 plus-- I'm getting 7.875.

678
00:35:02,370 --> 00:35:08,790
This is for the
average sugar, given

679
00:35:08,790 --> 00:35:11,430
that they received
medication B. Is that

680
00:35:11,430 --> 00:35:12,680
what other people are getting?

681
00:35:12,680 --> 00:35:13,130
AUDIENCE: Yeah.

682
00:35:13,130 --> 00:35:13,838
DAVID SONTAG: OK.

683
00:35:13,838 --> 00:35:16,170
What about for
the second number?

684
00:35:16,170 --> 00:35:20,070
Average sugar, given A?

685
00:35:24,820 --> 00:35:26,058
I want you to compute it.

686
00:35:26,058 --> 00:35:27,850
And I'm going to ask
everyone to say it out

687
00:35:27,850 --> 00:35:29,123
loud in literally one minute.

688
00:35:29,123 --> 00:35:30,540
And if you get it
wrong, of course

689
00:35:30,540 --> 00:35:33,360
you're going to be embarrassed.

690
00:35:33,360 --> 00:35:34,360
I'm going to try myself.

691
00:35:53,090 --> 00:35:53,910
OK.

692
00:35:53,910 --> 00:35:55,493
On the count of
three, I want everyone

693
00:35:55,493 --> 00:35:57,680
to read out what
that third number is.

694
00:35:57,680 --> 00:36:00,020
One, two, three.

695
00:36:00,020 --> 00:36:04,100
ALL: 7.125.

696
00:36:04,100 --> 00:36:05,250
DAVID SONTAG: All right.

697
00:36:05,250 --> 00:36:05,750
Good.

698
00:36:05,750 --> 00:36:08,830
We can all do arithmetic.

699
00:36:08,830 --> 00:36:10,370
All right.

700
00:36:10,370 --> 00:36:11,595
Good.

701
00:36:11,595 --> 00:36:17,000
So, again, we're just
looking at the red numbers

702
00:36:17,000 --> 00:36:18,590
here, just the red numbers.

703
00:36:18,590 --> 00:36:20,960
So we just computed
that difference,

704
00:36:20,960 --> 00:36:24,280
which is point what?

705
00:36:24,280 --> 00:36:25,690
AUDIENCE: 0.75.

706
00:36:25,690 --> 00:36:27,500
DAVID SONTAG: 0.75?

707
00:36:27,500 --> 00:36:29,030
Yeah, that looks about right.

708
00:36:29,030 --> 00:36:29,670
Good.

709
00:36:29,670 --> 00:36:30,170
All right.

710
00:36:30,170 --> 00:36:33,220
So that's a positive number.

711
00:36:33,220 --> 00:36:37,260
Now let's do
something different.

712
00:36:37,260 --> 00:36:42,600
Now let's compute the actual
average treatment effect, which

713
00:36:42,600 --> 00:36:50,310
is we're now going to average
every number in this column,

714
00:36:50,310 --> 00:36:53,400
and we're going to average
every number in this column.

715
00:36:53,400 --> 00:36:56,880
So this is the
average sugar level

716
00:36:56,880 --> 00:37:00,030
under the potential outcome
of had the individual received

717
00:37:00,030 --> 00:37:03,590
treatment B, and this is
the average sugar level

718
00:37:03,590 --> 00:37:06,080
under the potential outcome
that the individual received

719
00:37:06,080 --> 00:37:12,350
treatment A. All right.

720
00:37:12,350 --> 00:37:13,550
Who's doing it?

721
00:37:13,550 --> 00:37:14,960
AUDIENCE: 0.75.

722
00:37:14,960 --> 00:37:16,385
DAVID SONTAG: 0.75 is what?

723
00:37:16,385 --> 00:37:17,760
AUDIENCE: The difference.

724
00:37:17,760 --> 00:37:19,010
DAVID SONTAG: How do you know?

725
00:37:19,010 --> 00:37:21,160
AUDIENCE: [INAUDIBLE]

726
00:37:21,160 --> 00:37:22,910
DAVID SONTAG: Wow, you're fast.

727
00:37:22,910 --> 00:37:23,410
OK.

728
00:37:23,410 --> 00:37:24,240
Let's see if you're right.

729
00:37:24,240 --> 00:37:25,190
I actually don't know.

730
00:37:25,190 --> 00:37:25,760
OK.

731
00:37:25,760 --> 00:37:26,930
The first one is 0.75.

732
00:37:26,930 --> 00:37:27,500
Good, we got that right.

733
00:37:27,500 --> 00:37:29,917
I intentionally didn't post
the slides to today's lecture.

734
00:37:32,240 --> 00:37:38,340
And the second
one is minus 0.75.

735
00:37:38,340 --> 00:37:38,840
All right.

736
00:37:38,840 --> 00:37:43,330
So now let's put us in the
shoes of a policymaker.

737
00:37:43,330 --> 00:37:47,135
The policymaker has to
decide, is it a good idea to--

738
00:37:47,135 --> 00:37:49,010
or let's say it's a
health insurance company.

739
00:37:49,010 --> 00:37:50,843
A health insurance
company is trying decide,

740
00:37:50,843 --> 00:37:53,768
should I reimburse for
treatment B or not?

741
00:37:53,768 --> 00:37:55,310
Or should I simply
say, no, I'm never

742
00:37:55,310 --> 00:37:58,430
going to reimburse for treatment
because it doesn't work well?

743
00:37:58,430 --> 00:38:02,300
So if they had done the
naive estimator, that

744
00:38:02,300 --> 00:38:04,860
would have been
the first example,

745
00:38:04,860 --> 00:38:10,540
then it would look
like medication B is--

746
00:38:10,540 --> 00:38:12,380
we want lower
numbers here, so it

747
00:38:12,380 --> 00:38:18,610
would look like medication B
is worse than medication A.

748
00:38:18,610 --> 00:38:21,730
And if you properly
estimated what

749
00:38:21,730 --> 00:38:24,580
the actual average
treatment effect is,

750
00:38:24,580 --> 00:38:26,890
you get the absolute
opposite conclusion.

751
00:38:26,890 --> 00:38:29,950
You conclude that medication B
is much better than medication

752
00:38:29,950 --> 00:38:33,660
A. It's just a simple
example to really illustrate

753
00:38:33,660 --> 00:38:35,970
the difference
between conditioning

754
00:38:35,970 --> 00:38:39,035
and actually computing
that counterfactual.

755
00:38:42,890 --> 00:38:43,390
OK.

756
00:38:43,390 --> 00:38:45,170
So hopefully now you're
starting to get it.

757
00:38:45,170 --> 00:38:47,795
And again, you're going to have
many more opportunities to work

758
00:38:47,795 --> 00:38:52,700
through these things in your
homework assignment and so on.

759
00:38:52,700 --> 00:38:55,550
So by now you should be
starting to wonder, how the hell

760
00:38:55,550 --> 00:38:57,620
could I do anything in
this state of the world?

761
00:38:57,620 --> 00:39:00,620
Because you don't actually
observe those black numbers.

762
00:39:00,620 --> 00:39:02,540
These are all unobserved.

763
00:39:02,540 --> 00:39:05,600
And clearly there
is bias in what

764
00:39:05,600 --> 00:39:07,100
the values should
be because of what

765
00:39:07,100 --> 00:39:08,790
I've been saying all along.

766
00:39:08,790 --> 00:39:11,163
So what can we do?

767
00:39:11,163 --> 00:39:12,830
Well, the first thing
we have to realize

768
00:39:12,830 --> 00:39:15,247
is that typically, this is an
impossible problem to solve.

769
00:39:15,247 --> 00:39:18,920
So your instincts
aren't wrong, and we're

770
00:39:18,920 --> 00:39:20,740
going to have to make
a ton of assumptions

771
00:39:20,740 --> 00:39:23,950
in order to do anything here.

772
00:39:23,950 --> 00:39:26,430
So the first assumption
is called SUTVA.

773
00:39:26,430 --> 00:39:27,930
I'm not even going
to talk about it.

774
00:39:27,930 --> 00:39:29,725
You can read about
that in your readings.

775
00:39:29,725 --> 00:39:31,350
I'll tell you about
the two assumptions

776
00:39:31,350 --> 00:39:34,890
that are a little bit
easier to describe.

777
00:39:34,890 --> 00:39:37,920
The first critical assumption
is that there are no unobserved

778
00:39:37,920 --> 00:39:39,840
confounding factors.

779
00:39:39,840 --> 00:39:41,370
Mathematically
what that's saying

780
00:39:41,370 --> 00:39:44,610
is that your potential
outcomes, Y0 and Y1,

781
00:39:44,610 --> 00:39:47,340
are conditionally independent
of the treatment decision given

782
00:39:47,340 --> 00:39:52,780
what you observe on
the individual, X.

783
00:39:52,780 --> 00:39:55,900
Now, this could
be a bit hard to--

784
00:39:55,900 --> 00:39:57,300
and that's called ignorability.

785
00:39:57,300 --> 00:39:59,008
And this can be a bit
hard to understand,

786
00:39:59,008 --> 00:40:01,950
so let me draw a picture.

787
00:40:01,950 --> 00:40:04,200
So X is your covariance, T
is your treatment decision.

788
00:40:04,200 --> 00:40:06,600
And now I've drawn for you
a slightly different graph.

789
00:40:06,600 --> 00:40:10,860
Over here I said X goes
to T, X and T go to Y.

790
00:40:10,860 --> 00:40:14,610
But now I don't have Y.
Instead, I have Y0 and Y1,

791
00:40:14,610 --> 00:40:16,662
and I don't have any
edge from T to them.

792
00:40:16,662 --> 00:40:18,120
And that's because
now I'm actually

793
00:40:18,120 --> 00:40:20,850
using the potential
outcomes notation.

794
00:40:20,850 --> 00:40:22,225
Y0 is a potential
outcome of what

795
00:40:22,225 --> 00:40:24,558
would have happened to this
individual had they received

796
00:40:24,558 --> 00:40:26,040
treatment 0, and
Y1 is what would

797
00:40:26,040 --> 00:40:28,950
have happened to this individual
if they received treatment one.

798
00:40:28,950 --> 00:40:31,395
And because you already know
what treatment the individual

799
00:40:31,395 --> 00:40:32,853
has received, it
doesn't make sense

800
00:40:32,853 --> 00:40:35,560
to talk about an edge
from T to those values.

801
00:40:35,560 --> 00:40:37,150
That's why there's
no edge there.

802
00:40:37,150 --> 00:40:39,150
So then you might wonder,
how could you possibly

803
00:40:39,150 --> 00:40:41,192
have a violation of this
conditional independence

804
00:40:41,192 --> 00:40:42,180
assumption?

805
00:40:42,180 --> 00:40:43,680
Well, before I give
you that answer,

806
00:40:43,680 --> 00:40:45,970
let me put some names
to these things.

807
00:40:45,970 --> 00:40:48,870
So we might think about X as
being the age, gender, weight,

808
00:40:48,870 --> 00:40:50,850
diet, and so on
of the individual.

809
00:40:50,850 --> 00:40:54,300
T might be a medication, like
an anti-hypertensive medication

810
00:40:54,300 --> 00:40:56,820
to try to lower a
patient's blood pressure.

811
00:40:56,820 --> 00:40:58,770
And these would be
the potential outcomes

812
00:40:58,770 --> 00:41:00,990
after those two medications.

813
00:41:00,990 --> 00:41:04,470
So an example of a
violation of ignorability

814
00:41:04,470 --> 00:41:10,970
is if there is something else,
some hidden variable h, which

815
00:41:10,970 --> 00:41:13,490
is not observed
and which affects

816
00:41:13,490 --> 00:41:15,470
both the decision
of what treatment

817
00:41:15,470 --> 00:41:17,750
the individual in
your data set receives

818
00:41:17,750 --> 00:41:20,545
and the potential outcomes.

819
00:41:20,545 --> 00:41:22,170
Now it should be
really clear that this

820
00:41:22,170 --> 00:41:24,378
would be a violation of that
conditional independence

821
00:41:24,378 --> 00:41:25,100
assumption.

822
00:41:25,100 --> 00:41:29,010
In this graph, Y0 and
Y1 are not conditionally

823
00:41:29,010 --> 00:41:32,760
independent of T
given X. All right.

824
00:41:32,760 --> 00:41:34,800
So what are these
hidden confounders?

825
00:41:34,800 --> 00:41:37,710
Well, they might be things,
for example, which really

826
00:41:37,710 --> 00:41:40,020
affect treatment decisions.

827
00:41:40,020 --> 00:41:42,420
So maybe there's a
treatment guideline

828
00:41:42,420 --> 00:41:44,400
saying that for
diabetic patients,

829
00:41:44,400 --> 00:41:47,700
they should receive
treatment zero, that that's

830
00:41:47,700 --> 00:41:50,350
the right thing to do.

831
00:41:50,350 --> 00:41:54,270
And so a violation
of this would be

832
00:41:54,270 --> 00:41:56,700
if the fact that the
patient's diabetic

833
00:41:56,700 --> 00:41:59,950
were not recorded in the
electronic health record.

834
00:41:59,950 --> 00:42:01,660
So you don't know--

835
00:42:01,660 --> 00:42:02,540
that's not up there.

836
00:42:02,540 --> 00:42:05,610
You don't know that,
in fact, the reason

837
00:42:05,610 --> 00:42:08,700
the patient received treatment
T was because of this h factor.

838
00:42:08,700 --> 00:42:10,450
And there's critically
another assumption,

839
00:42:10,450 --> 00:42:12,540
which is that h actually
affects the outcome,

840
00:42:12,540 --> 00:42:15,435
which is why you have these
edges from h to the Y's.

841
00:42:15,435 --> 00:42:17,310
If h were something
which might have affected

842
00:42:17,310 --> 00:42:21,620
treatment decision but not the
actual potential outcomes--

843
00:42:21,620 --> 00:42:23,530
and that can happen, of course.

844
00:42:23,530 --> 00:42:26,880
Things like gender can often
affect treatment decisions,

845
00:42:26,880 --> 00:42:32,570
but maybe, for some diseases,
it might not affect outcomes.

846
00:42:32,570 --> 00:42:36,270
In that situation it wouldn't
be a confounding factor

847
00:42:36,270 --> 00:42:38,540
because it doesn't
violate this assumption.

848
00:42:38,540 --> 00:42:40,290
And, in fact, one would
be able to come up

849
00:42:40,290 --> 00:42:42,960
with consistent estimators
of average treatment effect

850
00:42:42,960 --> 00:42:44,130
under that assumption.

851
00:42:44,130 --> 00:42:47,656
Where things go to hell is when
you have both of those edges.

852
00:42:47,656 --> 00:42:49,950
All right.

853
00:42:49,950 --> 00:42:51,790
So there can't be
any of these h's.

854
00:42:51,790 --> 00:42:53,540
You have to observe
all things that affect

855
00:42:53,540 --> 00:42:55,055
both treatment and outcomes.

856
00:42:57,848 --> 00:42:59,390
The second big
assumption-- oh, yeah.

857
00:42:59,390 --> 00:42:59,930
Question?

858
00:42:59,930 --> 00:43:02,098
AUDIENCE: In practice, how
good of a model is this?

859
00:43:02,098 --> 00:43:03,890
DAVID SONTAG: Of what
I'm showing you here?

860
00:43:03,890 --> 00:43:04,610
AUDIENCE: Yeah.

861
00:43:04,610 --> 00:43:06,015
DAVID SONTAG: For hypertension?

862
00:43:06,015 --> 00:43:06,640
AUDIENCE: Sure.

863
00:43:06,640 --> 00:43:07,848
DAVID SONTAG: I have no idea.

864
00:43:10,248 --> 00:43:11,790
But I think what
you're really trying

865
00:43:11,790 --> 00:43:14,248
to get at here in asking your
question, how good of a model

866
00:43:14,248 --> 00:43:17,540
is this, is, well, oh,
my god, how do I know

867
00:43:17,540 --> 00:43:19,200
if I've observed everything?

868
00:43:19,200 --> 00:43:20,100
Right?

869
00:43:20,100 --> 00:43:20,600
All right.

870
00:43:20,600 --> 00:43:22,017
And that's where
you need to start

871
00:43:22,017 --> 00:43:24,100
talking to domain experts.

872
00:43:24,100 --> 00:43:27,700
So this is my
starting place where

873
00:43:27,700 --> 00:43:31,450
I said, no, I'm not
going to attempt

874
00:43:31,450 --> 00:43:32,965
to fit the causal graph.

875
00:43:35,470 --> 00:43:37,478
I'm going to assume I
know the causal graph

876
00:43:37,478 --> 00:43:39,020
and just try to
estimate the effects.

877
00:43:39,020 --> 00:43:41,228
That's where this starts to
become really irrelevant.

878
00:43:41,228 --> 00:43:44,053
Because if you notice, this
is another causal graph, not

879
00:43:44,053 --> 00:43:45,220
the one I drew on the board.

880
00:43:48,100 --> 00:43:50,110
And so that's something
where, really,

881
00:43:50,110 --> 00:43:52,100
talking with domain
experts would be relevant.

882
00:43:52,100 --> 00:43:57,120
So if you say, OK, I'm going
to be studying hypertension

883
00:43:57,120 --> 00:44:00,870
and this is the data I've
observed on patients,

884
00:44:00,870 --> 00:44:04,980
well, you can then go to a
clinician, maybe a primary care

885
00:44:04,980 --> 00:44:08,220
doctor who often treats
patients with hypertension,

886
00:44:08,220 --> 00:44:10,530
and you say, OK, what usually
affects your treatment

887
00:44:10,530 --> 00:44:11,850
decisions?

888
00:44:11,850 --> 00:44:13,370
And you get a set
of variables out,

889
00:44:13,370 --> 00:44:15,660
and then you check
to make sure, am I

890
00:44:15,660 --> 00:44:17,610
observing all of those
variables, at least

891
00:44:17,610 --> 00:44:20,988
the variables that would
also affect outcomes?

892
00:44:20,988 --> 00:44:22,530
So, often, there's
going to be a back

893
00:44:22,530 --> 00:44:24,900
and forth in that conversation
to make sure that you've

894
00:44:24,900 --> 00:44:26,195
set up your problem correctly.

895
00:44:26,195 --> 00:44:27,570
And again, this
is one area where

896
00:44:27,570 --> 00:44:29,400
you see a critical
difference between the way

897
00:44:29,400 --> 00:44:31,067
that we do causal
inference from the way

898
00:44:31,067 --> 00:44:32,400
that we do machine learning.

899
00:44:32,400 --> 00:44:36,580
Machine learning, if there's
some unobserved variables,

900
00:44:36,580 --> 00:44:37,080
so what?

901
00:44:37,080 --> 00:44:38,880
I mean, maybe your predictive
accuracy isn't quite as good

902
00:44:38,880 --> 00:44:40,740
as it could have
been, but whatever.

903
00:44:40,740 --> 00:44:43,920
Here, your conclusions
could be completely wrong

904
00:44:43,920 --> 00:44:48,810
if you don't get those
confounding factors right.

905
00:44:48,810 --> 00:44:50,610
Now, in some of the
optional readings

906
00:44:50,610 --> 00:44:52,710
for Thursday's lecture--

907
00:44:52,710 --> 00:44:55,290
and we'll touch on it
very briefly on Thursday,

908
00:44:55,290 --> 00:44:57,300
but there's not much
time in this course--

909
00:44:57,300 --> 00:45:00,690
I'll talk about ways and
you'll read about ways

910
00:45:00,690 --> 00:45:03,780
to try to assess
robustness to violations

911
00:45:03,780 --> 00:45:05,430
of these assumptions.

912
00:45:05,430 --> 00:45:08,075
And those go by the name
of sensitivity analysis.

913
00:45:08,075 --> 00:45:10,200
So, for example, the type
of question you might ask

914
00:45:10,200 --> 00:45:12,300
is, how would my
conclusions have

915
00:45:12,300 --> 00:45:15,060
changed if there were a
confounding factor which

916
00:45:15,060 --> 00:45:17,860
was blah strong?

917
00:45:17,860 --> 00:45:23,210
And that's something that one
could try to answer from data,

918
00:45:23,210 --> 00:45:25,420
but it's really starting
to get beyond the scope

919
00:45:25,420 --> 00:45:26,510
of this course.

920
00:45:26,510 --> 00:45:28,052
So I'll give you
some readings on it,

921
00:45:28,052 --> 00:45:32,000
but I won't be able to talk
about it in the lecture.

922
00:45:32,000 --> 00:45:34,660
Now, the second major
assumption that one needs

923
00:45:34,660 --> 00:45:36,943
is what's known
as common support.

924
00:45:36,943 --> 00:45:38,610
And by the way, pay
close attention here

925
00:45:38,610 --> 00:45:43,680
because at the end of today's
lecture-- and if I forget,

926
00:45:43,680 --> 00:45:45,030
someone must remind me--

927
00:45:45,030 --> 00:45:49,650
I'm going to ask you where did
these two assumptions come up

928
00:45:49,650 --> 00:45:52,870
in the proof that I'm
about to give you.

929
00:45:52,870 --> 00:45:55,370
The first one I'm going to give
you will be a dead giveaway.

930
00:45:55,370 --> 00:45:57,840
So I'm going to answer to you
where ignorability comes up,

931
00:45:57,840 --> 00:45:59,423
but it's up to you
to figure out where

932
00:45:59,423 --> 00:46:01,560
does common support show up.

933
00:46:01,560 --> 00:46:02,670
So what is common support?

934
00:46:02,670 --> 00:46:07,560
Well, what common support
says is that there always

935
00:46:07,560 --> 00:46:11,440
must be some stochasticity
in the treatment decisions.

936
00:46:11,440 --> 00:46:17,270
For example, if in
your data patients only

937
00:46:17,270 --> 00:46:21,780
receive treatment A and no
patient receives treatment B,

938
00:46:21,780 --> 00:46:24,420
then you would never be able to
figure out the counterfactual,

939
00:46:24,420 --> 00:46:29,008
what would have happened if
patients receive treatment B.

940
00:46:29,008 --> 00:46:31,050
But what happens if it's
not quite that universal

941
00:46:31,050 --> 00:46:34,260
but maybe there is
classes of people?

942
00:46:34,260 --> 00:46:37,350
Some individual is X, let's
say, people with blue hair.

943
00:46:37,350 --> 00:46:42,450
People with blue hair always
receive treatment zero

944
00:46:42,450 --> 00:46:45,200
and they never
see treatment one.

945
00:46:45,200 --> 00:46:49,340
Well, for those people,
if for some reason

946
00:46:49,340 --> 00:46:50,990
something about them
having blue hair

947
00:46:50,990 --> 00:46:53,600
was also going to affect
how they would respond

948
00:46:53,600 --> 00:46:55,250
to the treatment,
then you wouldn't

949
00:46:55,250 --> 00:46:57,470
be able to answer anything
about the counterfactual

950
00:46:57,470 --> 00:46:59,660
for those individuals.

951
00:46:59,660 --> 00:47:03,560
This goes by the name of what's
called a propensity score.

952
00:47:03,560 --> 00:47:07,310
It's the probability of
receiving some treatment

953
00:47:07,310 --> 00:47:09,230
for each individual.

954
00:47:09,230 --> 00:47:14,150
And we're going to assume that
this propensity score is always

955
00:47:14,150 --> 00:47:17,150
bounded between 0 and 1.

956
00:47:17,150 --> 00:47:20,000
So it's between 1 minus
epsilon and epsilon

957
00:47:20,000 --> 00:47:23,020
for some small epsilon.

958
00:47:23,020 --> 00:47:25,330
And violations of
that assumption

959
00:47:25,330 --> 00:47:28,000
are going to completely
invalidate all conclusions

960
00:47:28,000 --> 00:47:30,610
that we could draw
from the data.

961
00:47:30,610 --> 00:47:31,520
All right.

962
00:47:31,520 --> 00:47:35,190
Now, in actual clinical
practice, you might wonder,

963
00:47:35,190 --> 00:47:37,010
can this ever hold?

964
00:47:37,010 --> 00:47:40,880
Because there are
clinical guidelines.

965
00:47:40,880 --> 00:47:43,867
Well, a couple of places where
you'll see this are as follows.

966
00:47:43,867 --> 00:47:46,450
First, often, there are settings
where we haven't the faintest

967
00:47:46,450 --> 00:47:49,720
idea how to treat patients,
like second line diabetes

968
00:47:49,720 --> 00:47:51,010
treatments.

969
00:47:51,010 --> 00:47:54,370
You know that the first thing
we start with is metformin.

970
00:47:54,370 --> 00:47:57,310
But if metformin doesn't help
control the patient's glucose

971
00:47:57,310 --> 00:48:00,490
values, there are several
second line diabetic treatments.

972
00:48:00,490 --> 00:48:03,100
And right now, we don't
really know which one to try.

973
00:48:03,100 --> 00:48:06,340
So a clinician might start
with treatments from one class.

974
00:48:06,340 --> 00:48:08,570
And if that's not working,
you try a different class,

975
00:48:08,570 --> 00:48:09,040
and so on.

976
00:48:09,040 --> 00:48:10,582
And it's a bit random
which class you

977
00:48:10,582 --> 00:48:13,550
start with for any one patient.

978
00:48:13,550 --> 00:48:16,600
In other settings, there might
be good clinical guidelines,

979
00:48:16,600 --> 00:48:18,860
but there is randomness
in other ways.

980
00:48:18,860 --> 00:48:25,500
For example, clinicians who
are trained on the west coast

981
00:48:25,500 --> 00:48:28,350
might be trained that this is
the right way to do things,

982
00:48:28,350 --> 00:48:30,420
and clinicians who are
trained in the east coast

983
00:48:30,420 --> 00:48:33,630
might be trained that this is
the right way to do things.

984
00:48:33,630 --> 00:48:37,860
And so even if any one
clinician's treatment decisions

985
00:48:37,860 --> 00:48:40,260
are deterministic
in some way, you'll

986
00:48:40,260 --> 00:48:43,530
see some stochasticity
now across clinicians.

987
00:48:43,530 --> 00:48:45,790
It's a bit subtle how to
use that in your analysis,

988
00:48:45,790 --> 00:48:49,160
but trust me, it can be done.

989
00:48:49,160 --> 00:48:51,680
So if you want to
do causal inference

990
00:48:51,680 --> 00:48:53,290
from observational
data, you're going

991
00:48:53,290 --> 00:48:56,570
to have to first start to
formalize things mathematically

992
00:48:56,570 --> 00:49:01,190
in terms of what is your X, what
is your T, what is your Y. You

993
00:49:01,190 --> 00:49:04,830
have to think through,
do these choices

994
00:49:04,830 --> 00:49:09,310
satisfy these assumptions
of ignorability and overlap?

995
00:49:09,310 --> 00:49:11,310
Some of these things you
can check in your data.

996
00:49:11,310 --> 00:49:13,770
Ignorability you can't
explicitly check in your data.

997
00:49:13,770 --> 00:49:19,580
But overlap, this thing,
you can test in your data.

998
00:49:19,580 --> 00:49:20,550
By the way, how?

999
00:49:20,550 --> 00:49:21,050
Any idea?

1000
00:49:24,828 --> 00:49:26,370
Someone else who
hasn't spoken today.

1001
00:49:31,320 --> 00:49:33,690
So just think back to
the previous example.

1002
00:49:33,690 --> 00:49:41,220
You have this table of these X's
and treatment A or B and then

1003
00:49:41,220 --> 00:49:42,750
sugar values.

1004
00:49:42,750 --> 00:49:44,303
How would you test this?

1005
00:49:44,303 --> 00:49:46,220
AUDIENCE: You could use
a frequentist approach

1006
00:49:46,220 --> 00:49:48,550
and just count how
many things show up.

1007
00:49:48,550 --> 00:49:51,880
And if there is zero, then you
could say that it's violated.

1008
00:49:51,880 --> 00:49:52,770
DAVID SONTAG: Good.

1009
00:49:52,770 --> 00:49:54,705
So you have this table.

1010
00:49:54,705 --> 00:49:58,140
I'll just go back to that table.

1011
00:49:58,140 --> 00:50:03,420
We have this table,
and these are your X's.

1012
00:50:05,805 --> 00:50:07,680
Actually, we'll go back
to the previous slide

1013
00:50:07,680 --> 00:50:08,972
where it's a bit easier to see.

1014
00:50:13,930 --> 00:50:17,020
Here, we're going to ignore
the outcome, the sugar

1015
00:50:17,020 --> 00:50:19,150
levels because,
remember, this only

1016
00:50:19,150 --> 00:50:22,030
has to do with
probability of treatment

1017
00:50:22,030 --> 00:50:23,770
given your covariance.

1018
00:50:23,770 --> 00:50:25,588
The Y doesn't show
up here at all.

1019
00:50:25,588 --> 00:50:27,130
So this thing on
the right-hand side,

1020
00:50:27,130 --> 00:50:29,977
the observed sugar levels, is
irrelevant for this question.

1021
00:50:29,977 --> 00:50:31,810
All we care about is
what goes on over here.

1022
00:50:31,810 --> 00:50:32,740
So we look at this.

1023
00:50:32,740 --> 00:50:35,100
These are your X's, and
this is your treatment.

1024
00:50:35,100 --> 00:50:37,840
And you can look to
see, OK, here you

1025
00:50:37,840 --> 00:50:42,010
have one 75-year-old
male who does exercise

1026
00:50:42,010 --> 00:50:44,680
frequently and received
treatment A. Is there any one

1027
00:50:44,680 --> 00:50:48,370
else in the data set who
is 75 years old and male,

1028
00:50:48,370 --> 00:50:51,190
does exercise regularly
but received treatment B?

1029
00:50:51,190 --> 00:50:52,450
Yes or no?

1030
00:50:52,450 --> 00:50:53,580
No.

1031
00:50:53,580 --> 00:50:54,080
Good.

1032
00:50:54,080 --> 00:50:54,580
OK.

1033
00:50:54,580 --> 00:50:59,360
So overlap is not satisfied
here, at least not empirically.

1034
00:50:59,360 --> 00:51:03,190
Now, you might argue that I'm
being a bit too coarse here.

1035
00:51:03,190 --> 00:51:05,740
Well, what happens if
the individual is 74

1036
00:51:05,740 --> 00:51:06,850
and received treatment B?

1037
00:51:06,850 --> 00:51:08,200
Maybe that's close enough.

1038
00:51:08,200 --> 00:51:09,820
So there starts to
become subtleties

1039
00:51:09,820 --> 00:51:12,700
in assessing these things
when you have finite data.

1040
00:51:12,700 --> 00:51:14,710
But it is something at
the fundamental level

1041
00:51:14,710 --> 00:51:17,290
that you could start
to assess using data.

1042
00:51:17,290 --> 00:51:19,870
As opposed to ignorability,
which you cannot test using

1043
00:51:19,870 --> 00:51:21,290
data.

1044
00:51:21,290 --> 00:51:21,790
All right.

1045
00:51:21,790 --> 00:51:29,990
So you have to think about, are
these assumptions satisfied?

1046
00:51:29,990 --> 00:51:34,160
And only once you start to think
through those questions can

1047
00:51:34,160 --> 00:51:37,340
you start to do your analysis.

1048
00:51:37,340 --> 00:51:41,460
And so that now brings me to
the next part of this lecture,

1049
00:51:41,460 --> 00:51:45,260
which is how do we actually--
let's just now believe David,

1050
00:51:45,260 --> 00:51:46,760
believe that these
assumptions hold.

1051
00:51:46,760 --> 00:51:49,720
How do we do that
causal inference?

1052
00:51:49,720 --> 00:51:50,220
Yeah?

1053
00:51:50,220 --> 00:51:51,802
AUDIENCE: I just had a
question on [INAUDIBLE]..

1054
00:51:51,802 --> 00:51:54,687
If you know that some patients,
for instance, healthy patients,

1055
00:51:54,687 --> 00:51:56,270
are not tracking to
get any treatment,

1056
00:51:56,270 --> 00:51:58,890
should we just remove
them, basically?

1057
00:51:58,890 --> 00:52:00,500
DAVID SONTAG: So
the question is,

1058
00:52:00,500 --> 00:52:04,710
what happens if you have
a violation of overlap?

1059
00:52:04,710 --> 00:52:08,240
For example, you know that
healthy individuals never

1060
00:52:08,240 --> 00:52:09,770
receive any treatment.

1061
00:52:09,770 --> 00:52:11,520
Should you remove them
from your data set?

1062
00:52:11,520 --> 00:52:14,020
Well, first of all, that has
to do with how do you formalize

1063
00:52:14,020 --> 00:52:16,160
the question because not
receiving a treatment

1064
00:52:16,160 --> 00:52:18,250
is a treatment.

1065
00:52:18,250 --> 00:52:21,880
So that might be your control
arm, just to be clear.

1066
00:52:21,880 --> 00:52:24,160
Now, if you're asking about
the difference between two

1067
00:52:24,160 --> 00:52:26,200
treatments-- two different
classes of treatment

1068
00:52:26,200 --> 00:52:34,000
for a condition, then often one
defines the relevant inclusion

1069
00:52:34,000 --> 00:52:40,990
criteria in order to have
these conditions hold.

1070
00:52:40,990 --> 00:52:44,830
For example, we could try to
redefine the set of individuals

1071
00:52:44,830 --> 00:52:47,140
that we're asking about
so that overlap does hold.

1072
00:52:47,140 --> 00:52:48,640
But then in that
situation, you have

1073
00:52:48,640 --> 00:52:51,520
to just make sure that your
policy is also modified.

1074
00:52:51,520 --> 00:52:54,640
You say, OK, I conclude that
the average treatment effect is

1075
00:52:54,640 --> 00:52:57,740
blah for this type of people.

1076
00:52:57,740 --> 00:52:59,830
OK?

1077
00:52:59,830 --> 00:53:01,730
OK.

1078
00:53:01,730 --> 00:53:05,560
So how could we possibly compute
the average treatment effect

1079
00:53:05,560 --> 00:53:07,835
from data?

1080
00:53:07,835 --> 00:53:09,960
Remember, average treatment
effect, mathematically,

1081
00:53:09,960 --> 00:53:13,635
is the expectation between
potential outcome Y1 minus Y0.

1082
00:53:16,860 --> 00:53:20,250
The key tool which we'll use
in order to estimate that

1083
00:53:20,250 --> 00:53:22,203
is what's known as the
adjustment formula.

1084
00:53:22,203 --> 00:53:24,370
This goes by many names in
the statistics community,

1085
00:53:24,370 --> 00:53:26,960
such as the G-formula as well.

1086
00:53:26,960 --> 00:53:30,460
Here, I'll give you
a derivation of it.

1087
00:53:30,460 --> 00:53:34,790
We're first going to recognize
that this expectation is

1088
00:53:34,790 --> 00:53:36,830
actually two
expectations in one.

1089
00:53:36,830 --> 00:53:39,770
It's the expectation
over individuals X

1090
00:53:39,770 --> 00:53:43,575
and it's the expectation over
potential outcomes Y given X.

1091
00:53:43,575 --> 00:53:45,200
So I'm first just
going to write it out

1092
00:53:45,200 --> 00:53:47,330
in terms of those
two expectations,

1093
00:53:47,330 --> 00:53:50,870
and I'll write the expectations
related to X on the outside.

1094
00:53:50,870 --> 00:53:54,760
That goes by name of law
of total expectation.

1095
00:53:54,760 --> 00:53:58,750
This is trivial at this stage.

1096
00:53:58,750 --> 00:54:02,230
And by the way, I'm just
writing out expectation of Y1.

1097
00:54:02,230 --> 00:54:04,900
In a few minutes, I'll
show you expectation of Y0,

1098
00:54:04,900 --> 00:54:07,840
but it's going to be
exactly analogous.

1099
00:54:07,840 --> 00:54:11,980
Now, the next step is
where we use ignorability.

1100
00:54:11,980 --> 00:54:15,540
I told you I was going
to give that one away.

1101
00:54:15,540 --> 00:54:19,000
So remember, we said
that we're assuming

1102
00:54:19,000 --> 00:54:23,740
that Y1 is conditionally
independent of the treatment

1103
00:54:23,740 --> 00:54:34,210
T given X. What that
means is probability of Y1

1104
00:54:34,210 --> 00:54:39,910
given X is equal to
probability of Y1

1105
00:54:39,910 --> 00:54:43,750
given X comma T equals
whatever-- in this case

1106
00:54:43,750 --> 00:54:46,750
I'll just say T equals 1.

1107
00:54:46,750 --> 00:54:52,220
This is implied by Y1 being
conditionally independent of T

1108
00:54:52,220 --> 00:54:54,050
given X.

1109
00:54:54,050 --> 00:54:59,640
So I can just stick n
comma T equals 1 here,

1110
00:54:59,640 --> 00:55:05,090
and that's explicitly because
of ignorability holding.

1111
00:55:05,090 --> 00:55:08,570
But now we're in a really good
place because notice that--

1112
00:55:08,570 --> 00:55:10,760
and here I've just done
some short notation.

1113
00:55:10,760 --> 00:55:14,391
I'm just going to
hide this expectation.

1114
00:55:17,550 --> 00:55:19,840
And by the way, you could
do the same for Y0--

1115
00:55:19,840 --> 00:55:21,640
Y1, Y0.

1116
00:55:21,640 --> 00:55:26,330
And now notice
that we can replace

1117
00:55:26,330 --> 00:55:30,140
this average human effect
with now this expectation

1118
00:55:30,140 --> 00:55:32,440
with respect to
all individuals X

1119
00:55:32,440 --> 00:55:35,720
of the expectation of Y1 given
X comma T equals 1, and so on.

1120
00:55:39,500 --> 00:55:43,600
And these are mostly
quantities that we can now

1121
00:55:43,600 --> 00:55:45,560
observe from our data.

1122
00:55:45,560 --> 00:55:50,980
So, for example, we can
look at the individuals who

1123
00:55:50,980 --> 00:55:54,550
received treatment one,
and for those individuals

1124
00:55:54,550 --> 00:55:56,690
we have realizations of Y1.

1125
00:55:56,690 --> 00:55:58,940
We can look at individuals
who receive treatment zero,

1126
00:55:58,940 --> 00:56:02,497
and for those individuals
we have realizations of Y0.

1127
00:56:02,497 --> 00:56:04,330
And we could just average
those realizations

1128
00:56:04,330 --> 00:56:07,810
to get estimates of the
corresponding expectations.

1129
00:56:07,810 --> 00:56:10,150
So these we can easily
estimate from our data.

1130
00:56:13,020 --> 00:56:14,660
And so we've made progress.

1131
00:56:14,660 --> 00:56:18,665
We can now estimate some
part of this from our data.

1132
00:56:18,665 --> 00:56:20,040
But notice, there
are some things

1133
00:56:20,040 --> 00:56:22,123
that we can't yet directly
estimate from our data.

1134
00:56:22,123 --> 00:56:27,450
In particular, we can't
estimate expectation of Y0

1135
00:56:27,450 --> 00:56:31,500
given X comma T equals 1
because we have no idea what

1136
00:56:31,500 --> 00:56:34,620
would have happened to this
individual who actually

1137
00:56:34,620 --> 00:56:37,170
got treatment one if they
had gotten treatment zero.

1138
00:56:37,170 --> 00:56:39,060
So these we don't know.

1139
00:56:42,210 --> 00:56:45,540
So these we don't know.

1140
00:56:45,540 --> 00:56:47,620
Now, what is the trick
I'm planning on you?

1141
00:56:47,620 --> 00:56:50,030
How does it help
that we can do this?

1142
00:56:50,030 --> 00:56:52,300
Well, the key point is
that these quantities

1143
00:56:52,300 --> 00:56:56,790
that we can estimate from
data show up in that term.

1144
00:56:56,790 --> 00:56:59,910
In particular, if you
look at the individuals X

1145
00:56:59,910 --> 00:57:04,230
that you've sampled from the
full set of individuals P of X,

1146
00:57:04,230 --> 00:57:07,650
for that individual
X for which, in fact,

1147
00:57:07,650 --> 00:57:11,310
we observed T equals 1, then we
can estimate expectation of Y1

1148
00:57:11,310 --> 00:57:16,430
given X comma T equals
1, and similarly for Y0.

1149
00:57:16,430 --> 00:57:19,080
But what we need to be able
to do is to extrapolate.

1150
00:57:19,080 --> 00:57:22,995
Because empirically, we only
have samples from P of X

1151
00:57:22,995 --> 00:57:24,620
given T equals 1, P
of X given T equals

1152
00:57:24,620 --> 00:57:27,830
0 for those two potential
outcomes correspondingly.

1153
00:57:27,830 --> 00:57:31,670
But we are going to also
get samples of X such

1154
00:57:31,670 --> 00:57:33,620
that for those individuals
in your data set,

1155
00:57:33,620 --> 00:57:36,650
you might have only
observed T equals 0.

1156
00:57:36,650 --> 00:57:41,180
And to compute this formula,
you have to answer, for that X,

1157
00:57:41,180 --> 00:57:44,360
what would it have been if
they got treatment equals one?

1158
00:57:44,360 --> 00:57:46,283
So there are going to
be a set of individuals

1159
00:57:46,283 --> 00:57:47,950
that we have to
extrapolate for in order

1160
00:57:47,950 --> 00:57:50,275
to use this adjustment
formula for estimate.

1161
00:57:52,780 --> 00:57:53,280
Yep?

1162
00:57:53,280 --> 00:57:55,405
AUDIENCE: I thought because
common support is true,

1163
00:57:55,405 --> 00:57:58,010
we have some patients that
received each treatment

1164
00:57:58,010 --> 00:58:00,150
or a given type of X.

1165
00:58:00,150 --> 00:58:02,110
DAVID SONTAG: Yes.

1166
00:58:02,110 --> 00:58:06,850
But now-- so, yes, that's true.

1167
00:58:09,460 --> 00:58:13,920
But that's a statement
about infinite data.

1168
00:58:13,920 --> 00:58:17,010
And in reality, one
only has finite data.

1169
00:58:17,010 --> 00:58:22,590
And so although common support
has to hold to some extent,

1170
00:58:22,590 --> 00:58:25,500
you can't just build on
that to say that you always

1171
00:58:25,500 --> 00:58:29,240
observe the counterfactual
for every individual,

1172
00:58:29,240 --> 00:58:30,990
such as the pictures
I showed you earlier.

1173
00:58:33,740 --> 00:58:36,340
So I'm going to leave this slide
up for just one more second

1174
00:58:36,340 --> 00:58:38,260
to let it sink in and
see what it's saying.

1175
00:58:41,660 --> 00:58:44,480
We started out from the goal of
computing the average treatment

1176
00:58:44,480 --> 00:58:48,330
effect, expected
value of Y1 minus Y0.

1177
00:58:48,330 --> 00:58:50,970
Using the adjustment
formula, we've

1178
00:58:50,970 --> 00:58:54,750
gotten to now an equivalent
representation, which

1179
00:58:54,750 --> 00:58:58,080
is now an expectation with
respect to all individuals

1180
00:58:58,080 --> 00:59:03,310
sampling from P of X
of expected value of Y1

1181
00:59:03,310 --> 00:59:05,800
given X comma T equals
1, expected value of Y0

1182
00:59:05,800 --> 00:59:08,060
given X comma T equals 0.

1183
00:59:08,060 --> 00:59:10,223
For some of the individuals,
you can observe this,

1184
00:59:10,223 --> 00:59:12,140
and for some of them,
you have to extrapolate.

1185
00:59:14,670 --> 00:59:18,547
So from here, there are
many ways that one can go.

1186
00:59:18,547 --> 00:59:20,130
Hold your question
for a little while.

1187
00:59:23,180 --> 00:59:25,940
So types of causal
inference methods

1188
00:59:25,940 --> 00:59:27,500
that you will have
heard of include

1189
00:59:27,500 --> 00:59:29,090
things like
covariance adjustment,

1190
00:59:29,090 --> 00:59:32,120
propensity score re-weighting,
doubly robust estimators,

1191
00:59:32,120 --> 00:59:34,830
matching, and so on.

1192
00:59:34,830 --> 00:59:37,520
And those are the tools of
the causal inference trade.

1193
00:59:37,520 --> 00:59:39,320
And in this course,
we're only going

1194
00:59:39,320 --> 00:59:40,520
to talk about the first two.

1195
00:59:40,520 --> 00:59:41,750
And in today's
lecture, we're only

1196
00:59:41,750 --> 00:59:44,083
going to talk about the first
one, covariate adjustment.

1197
00:59:44,083 --> 00:59:47,610
And on Thursday, we'll
talk about the second one.

1198
00:59:47,610 --> 00:59:50,690
So covariate adjustment
is a very natural way

1199
00:59:50,690 --> 00:59:54,505
to try to do that extrapolation.

1200
00:59:54,505 --> 00:59:56,880
It also goes by the name, by
the way, of response surface

1201
00:59:56,880 --> 00:59:57,500
modeling.

1202
00:59:57,500 --> 00:59:59,042
What we're going to
do is we're going

1203
00:59:59,042 --> 01:00:04,010
to learn a function f, which
takes as an input X and T,

1204
01:00:04,010 --> 01:00:06,500
and its goals is to predict
Y. So intuitively, you

1205
01:00:06,500 --> 01:00:10,790
should think about f as
this conditional probability

1206
01:00:10,790 --> 01:00:12,620
distribution.

1207
01:00:12,620 --> 01:00:19,140
It's predicting Y
given X and T. So

1208
01:00:19,140 --> 01:00:23,340
T is going to be an input
to the machine learning

1209
01:00:23,340 --> 01:00:25,830
algorithm, which is going
to predict what would be

1210
01:00:25,830 --> 01:00:30,850
the potential outcome Y for this
individual described by feature

1211
01:00:30,850 --> 01:00:42,720
as X1 through Xd
under intervention T.

1212
01:00:42,720 --> 01:00:44,710
So this is just from
the previous slide.

1213
01:00:44,710 --> 01:00:46,290
And what we're going
to do now are--

1214
01:00:46,290 --> 01:00:50,640
this is now where we get the
reduction to machine learning--

1215
01:00:50,640 --> 01:00:53,820
is we're going to use empirical
risk minimization, or maybe

1216
01:00:53,820 --> 01:00:57,480
some regularized empirical risk
minimization, to fit a function

1217
01:00:57,480 --> 01:01:02,070
f which approximates the
expected value of YT given

1218
01:01:02,070 --> 01:01:03,910
capital T equals little t.

1219
01:01:03,910 --> 01:01:07,830
Got my X. And then once
you have that function,

1220
01:01:07,830 --> 01:01:10,260
we're going to be able
to use that to estimate

1221
01:01:10,260 --> 01:01:15,420
the average treatment effect
by just implementing now

1222
01:01:15,420 --> 01:01:16,863
this formula here.

1223
01:01:16,863 --> 01:01:19,196
So we're going to first take
an expectation with respect

1224
01:01:19,196 --> 01:01:20,790
to the individuals
in the data set.

1225
01:01:20,790 --> 01:01:22,440
So we're going to
approximate that

1226
01:01:22,440 --> 01:01:25,890
with an empirical expectation
where we sum over the little n

1227
01:01:25,890 --> 01:01:28,370
individuals in your data set.

1228
01:01:28,370 --> 01:01:29,870
Then what we're
going to do is we're

1229
01:01:29,870 --> 01:01:36,620
going to estimate the first
term, which is f of Xi comma 1

1230
01:01:36,620 --> 01:01:39,590
because that is approximating
the expected value of Y1

1231
01:01:39,590 --> 01:01:41,330
given T comma X--

1232
01:01:41,330 --> 01:01:43,970
T equals 1 comma
X. And we're going

1233
01:01:43,970 --> 01:01:47,240
to approximate the second
term, which is just plugging

1234
01:01:47,240 --> 01:01:49,750
now 0 for T instead of 1.

1235
01:01:49,750 --> 01:01:51,950
And we're going to take the
difference between them,

1236
01:01:51,950 --> 01:01:54,630
and that will be our estimator
of the average treatment

1237
01:01:54,630 --> 01:01:55,130
effect.

1238
01:02:00,357 --> 01:02:02,065
Here's a natural place
to ask a question.

1239
01:02:07,210 --> 01:02:12,578
One thing you might wonder
is, in your data set,

1240
01:02:12,578 --> 01:02:14,870
you actually did observe
something for that individual,

1241
01:02:14,870 --> 01:02:15,660
right.

1242
01:02:15,660 --> 01:02:20,550
Notice how your raw data
doesn't show up in this at all.

1243
01:02:20,550 --> 01:02:23,250
Because I've done
machine learning,

1244
01:02:23,250 --> 01:02:27,030
and then I've thrown
away the observed Y's,

1245
01:02:27,030 --> 01:02:30,330
and I used this estimator.

1246
01:02:30,330 --> 01:02:33,120
So what you could have done--
an alternative formula, which,

1247
01:02:33,120 --> 01:02:35,530
by the way, is also a
consistent estimator,

1248
01:02:35,530 --> 01:02:38,280
would have been to
use the observed

1249
01:02:38,280 --> 01:02:41,760
Y for whatever the factual
is and the imputed Y

1250
01:02:41,760 --> 01:02:44,642
for the counterfactual using f.

1251
01:02:44,642 --> 01:02:46,350
That would have been
that would have also

1252
01:02:46,350 --> 01:02:48,690
been a consistent estimator for
the average treatment effect.

1253
01:02:48,690 --> 01:02:49,732
You could've done either.

1254
01:02:53,790 --> 01:02:54,290
OK.

1255
01:02:57,050 --> 01:02:59,360
Now, sometimes you're
not interested in just

1256
01:02:59,360 --> 01:03:00,680
the average treatment
effect, but you're actually

1257
01:03:00,680 --> 01:03:02,555
interested in understanding
the heterogeneity

1258
01:03:02,555 --> 01:03:03,647
in the population.

1259
01:03:03,647 --> 01:03:05,480
Well, this also now
gives you an opportunity

1260
01:03:05,480 --> 01:03:08,460
to try to explore
that heterogeneity.

1261
01:03:08,460 --> 01:03:10,520
So for each
individual Xi, you can

1262
01:03:10,520 --> 01:03:12,530
look at just the
difference between what

1263
01:03:12,530 --> 01:03:16,580
f predicts for
treatment one and what X

1264
01:03:16,580 --> 01:03:17,930
predicts given treatment zero.

1265
01:03:17,930 --> 01:03:19,070
And the difference
between those is

1266
01:03:19,070 --> 01:03:21,195
your estimate of your
conditional average treatment

1267
01:03:21,195 --> 01:03:21,695
effect.

1268
01:03:21,695 --> 01:03:23,445
So, for example, if
you want to figure out

1269
01:03:23,445 --> 01:03:25,460
for this individual, what
is the optimal policy,

1270
01:03:25,460 --> 01:03:27,667
you might look to see is
CATE positive or negative,

1271
01:03:27,667 --> 01:03:29,750
or is it greater than some
threshold, for example?

1272
01:03:32,148 --> 01:03:33,440
So let's look at some pictures.

1273
01:03:36,300 --> 01:03:39,030
Now what we're using is we're
using that function f in order

1274
01:03:39,030 --> 01:03:41,190
to impute those counterfactuals.

1275
01:03:41,190 --> 01:03:43,920
And now we have those
observed, and we can actually

1276
01:03:43,920 --> 01:03:45,540
compute the CATE.

1277
01:03:45,540 --> 01:03:48,120
And averaging over those,
you can estimate now

1278
01:03:48,120 --> 01:03:51,060
the average treatment effect.

1279
01:03:51,060 --> 01:03:51,560
Yep?

1280
01:03:51,560 --> 01:03:53,180
AUDIENCE: How is f non-biased?

1281
01:03:54,968 --> 01:03:55,760
DAVID SONTAG: Good.

1282
01:03:55,760 --> 01:03:57,008
So where can this go wrong?

1283
01:03:57,008 --> 01:03:58,550
So what do you mean
by biased, first?

1284
01:03:58,550 --> 01:03:59,168
I'll ask that.

1285
01:03:59,168 --> 01:04:00,710
AUDIENCE: For
instance, as we've seen

1286
01:04:00,710 --> 01:04:04,820
in the paper like pneumonia
and people who have asthma,

1287
01:04:04,820 --> 01:04:07,830
[INAUDIBLE]

1288
01:04:08,717 --> 01:04:11,300
DAVID SONTAG: Oh, thank you so
much for bringing that back up.

1289
01:04:11,300 --> 01:04:15,350
So you're referring
to one of the readings

1290
01:04:15,350 --> 01:04:17,000
for the course
from several weeks

1291
01:04:17,000 --> 01:04:20,180
ago, where we talked about using
just a pure machine learning

1292
01:04:20,180 --> 01:04:24,903
algorithm to try to predict
outcomes in a hospital setting.

1293
01:04:24,903 --> 01:04:26,570
In particular, what
happens for patients

1294
01:04:26,570 --> 01:04:29,780
who have pneumonia in
the emergency department?

1295
01:04:29,780 --> 01:04:32,600
And if you all remember,
there was this asthma example,

1296
01:04:32,600 --> 01:04:36,320
where patients with
asthma were predicted

1297
01:04:36,320 --> 01:04:41,090
to have better outcomes than
patients without asthma.

1298
01:04:43,700 --> 01:04:45,188
And you're calling that bias.

1299
01:04:45,188 --> 01:04:46,980
But you remember, when
I taught about this,

1300
01:04:46,980 --> 01:04:48,610
I called it biased due
to a particular thing.

1301
01:04:48,610 --> 01:04:49,735
What's the language I used?

1302
01:04:52,990 --> 01:04:58,978
I said bias due to
intervention, maybe, is what I--

1303
01:04:58,978 --> 01:05:00,520
I can't remember
exactly what I said.

1304
01:05:00,520 --> 01:05:02,170
[LAUGHTER]

1305
01:05:02,170 --> 01:05:03,400
I don't know.

1306
01:05:03,400 --> 01:05:06,240
Make it up.

1307
01:05:06,240 --> 01:05:08,960
Now a textbook will be written
with bias by intervention.

1308
01:05:08,960 --> 01:05:09,460
OK.

1309
01:05:09,460 --> 01:05:12,160
So the problem
there is that they

1310
01:05:12,160 --> 01:05:16,193
didn't formulize the
prediction problem correctly.

1311
01:05:16,193 --> 01:05:17,860
The question that
they should have asked

1312
01:05:17,860 --> 01:05:20,920
is, for asthma patients--

1313
01:05:23,440 --> 01:05:31,150
what you really want to ask is a
question of X and then T and Y,

1314
01:05:31,150 --> 01:05:39,360
where T are the interventions
that are done for asthmatics.

1315
01:05:45,090 --> 01:05:48,450
So the failure of that
paper is that it ignored

1316
01:05:48,450 --> 01:05:51,360
the causal inference question
which was hidden in the data,

1317
01:05:51,360 --> 01:05:54,120
and it just went to predict
Y given X marginalizing

1318
01:05:54,120 --> 01:05:55,320
over T altogether.

1319
01:05:55,320 --> 01:05:59,070
So T was never in
the predictive model.

1320
01:05:59,070 --> 01:06:01,870
And said differently, they never
asked counterfactual questions

1321
01:06:01,870 --> 01:06:04,200
of what would have happened
had you done a different T.

1322
01:06:04,200 --> 01:06:06,640
And then they still used it
to try to guide some treatment

1323
01:06:06,640 --> 01:06:07,140
decisions.

1324
01:06:07,140 --> 01:06:09,840
Like, for example, should
you send this person home,

1325
01:06:09,840 --> 01:06:12,173
or should you keep them for
careful monitoring or so on?

1326
01:06:12,173 --> 01:06:14,415
So this is exactly
the same example

1327
01:06:14,415 --> 01:06:16,260
as I gave in the
beginning of the lecture,

1328
01:06:16,260 --> 01:06:19,020
where I said if you just
use a risk stratification

1329
01:06:19,020 --> 01:06:23,430
model to make some decisions,
you run the risk that you're

1330
01:06:23,430 --> 01:06:27,720
making the wrong decisions
because those predictions were

1331
01:06:27,720 --> 01:06:30,360
biased by decisions
in your data.

1332
01:06:30,360 --> 01:06:32,580
So that doesn't happen here
because we're explicitly

1333
01:06:32,580 --> 01:06:35,320
accounting for T in
all of our analysis.

1334
01:06:35,320 --> 01:06:35,980
Yep?

1335
01:06:35,980 --> 01:06:38,330
AUDIENCE: In the data sets
that we've used, like MIMIC,

1336
01:06:38,330 --> 01:06:39,922
how much treatment
information exists?

1337
01:06:39,922 --> 01:06:41,880
DAVID SONTAG: So how much
treatment information

1338
01:06:41,880 --> 01:06:42,380
is in MIMIC?

1339
01:06:42,380 --> 01:06:44,880
A ton.

1340
01:06:44,880 --> 01:06:48,240
In fact, one of the
readings for next week

1341
01:06:48,240 --> 01:06:52,350
is going to be about trying to
understand how one could manage

1342
01:06:52,350 --> 01:06:58,920
sepsis, which is a condition
caused by infection, which

1343
01:06:58,920 --> 01:07:02,670
is managed by, for example,
giving broad spectrum

1344
01:07:02,670 --> 01:07:05,850
antibiotics, giving
fluids, giving

1345
01:07:05,850 --> 01:07:07,602
pressers and ventilators.

1346
01:07:07,602 --> 01:07:09,060
And all of those
are interventions,

1347
01:07:09,060 --> 01:07:11,227
and all those interventions
are recorded in the data

1348
01:07:11,227 --> 01:07:13,590
so that one could then ask
counterfactual questions

1349
01:07:13,590 --> 01:07:14,880
from the data, like
what would have happened

1350
01:07:14,880 --> 01:07:16,170
if this patient
had they received

1351
01:07:16,170 --> 01:07:17,545
a different set
of interventions?

1352
01:07:17,545 --> 01:07:20,010
Would we have prolonged
their life, for example?

1353
01:07:20,010 --> 01:07:24,383
And so in an intensive care unit
setting, most of the questions

1354
01:07:24,383 --> 01:07:26,550
that we want to ask about,
not all, but many of them

1355
01:07:26,550 --> 01:07:29,010
are about dynamic treatments
because it's not just

1356
01:07:29,010 --> 01:07:30,698
a single treatment
but really about

1357
01:07:30,698 --> 01:07:32,490
a service sequence of
treatments responding

1358
01:07:32,490 --> 01:07:34,053
to the current
patient condition.

1359
01:07:34,053 --> 01:07:36,720
And so that's where we'll really
start to get into that material

1360
01:07:36,720 --> 01:07:40,310
next week, not in
today's lecture.

1361
01:07:40,310 --> 01:07:41,300
Yep?

1362
01:07:41,300 --> 01:07:44,022
AUDIENCE: How do you make sure
that your f function really

1363
01:07:44,022 --> 01:07:46,388
learned from the relationship
between T and the outcome?

1364
01:07:46,388 --> 01:07:48,180
DAVID SONTAG: That's
a phenomenal question.

1365
01:07:48,180 --> 01:07:50,810
Where were you
this whole course?

1366
01:07:50,810 --> 01:07:51,810
Thank you for asking it.

1367
01:07:51,810 --> 01:07:53,640
So I'll repeat it.

1368
01:07:53,640 --> 01:07:56,100
How do you know that
your function f actually

1369
01:07:56,100 --> 01:07:59,850
learned something about the
relationship between the input

1370
01:07:59,850 --> 01:08:04,730
X and the treatment
T and the outcome?

1371
01:08:04,730 --> 01:08:07,070
And that really gets
to the question of,

1372
01:08:07,070 --> 01:08:09,410
is my reduction actually valid?

1373
01:08:09,410 --> 01:08:19,979
So I've taken this
problem and I've

1374
01:08:19,979 --> 01:08:23,350
reduced it to this machine
learning problem, where

1375
01:08:23,350 --> 01:08:27,340
I take my data, and
literally I just

1376
01:08:27,340 --> 01:08:29,770
learn a function f to
try to predict well

1377
01:08:29,770 --> 01:08:32,550
the observations in the data.

1378
01:08:32,550 --> 01:08:34,705
And how do we know that
that function f actually

1379
01:08:34,705 --> 01:08:36,330
does a good job at
estimating something

1380
01:08:36,330 --> 01:08:38,682
like average treatment effect?

1381
01:08:38,682 --> 01:08:41,250
In fact, it might not.

1382
01:08:41,250 --> 01:08:44,250
And this is where
things start to get

1383
01:08:44,250 --> 01:08:47,460
really tricky, particularly
with high dimensional data.

1384
01:08:47,460 --> 01:08:51,520
Because it could happen, for
example, that your treatment

1385
01:08:51,520 --> 01:08:55,470
decision is only one of a huge
number of factors that affect

1386
01:08:55,470 --> 01:08:59,130
the outcome Y. And it
could be that a much more

1387
01:08:59,130 --> 01:09:02,130
important factor is hidden in
X. And because you don't have

1388
01:09:02,130 --> 01:09:05,640
much data, and because you have
to regularize your learning

1389
01:09:05,640 --> 01:09:08,100
algorithm, let's say, with L1
or L2 regularization or maybe

1390
01:09:08,100 --> 01:09:10,590
early stopping if you're
using deep neural network,

1391
01:09:10,590 --> 01:09:15,790
your algorithm might never learn
the actual dependence on T.

1392
01:09:15,790 --> 01:09:19,859
It might learn just to
throw away T and just

1393
01:09:19,859 --> 01:09:23,649
use X to predict Y.
And if that's the case,

1394
01:09:23,649 --> 01:09:26,160
you will never be able to
infer these average treatment

1395
01:09:26,160 --> 01:09:27,750
effects accurately.

1396
01:09:27,750 --> 01:09:29,545
You'll have huge errors.

1397
01:09:29,545 --> 01:09:31,170
And that gets back
to one of the slides

1398
01:09:31,170 --> 01:09:33,990
that I skipped, where I
started out from this picture.

1399
01:09:33,990 --> 01:09:36,990
This is the machine learning
picture saying, OK, a reduction

1400
01:09:36,990 --> 01:09:38,729
to machine learning is--

1401
01:09:38,729 --> 01:09:40,229
now you add an
additional feature,

1402
01:09:40,229 --> 01:09:41,760
which is your
treatment decision,

1403
01:09:41,760 --> 01:09:44,795
and you learn that
black box function f.

1404
01:09:44,795 --> 01:09:46,920
But this is where machine
learning causal inference

1405
01:09:46,920 --> 01:09:50,100
starts to differ because
we don't actually

1406
01:09:50,100 --> 01:09:55,703
care about the quality
of predicting Y.

1407
01:09:55,703 --> 01:09:57,495
We can measure your
root mean squared error

1408
01:09:57,495 --> 01:10:00,630
in predicting Y given your
X's and T's, and that error

1409
01:10:00,630 --> 01:10:02,550
might be low.

1410
01:10:02,550 --> 01:10:05,400
But you can run into
these failure modes

1411
01:10:05,400 --> 01:10:08,130
where it just completely
ignores T, for example.

1412
01:10:08,130 --> 01:10:10,263
So T is special here.

1413
01:10:10,263 --> 01:10:12,180
So really, the picture
we want to have in mind

1414
01:10:12,180 --> 01:10:15,760
is that T is some
parameter of interest.

1415
01:10:15,760 --> 01:10:19,500
We want to learn a model f
such that if we twiddle T,

1416
01:10:19,500 --> 01:10:22,500
we can see how there is a
differential effect on Y based

1417
01:10:22,500 --> 01:10:24,297
on twiddling T.
That's what we truly

1418
01:10:24,297 --> 01:10:26,130
care about when we're
using machine learning

1419
01:10:26,130 --> 01:10:28,290
for causal inference.

1420
01:10:28,290 --> 01:10:30,150
And so that's really
the gap, that's

1421
01:10:30,150 --> 01:10:32,930
the gap in our
understanding today.

1422
01:10:32,930 --> 01:10:34,680
And it's really an
active area of research

1423
01:10:34,680 --> 01:10:37,320
to figure out how do you change
the whole machine learning

1424
01:10:37,320 --> 01:10:40,938
paradigm to recognize that when
you're using machine learning

1425
01:10:40,938 --> 01:10:42,480
for causal inference,
you're actually

1426
01:10:42,480 --> 01:10:44,995
interested in something
a little bit different.

1427
01:10:44,995 --> 01:10:47,370
And by the way, that's a major
area of my lab's research,

1428
01:10:47,370 --> 01:10:49,037
and we just published
a series of papers

1429
01:10:49,037 --> 01:10:50,503
trying to answer that question.

1430
01:10:50,503 --> 01:10:52,170
Beyond the scope of
this course, but I'm

1431
01:10:52,170 --> 01:10:56,370
happy to send you those
papers if anyone's interested.

1432
01:10:56,370 --> 01:11:00,880
So that type of question
is extremely important.

1433
01:11:00,880 --> 01:11:04,740
It doesn't show up quite as much
when your X's aren't very high

1434
01:11:04,740 --> 01:11:07,560
dimensional and where
things like regularization

1435
01:11:07,560 --> 01:11:09,090
don't become important.

1436
01:11:09,090 --> 01:11:11,310
But once your X becomes
high dimensional

1437
01:11:11,310 --> 01:11:14,160
and once you want to start to
consider more and more complex

1438
01:11:14,160 --> 01:11:16,050
f's during your
fitting, like you

1439
01:11:16,050 --> 01:11:18,510
want to use deep neural
networks, for example,

1440
01:11:18,510 --> 01:11:22,650
these differences in goals
become extremely important.

1441
01:11:35,790 --> 01:11:37,930
So there are other ways
in which things can fail.

1442
01:11:37,930 --> 01:11:43,205
So I want to give you
here an example where--

1443
01:11:43,205 --> 01:11:44,580
shoot, I'm answering
my question.

1444
01:11:46,930 --> 01:11:47,430
OK.

1445
01:11:50,840 --> 01:11:52,520
No one saw that slide.

1446
01:11:52,520 --> 01:11:54,650
Question-- where did
the overlap assumptions

1447
01:11:54,650 --> 01:11:59,930
show up in our approach for
estimating average treatment

1448
01:11:59,930 --> 01:12:02,705
effect using
covariate adjustment?

1449
01:12:17,580 --> 01:12:19,115
Let me go back to the formula.

1450
01:12:24,630 --> 01:12:27,743
Someone who hasn't
spoken today, hopefully.

1451
01:12:27,743 --> 01:12:28,910
You can be wrong, it's fine.

1452
01:12:31,430 --> 01:12:32,415
Yeah, in the back?

1453
01:12:32,415 --> 01:12:34,290
AUDIENCE: Is it the
version with the same age

1454
01:12:34,290 --> 01:12:37,520
in receiving treatment
B and treatment B?

1455
01:12:37,520 --> 01:12:43,917
DAVID SONTAG: So maybe you have
an individual with some age--

1456
01:12:43,917 --> 01:12:45,500
we're going to want
to be able to look

1457
01:12:45,500 --> 01:12:48,358
at the difference between what
f predicts for that individual

1458
01:12:48,358 --> 01:12:50,150
if they got treatment
A versus treatment B,

1459
01:12:50,150 --> 01:12:52,100
or one versus zero.

1460
01:12:52,100 --> 01:12:57,500
And let me try to lead
this a little bit.

1461
01:12:57,500 --> 01:12:59,000
And it might happen
in your data set

1462
01:12:59,000 --> 01:13:04,310
that for individuals
like them, you only ever

1463
01:13:04,310 --> 01:13:07,100
observe treatment one and
there's no one even remotely

1464
01:13:07,100 --> 01:13:09,800
like them who you
observe treatment zero.

1465
01:13:09,800 --> 01:13:13,690
So what's this function
going to output then

1466
01:13:13,690 --> 01:13:17,290
when you input zero for
that second argument?

1467
01:13:17,290 --> 01:13:19,850
Everyone say out loud.

1468
01:13:19,850 --> 01:13:22,200
Garbage?

1469
01:13:22,200 --> 01:13:22,700
Right?

1470
01:13:22,700 --> 01:13:27,710
If in your data set you never
observed anyone even remotely

1471
01:13:27,710 --> 01:13:31,730
similar to Xi who
received treatment zero,

1472
01:13:31,730 --> 01:13:34,428
then this function is basically
undefined for that individual.

1473
01:13:34,428 --> 01:13:36,470
I mean, yeah, your function
will output something

1474
01:13:36,470 --> 01:13:41,610
because you fit it, but it's not
going to be the right answer.

1475
01:13:41,610 --> 01:13:45,530
And so that's where this
assumption starts to show up.

1476
01:13:45,530 --> 01:13:49,910
When one talks about the
sample complexity of learning

1477
01:13:49,910 --> 01:13:53,030
these functions f to do
covariate adjustment,

1478
01:13:53,030 --> 01:13:55,730
and when one talks
about the consistency

1479
01:13:55,730 --> 01:13:57,140
of these arguments--
for example,

1480
01:13:57,140 --> 01:13:58,640
you'd like to be
able to make claims

1481
01:13:58,640 --> 01:14:01,220
that as the amount of
data grows to, let's

1482
01:14:01,220 --> 01:14:04,430
say, infinity, that this is
the right answer-- gives you

1483
01:14:04,430 --> 01:14:05,480
the right estimate.

1484
01:14:05,480 --> 01:14:07,490
So that's the type
of proof which

1485
01:14:07,490 --> 01:14:10,560
is often given in the
causal inference literature.

1486
01:14:10,560 --> 01:14:13,920
Well, if you have overlap,
then as the amount of data

1487
01:14:13,920 --> 01:14:18,138
goes to infinity, you
will observe someone,

1488
01:14:18,138 --> 01:14:19,930
like the person who
received treatment one,

1489
01:14:19,930 --> 01:14:22,120
you'll observe someone who
also received treatment zero.

1490
01:14:22,120 --> 01:14:23,920
It might have taken you a huge
amount of data to get there

1491
01:14:23,920 --> 01:14:26,110
because treatment zero
might have been much less

1492
01:14:26,110 --> 01:14:27,520
likely than treatment one.

1493
01:14:27,520 --> 01:14:30,970
But because the probability
of treatment zero is not zero,

1494
01:14:30,970 --> 01:14:32,810
eventually you'll see
someone like that.

1495
01:14:32,810 --> 01:14:34,477
And so eventually
you'll get enough data

1496
01:14:34,477 --> 01:14:37,120
in order to learn a function
which can extrapolate correctly

1497
01:14:37,120 --> 01:14:39,700
for that individual.

1498
01:14:39,700 --> 01:14:43,930
And so that's where
overlap comes in

1499
01:14:43,930 --> 01:14:46,450
in giving that type of
consistency argument.

1500
01:14:46,450 --> 01:14:51,100
Of course, in reality, you
never have infinite data.

1501
01:14:51,100 --> 01:14:54,280
And so these questions
about trade-offs

1502
01:14:54,280 --> 01:14:56,050
between the amount
of data you have

1503
01:14:56,050 --> 01:14:59,470
and the fact that
you never truly have

1504
01:14:59,470 --> 01:15:02,350
empirical overlap with
a small amount of data,

1505
01:15:02,350 --> 01:15:05,380
and answering when can
you extrapolate correctly

1506
01:15:05,380 --> 01:15:07,750
despite that is the
critical question

1507
01:15:07,750 --> 01:15:09,800
that one needs to answer,
but is, by the way,

1508
01:15:09,800 --> 01:15:11,662
not studied very well
in the literature

1509
01:15:11,662 --> 01:15:13,870
because people don't usually
think in terms of sample

1510
01:15:13,870 --> 01:15:16,148
complexity in that field.

1511
01:15:16,148 --> 01:15:18,190
That's where computer
scientists can start really

1512
01:15:18,190 --> 01:15:20,560
to contribute to this
literature and bringing things

1513
01:15:20,560 --> 01:15:22,060
that we often think
about in machine

1514
01:15:22,060 --> 01:15:26,120
learning to this new topic.

1515
01:15:26,120 --> 01:15:30,110
So I've got a couple
of minutes left.

1516
01:15:30,110 --> 01:15:31,860
Are there any other
questions, or should I

1517
01:15:31,860 --> 01:15:33,840
introduce some new
material in one minute?

1518
01:15:33,840 --> 01:15:35,550
Yeah?

1519
01:15:35,550 --> 01:15:38,160
AUDIENCE: So you said that
the average treatment effect

1520
01:15:38,160 --> 01:15:40,020
estimator here is consistent.

1521
01:15:40,020 --> 01:15:43,070
But does it matter if
we choose the wrong--

1522
01:15:43,070 --> 01:15:46,830
do we have to choose some
functional form of the features

1523
01:15:46,830 --> 01:15:47,413
to the effect?

1524
01:15:47,413 --> 01:15:48,622
DAVID SONTAG: Great question.

1525
01:15:48,622 --> 01:15:51,710
AUDIENCE: Is it consistent even
if we choose a completely wrong

1526
01:15:51,710 --> 01:15:52,582
function or formula?

1527
01:15:52,582 --> 01:15:53,290
DAVID SONTAG: No.

1528
01:15:53,290 --> 01:15:53,910
AUDIENCE: That's
a different thing?

1529
01:15:53,910 --> 01:15:54,520
DAVID SONTAG: No, no.

1530
01:15:54,520 --> 01:15:56,103
You're asking all
the right questions.

1531
01:15:56,103 --> 01:15:58,050
Good job today, everyone.

1532
01:15:58,050 --> 01:16:00,090
So, no.

1533
01:16:00,090 --> 01:16:03,750
If you walk through
that argument I made,

1534
01:16:03,750 --> 01:16:04,830
I assume two things.

1535
01:16:04,830 --> 01:16:06,660
First, that you observe
enough data such

1536
01:16:06,660 --> 01:16:12,330
that you can have any chance
of extrapolating correctly.

1537
01:16:12,330 --> 01:16:13,943
But then implicit
in that statement

1538
01:16:13,943 --> 01:16:15,360
is that you're
choosing a function

1539
01:16:15,360 --> 01:16:17,130
family which is
powerful enough that it

1540
01:16:17,130 --> 01:16:19,060
can extrapolate correctly.

1541
01:16:19,060 --> 01:16:21,255
So if your true function is--

1542
01:16:24,010 --> 01:16:28,970
if you think back to this
figure I showed you here,

1543
01:16:28,970 --> 01:16:30,830
if the true potential
outcome functions are

1544
01:16:30,830 --> 01:16:34,250
these quadratic functions
and you're fitting them

1545
01:16:34,250 --> 01:16:36,340
with a linear function,
then no matter

1546
01:16:36,340 --> 01:16:37,840
how much data you
have you're always

1547
01:16:37,840 --> 01:16:42,230
going to get wrong estimates
because this argument really

1548
01:16:42,230 --> 01:16:45,080
requires that you're considering
more and more complex

1549
01:16:45,080 --> 01:16:48,710
non-linearity as your
amount of data grows.

1550
01:16:48,710 --> 01:16:51,050
So now here's a visual
depiction of what can go wrong

1551
01:16:51,050 --> 01:16:53,490
if you don't have overlap.

1552
01:16:53,490 --> 01:16:55,070
So now I've taken out--

1553
01:16:55,070 --> 01:16:57,530
previously, I had one or two
red points over here and one

1554
01:16:57,530 --> 01:17:00,080
or two blue points over here,
but I've taken those out.

1555
01:17:00,080 --> 01:17:02,460
So in your data all you
have are these blue points

1556
01:17:02,460 --> 01:17:03,335
and those red points.

1557
01:17:06,500 --> 01:17:09,350
So all you have are
the points, and now one

1558
01:17:09,350 --> 01:17:12,140
can learn as good functions,
as you can imagine, to try to,

1559
01:17:12,140 --> 01:17:14,840
let's say, minimize the mean
squared error of predicting

1560
01:17:14,840 --> 01:17:17,840
these blue points and minimize
the mean squared error

1561
01:17:17,840 --> 01:17:19,400
of predicting those red points.

1562
01:17:19,400 --> 01:17:21,530
And what you might get
out is something-- maybe

1563
01:17:21,530 --> 01:17:23,130
you'll decide on
a linear function.

1564
01:17:23,130 --> 01:17:26,090
That's as good as you
could do if all you

1565
01:17:26,090 --> 01:17:28,940
have are those red points.

1566
01:17:28,940 --> 01:17:30,910
And so even if you were
willing to consider

1567
01:17:30,910 --> 01:17:34,010
more and more complex
hypothesis classes,

1568
01:17:34,010 --> 01:17:36,950
here, if you tried to consider
a more complex hypothesis

1569
01:17:36,950 --> 01:17:39,620
class than this line, you'd
probably just over-fitting

1570
01:17:39,620 --> 01:17:41,360
to the data you have.

1571
01:17:41,360 --> 01:17:44,750
And so you decide
on that line, which,

1572
01:17:44,750 --> 01:17:47,480
because you had
no data over here,

1573
01:17:47,480 --> 01:17:51,680
you don't even know that it's
not a good fit to the data.

1574
01:17:51,680 --> 01:17:53,360
And then you notice
that you're getting

1575
01:17:53,360 --> 01:17:54,485
completely wrong estimates.

1576
01:17:54,485 --> 01:17:58,050
For example, if you asked about
the CATE for a young person,

1577
01:17:58,050 --> 01:18:01,610
it would have the wrong sign
over here because they flipped,

1578
01:18:01,610 --> 01:18:03,200
the two lines.

1579
01:18:03,200 --> 01:18:07,760
So that's an example of how
one can start to get errors.

1580
01:18:07,760 --> 01:18:10,790
And when we begin on
Thursday's lecture,

1581
01:18:10,790 --> 01:18:13,610
we're going to pick up right
where we left off today,

1582
01:18:13,610 --> 01:18:17,370
and I'll talk about this issue
a little bit more in detail.

1583
01:18:17,370 --> 01:18:21,290
I'll talk about how, if one were
to learn a linear function, how

1584
01:18:21,290 --> 01:18:23,090
one could actually,
under the assumption

1585
01:18:23,090 --> 01:18:25,130
that the true potential
outcomes are linear,

1586
01:18:25,130 --> 01:18:27,500
how one could actually
interpret the coefficients

1587
01:18:27,500 --> 01:18:29,435
of that linear function
in a causal way

1588
01:18:29,435 --> 01:18:31,310
under the very strong
assumption that the two

1589
01:18:31,310 --> 01:18:32,700
potential outcomes are linear.

1590
01:18:32,700 --> 01:18:35,180
So that's what we'll
return to on Thursday.