This video will focus on randomized trials with noncompliance. So we'll begin by discussing the difference between the assigned treatment, which will think of as an instrument, and treatment received, which will think of as a observed treatment in a randomized trial. We'll also discuss what potential treatments are and we'll think about the causal effects of treatment assignment. So we'll begin by thinking about randomized trials and here Z will end up being the thing that we actually randomized. So Z is randomization to treatment or you could think of it as randomization to some kind of treatment assignment. So, for example, Z could take value 1 if you randomized to treatment and could take 0 if you randomized to, say placebo, or no treatment, or the control condition. So, essentially we were thinking of Z as something that we'll randomize and then we hope that the person will do. So, if somebody is randomized to treatment, then we'll basically ask them to take the treatment so that's what Z is. Whereas A is what people actually receive, so it's treatment received. So it's 1 if you receive the treatment and 0 otherwise, and then we'll have some outcome we're interested in Y, and so, in practice in randomized trials, Z and A will not always be equal to each other. So sometimes people will be randomized to treatment but they do not actually take the treatment. So for example, if you are in a drug trial, you might be randomized to take some drug and then you might just not take it, alright. So sometimes there's noncompliance, and so, we want to distinguish between this treatment received, which is A, which is really the thing we're interested in is the effects of actual treatment versus treatment assignment which is where the randomization happens. So, if you have noncompliance, then essentially this randomized trial is starting to look like an observational study in the sense that we might have confounding, and here what we mean by that is confounding in terms of actual treatment received. So there could be variables that both affect what treatment you receive and the outcome. So, we'll call those variables X. So X here are confounders and so you'll be assigned some treatment based on Z, right, so Z is the thing that we randomize, and you know, we're hoping that if you randomize to treatment that you'll take it, but whether or not you actually do might depend on other variables. So X are these confounders. It could be, you know, your age or how healthy you are, and for example, maybe people who are less healthy might be more likely to take the treatment, hypothetically. So that's the kind of thing we have to worry about. So the part of this DAG over to the right looks very familiar where the sort of classic confounding kind of situation. What makes the setup different is we have this randomization Z that's also affecting treatment. So, to think about this problem, what we'll do is we'll begin by first pointing out the observed data are this set Z, A, and Y. So for every person, we will observe what their treatment assignment is. We'll assume here that we observe their treatment received A and we also then eventually will observe some outcome for them. So that's the observed data, Z, A, and Y. So essentially, you know, for a given subject, you know, they are assigned treatment Z and they receive treatment A, but of course, their treatment might be different than... their observed treatment might be different than the one they were actually assigned. But what we can do now is just slightly define potential outcomes in a separate video. We can also define potential values of treatment here. It's the same kind of idea. So we could imagine that each person has two potential values of treatment. So the first one is Aᶻ=¹, and so, you could think of this as a sort of happening before the study even begins. So you have, there's some patient or some subject in your study and if you happen to reassign them, you randomize them to receive the treatment, so Z =1, then they would end up actually receiving some value of the treatment which what we're calling A¹. So that could be either the treatment or no treatment, all right, so A here, A¹ just means, it's the value of the treatment that would occur if they were prescribed or assigned to the Z =1 group. So A¹ here could end up being treatment or no treatment. It's just the value of that variable if they happen to get randomize to Z =1, and so, also we could think of potential treatment under the control condition. So if somebody, if the same person instead had been assigned to Z=0, what would their treatment be? Well, we'll call that A⁰. So that's the value of the treatment of the randomized Z=0. So basically, think of each person as having a value for A⁰ and A¹. Even before the study begins, we'll imagine that that exists. You know, we'll never see them both simultaneously but we still imagine that they exist. So now we could actually if using a sort of potential treatment idea, we can actually then think of a causal effect of treatment assignment and treatment received. So, this is going to look familiar, when we were looking at potential outcomes and causal effects, but now we're just thinking of the causal effect on treatment received. So here, we have a causal effect. So we're just contrasting the treatment that you would have received if you were assigned Z=1 versus the treatment you would have received if you were assigned treatment Z=0, and then we take the means, so we average over the whole population. So this, in words, it's a proportion treated if everyone in the whole population had been assigned to receive treatment which is equal 1 minus the proportion who would have received treatment if no one had been assigned to receive the treatment where Z=0. So this contrast, if there's perfect compliance, if everybody did what they were told, would be equal to 1. So A¹ would equal 1 for everybody and A⁰ would equal 0 for everybody. So, in the world of perfect compliance that would be what you would get. So this is a causal effect, right, because we're contrasting potential outcomes on a common population. So that's all causal effect ever is as a contrast to potential outcomes on a common population. So, this is the population level causal effect because we're not conditioning on anything. So we're looking at the population as a whole. Now, turns out we can actually estimate this from observed data. Since we have, we've randomized Z, and as long as we can assume consistency again where consistency just means, for example, that A¹ would just be equal to your value of A if you happen to be assigned Z=1. So the potential outcome that you would receive if you were assigned Z=1 corresponds to your treatment received if you did happen to be randomized to Z equal 1, okay. So that's a pretty basic assumption there, and we also have randomized, so assuming that we've actually randomized, then we can estimate this from observed data. So for example, the expected value of the potential treatment under Z=1, would just be equal to the expected value of treatment given Z=1. So if we restrict to the subpopulation of people who are actually assigned Z=1 and we just take the mean of A for that group and that should be the same as the expected value of A¹ in the whole population and that's because of randomization. So if I stratify, if I only look at the subpopulation who were assigned treatment, that subpopulation should have the same characteristics as the whole population because we randomized, and I say it's estimable because you'll notice this expectation only involves observed data. So we observe A, we observe Z. So a simple way to estimate this, for example, would just be to look at the subpopulation that has Z equal 1 and take the sample mean of A. So this is something we can estimate, same kind of thing for A⁰. We could just look at the subpopulation that were assigned to control condition and take the expected value of A. So that's the causal effect of assignment on receipt of treatment and this is observable from the data. So you can estimate that, you can identify it. We also can think of the causal effect of treatment assignment on the outcome itself, so then this would be known as an intention to treat effect, so we could contrast the outcomes had everyone been prescribed or assigned Z=1 versus the outcome had everyone been assigned Z=0. So this is an intention to treat effect because it's basically contrasting outcomes based on what you intended to do. So you know, our intention was for people assign Z=1 to get treatment, you know, so whether they did or not that was our intention. So that's what's meant by intention to treat effect, and if there was perfect compliance, then this would be an actual effective, causal effect of treatment itself, right, because Z=1 would always directly exactly correspond with A=1. And again, so this is actually something we can estimate from observed data as well. So this is identifiable as long as we have randomization. So for example, the expected value Yᶻ=¹, so this is the average value of the outcome had everyone in the population been assigned Z=1, but we've randomized Z=1, so therefore, we could just condition on Z=1 and then take the mean of the observed outcome. And so again it's identifiable because both Y and Z are just observed data and so we could estimate that. So because we've directly randomized Z, then we can estimate a causal effect of Z on anything really, so one treatment received or on the outcome. So those two things are relatively easy to do. So getting a causal effect of assignment on treatment and getting a causal effect of assignment on the outcome. But what about the causal effect of treatment received on the outcomes? So rather than treatment assignment on the outcome, what about treatment received on the outcome? So this might be of greater interest to us is what was the causal effect of treatment itself and the outcome? So we could think of Z as encouragement to receive the treatment and then a randomized trial like this you can think of it as strong encouragement because typically even though there will be some noncompliance in a randomized trial, a lot of times there's various kinds of blinding and so on, and you know, compliance tends to be pretty good. So you can think of this as strong encouragement and that most people will end up doing what they're told. So Z here is an instrumental variable. We think that it certainly should affect treatment received, so treatment assignment should affect treatment received, but we don't think the random assignment necessarily would affect the outcome directly. So this sort of process of randomizing treatment, we don't think in general should affect the outcome directly. So we could think of this as an instrumental variable, but the question that is does that actually help us estimate the causal effect of treatment itself? So this is really the main goal of instrumental variable methods is to use this instrument, this randomization to exploit that to try and estimate the effect of treatment itself. So this will be a topic for another video.
