Okay, so, so far, we've discussed identification and estimation of various kinds of mediated effects, average direct and indirect effects, controlled direct effects. Okay, now, you recall from way back when module one, identification, you recall that this mediation analysis require rather heroic assumptions, and therefore, it's useful to try to asses what happens if these assumptions aren't met. Now, I'm going to talk a bit about sensitivity analysis for mediation, and hopefully you'll recall that we discussed that a little bit in one. Okay, because if you recall the discussion there, then what we're going to do here is just a very straightforward extension of that. It involves just taking in the mediator. Okay, so I'm going to omit this subscript i in this lesson, and I'm going to review the approach in part one. Okay, so you have observed confounders, X, and you have an unobserved confounder, U, and you have treatment assignment, Z, and you have your outcome. So it's just like in one. Now, the goal is to estimate the average treatment effect at the value X. It could be something else, but you'll get the idea here. Now, the assumption is the treatment assignment is unconfounded given, X, and the unobserved covariate, U, but the investigator assumes unconfoundedness given X. So now, the bias from ignoring U, it's pretty simple to write out, is what the actual difference in expectations in the treatment group and the control group condition on X. Okay, because if we didn't have U in the way, that would be the average treatment effect at X, but it's not because of the U. And then, we have to difference out what we really want which is the average treatment effect. So you see that that's just the straightforward expression for bias. Now, we're going to rewrite it a little bit. Now, we're going to have this unobserved continuous confounder, U, the bias at X can be written as follows, you can just do that yourself, it's pretty simple. If you look at the first expression, that's just the expected value of Y, given X and Z equals 1, minus the expected value Y1 given X. And then the second expression is the same for in the control group. You can write that out for yourself using the previous definition of bias. Now, if I integrate the bias over the distribution of X, then I can get the unconditional bias. But we're not going to worry about that right now. But I mean, that's how you would do. Now, we saw, we expressed the bias. And so, the utility of the sensitivity analysis will hinge on whether or not the investigator can make some reasonable assumptions about the unknowns in the previous equation. So here's what VanderWeele and Arah assume about this. So they make this assumption that the expected value of Y, given X and Z equals 1, and U is X in value U versus the same thing when U takes on some different value, but those guys are equal in the treatment group and the control group, BC. And I'm going to come back to that momentarily. Now, then they say in this special case where U is binary, the integral of course is replaced with the sum a little arithmetic gives you this nice little expression. And then, a sensitivity analysis would then proceed under various assumptions about the magnitude of these two components. And you can obtain similar results for risk ratios, but I want you to see something, and that's the assumption that they've made that those two guys are equal in the treatment group and the control group. That's a pretty strong assumption. So rearranging and using the unconfoundedness assumption will give you the following. So what we're seeing is the expected, the average treatment effect at X and the value U is equal to the average treatment effect at the value X, and U equals u prime. And a sufficient condition for this to hold is that the difference between Y(1) and Y(0) is independent of U, given X. So given the covariates, X, the difference in the potential outcomes does not depend on the unobserved confounder. Now, this is certainly more credible than assuming the potential outcomes are independent of U, given the observed confounders, but the condition still seems quite strong. An investigator who is unwilling to make this strong assumption may also generally be unwilling to make the weaker assumption above. A minor modification of the results above leads immediately to sensitivity analysis for the control direct effects. To that end, suppose the previous assumptions we made to identify these effects hold, once we add U to the conditioning set. So now, we just go through and essentially now we have a mediator in there. And the bias can now be written, okay. And you'll see that it's essentially the same expression, except now we also have the mediator M in there, as before, if you integrate over the distribution of X, you get the unconditional. Now, VanderWeele is considers the case of the control direct effects, and then if we assume the same kind of assumption, you'll see that Z equals 1 in the top and Z equals 0 in the bottom, so we're assuming inequality between the treatment groups and the control groups. Then in the special case where U is binary, the integral again is replaced with the sum, and we get a nice, tidy, little expression for the bias, parallel Z expression we had previously, except now, M is now in there as well. And as before, sensitivity analysis would then proceeded under varying assumptions about the magnitude of these two components. Now, as before, we can rewrite that condition that's assumed in the following sort of way. We see that we're making again a pretty strong assumption in order to do the sensitivity analysis, an assumption we may not be comfortable with. Under additional types of conditions given for the identification of average direct and indirect effects, the bias due to ignoring U for these quantities can also be obtained, and I refer the reader to the book by VanderWeele that's previously been mentioned, the sort of encyclopedic book on mediation. And I should also mention that, for the case of linear models, Imai et al has some nice results. In a randomized study where U is not a confounder for the relationship between outcomes and treatment assignment Z, the confounder U is needed to make the relationship between the potential outcomes Y(z, m), and the mediator M conditionally unconfounded. A crucial point is that in the approaches above, U is either a pre-treatment covariate, or a post-treatment variable not affected by Z, the more likely case of cofounding by post-treatment variables affected by Z is not covered above. All right, so now, I'm going to turn to different approaches to dealing with intermediate outcomes, namely instrumental variables and principle stratification.