In this video, we'll discuss assumptions related to instrumental variables. So we'll begin with assumptions about what defines an instrumental variable. How do you know if it's a valid instrument? What assumptions do you have to make? And then we'll also introduce the monotonicity assumption, discuss what it is, and why it's helpful for identifying causal effects. So we'll look in some detail here at assumptions about instrumental variables, meaning what makes a variable a valid instrumental variable. So what is the definition of an instrumental variable? And so, it has to be associated with the treatment and it has to affect the outcome, I mean, it can only affect the outcome through the effect that it has on treatment, so it cannot directly affect the outcome. So the second assumption is known as the exclusion restriction. So here's this kind of common picture that you, kind of DAG, that you might look at if you – when you think about instrumental variable problems. So here, Z is the instrument, A is treatment, Y is the outcome, and X could be – you can think of X as confounders, confounding the relationship between A and Y. So the first assumption was that Z must be associated with treatment, so that you'll see that there is this arrow from Z to A. So Z is affecting treatment. So that's our first assumption. So if you were trying to decide if a variable is an instrument, you know, one of the first things you would want to do is think about, well, does it affect treatment? And hopefully, it strongly effects treatment. And then it must not, it cannot directly affect Y. So you'll see that I have the X through this arrow, so you can't have a direct arrow from Z to Y. So we're excluding that as an option – this is the exclusion restriction. So, if we think about a randomized trial, for example, we are randomizing treatment and then A is treatment received. Randomization should not be affecting the outcome, except possibly through the impact that randomization had on what treatment somebody received. So that's an assumption that we have to make, is this exclusion restriction. This exclusion restriction can also take another form or a part or it can be sort of viewed in another way. So imagine we have some unmeasured confounders U. So, the part of the reason we're interested in instrumental variable analysis is because we think there might be unmeasured confounders U. While in exclusion restriction also says that Z cannot be directly affecting U. OK, so you'll notice here that there is an arrow from U to Y and if there was also an arrow from Z to U, what that would mean is that Z is affecting Y through some path that's not A, right, so we could go from Z to U to Y. So Z could be then affecting Y, not through its effect on treatment, and that we're saying cannot happen if it's a valid instrument. So you could think of the exclusion restriction as satisfying both of the two, from the two slides that I just showed you. So, there's no direct arrow from Z to Y so I had an X through that. And also, if you have unmeasured confounding, there can't be an arrow from Z to U in that case, in this case. So Z cannot be directly affecting Y and it also can't be indirectly affecting it through its impact on some unmeasured confounders. So basically think of – if Z is a valid instrument, that it's affecting treatment, and if it is affecting the outcome, it's only affecting it through its effect on treatment, not through some other path. So that's the exclusion restriction. And so, one of the things if you're going to do an instrumental variable analysis, you'll need to think carefully about is whether the assumptions are reasonable. And most of the debate about whether the reasonable sort of centers around the exclusion restriction. That's the one that takes more justification. But let's think about the case where we're in a randomized trial. So imagine Z as random treatment assignment. Well, in that case, we should be really confident that it affects treatment received. It would be a pretty terrible randomized trial if treatment assignment did not have any impact on treatment received. But this is also something that you can check with data, you can see if treatment assignment is associated with treatment received. So one nice thing about this assumption is it's checkable, so you can use your observed data and really see if it's true or not. But let's think about the exclusion restriction. This is one that people worry about more in general. So one argument that it would be valid is if you think of randomization as just a coin flip. Well, there's no reason a coin flip should affect the outcome or affect unmeasured confounders or anything like that, right? We're just flipping a coin, we're just randomly generating a number. So, in general, we don't see why this would impact the outcome, other then through maybe the impact it had on treatment. However, you can imagine that in some randomized trials there might, maybe there's not blinding. So blinding would mean that – so if there's blinding in a randomized trial like, for example, the subjects or the patients wouldn't know which treatment they're receiving. So in a like a placebo-controlled trial, hopefully, the people don't even know what treatment they're receiving. So in that case, there's no reason to think that whether they're getting the active treatment or the placebo would be, I mean that sort of treatment decision itself would be affecting an outcome. But imagine that they're not blinded, so they participate in a randomized trial, but they're told what treatment they're receiving, or perhaps that knowledge could affect them in some way. So maybe it affects their behavior, maybe it affects other things they do. So it's possible then that it could impact the outcome in ways that aren't because of the treatment itself. Or if, you know, if you're in a clinical kind of setting where the clinician is sort of administering the treatment, if they're not blinded, knowledge of what they're sort of giving the patient might affect other ways in which they treat this patient. So it's possible that the exclusion restriction could be blinded in that case. So whether or not this assumption holds really just needs to be examined within the context of any given study. So you need to think about what is the exclusion restriction, and are there reasons to think that it might be violated? So unfortunately, this one, it's not something that is technically checkable from data. There are sort of – there's some ideas that people have come up with to kind of get at whether the assumption is met, but it's really more of the kind of assumption that you just have to use, sort of your subject matter knowledge and then have a degree of faith about it. So this is one of the kinds of assumptions, just like unmeasured confounding where you never can know for sure if you have captured all of the confounders. This is the same kind of thing with the exclusion restriction – you can't know for sure if it's met. So how confident you are about the results will depend on how confident you are about the assumption being met, so you have to think very carefully about it, whether it sort of makes sense in your particular case. So, as a reminder, if we have a valid instrumental variable, we want to use that to help us estimate what's known as a local treatment effect or a complier average causal effect in the case of randomized trials with noncompliance. So we're interested in the causal effect of treatment itself among the subpopulation of compliers, so this subpopulation of people who will take whatever treatment they're assigned. OK, so there's a challenge with identification here of the causal effects. So as a reminder, we can take these pairs of potential treatment values A^0 and A^1 and then we can label individuals based on that combination. So for example, if A^0 is equal to zero and A^1 is equal to zero, those are the never-takers. And as we've discussed, we're interested in this group. So in an instrumental variable analysis, we're interested in the causal effect of treatment received among compliers. But the challenge is that we don't actually know who the compliers are. So, again, we can write down the observed data Z and A, and based on that, we can narrow it down to two groups of what an individual falls in. So for example, if somebody was assigned no treatment and they didn't take the treatment, then we also know one of the potential treatment values, we know what would have happened, you know, if they had not been prescribed the treatment or assigned the treatment. All right, so we know that A^0 is equal to zero. So then we can narrow it down to that they're either a never-taker or a complier. And so we can do that for each possible combination of Z and A, so we can narrow it down to two, but we don't actually know for any given person which group they're in. And, you know, recall this is related to the fundamental problem of causal inference, which says that we never observe both potential outcomes here, we never observe both potential values of treatment, so we can't know for sure which compliance class people are in unless we make additional assumptions. So that motivates the need for an additional assumption – so a common assumption that people make is something known as monotonicity. And you could think of the monotonicity assumption in this setting as just meaning that there's no defiers. So we could make the assumption that the defiers don't exist in our particular study, so there's no one who consistently does the opposite of what they are told. So if we think of the instrumental variable as encouragement, nobody takes treatment when encouraged not to in a sense and doesn't take the treatment when they're encouraged to. All right, so nobody does the opposite of what they're told. So that's something that you might be willing to assume – that's known as a monotonicity assumption. In general, it's called monotonicity because what we're saying is that if you increase the amount of encouragement, that should increase the probability of treatment. So, so far we've thought about instrumental variables just in cases, in settings where it's binary, you know, either encourage or not, but you could also think of instrumental variables as being continuous where it's like a dose of encouragement and the monotonicity assumption there would have to deal with, as you increase the dose of encouragement, the probability of treatment should go up. So this is an assumption we could make, it's just that there's no defiers. And, you know, if you're in a randomized trial, you know, let's say it's a placebo-controlled trial so there's some active drug and some placebo, the people who were prescribed the active drug might not have access to the placebo for example, or people prescribed, assigned the placebo, they might not have access to the active drug and they might not even, you know, hopefully they don't even know which group they're in. So it might be a reasonable assumption in a lot of situations. So let's go back to our identifiability problem – so, the standard case where there's no monotonicity. So, so far all we've done is made standard instrumental variable assumptions, which, you know, that the variable affects treatment. And we also made the exclusion restriction assumption, that the variable does not affect the outcome except through its effect on treatment. So we've made the IV assumptions, but we haven't assumed the monotonicity. In that case, we're stuck in this situation where we don't know what group people are in, but we can narrow it down to two. However, if we make the monotonicity assumption, the problem gets a little easier. So you'll notice that, the monotonicity assumption says there's no defiers, so in that case I'll know this value, which is, these are people who were assigned to the control group but they actually took treatment. So we already knew that A^0 is equal to one, but since there's no defiers, we also know that they would, they must be an always-taker; all right, they can't be defier, so A^1 couldn't be zero in that case, so they have to be an always-taker. So now, if anybody who has Z equals zero and A equal one, we know they are an always-taker. So as long as we make the no-defier assumption or the monotonicity assumption, then if we observe a Z equal zero and an A equal one together, we know they're an always-taker. It also works the same where if you observe somebody to have Z equal one but they don't take treatment, so these are people who were assigned treatment but didn't take it, right? So we already knew then that A^1 was equal to zero, but now we also know that A^0 would have to be equal to zero, right? So these are people who were encouraged to take treatment but didn't, and so therefore if they weren't encouraged, we'll assume that they also wouldn't take treatment. So therefore, they're never-takers. So you'll notice now that we've – for some people, we're able to identify what group they're in. We're still interested in compliers, but by making this monotonicity assumption, we've made the problem much easier. And in fact, we've made the problem easier enough to the point where we can actually identify the causal effects. So we'll look at how we actually estimate the causal affects in another video. But the key point here is is that if you make this monotonicity assumption, now we actually will have enough information to identify the causal effect that we're interested in, which is the causal effect among compliers.
