Hi. So in this module I'm going to talk a little bit more about effective connectivity and explore the link between effective connectivity and causation. So the goal of functional connectivity analysis, is to make inferences on the structure of relationships among different brain regions. A goal of effective connectivity analysis on the other hand, is to make statements about causal effects among tasks and regions. So effective connectivity provides much more theoretically powerful inference. But also much stronger assumptions. And so the validity of the conclusions depends strongly on these assumptions being correct. And so the necessary assumptions are often poorly specified and difficult to check. Which I think is a major short coming of the field. So ultimately, personally I feel that the distinction between functional and effective connectivity is not always entirely clear. So if the discriminating factors between the two are a) a directional model in which causal influences are specified, and b) the willingness to make claims about direct versus indirect connections. Then many analyses, including multiple regression, might count as effective connectivity. So, to me, in the end, it's not the label or function or effect that's necessarily important. But the specific assumptions, robustness and validity of inference supported by each of these methods. And more care needs to be taken in discussing these concepts in connectivity studies. And at the end of the day, connectivity should be discussed in terms of carefully defined estimands of interest, and not on the applied estimation algorithm. So how should one talk about causation and effective connectivity? Well there's lots of ways of doing this, but as a statistician, I like to use what's called the potential outcomes notation. So I want to briefly describe this to you, as I hope maybe it will be useful to you and your research in thinking about causation and connectivity as well. So in statistics there's a growing literature on causal inference. And it's based on ideas from experimental design. So this requires that causal relationships sustain what's called counterfactual conditional statements. And this can be expressed using potential outcomes notation. And this gives conditions under which commonly used estimates actually estimate causal effects. And so most connectivity methods can be expressed in this potential outcomes notation. And the assumptions for making causal interpretations can be derived. So let's illustrate it here. So let’s consider that you have an experiment with six different subjects. Half in a treatment group and half in a control group. So we have six people in our study, and this is a study of say a blood pressure lowing medication. Three of the subjects we give the blood pressure lowering medication. The other three subjects we give a placebo. Now supposed we measure an outcome for each subject. And we want to determine what effect the treatment had on the outcome. So here the treatment is z. You either received the medicine or you received the placebo. And y is the outcome. And this is maybe your decrease in blood pressure or something like that. Like that. So let's define this using mathematical notation. so let's let Zi be the treatment given to subject i. So Zi is equal to one, if subject i is treated, and equal to zero if subject i was untreated. So let's say Z1 is equal to one if the first subject received the drug, and Z1 is equal to zero if he instead received the placebo. Okay, lets let Y i denote the outcome, so this is how much the blood pressure lowered for subject i so Y one would be how much the blood pressure lowered for subject one. Now we're going to define something that's a little bit different. These are called potential outcomes, so there are two possible potential outcomes that could have occurred in this study. So Yi one is the outcome if subject i is treated, so Yi one is how much the person's blood pressure is lowered if he received the treatment. Yi zero is the same outcome if subject i is untreated, i.e. He received the placebo. So here there's two possible outcomes of this experiment. Either we give the person the treatment and we observe Y1. Or we give the person the placebo and we observe Y0.one Having defined these two notations, we can now define a cause and effect. So what is the effect of the treatment on the subject well the that would be in mathematical notation, Y,(1)- U,(0). Because this is simply just looking at what is the outcome for this subject. How do you receive the treatment minus the outcome for the same subject had he received the placebo. So basically the only thing that's different between these two outcomes, is that in one case they received the treatment, and in the other they did not. So this is a true causal effect. And so what we want to do is we want to estimate this. However the problem here is that we can only observe one possible potential outcome. because we either gave the person the treatment, in which case we observed Y one. Or we gave them the placebo in which case we observe Y zero. So this becomes a missing data problem. And so, if you look at the table at the bottom, we see the six subjects here. The three of the subjects that received the placebo we observed their Y zero, and for the subjects four to six, we gave them the treatment in which case we observed Y one. So for none of the subjects we can actually estimate why one minus y zero because we have this missing data problem here. Now maybe we can get around this by instead estimating the average treatment effect across the whole population. In this case it would be the expected value of Y one minus y zero over the population. However unfortunately, since we can't estimate Y one and Y zero for any of the subjects in the population, we can't actually estimate this delta as well. However, we could estimate delta from the observed data using the following equation, which I'm going to call d. So d is equal to the expected value of y given z equal to one minus the expected value of y given z equal to zero. So basically what we're doing here is we're basically saying that, well the people for which we gave treatment, you know, those guys can be used to sort of. We're going to take the average of the people who receive the treatment, and then take the average of the people that didn't receive the treatment and take the difference between the two. That's what d is doing. So that's not really equal to delta, but it's equal to delta in the sense that we're now using the values of the other subjects to sort of fill in the blanks here. So in general, d need not equal delta, of course. So a non-zero value of d simply implies that Z and Y are associated with one another, but not that Z causes Y. However, we can equate these two terms by making assumptions. So if the potential outcomes are independent of z, then d and delta are equal to each other and association is causation. Because if they are independent we can get rid of this conditioning and they become the same equation. And so when is this true? Well this is true if subjects are exchangeable with respect to their treatment. So basically here we're saying that the people that received the treatment, well they're very representative of what the people who didn't receive the treatment would have gotten, had they received the treatment, and vice versa. And when will this be true? Well, this will generally be true if the treatment is randomly assigned. And so if we randomly assign the treatment, we can start making causal conclusions, based on the observed data here. So this sort of suggests the framework for performing causal inference here. So what we do is, we first construct the causal model using potential outcomes to spellout the causal effects of interest. Then we construct an associative model from which we can estimate equivalent effects from the observable data. So the first step would be the delta that we want to estimate. D is now the term that we can estimate but that we're not really interested in. And then the final step is to find conditions under which the effects from the causal and associative models are equivalent to each other, i.e. When we can say that d and delta are the same. All right, so how does the potential outcome notation allow us to think about connectivity and causality? Let's just do a simple example here. Let's suppose subjects are randomized to perform either a stress task, in which case Z = 1, or a control task in which case Z = 0. Let's have X be the brain response in the key stress related area of the brain, and let Y be the performance on some sort of stress task. Now if you want to study the relationship between Z and Y, you might now be interested in using the mediation framework that were discussed in previous lecture, and see whether or not the brain responds, mediates this relationship between Z and Y. Fine. And let's say that we're interested in particular on this link beta here, which is what happens if, for the treated subjects, we move our brain response from value x to value x plus 1. So what we're interested here in beta is the effect in the treated group If the brain responses changed one unit from X to X + Y. So this is important, because basically what we're interested in here is seeing, what is the causal effect of changing your brain response one unit? So what is the causal effect on the outcome? So this is something that we might be interested in estimating. So now that we've defined something that we're interested in estimating, and we have to figure out a way to estimate it. So a meditation model is often used to make inferences on the parameters of this type. And the validity of these inferences, equating association and causation, often rest on assumptions that are not part of the model. So in order to estimate beta, a path model would implicitly compute b, which is the difference in response between treated subjects with a brain response x and those with brain response x+y=. So we'll be comparing different subjects to one another. So in this case, for beta to be equal to b, subjects must be exchanged, but with respect to both their treatment and their brain response. And while we can randomize people to treatment, we can't randomize people to have a certain brain response. And so that's one of the shortcomings here, is that we can't really make causal conclusions using sort of the mediation framework. Unless we can sort of randomize the brain response, or assume that they behave as if there were randomly assigned, and that's a very strong assumption to make. So, in general, what I was trying to capture here is that effective connectivity is thought to provide more powerful conclusions than functional connectivity. But it also makes stronger assumptions. And so sometimes the difference between the two aren't so obvious. And the validity of the causal conclusions depend strongly on the assumptions being correct. And often the necessary assumptions are poorly specified and difficult to check, which is a major shortcoming of the field. And so in general, I believe that more care needs to be taken in discussing these concepts in connectivity studies. And in general, connectivity should be based on carefully defined estimates, not on applied estimation algorithm. So that's why I think a technique such as potential outcomes can potentially be quite useful in neuroimaging research. And that's why I wanted to sort of include this module as well. Okay, that's the end of this module and this is the end of our modules of effective conductivity. And in the coming module we'll be talking about multiboxal pattern analysis. Okay I'll see you then. Bye.
