Hi everyone, welcome back, today we're going to continue our discussion on
methods for treatment heterogeneity.
So, as Susan emphasized last week, and
Kate give many examples on treatment heterogeneity,
is a problem that comes up in a number of application areas.
And for this reason, it's kind of one of the big focuses of this class.
Just to kind of give one example to get everyone back back up to speed.
Suppose you have a new medical treatment, that you're interested in deploying.
You think people benefit from it, but maybe some people benefit
from it more than others, who would want to kind of based on medical history or
other aspects prioritize the treatment, to people who benefit from it the most.
You ended up in a problem where you want to kind of understand
the heterogeneous effect of the intervention.
The flip side of this is again, I suppose you have some treatment that has
severe side effects, and the side effects actually kind of a problem.
So you'd want to avoid giving intervention to people
who are at high risk of side effects.
Now understanding who are people who are the most vulnerable to side effects
giving a new treatment that's again, a question and treating heterogeneity
over now the outcome isn't configured thing you want to promote up.
It's the side effect you want to avoid, but statistically the problem is the same.
Other areas where this comes up to an industry, include targeted advertising,
people working in these settings often have very large data sets to work with.
So we are able to kind of tease out very subtle signals by using machine
learning in these problems.
All right, so is the theme of this class, today I'm going to focus
on machine learning based methods for treatment heterogeneity.
Something I just want to emphasize before going there though,
is that there's a lot of discussion these days about machine learning based
methods for treatment heterogeneity.
But this doesn't mean that you have to use machine learning based methods to do so.
There are also classical things you could do so run linear regression, and
put an interactions between the treatment and the measured covariance.
And this could also get our treatment heterogeneity in a useful way in some
applications, and there's nothing wrong with that.
Using machine learning methods doesn't change the nature of the problem.
It just enables you to work with the larger, more complex data sets,
and to end to automate some of your analysis.
Just kind of to pick an extreme example, let's say you wanted to do personalized
medicine, and one of the inputs to algorithm was a CT scan of your brain.
You want to do kind of based on CT scan,
figure out who's going to benefit from some treatment or not.
It's a problem and treatment heterogenerity, good like doing this with
or approaching this with linear regression though, you're probably going
to need kind of image recognition or based machine learning tools for that.
All right, so last week, Susan gave an introduction on treatment heterogeity,
with a focus on problems that come up in randomized trials.
She also, discussed some methods for treatment with heterogeneity,
including causal treat and forests.
This week, what I want to do is kind of continue that discussion, but
now no longer in the setting of randomized trials, but
in an observational study setting.
Where you also need to worry about some units selecting
into treatment, and this meaning that not all units
a priori have the same probability of getting treated.
And this ends up being quite a delicate problem, that essentially
combines most of the challenges we've seen in this class so far.
There's, in week two,
we discuss challenges in estimating average treatment effects,
the average treatment effect and itself is a very simple parameter.
It's just one number, but the difficulty comes from the fact that if you're
in an observational study setting, you need to control covariates,
you need to estimate propensity scores, and so forth.
And doing that well make some effort.
On the other hand, last week the focus was on treatment heterogeneity
estimation in randomized trials, here you know In a randomized trial.
So there's no difficulty from confounding, but the difficulty comes from the fact
that the heterogeneous treatment defect function might be quite complicated.
And so you need to kind of work harder to represent,
de-represent it in terms of the treat or something else.
Today we're going to have all of these difficulties, our target,
is still going to be the heterogeneous treatment effect function, so
it's going to be complicated.
But at the same time, we're going to also have to worry about confounding
estimating propensity scores using that to make ourselves robust,
so we're going to have that dimension to the problem.
It turns out though that, luckily you can combine ideas from the class so
far in the right way, you can make everything work.
So treatment heterogeneity and
observational studies is our topic for today.
And they're kind of, be three aspects of this I'm going to focus on this week.
The first is kind of a review of key difficulties that come up, when estimating
treatment heterogeneity, kind of regularization bias, you always get this.
This also came up in randomized trials, and now we also have the problem
of confounding bias, which is going to be our main focus today.
Then after that, I'm going to talk about, and
this might seem a little bit like a detour, but it'll be very important.
I'm going to talk a little bit about confounding robust methods, for
estimating constant treatment effects and observational studies.
You might ask, why do we care about estimating costs and treatment effects,
if we want to kind of understand how treatment effects vary and
focus on the heterogeneity?
But if you kind of think back to what does a causal tree do, for
example, a call tree tries to partition your space, so
you can find regions where the treatment effect is essentially constant.
And then within each leaf,
hoping that the treatment effect is roughly constant in the leaf.
You can estimate the treatment of the leaf,
if we're an observational study setting, this means you can't run a causal tree
unless you have some idea about how to estimate a constant treatment effect in
a leaf, but we hope the treatment effect is constant.
So we're going to talk about concentrating the effect, and
then the last part for lecture, we're going to talk about,
how to build on these ideas to build trees and forests for.
So that's our plan for this week, and then next week,
we'll continue and I'll discuss kind of how to go beyond these ideas for
trees and forests that are compounding robust.
And kind of the old methods that really mesh with any
generic machine learning method, so that's a plan.
In terms of the statistical setup, it'll be the same as usual.
We're going to assume we get to work with a setup and independent, and
identically distributed unit.
Each of them has kind of a feature vector X and outcome y treatment assignment W.
We have potential outcomes, corresponding to the outcome whether they're with or
without treatment, that so kind of,
the actual outcome you observe is the potential outcome,
corresponding to the treatment assignment you actually received.
And then, okay, what do you want to estimate?
In lecture two, we were in this setting, and we want to estimate
the average treatment effect on average difference in potential outcome.
But today, we want to focus on treatment heterogeneity.
So we're still going to focus on causal effects, this difference y one minus y
zero, but we're not going to just want to have the average of this quantity.
We're going to kind of want to have more nuanced picture of
what's going on, and, all right, how could one do so?
Just kind of quick thing before I talk about
the thing we actually want to estimate.
It's kind of if you say you want to estimate kind of personalized
treatment effects, then it's very tempting to think
that the quantity you want to estimate this is Yi(1)- Yi(0).
Remember Yi(1)?
Is the outcome you would've experienced with treatment.
Y(0) is the outcome you would've experienced with control.
So delta I, as defined here, is kind of literally
the effect of the treatment on the I person.
As defined in the potential outcomes, it's called the individual treatment effect.
And, of course, if you could know the individual treatment effects,
then you would want to estimate but the problem is that
these individual treatment effects are fundamentally Unknown.
For any specific person, either I give you the treatment or
I don't give you the treatment.
I can never both give you the treatment, and not give you the treatment.
And that means that kind of for any specific person,
either you observe Yi(1) or you observe Yi(0).
You can never observe Yi 1 minus Yi 0.
So you can never observe, Delta I, for
a single person that's fundamentally unknowable and
kind of to estimate treatment effects.
We always need to kind of make comparisons across some people,
who got treated and other comparable people who didn't get treated.
In kind of targeting the average treatment effect in a randomized trial,
for example, is a very kind of simple way of making this and
direct way of making these comparisons work.
So anyways, we don't want to estimate delta i, what do we want to estimate,
we want to estimate what's called the conditional average treatment effect.
And this is essentially like the average treatment effect.
So it's looking at an average of treatment minus control potential outcomes,
except instead of estimating an average overall,
we're estimating an average conditional on covariance.
So, we're not asking kind of for this specific person,
what was their treatment effect?
But if there are say some covariance you imagine the day data set,
kind of based on gender age medical history and so forth.
You can ask well for
people with this type of covariates on average what was the impact, and
kind of when when I'm going to say we want to measure treatment heterogeneity.
What I mean is, I want to understand how the conditional average
treatment effect varies with kind of access, you can measure.
All right, this was just kind of like quick a quick, you've already
talked about the conditional average treatment effect a bunch last week.
And then just to kind of have a clear target in mind,
today when I say we want to estimate the conditional average treatment
effect function when I'm going to mean.
Specifically is that we want to capture it accurately and
mean squared error, that is kind of.
We have a distribution of people, every person has their x covariance in terms of
your x's you have your true tower x, your true conditional average treatment effect.
And then we're going to estimate how that of x, a conditional
average treatment effect for you, and we want that kind of for the typical person.
These are close, so that's the conditional average treatment effect.
And like I said earlier, we want to estimate conditional average treatment
effects in observational studies.
This is the same setting we saw in week two.
We're not going to assume that the treatment is randomized.
But we're going to assume that they are axes that you can measure,
so that conditionally on x the treatment is as good as.
This is the assumption that we saw let you identify average treatment effects.
And also in very much the same way,
I'll let you identify conditional average treatment effects.
And again, the statistical difficulty when working under this
assumption is always just how do you efficiently get rid of
confounding effects due to X without pricing your estimates.
So all right, just the last thing I wanted to
say before diving into kind of the details for
this lecture, is just to say a few words of kind of
why is estimating kind of treatment heterogeneity
difficult, what are things to look out for.
And maybe also, what are situations where you think estimating treatment effect
heterogen might be easier or harder, and kind of just so what to pay attention to.
The reality is that kind of accurately estimating heterogeneity treatment effect
functions is much harder.
Then accurately estimating just an average treatment effect,
you often need more data.
And there are many problems settings,
you'll see this if you work with the datasets provided in the class,
where you might go look for heterogeneous treatment effects, and
just kind of with the available data, you can't find anything.
This is normal estimating treatment heterogeneity the hard task.
It's kind of one order of magnitude harder,
than estimating an average treatment effect, you're going to need more data.
But also kind of it doesn't mean that it's not worth looking for
treatment heterogeneity, because when you do active medical treatment,
you find that the treatment not only helps on average.
But that the treatment of some people more than others, then you can kind of be
smart about prioritization or who should be getting the treatment or not.
And this can have huge value.
So when you can find heterogeneity is often very good, but
you're not always going to be able to do so.
And the reason, often all it takes is to think in terms of
an application to see why this would be quite difficult.
So imagine, we're looking at victims of hemorrhagic shock that's heavy heavy, and
bleeding and we're looking at the effect of a treatment administering fibrinogen.
It's something people believe helps.
And we're looking at the effect of fibrinogen on mortality among victims of
hemorrhagic shock.
And let's say we also wanted to look for heterogeneity here, there some people who
benefit more from fibrinogen than others and preventing mortality.
So, it's an important question.
But there are some things that kind of stand out immediately when you think
about working the problem, first, right?
The outcome will care about it's mortality,
that's getting very drastic outcome.
The condition we're looking at Hemorrhagic shock, that's also very bad.
So there's kind of a lot of stuff happening in this problem,
relative to that administering fibrinogen.
Maybe it helps a little bit, but
there's kind of a clear upper bound to how much it can help, right?
This is data collected by first responders,
you could imagine that there's some people you get to them.
You'd see that they're so severely injured, that there's nothing you can do
to help them, and they're very likely to die regardless.
On the other hand, you might come to other people and
see that they're bleeding, but essentially they're mostly fine.
And they're going to survive kind of regardless.
There might be some group of people in between who are kind of on the fence, and
where administering paper written early might really make a difference.
But the thing is, there are some people who kind of might be at 90
A 90 plus percent, chance survival others but were 10.
If administering paper increases survival chances for some people by 5%,
that's very, very valuable.
But the kind of magnitude of the treatment effects, is just going to be much,
much weaker than base.
So this kind of shows that kind of relative to everything else going on
the problem, treatment effects even when very important.
Might just kind of in absolute magnitude be very small.
The other thing again in this problem is that, again you have data from first
responders, they might be more likely to administer fibrinogen to people who
kind of based on observed characteristic, they think might need it more.
So they may be selection and measured features excite,
you need to deal with this.
And again, when paired with the fact that treatment effects are weak especially.
Hey Britain, heterogeneity may be weak,
you might really be worried that kind of confounding effects might be
big relative to the size of the treatment effects you're interested in.
So you really need to be careful in controlling for confounding.
So these are these first two challenges,
how to accurately estimate treatment effects.
While recognizing that treatment effects are probably weak and
there may be up confounding to the measured features X.
This is what we're going to talk about today and kind of design
methods that are kind of our best attempt to address these difficulties.
There are other difficulties too, there may be confounding due to unmeasured
features, there may be interference were kind of.
Say you gave your dose of fibrinogen to one person which means you can't give it
to someone else.
And you have come this cross subject effects,
these are both kind of very delicate topics.
They're important, not going to talk about them in this lecture,
but I just wanted to flag them.
So you're aware to look out for them in case, they show up in your application.