WEBVTT 1 00:00:33.500 --> 00:00:36.179 David Stephens, Prof: Okay, so we'll pick up with the… 2 00:00:37.820 --> 00:00:44.579 David Stephens, Prof: Rest of the material, and, build on what we saw in the first half session. 3 00:00:45.430 --> 00:00:52.259 David Stephens, Prof: And, we're gonna put… To use the basic ideas 4 00:00:52.830 --> 00:00:57.320 David Stephens, Prof: Of these simple graphs into more complicated structures that allow us to 5 00:00:58.570 --> 00:01:04.430 David Stephens, Prof: construct models that will allow us to uncover causal effects. So. 6 00:01:11.840 --> 00:01:19.040 David Stephens, Prof: Our primary interest in causal inference is really assessing the association between, or the impact. 7 00:01:19.170 --> 00:01:22.099 David Stephens, Prof: of, Treatment Z on an outcome Y. 8 00:01:23.130 --> 00:01:28.110 David Stephens, Prof: What we… Ultimately want to be able to do is to assess whether… 9 00:01:28.670 --> 00:01:32.649 David Stephens, Prof: as a sort of null hypothesis, Z and Y are independent. 10 00:01:33.010 --> 00:01:40.589 David Stephens, Prof: or whether Z and Y are independent, once we've conditioned upon a collection of other variables X, other predictors. 11 00:01:43.140 --> 00:01:54.040 David Stephens, Prof: So… In order to do this, we're gonna use the graphical ideas, and… really assess… the… 12 00:01:54.520 --> 00:01:57.300 David Stephens, Prof: Flow of information through the graph. 13 00:01:58.360 --> 00:02:07.630 David Stephens, Prof: So what we're going to do is to see whether there's… once we, make an intervention, let's say, to change Z, the treatment variable. 14 00:02:07.820 --> 00:02:13.909 David Stephens, Prof: How can that… the impact of that change flow through the graph? 15 00:02:14.270 --> 00:02:17.570 David Stephens, Prof: In order to impact Y at the end of a path. 16 00:02:20.760 --> 00:02:21.430 David Stephens, Prof: Okay? 17 00:02:23.590 --> 00:02:28.530 David Stephens, Prof: So what we're going to be thinking about is whether we can assess the direct impact. 18 00:02:29.280 --> 00:02:32.839 David Stephens, Prof: of Z on Y, whether we can assess the impact 19 00:02:33.540 --> 00:02:39.070 David Stephens, Prof: Accounting for the presence of other variables, whether there are problematic paths in the 20 00:02:40.470 --> 00:02:43.630 David Stephens, Prof: In the graph, that distort 21 00:02:43.900 --> 00:02:47.100 David Stephens, Prof: What our statistical analysis would otherwise tell us. 22 00:02:52.190 --> 00:02:59.250 David Stephens, Prof: So… Really what, and this is going to lead us to the concept of de-separation. 23 00:02:59.830 --> 00:03:07.499 David Stephens, Prof: The… what we saw from the… from the graphical formulation, and then from the structural model formulation, is that… 24 00:03:07.960 --> 00:03:15.900 David Stephens, Prof: a… The cause and effect relationship is captured through… 25 00:03:16.020 --> 00:03:23.250 David Stephens, Prof: Structural equations, or through directed paths That lie connecting. 26 00:03:24.250 --> 00:03:25.710 David Stephens, Prof: One variable to another. 27 00:03:25.880 --> 00:03:34.310 David Stephens, Prof: So, we're really trying to look at the Causal effect of Z on Y, By studying the directed paths. 28 00:03:34.800 --> 00:03:36.820 David Stephens, Prof: between Z and Y in the DAG. 29 00:03:36.950 --> 00:03:37.740 David Stephens, Prof: Right? 30 00:03:38.290 --> 00:03:43.020 David Stephens, Prof: Because then… changing Z by intervention. 31 00:03:43.510 --> 00:03:51.269 David Stephens, Prof: That change will lead to… subsequent changes through the directed path that end up at Y. 32 00:03:54.590 --> 00:03:56.660 David Stephens, Prof: So our focus will be on directed. 33 00:03:59.860 --> 00:04:03.439 David Stephens, Prof: Now… It's pretty clear that… 34 00:04:04.630 --> 00:04:10.329 David Stephens, Prof: Once we, acknowledge the concept of colliders, so remember this is what a collider 35 00:04:10.530 --> 00:04:18.829 David Stephens, Prof: Collider subgraph looks like, so… Here, there's a collider at, X, because there are arrows going in from 36 00:04:19.500 --> 00:04:20.670 David Stephens, Prof: Both sides. 37 00:04:21.170 --> 00:04:22.630 David Stephens, Prof: So if there's a… 38 00:04:23.740 --> 00:04:28.670 David Stephens, Prof: collider there. That tells us straight away that, in this case, Y and Z are independent. 39 00:04:28.780 --> 00:04:40.279 David Stephens, Prof: But Y and Z are not conditionally independent given X. If I condition on that value, then that, in some way, imparts information in both directions to the input variables. 40 00:04:43.070 --> 00:04:47.600 David Stephens, Prof: And the key thing is that a directed path Can't contain a collider. 41 00:04:48.070 --> 00:04:52.219 David Stephens, Prof: Because the directed path, of course, has all the arrows going in the same direction. 42 00:04:55.260 --> 00:04:56.210 David Stephens, Prof: So… 43 00:04:59.980 --> 00:05:06.809 David Stephens, Prof: the directed paths that we're interested in, of course, will have arrows going in the same direction. If there are colliders on any 44 00:05:07.070 --> 00:05:13.360 David Stephens, Prof: Undirected paths, then, in a sense, that, 45 00:05:13.620 --> 00:05:17.280 David Stephens, Prof: Sort of breaks the path at that point, and the… 46 00:05:17.700 --> 00:05:20.059 David Stephens, Prof: The dependence can't flow across that. 47 00:05:23.720 --> 00:05:26.250 David Stephens, Prof: The other thing to say about the Collider, though, is that 48 00:05:26.500 --> 00:05:30.009 David Stephens, Prof: It becomes… the notion of a collider becomes path-specific. 49 00:05:30.310 --> 00:05:36.210 David Stephens, Prof: So, you can have the same variable that is a collider on one path, in this case. 50 00:05:37.040 --> 00:05:41.399 David Stephens, Prof: And in this case, the path between Z and U has a collider at X. 51 00:05:42.220 --> 00:05:44.229 David Stephens, Prof: Right, because there are arrows… 52 00:05:45.590 --> 00:05:54.500 David Stephens, Prof: Both going into X along that path. However, the same variable is not a collider on the path from U to Y, or from Z to Y. 53 00:05:54.870 --> 00:05:59.199 David Stephens, Prof: Because they're there, we're just following this sort of chain structure. 54 00:06:00.600 --> 00:06:03.539 David Stephens, Prof: So the notion of being a collider is path-specific. 55 00:06:09.850 --> 00:06:11.480 David Stephens, Prof: Now some more terminology. 56 00:06:12.390 --> 00:06:17.929 David Stephens, Prof: And we're going to think about this concept of de-separation in two slightly different ways. 57 00:06:19.350 --> 00:06:22.340 David Stephens, Prof: The first one is unconditional de-separation. 58 00:06:23.960 --> 00:06:28.279 David Stephens, Prof: So, if we consider a general path between two variables Z and Y in the graph. 59 00:06:28.760 --> 00:06:33.990 David Stephens, Prof: Then, that path is termed open, or unblocked. 60 00:06:34.220 --> 00:06:36.049 David Stephens, Prof: If there's no collider on the path. 61 00:06:42.760 --> 00:06:46.600 David Stephens, Prof: If there is a collider, The path is closed, or blocked. 62 00:06:49.030 --> 00:06:55.929 David Stephens, Prof: So, in this unconditional setting, Z and Y are de-separated. 63 00:06:56.520 --> 00:07:00.720 David Stephens, Prof: If there is no open path between those variables. 64 00:07:03.680 --> 00:07:05.449 David Stephens, Prof: Okay, so imagine the DAG. 65 00:07:06.190 --> 00:07:09.449 David Stephens, Prof: Imagine identifying where Z is and where Y is. 66 00:07:10.670 --> 00:07:15.210 David Stephens, Prof: And imagine tracing all the paths Between them. 67 00:07:16.660 --> 00:07:19.859 David Stephens, Prof: Forgetting about the directions of the arrows, okay? 68 00:07:20.860 --> 00:07:23.020 David Stephens, Prof: Z and Y are D separated. 69 00:07:23.470 --> 00:07:26.309 David Stephens, Prof: If there's no open path between Z and Y. 70 00:07:29.760 --> 00:07:30.420 David Stephens, Prof: Okay. 71 00:07:31.990 --> 00:07:36.150 David Stephens, Prof: If there is an open path, than ZMYR. 72 00:07:37.990 --> 00:07:38.920 David Stephens, Prof: Deconnected. 73 00:07:42.550 --> 00:07:45.870 David Stephens, Prof: So, an open path, of course, must be… 74 00:07:45.980 --> 00:07:49.889 David Stephens, Prof: Could be open directed, and open undirected. 75 00:07:50.280 --> 00:07:53.919 David Stephens, Prof: So that means the path must comprise chains and forks only. 76 00:07:56.390 --> 00:07:57.880 David Stephens, Prof: This goes back to this. 77 00:07:57.980 --> 00:08:04.440 David Stephens, Prof: Notion here that a directed path can't have a collider. 78 00:08:06.620 --> 00:08:14.210 David Stephens, Prof: Okay? So… Our analysis is going to consist of, drawing the DAG, 79 00:08:14.450 --> 00:08:20.520 David Stephens, Prof: Identifying Z, identifying Y, looking at all the paths between Z and Y, 80 00:08:23.550 --> 00:08:27.460 David Stephens, Prof: And assessing whether those paths are open. 81 00:08:28.320 --> 00:08:34.979 David Stephens, Prof: And in the unconditional setting, The only reason… the only way they can be not open 82 00:08:35.220 --> 00:08:37.910 David Stephens, Prof: Is if they don't have a collider. 83 00:08:39.370 --> 00:08:41.480 David Stephens, Prof: Sorry, the only way there can be… 84 00:08:42.409 --> 00:08:46.370 David Stephens, Prof: closed, I guess, as if they… if they do have a collider. 85 00:08:49.420 --> 00:08:51.879 David Stephens, Prof: Okay, everybody got the idea. So… 86 00:08:53.870 --> 00:08:56.890 David Stephens, Prof: Trace out all the paths, identify the… 87 00:08:57.150 --> 00:09:04.689 David Stephens, Prof: individual paths, check whether there's a collider on those paths. If there's no colliders on any of those paths, then those paths are open. 88 00:09:09.340 --> 00:09:16.049 David Stephens, Prof: So, this is an example taken from the textbook by Rothman, et al, the book on modern epidemiology. 89 00:09:17.540 --> 00:09:23.070 David Stephens, Prof: So, the example they have is that this 5-node Graph? 90 00:09:23.730 --> 00:09:28.720 David Stephens, Prof: X1 and X2, family income and genetic risk. Those are the founders. 91 00:09:29.420 --> 00:09:31.999 David Stephens, Prof: W is parental diabetes. 92 00:09:36.860 --> 00:09:46.399 David Stephens, Prof: Zed is, low educational attainment. Why is the diabetes of the subject? 93 00:09:47.440 --> 00:09:51.590 David Stephens, Prof: And we might be interested in assessing the association between Z and Y. 94 00:09:53.080 --> 00:09:56.259 David Stephens, Prof: So imagine a collecting data set that would allow you to do this. 95 00:09:57.500 --> 00:10:04.320 David Stephens, Prof: So, in this DAG, We can see that… Z and Y are independent. 96 00:10:07.650 --> 00:10:09.310 David Stephens, Prof: They're de-separated. 97 00:10:10.020 --> 00:10:12.769 David Stephens, Prof: There is only one path between Z and Y. 98 00:10:17.620 --> 00:10:19.190 David Stephens, Prof: But it's not open. 99 00:10:19.460 --> 00:10:22.360 David Stephens, Prof: Because it's blocked at the collider W. 100 00:10:24.690 --> 00:10:29.210 David Stephens, Prof: So, that means that… Z and Y are D separated. 101 00:10:29.340 --> 00:10:41.080 David Stephens, Prof: They're independent. The factorization of the joint distribution looks something like this, so X1, X2, and then W given X1, X2, and then Z given X1, Y given X2. 102 00:10:42.620 --> 00:10:43.380 David Stephens, Prof: Okay. 103 00:10:43.770 --> 00:10:45.450 David Stephens, Prof: That's what that structure is. 104 00:10:46.020 --> 00:10:48.610 David Stephens, Prof: So, if we were to collect this dataset. 105 00:10:49.450 --> 00:10:56.590 David Stephens, Prof: of all of these variables, and when we were to look at the association between Z and Y, we would see that there was none. 106 00:10:58.170 --> 00:11:01.639 David Stephens, Prof: Z and Y would be uncorrelated, essentially. 107 00:11:18.250 --> 00:11:21.040 David Stephens, Prof: That's the idea of… 108 00:11:21.410 --> 00:11:34.750 David Stephens, Prof: unconditional de-separation. And this is a simple example, but you can imagine having, multiple more variables, multiple paths between them. Our analysis is always the same. We trace out the paths. 109 00:11:35.280 --> 00:11:37.159 David Stephens, Prof: And we look for colliders. 110 00:11:41.190 --> 00:11:46.710 David Stephens, Prof: What'll be more important later on when we come to do the analysis is seeing whether the… 111 00:11:47.010 --> 00:11:50.420 David Stephens, Prof: Remaining paths are directed or undirected. 112 00:11:51.470 --> 00:11:56.620 David Stephens, Prof: But in this case, all we're looking at to see is whether there's a directed path or not. 113 00:11:57.120 --> 00:12:02.420 David Stephens, Prof: Sorry, whether there's an open path or not. Here, there is no open path. 114 00:12:03.290 --> 00:12:05.550 David Stephens, Prof: The two variables are de-separated. 115 00:12:06.530 --> 00:12:08.370 David Stephens, Prof: The two variables are independent. 116 00:12:10.560 --> 00:12:15.469 David Stephens, Prof: Okay, you can think of this as saying, well, if I imagine information flowing from Z to Y, I… 117 00:12:15.820 --> 00:12:20.789 David Stephens, Prof: perturb Z. I imagine the information flowing down this path. 118 00:12:21.230 --> 00:12:22.919 David Stephens, Prof: it gets blocked at W. 119 00:12:24.720 --> 00:12:28.190 David Stephens, Prof: So any change I make in Z has no effect on Y. 120 00:12:32.410 --> 00:12:33.110 David Stephens, Prof: Okay. 121 00:12:36.800 --> 00:12:39.819 David Stephens, Prof: So, now let's think about conditional de-separation. 122 00:12:40.430 --> 00:12:46.159 David Stephens, Prof: So, conditional desceparation is what we consider when we imagine conditioning on variables. 123 00:12:48.260 --> 00:12:54.869 David Stephens, Prof: So, if we have a non-collider at X, in the fork and the chain graphs. 124 00:12:55.220 --> 00:13:02.550 David Stephens, Prof: What we saw before was that conditioning on the X variable, Blocks the path. 125 00:13:03.120 --> 00:13:10.530 David Stephens, Prof: In the sense that, If I condition on X, in this… fork. 126 00:13:11.370 --> 00:13:14.180 David Stephens, Prof: Z and Y are independent. 127 00:13:15.240 --> 00:13:21.420 David Stephens, Prof: Similarly, for the chain graph, if I condition on X, Z and Y are independent. 128 00:13:23.140 --> 00:13:23.910 David Stephens, Prof: Okay? 129 00:13:24.100 --> 00:13:26.439 David Stephens, Prof: So what we're going to think about doing is. 130 00:13:26.920 --> 00:13:30.409 David Stephens, Prof: Turning an open path into a closed path. 131 00:13:30.830 --> 00:13:33.160 David Stephens, Prof: Or an unblocked path into a blocked path. 132 00:13:34.090 --> 00:13:36.369 David Stephens, Prof: By conditioning on variables. 133 00:13:38.700 --> 00:13:42.880 David Stephens, Prof: So, in this type of setup, that conditioning blocks the path. 134 00:13:44.150 --> 00:13:48.760 David Stephens, Prof: Alright? This path between Z and Y is… Open. 135 00:13:48.860 --> 00:13:52.100 David Stephens, Prof: Until I condition on X, when it becomes blocked. 136 00:13:53.210 --> 00:13:56.620 David Stephens, Prof: This path from Z to Y, this is open. 137 00:13:56.750 --> 00:13:59.789 David Stephens, Prof: Until I condition on X, in which case it becomes blocked. 138 00:14:03.480 --> 00:14:05.280 David Stephens, Prof: Those are for non-colliders. 139 00:14:07.130 --> 00:14:13.280 David Stephens, Prof: If I have a collider, though, The situation is reversed. 140 00:14:13.660 --> 00:14:19.040 David Stephens, Prof: So the… the, path… Between Z and Y here. 141 00:14:20.510 --> 00:14:23.309 David Stephens, Prof: is… blocked at X. 142 00:14:24.260 --> 00:14:25.859 David Stephens, Prof: So there is no open path. 143 00:14:26.810 --> 00:14:27.870 David Stephens, Prof: Unconditionally. 144 00:14:28.530 --> 00:14:35.269 David Stephens, Prof: However, if I condition on X, That opens the path Because X is a collider. 145 00:14:39.370 --> 00:14:41.699 David Stephens, Prof: Okay? And we'll think about what 146 00:14:42.410 --> 00:14:45.519 David Stephens, Prof: How conditioning, conditioning on a variable. 147 00:14:45.650 --> 00:14:52.580 David Stephens, Prof: impact in a statistical analysis later on. Essentially, you can think of it as including a variable in a regression model. 148 00:14:54.760 --> 00:14:57.269 David Stephens, Prof: That's gonna be the equivalent of conditioning. 149 00:14:57.800 --> 00:15:03.439 David Stephens, Prof: Right? So, for a non-collider, conditioning blocks open paths. 150 00:15:04.840 --> 00:15:09.750 David Stephens, Prof: For a collider, conditioning opens block paths. 151 00:15:12.410 --> 00:15:15.770 David Stephens, Prof: Gets a bit more complicated with colliders. 152 00:15:16.450 --> 00:15:19.309 David Stephens, Prof: Let's imagine this… we've got this DAG. 153 00:15:20.920 --> 00:15:24.360 David Stephens, Prof: And we're still investigating the relationship between Z and Y. 154 00:15:24.500 --> 00:15:29.120 David Stephens, Prof: Now, Z… to Y. There's only one path. 155 00:15:30.220 --> 00:15:33.329 David Stephens, Prof: goes through X, X is a collider, that path is blocked. 156 00:15:35.130 --> 00:15:41.010 David Stephens, Prof: What we have in this DAG, though, is that W is a descendant of X. 157 00:15:43.780 --> 00:15:49.890 David Stephens, Prof: And what we're going to think about is, what happens in the relationship between Z and Y 158 00:15:50.390 --> 00:15:52.260 David Stephens, Prof: If we condition on W. 159 00:15:54.180 --> 00:15:56.080 David Stephens, Prof: Now, if we condition on X, 160 00:15:57.540 --> 00:16:01.030 David Stephens, Prof: As we saw before, the classic definition of the collider. 161 00:16:01.960 --> 00:16:07.040 David Stephens, Prof: Conditioning on X renders Z and Y dependent. 162 00:16:08.400 --> 00:16:10.719 David Stephens, Prof: That's clear, that's from the definition, basically. 163 00:16:11.790 --> 00:16:16.950 David Stephens, Prof: But what happens if we condition on W? Now, W is not even on the path between Z and Y. 164 00:16:17.330 --> 00:16:21.530 David Stephens, Prof: So is it harmless to do that? Does it have… has it no effect at all? 165 00:16:21.880 --> 00:16:26.620 David Stephens, Prof: Well, no. It turns out that if you condition on a descendant of a collider. 166 00:16:27.800 --> 00:16:30.749 David Stephens, Prof: That has the same effect as conditioning on the collider. 167 00:16:31.230 --> 00:16:33.770 David Stephens, Prof: This is the mathematical result that shows that. 168 00:16:36.060 --> 00:16:41.750 David Stephens, Prof: Essentially, what we're doing, if we condition on W, which is a descendant of the collider X, 169 00:16:42.220 --> 00:16:43.800 David Stephens, Prof: is conditioning on X. 170 00:16:46.110 --> 00:16:48.650 David Stephens, Prof: And you can think of that as being, well, 171 00:16:50.350 --> 00:16:55.940 David Stephens, Prof: because of the relationship between X and W here, so this clearly says X causes W. 172 00:16:57.060 --> 00:17:01.359 David Stephens, Prof: Meaning that X and W are… Correlated, associated. 173 00:17:01.950 --> 00:17:07.269 David Stephens, Prof: So conditioning on W has more or less the same effect as conditioning on X. 174 00:17:07.569 --> 00:17:09.279 David Stephens, Prof: unconditioning on X. 175 00:17:09.510 --> 00:17:10.599 David Stephens, Prof: Opens the path. 176 00:17:16.980 --> 00:17:23.280 David Stephens, Prof: So, conditioning on a non-collider, Blocks the path at that variable. 177 00:17:23.910 --> 00:17:28.319 David Stephens, Prof: Conditioning on a collider or a descendant of a collider opens the path. 178 00:17:28.569 --> 00:17:29.790 David Stephens, Prof: At the collider. 179 00:17:32.490 --> 00:17:33.400 David Stephens, Prof: Now… 180 00:17:33.610 --> 00:17:44.130 David Stephens, Prof: These points are important when we come to statistical modeling, because often in practice, what we have to do is to decide on which variables go into our regression model. 181 00:17:46.340 --> 00:17:54.880 David Stephens, Prof: Conditioning on a variable is essentially, or at least for our purposes, the equivalent of including that variable in the regression model. 182 00:17:56.390 --> 00:17:58.449 David Stephens, Prof: So what these results tell us 183 00:18:02.340 --> 00:18:06.379 David Stephens, Prof: is that… The choice that we make 184 00:18:07.860 --> 00:18:13.240 David Stephens, Prof: In including or not including a variable in a regression model can have profound effects. 185 00:18:13.700 --> 00:18:15.720 David Stephens, Prof: Depending on the underlying DAG. 186 00:18:21.290 --> 00:18:23.910 David Stephens, Prof: So now we've got this concept of de-separation. 187 00:18:25.660 --> 00:18:28.070 David Stephens, Prof: And we'll recap it again as we go through a bit. 188 00:18:29.180 --> 00:18:35.169 David Stephens, Prof: We can start thinking about the models that we should use in a statistical analysis. 189 00:18:35.570 --> 00:18:38.100 David Stephens, Prof: So if you've got two nodes, X and Y, 190 00:18:38.470 --> 00:18:41.949 David Stephens, Prof: And so, generically, X and Y, with possible 191 00:18:42.230 --> 00:18:44.739 David Stephens, Prof: Several possible open paths connecting them. 192 00:18:47.080 --> 00:18:49.789 David Stephens, Prof: Suppose we've got another set of variables S, 193 00:18:53.060 --> 00:18:58.420 David Stephens, Prof: Then, we say that If we condition on S, 194 00:19:04.130 --> 00:19:05.599 David Stephens, Prof: One of two things could happen. 195 00:19:09.130 --> 00:19:11.660 David Stephens, Prof: Conditioning on S could block a path. 196 00:19:13.640 --> 00:19:15.830 David Stephens, Prof: If it's a… if it's currently open? 197 00:19:17.690 --> 00:19:19.810 David Stephens, Prof: So the path will ultimately be closed. 198 00:19:20.970 --> 00:19:22.590 David Stephens, Prof: After we condition on S. 199 00:19:22.690 --> 00:19:26.169 David Stephens, Prof: Or it could be that conditioning on S opens a path. 200 00:19:35.200 --> 00:19:39.249 David Stephens, Prof: If that collection of variables blocks every path. 201 00:19:40.970 --> 00:19:47.890 David Stephens, Prof: then X and Y are desceparated by S, that's the… That's the… Bottom line. 202 00:19:51.460 --> 00:19:56.400 David Stephens, Prof: And the critical implication of that is that… 203 00:19:56.810 --> 00:20:04.920 David Stephens, Prof: If, the set S, set of variables S, D separates X and Y, then X and Y are independent, conditionally on S. 204 00:20:05.060 --> 00:20:11.470 David Stephens, Prof: That is, the distribution of X given Y and S Doesn't depend on why. 205 00:20:15.930 --> 00:20:19.220 David Stephens, Prof: We'll see the implications of this for Causal modeling. 206 00:20:19.490 --> 00:20:20.899 David Stephens, Prof: In a few slides. 207 00:20:23.010 --> 00:20:26.209 David Stephens, Prof: Essentially, what we're gonna be doing is… 208 00:20:26.440 --> 00:20:34.930 David Stephens, Prof: to trying to identify a collection of variables S, that leaves… Desirable paths open. 209 00:20:36.120 --> 00:20:39.550 David Stephens, Prof: And blocks undesirable paths. 210 00:20:40.520 --> 00:20:42.530 David Stephens, Prof: In the way that we'll describe later. 211 00:20:45.000 --> 00:20:45.830 David Stephens, Prof: So… 212 00:20:46.130 --> 00:20:52.859 David Stephens, Prof: As an example of this conditional de-separation idea, we'll go back to this 5-node graph that was presented earlier on. 213 00:20:54.940 --> 00:21:05.950 David Stephens, Prof: So, let's think about the relationship between… Ex… X2 and X3. 214 00:21:07.240 --> 00:21:14.489 David Stephens, Prof: So… There's X2, there's X3, So, to understand 215 00:21:14.940 --> 00:21:23.219 David Stephens, Prof: the dependence relationship between them. We need to identify all of the paths Between those variables. 216 00:21:23.750 --> 00:21:24.890 David Stephens, Prof: And there are two. 217 00:21:25.600 --> 00:21:35.390 David Stephens, Prof: There's the path that goes, if I start at X2, It goes X1, X3, or X4X3. 218 00:21:42.990 --> 00:21:43.690 David Stephens, Prof: Okay. 219 00:21:44.760 --> 00:21:49.680 David Stephens, Prof: So, let's think about… First of all, the set of variables X1. 220 00:21:50.820 --> 00:21:57.899 David Stephens, Prof: If I condition on X1, That blocks the path between X2 and X3 along this arm. 221 00:21:59.210 --> 00:22:01.780 David Stephens, Prof: That's an open and directed path there. 222 00:22:03.500 --> 00:22:05.950 David Stephens, Prof: If I condition on X1, it becomes blocked. 223 00:22:07.900 --> 00:22:09.879 David Stephens, Prof: The only other path between X… 224 00:22:10.050 --> 00:22:15.350 David Stephens, Prof: 2 and X3 goes through X4, but X4 is a collider on that path, so that path's not open. 225 00:22:16.000 --> 00:22:16.920 David Stephens, Prof: Unconditionally. 226 00:22:20.590 --> 00:22:22.480 David Stephens, Prof: So what this says is that… 227 00:22:24.060 --> 00:22:27.600 David Stephens, Prof: X2 and X3 are de-separated by the variable X1. 228 00:22:29.090 --> 00:22:35.140 David Stephens, Prof: If I condition on X1, Then there are no remaining open paths. 229 00:22:35.340 --> 00:22:37.739 David Stephens, Prof: between X2 and X3. 230 00:22:41.490 --> 00:22:42.849 David Stephens, Prof: Everybody get the idea? 231 00:22:44.610 --> 00:22:56.369 David Stephens, Prof: There are only two paths in the first place. One path is automatically blocked by the collider at X4. The only path we have to worry about is the path through X1, so if I condition on X1, that blocks that path. 232 00:23:08.010 --> 00:23:15.130 David Stephens, Prof: If I consider… so, in this case, the set of variables S is merely the set X1. 233 00:23:15.480 --> 00:23:16.759 David Stephens, Prof: Just a single variable. 234 00:23:19.220 --> 00:23:28.230 David Stephens, Prof: If I extend the set, to being the two variables X1 and X5, X1, X5. 235 00:23:29.160 --> 00:23:37.940 David Stephens, Prof: Then… It turns out that X2 and X3 are not de-separated by X1, X5, 236 00:23:38.700 --> 00:23:42.210 David Stephens, Prof: I mean, X1 does the job on the left-hand side, that's fine. 237 00:23:42.460 --> 00:23:45.670 David Stephens, Prof: But, if I condition on X5, 238 00:23:45.900 --> 00:23:51.270 David Stephens, Prof: the path that was previously blocked at X4 becomes unblocked. 239 00:23:53.780 --> 00:23:57.180 David Stephens, Prof: Right? X4… is a collider. 240 00:23:59.170 --> 00:24:02.059 David Stephens, Prof: Unconditionally, the path is blocked at X4. 241 00:24:02.510 --> 00:24:04.979 David Stephens, Prof: But if I condition on X5, 242 00:24:05.250 --> 00:24:07.999 David Stephens, Prof: That opens the path at the collider. 243 00:24:22.400 --> 00:24:27.810 David Stephens, Prof: So there is an example in the handout that we'll see later on to do with selection bias. 244 00:24:28.070 --> 00:24:36.229 David Stephens, Prof: Selection bias is, again, about inducing association. By conditioning. 245 00:24:36.670 --> 00:24:38.500 David Stephens, Prof: Which is otherwise not there. 246 00:24:40.720 --> 00:24:43.010 David Stephens, Prof: In particular, it's a gen… well, really, it's a… 247 00:24:43.220 --> 00:24:45.940 David Stephens, Prof: A generalization of this collider effect. 248 00:24:46.490 --> 00:24:51.480 David Stephens, Prof: So, if selection blast occurs when we have conditioning on the common effect of two causes. 249 00:24:51.810 --> 00:24:54.500 David Stephens, Prof: Rendering those two causes dependent. 250 00:24:55.540 --> 00:24:56.640 David Stephens, Prof: As in here. 251 00:24:59.350 --> 00:25:03.709 David Stephens, Prof: So a more complicated version of that is in this DAG here. 252 00:25:04.780 --> 00:25:08.400 David Stephens, Prof: So, imagine the relationship between Z1 and Z2. 253 00:25:09.500 --> 00:25:12.619 David Stephens, Prof: There are only two paths between Z1 and Z2. 254 00:25:13.070 --> 00:25:15.349 David Stephens, Prof: There's the path that goes through W, 255 00:25:15.560 --> 00:25:18.079 David Stephens, Prof: and the path that goes through X and W. 256 00:25:20.830 --> 00:25:23.960 David Stephens, Prof: But W is a collider on both of those paths. 257 00:25:28.700 --> 00:25:35.200 David Stephens, Prof: So, whereas Z1 and Z2 are… Independent, unconditionally. 258 00:25:35.560 --> 00:25:40.049 David Stephens, Prof: If I condition on W, That renders them dependent. 259 00:25:46.250 --> 00:25:51.460 David Stephens, Prof: So, Z1 and Z2 are unconditionally desceparated. 260 00:25:52.400 --> 00:26:00.070 David Stephens, Prof: But if I condition on W, They become conditionally not desceparated. 261 00:26:01.360 --> 00:26:02.630 David Stephens, Prof: But deconnected. 262 00:26:12.400 --> 00:26:14.180 David Stephens, Prof: Here's another example of that. 263 00:26:14.960 --> 00:26:22.000 David Stephens, Prof: Again, if we examine the relationship between Z1 and Z2, There are… 264 00:26:22.390 --> 00:26:26.190 David Stephens, Prof: essentially two paths to consider between Z1 and Z2. 265 00:26:28.270 --> 00:26:39.110 David Stephens, Prof: Z1, X, V, W, Z2, But X is a collider, so that path is unconditionally open. 266 00:26:39.610 --> 00:26:41.599 David Stephens, Prof: I'm sorry, unconditionally closed. 267 00:26:45.720 --> 00:26:47.660 David Stephens, Prof: That path is blocked at the X. 268 00:26:49.430 --> 00:26:53.710 David Stephens, Prof: The other path is Z1, X. 269 00:26:53.880 --> 00:26:55.920 David Stephens, Prof: W. Z2? 270 00:26:59.300 --> 00:27:03.459 David Stephens, Prof: That's also blocked at X, and also blocked at the collider W. 271 00:27:08.110 --> 00:27:12.569 David Stephens, Prof: So… Z1 and Z2 are independent. 272 00:27:13.430 --> 00:27:16.039 David Stephens, Prof: Because there are no open paths. 273 00:27:17.870 --> 00:27:22.010 David Stephens, Prof: connecting them. They are unconditionally de-separated. 274 00:27:23.490 --> 00:27:26.039 David Stephens, Prof: Whereas if I condition on X, 275 00:27:29.360 --> 00:27:36.800 David Stephens, Prof: that opens both of the previously blocked paths, and renders the two variables deconnected. 276 00:27:44.490 --> 00:27:47.280 David Stephens, Prof: So, why is this de-separation? 277 00:27:47.430 --> 00:27:51.870 David Stephens, Prof: important. Well, we'll come to see the specific details in a… in a short while. 278 00:27:52.400 --> 00:28:00.250 David Stephens, Prof: But it's clear that if… If X and Y are desceparated by this set of variables S, 279 00:28:00.400 --> 00:28:03.060 David Stephens, Prof: Then X and Y are independent, given S. 280 00:28:03.410 --> 00:28:08.819 David Stephens, Prof: And that happens for all Probability distributions that we would associate with the graph. 281 00:28:12.530 --> 00:28:14.559 David Stephens, Prof: If they're not de-separated. 282 00:28:15.150 --> 00:28:23.080 David Stephens, Prof: then X and Y are dependent for at least one distribution compatible with the graph. There is… 283 00:28:23.550 --> 00:28:28.330 David Stephens, Prof: ape… A path that leads to 284 00:28:28.570 --> 00:28:31.690 David Stephens, Prof: Information flow between the two variables. 285 00:28:32.980 --> 00:28:34.829 David Stephens, Prof: An open, undirected. 286 00:28:35.020 --> 00:28:36.320 David Stephens, Prof: Path, it turns out to be. 287 00:28:38.440 --> 00:28:43.450 David Stephens, Prof: So, finding this set of variables S, To condition upon. 288 00:28:43.560 --> 00:28:45.550 David Stephens, Prof: It's gonna turn out to be crucial. 289 00:28:47.510 --> 00:28:53.190 David Stephens, Prof: And we'll see this in the most elementary way in the construction of the confounding example. 290 00:28:53.900 --> 00:28:58.590 David Stephens, Prof: Where… We've got a simple 3-node graph. 291 00:28:59.330 --> 00:29:09.290 David Stephens, Prof: We're interested in the connection between treatment Z and outcome Y, but there's a third variable, X, that connects the two. What we'll see is that conditioning on X 292 00:29:11.350 --> 00:29:16.860 David Stephens, Prof: Allows us to study the effect of Zettomoi. 293 00:29:26.210 --> 00:29:29.179 David Stephens, Prof: So, to start thinking about this notion of 294 00:29:29.830 --> 00:29:34.470 David Stephens, Prof: a Causal effect. We go back to the concept of interventions to start off with. 295 00:29:36.210 --> 00:29:42.099 David Stephens, Prof: So, how do interventions… Affect how we think about the graph. 296 00:29:42.850 --> 00:29:49.589 David Stephens, Prof: So the idea at the moment is we've reached the point where we've sat down, And… Collected… 297 00:29:50.420 --> 00:29:54.209 David Stephens, Prof: our variables, and we've written down a DAG that we believe 298 00:29:55.050 --> 00:29:58.629 David Stephens, Prof: Encapsulates the connections between all of the variables. 299 00:29:59.150 --> 00:29:59.940 David Stephens, Prof: Okay? 300 00:30:00.260 --> 00:30:02.899 David Stephens, Prof: The treatment variable Z is in one position. 301 00:30:03.220 --> 00:30:10.369 David Stephens, Prof: The outcome variable Y is in another position. There are possibly multiple paths through different X's that connect. 302 00:30:11.280 --> 00:30:12.530 David Stephens, Prof: Z to Y. 303 00:30:13.170 --> 00:30:17.969 David Stephens, Prof: Some of them will be directed, some of them will be undirected, some of them will be… 304 00:30:18.300 --> 00:30:20.690 David Stephens, Prof: Open, some of them will be closed. 305 00:30:21.870 --> 00:30:25.810 David Stephens, Prof: We have to pick all of these apart to understand 306 00:30:26.860 --> 00:30:33.470 David Stephens, Prof: The Causal effect, to get back to that idea earlier on, which said that a causal effect of one variable on another 307 00:30:33.640 --> 00:30:46.470 David Stephens, Prof: is… Captured by… The function that sets the second variable's value as a function of the first. 308 00:30:48.350 --> 00:30:49.810 David Stephens, Prof: So, interventions. 309 00:30:52.620 --> 00:30:57.919 David Stephens, Prof: The effect of an intervention on a graph is pretty straightforward. 310 00:31:00.010 --> 00:31:11.630 David Stephens, Prof: So, let's say we've got the treatment Z, and we want to intervene to set that value to little red Z. So that is, we're not saying we're now working with the observational data anymore. 311 00:31:12.020 --> 00:31:21.660 David Stephens, Prof: We are perturbing the system, to say, instead of Z arising from some natural mechanism. 312 00:31:21.820 --> 00:31:25.319 David Stephens, Prof: We are intervening to set the value of that variable. 313 00:31:28.010 --> 00:31:30.860 David Stephens, Prof: Equal to some numerical value, little red z. 314 00:31:31.630 --> 00:31:34.359 David Stephens, Prof: So what effect does that have? Well, because 315 00:31:34.510 --> 00:31:37.199 David Stephens, Prof: The researcher, or we, are the only… 316 00:31:37.970 --> 00:31:42.930 David Stephens, Prof: cause of the change in Z now. No other variable is involved. 317 00:31:43.760 --> 00:31:48.019 David Stephens, Prof: we can think of representing that by removing all the incoming arrows to Z. 318 00:31:53.150 --> 00:31:55.410 David Stephens, Prof: Arrows represent causes. 319 00:31:57.650 --> 00:32:02.690 David Stephens, Prof: So maybe we thought before another variable caused Z in an observational setting. 320 00:32:03.000 --> 00:32:13.759 David Stephens, Prof: In the interventional world, in the experimental world, what we're saying is that, no, no, we are now the person responsible for setting the value of Z. No other variable is doing that. 321 00:32:14.000 --> 00:32:16.380 David Stephens, Prof: So there is no other cause apart from us. 322 00:32:17.390 --> 00:32:20.709 David Stephens, Prof: So that has the effect of removing all the incoming arrows. 323 00:32:21.630 --> 00:32:33.280 David Stephens, Prof: The other thing we have to do in the probability distribution is to say, well, okay, now Z doesn't have a distribution, there's some randomness in that. We are setting it equal to a specific value. 324 00:32:33.710 --> 00:32:34.720 David Stephens, Prof: Treated. 325 00:32:34.920 --> 00:32:36.590 David Stephens, Prof: Or… untreated. 326 00:32:37.000 --> 00:32:40.479 David Stephens, Prof: Or, treatment to level 2.4, or whatever. 327 00:32:42.970 --> 00:32:47.100 David Stephens, Prof: So what that has the effect of doing is switching the marginal distribution 328 00:32:48.130 --> 00:32:52.629 David Stephens, Prof: To a degenerate distribution, where all the probability is placed at one value. 329 00:32:53.820 --> 00:33:03.689 David Stephens, Prof: That's what this notation means, but it's not particularly important, right? It just says, instead of Z now arising randomly, no, no, we are setting it to be a particular value. 330 00:33:05.310 --> 00:33:13.169 David Stephens, Prof: Okay? And the notion of independent… Sorry, of intervention is important, of course, because that is critical to the causal effect idea. 331 00:33:15.100 --> 00:33:22.699 David Stephens, Prof: Causal effect is the effect of intervening to change the treatment level from one level to another. In both of those 332 00:33:23.260 --> 00:33:26.820 David Stephens, Prof: Scenarios, though, the first level or the second level. 333 00:33:27.550 --> 00:33:35.670 David Stephens, Prof: We've intervened to set the treatment to that level, so it's critical that we, we, 334 00:33:37.950 --> 00:33:41.379 David Stephens, Prof: can distinguish that from the case where Z is arising randomly. 335 00:33:43.880 --> 00:33:50.840 David Stephens, Prof: Alright, so… This is the simple example I was referring to earlier on. This is a… 336 00:33:51.000 --> 00:33:53.919 David Stephens, Prof: Imagine this is a data generating mechanism. 337 00:33:54.210 --> 00:33:59.030 David Stephens, Prof: And it turns out to be the classical representation of confounding. 338 00:33:59.500 --> 00:34:01.000 David Stephens, Prof: In a statistical model. 339 00:34:02.640 --> 00:34:04.439 David Stephens, Prof: So we have a variable X, 340 00:34:04.890 --> 00:34:11.049 David Stephens, Prof: X is a cause of Z, X is a cause of Y, and Z is a cause of Y. 341 00:34:11.630 --> 00:34:17.889 David Stephens, Prof: We are interested, in the causal setting, in studying the relationship between Z and Y. 342 00:34:18.139 --> 00:34:21.120 David Stephens, Prof: So really, we want to feel the force of that arrow. 343 00:34:22.070 --> 00:34:29.999 David Stephens, Prof: But we recognize that there are multiple paths that connect Z to Y, There is the direct path. 344 00:34:30.290 --> 00:34:33.840 David Stephens, Prof: but also the direct… indirect path that goes through X. 345 00:34:37.620 --> 00:34:39.679 David Stephens, Prof: If we think about an intervention. 346 00:34:40.800 --> 00:34:49.719 David Stephens, Prof: If we're intervening on the value Z… sorry, on the variable Z to set it equal to some little red value, I can denote it this way, there are several ways we can do it. 347 00:34:50.280 --> 00:34:55.439 David Stephens, Prof: Okay? But… what that has the effect of doing is removing this arrow. 348 00:34:56.340 --> 00:35:02.850 David Stephens, Prof: Okay? Because it's no longer the case that Z is causing… sorry, X is causing Z. 349 00:35:04.110 --> 00:35:05.800 David Stephens, Prof: We are causing Z. 350 00:35:07.150 --> 00:35:11.009 David Stephens, Prof: So that arrow is removed. So in the interventional world. 351 00:35:11.550 --> 00:35:14.460 David Stephens, Prof: The relevant DAG is this one. 352 00:35:16.270 --> 00:35:20.530 David Stephens, Prof: Whereas in the observational world, the original world, the DAG was this one. 353 00:35:23.330 --> 00:35:29.020 David Stephens, Prof: Now, in this interventional world, It's clear… that… 354 00:35:29.760 --> 00:35:34.709 David Stephens, Prof: Studying the relationship between Z and Y really just involves 355 00:35:34.960 --> 00:35:38.129 David Stephens, Prof: Studying the force down this arrow. 356 00:35:40.710 --> 00:35:44.179 David Stephens, Prof: I change Z, what effect does that have on Y? 357 00:35:45.750 --> 00:35:47.380 David Stephens, Prof: X has no influence. 358 00:35:48.320 --> 00:35:49.630 David Stephens, Prof: So this… 359 00:35:49.880 --> 00:35:57.189 David Stephens, Prof: experimental world, or interventional world, is the one that I'm really interested in when I'm thinking about causal effects. 360 00:35:59.120 --> 00:36:04.350 David Stephens, Prof: But I have to recognize that, in the real world, the DAG looks like this. 361 00:36:09.340 --> 00:36:11.860 David Stephens, Prof: So the effect of changing Z 362 00:36:12.540 --> 00:36:17.130 David Stephens, Prof: Sure, it might be felt down this path, But also. 363 00:36:17.710 --> 00:36:20.649 David Stephens, Prof: the effect of Z can leak out through this path. 364 00:36:23.040 --> 00:36:26.349 David Stephens, Prof: And so what's gonna happen is I'm gonna get a distorted view. 365 00:36:26.540 --> 00:36:28.550 David Stephens, Prof: of the effect of Z on Y. 366 00:36:29.160 --> 00:36:30.169 David Stephens, Prof: Because of that. 367 00:36:31.560 --> 00:36:33.220 David Stephens, Prof: That's gonna be the key thing. 368 00:36:34.960 --> 00:36:37.210 David Stephens, Prof: We want to focus on these. 369 00:36:37.540 --> 00:36:41.059 David Stephens, Prof: Directed paths between treatment and outcome. 370 00:36:43.430 --> 00:36:44.390 David Stephens, Prof: But… 371 00:36:48.340 --> 00:36:52.490 David Stephens, Prof: We have to acknowledge the presence of these undirected paths. 372 00:36:54.300 --> 00:36:56.339 David Stephens, Prof: And we have to handle them in some way. 373 00:36:59.870 --> 00:37:04.639 David Stephens, Prof: Okay, now, this DAG… Leads to this factorization. 374 00:37:06.300 --> 00:37:07.590 David Stephens, Prof: Founder X. 375 00:37:08.270 --> 00:37:16.950 David Stephens, Prof: conditional distribution of Z given X, but then Z and X get together, to cause Y, so the… Daggers. 376 00:37:17.530 --> 00:37:21.129 David Stephens, Prof: Encapsulated in this probability distribution factorization. 377 00:37:21.690 --> 00:37:23.419 David Stephens, Prof: Whereas in this one. 378 00:37:26.080 --> 00:37:32.479 David Stephens, Prof: The only thing that changes is that I now have a distribution of Z which is degenerate at 379 00:37:32.630 --> 00:37:35.579 David Stephens, Prof: Some fixed numerical value, little red z. 380 00:37:36.280 --> 00:37:46.169 David Stephens, Prof: So… All the inferences, all the conclusions that I want to be able to make, Are in this world. 381 00:37:49.020 --> 00:37:53.130 David Stephens, Prof: But… My data… Lie in this world. 382 00:37:56.850 --> 00:37:57.520 David Stephens, Prof: Okay? 383 00:37:58.180 --> 00:38:05.139 David Stephens, Prof: So, then the question is, well, how can I make sure that the inferences I get from the real data 384 00:38:05.610 --> 00:38:12.270 David Stephens, Prof: Allow me to get at the conclusions I want from this hypothetical data-generating situation. 385 00:38:15.870 --> 00:38:20.790 David Stephens, Prof: And this graph really tells the story. We have to account for this open… 386 00:38:21.990 --> 00:38:28.840 David Stephens, Prof: undirected path between Z and Y. That's the problematic path, that's, as we'll see, that's termed the biasing path. 387 00:38:28.990 --> 00:38:31.659 David Stephens, Prof: We have to handle the biasing path in some way. 388 00:38:33.190 --> 00:38:38.170 David Stephens, Prof: Now, From what I've discussed earlier on, The conclusion is obvious. 389 00:38:39.090 --> 00:38:43.689 David Stephens, Prof: For this DAG, to block this path at X, we just need to condition on X. 390 00:38:46.840 --> 00:38:56.950 David Stephens, Prof: So that mimics what we might do in practice from common regression experiences. If we wanted to study the effect of Z on Y in a model. 391 00:38:57.060 --> 00:39:02.629 David Stephens, Prof: But acknowledge the presence of X, In this kind of relationship. 392 00:39:03.090 --> 00:39:06.389 David Stephens, Prof: What we would have to do would be to throw X into the regression model. 393 00:39:08.380 --> 00:39:11.180 David Stephens, Prof: If we just regressed Y on Z, 394 00:39:11.460 --> 00:39:17.789 David Stephens, Prof: then we would get the wrong answer. We have to acknowledge the presence of X, and we do that by throwing it into the regression model. 395 00:39:20.940 --> 00:39:26.370 David Stephens, Prof: So what we're going to see, basically much of what we're going to see, is really generalizing that idea. 396 00:39:31.140 --> 00:39:35.859 David Stephens, Prof: Just thinking, finally, about interventions again, so this is the 5-node DAG we had. 397 00:39:37.600 --> 00:39:38.970 David Stephens, Prof: Previously. 398 00:39:41.210 --> 00:39:43.200 David Stephens, Prof: Okay, and there's the factorization. 399 00:39:45.150 --> 00:39:58.010 David Stephens, Prof: Intervening to set X3 to be some value changes the DAG to this one. Previously, there was, an arrow between X1 and X3, so X1 is a cause of X3. 400 00:39:59.500 --> 00:40:01.580 David Stephens, Prof: But in the interventional world. 401 00:40:02.980 --> 00:40:09.029 David Stephens, Prof: Where we're setting the value of X3, X1 no longer causes X3, so that arrow is removed. 402 00:40:10.240 --> 00:40:13.209 David Stephens, Prof: the joint distribution… Changes to this. 403 00:40:14.520 --> 00:40:20.109 David Stephens, Prof: The only thing that changes is that here we have a conditional distribution of X3 given X1, 404 00:40:20.780 --> 00:40:24.039 David Stephens, Prof: And here we have only a marginal for X3. 405 00:40:24.620 --> 00:40:30.130 David Stephens, Prof: The star is just a… Notation to draw the eye, basically. 406 00:40:34.080 --> 00:40:34.970 David Stephens, Prof: So… 407 00:40:36.090 --> 00:40:44.819 David Stephens, Prof: That's the effect of an intervention on the DAG. And you can always think of interventions in acting that way. Essentially, removing errors. Arrows. 408 00:40:44.960 --> 00:40:47.569 David Stephens, Prof: Into the… intervened upon variable. 409 00:40:47.960 --> 00:40:52.669 David Stephens, Prof: So this is equivalent to the do operator that's sometimes used in the causal world. 410 00:40:55.530 --> 00:40:56.820 David Stephens, Prof: Alright, so… 411 00:40:59.870 --> 00:41:07.110 David Stephens, Prof: Finally, we're going to think about the, impact of these other variables, these, 412 00:41:07.400 --> 00:41:12.020 David Stephens, Prof: problematic paths in the DAG when it comes to causal inference. 413 00:41:14.140 --> 00:41:19.340 David Stephens, Prof: Our task is to understand the effect of Z on Y, The treatment on the outcome. 414 00:41:22.680 --> 00:41:26.290 David Stephens, Prof: Any open, undirected path between Z and Y 415 00:41:27.300 --> 00:41:32.449 David Stephens, Prof: allows for the association between Z and Y to be modified by the presence of those other variables. 416 00:41:39.240 --> 00:41:40.999 David Stephens, Prof: Back in this setting here. 417 00:41:41.500 --> 00:41:53.450 David Stephens, Prof: We want to study the direct relationship between Z and Y, but that's complicated by the fact that there is an open path that goes through X to Y. 418 00:41:57.640 --> 00:41:59.879 David Stephens, Prof: What essentially we have to do is to block that path. 419 00:42:01.980 --> 00:42:05.979 David Stephens, Prof: Alright, so… such paths are known as biasing paths. 420 00:42:08.210 --> 00:42:14.500 David Stephens, Prof: And we can associate that with the usual concept of statistical bias, so… 421 00:42:14.760 --> 00:42:19.059 David Stephens, Prof: Not getting an accurate representation of the true relationship. 422 00:42:21.320 --> 00:42:36.069 David Stephens, Prof: It needn't literally be bias in the technical sense, but it's something that changes the pattern of association between the variables. Association, well, usually that means some sort of correlation, some sort of observational dependence. 423 00:42:36.420 --> 00:42:40.729 David Stephens, Prof: And we would normally get at that by using a regression model. 424 00:42:41.610 --> 00:42:43.770 David Stephens, Prof: In the usual way that we use regression. 425 00:42:49.050 --> 00:42:51.769 David Stephens, Prof: So, to categorize the situations we're gonna be in. 426 00:42:53.650 --> 00:42:55.580 David Stephens, Prof: Let's think about the good cases. 427 00:42:57.320 --> 00:43:01.709 David Stephens, Prof: If we think about measuring the association between Z and Y, 428 00:43:02.220 --> 00:43:07.339 David Stephens, Prof: Then that association is unbiased, or can be estimated in an unbiased fashion. 429 00:43:10.590 --> 00:43:16.490 David Stephens, Prof: If the only open paths between them are directed paths. 430 00:43:17.440 --> 00:43:22.000 David Stephens, Prof: That means a direct path from Z to Y. 431 00:43:22.180 --> 00:43:26.090 David Stephens, Prof: or a chain from Z to Y that goes through some other variable X. 432 00:43:26.550 --> 00:43:28.509 David Stephens, Prof: Or multiple variables, X. 433 00:43:29.410 --> 00:43:36.869 David Stephens, Prof: So, in this situation, Essentially, regressing Y on Z will give you the right answer. 434 00:43:47.730 --> 00:43:49.640 David Stephens, Prof: Thinking about a de-separation. 435 00:43:50.810 --> 00:44:02.360 David Stephens, Prof: If we have another set of nodes… so let's say we're not in this wonderfully simple situation where all the paths… the only open paths between the two variables are directed paths. Let's say we have some open undirected paths. 436 00:44:04.570 --> 00:44:08.399 David Stephens, Prof: We want to… Be able to account for those. 437 00:44:09.550 --> 00:44:14.110 David Stephens, Prof: So… this set of nodes S that we considered before. 438 00:44:14.350 --> 00:44:17.430 David Stephens, Prof: S is sufficient to control the bias. 439 00:44:18.190 --> 00:44:22.480 David Stephens, Prof: in the association between Z and Y, if, after conditioning on S, 440 00:44:23.210 --> 00:44:31.800 David Stephens, Prof: The only remaining open paths between Z and Y are the directed paths. That is, by conditioning on S, 441 00:44:32.070 --> 00:44:35.949 David Stephens, Prof: We block the other open paths. 442 00:44:36.390 --> 00:44:38.980 David Stephens, Prof: The open, undirected paths. 443 00:44:43.300 --> 00:44:47.090 David Stephens, Prof: And then minimally sufficient if S is the smallest set of such variables. 444 00:44:53.670 --> 00:45:00.260 David Stephens, Prof: There's one thing to note in this, which has a nod to statistical formulations. 445 00:45:00.720 --> 00:45:02.240 David Stephens, Prof: Which is that… 446 00:45:03.380 --> 00:45:13.929 David Stephens, Prof: let's say we're still in this situation where we've got Z and Y that we want to study, the relationship between the treatment and the outcome, but we believe there's this chain relationship. 447 00:45:15.720 --> 00:45:21.620 David Stephens, Prof: Well… Let's say… so Z… so X is entirely a descendant of Z. 448 00:45:24.390 --> 00:45:28.870 David Stephens, Prof: Then, if we condition on X, that blocks the dependence between Z and Y. 449 00:45:29.730 --> 00:45:32.540 David Stephens, Prof: What that means is, in a regression setting. 450 00:45:32.860 --> 00:45:36.130 David Stephens, Prof: If we were to regress Y on Z, 451 00:45:36.240 --> 00:45:38.880 David Stephens, Prof: Then we would get a correct relationship. 452 00:45:39.490 --> 00:45:40.530 David Stephens, Prof: measure. 453 00:45:41.450 --> 00:45:43.559 David Stephens, Prof: for the association between Z and Y. 454 00:45:44.250 --> 00:45:48.710 David Stephens, Prof: If we regressed Y on X and Z, 455 00:45:49.810 --> 00:45:55.279 David Stephens, Prof: then we would only pick up the relationship between X and Z. Sorry, between X and Y. 456 00:45:56.160 --> 00:46:02.970 David Stephens, Prof: Right? Because conditioning on X renders Y… And Z independent. 457 00:46:07.170 --> 00:46:14.919 David Stephens, Prof: So that might give us a misleading representation of the relationship between Z and Y, right? If we condition on X in this setting. 458 00:46:15.590 --> 00:46:23.740 David Stephens, Prof: Then we're gonna pick up a non-zero coefficient in the regression model for X, and essentially a zero coefficient 459 00:46:24.200 --> 00:46:25.300 David Stephens, Prof: for Zed. 460 00:46:28.060 --> 00:46:29.959 David Stephens, Prof: So that might tell us that… 461 00:46:31.680 --> 00:46:35.190 David Stephens, Prof: Z and Y were not related, but… whereas, in fact, they are. 462 00:46:36.660 --> 00:46:40.189 David Stephens, Prof: So, the idea here is that conditioning on X is unnecessary. 463 00:46:44.560 --> 00:46:51.699 David Stephens, Prof: In a similar situation, we might also create problems for ourselves by conditioning on that descendant of Z. 464 00:46:52.320 --> 00:46:56.160 David Stephens, Prof: So here's another setting. There is a direct effect of Z on Y. 465 00:46:58.500 --> 00:47:07.130 David Stephens, Prof: There are two paths from Z to Y. One is the direct path, one is the indirect path that goes through X and V, 466 00:47:07.950 --> 00:47:11.930 David Stephens, Prof: But… X is a collider. 467 00:47:13.880 --> 00:47:16.010 David Stephens, Prof: So, if we had this situation. 468 00:47:16.260 --> 00:47:21.980 David Stephens, Prof: There is only one open path between Z and Y, That's the direct path. 469 00:47:23.530 --> 00:47:27.829 David Stephens, Prof: So regressing Y on Z will give us precisely the right answer. 470 00:47:30.710 --> 00:47:33.430 David Stephens, Prof: Whereas if we include X in the model. 471 00:47:33.940 --> 00:47:36.520 David Stephens, Prof: That opens the path at the collider. 472 00:47:38.290 --> 00:47:42.750 David Stephens, Prof: Now there is an open undirected path, conditional on X. 473 00:47:43.570 --> 00:47:45.129 David Stephens, Prof: And that leads to bias. 474 00:47:47.110 --> 00:47:53.159 David Stephens, Prof: So again, we're in a situation where conditioning unnecessarily on X is causing us problems. 475 00:47:57.170 --> 00:48:08.640 David Stephens, Prof: We might also be in this situ… alternatively be in this situation, so let's say Z has a direct impact on Y. It's also got a direct impact on X, but X and Y are unrelated, conditional on Z. 476 00:48:11.250 --> 00:48:18.789 David Stephens, Prof: So again, thinking in a regression setting, if you were faced with this situation, you would again only want to regress Y on Z, 477 00:48:19.090 --> 00:48:23.070 David Stephens, Prof: Because including Acts would be completely unnecessary. 478 00:48:25.090 --> 00:48:32.630 David Stephens, Prof: So you might lose statistical power having to estimate, let's say, a coefficient for X if you did this. 479 00:48:35.320 --> 00:48:38.819 David Stephens, Prof: So these three points lead us to argue that 480 00:48:39.380 --> 00:48:43.979 David Stephens, Prof: Conditioning on descendants of the treatment variable of interest should not be done. 481 00:48:48.700 --> 00:48:53.639 David Stephens, Prof: It's either… Blocking a relationship that you're interested in. 482 00:48:54.040 --> 00:48:57.300 David Stephens, Prof: Or opening a biasing path unnecessarily. 483 00:48:57.860 --> 00:49:00.110 David Stephens, Prof: Or statistically spurious. 484 00:49:03.270 --> 00:49:09.369 David Stephens, Prof: So this says in our statistical models, what we should try to do is to avoid conditioning on descendants of our identified treatment. 485 00:49:17.000 --> 00:49:20.769 David Stephens, Prof: To drill a little bit more into this concept of, bias. 486 00:49:22.750 --> 00:49:25.949 David Stephens, Prof: Undirected patterns led to why I turn backdoor paths. 487 00:49:26.370 --> 00:49:34.080 David Stephens, Prof: If they start with an arrow pointing into Z, Right? So, remember, we've just… 488 00:49:34.790 --> 00:49:37.559 David Stephens, Prof: Essentially ruled out the possibility of 489 00:49:37.800 --> 00:49:41.040 David Stephens, Prof: Conditioning on things that… where the arrow is in the other direction. 490 00:49:42.500 --> 00:49:44.040 David Stephens, Prof: So in this setting. 491 00:49:45.310 --> 00:49:56.199 David Stephens, Prof: Z and Y are, as in the example we saw before, the diabetes example, the path from Z to Y is blocked at the collider W, 492 00:49:56.690 --> 00:50:05.389 David Stephens, Prof: So Z and Y are not associated. That is, the true relationship between Z and Y is that they're uncorrelated, they are independent. 493 00:50:11.360 --> 00:50:12.680 David Stephens, Prof: There's only one. 494 00:50:13.420 --> 00:50:16.829 David Stephens, Prof: Backdoor path, But it's not open. 495 00:50:25.970 --> 00:50:31.809 David Stephens, Prof: There's a correspondence between the biasing paths and the backdoor paths. 496 00:50:33.070 --> 00:50:36.239 David Stephens, Prof: All biasing paths are backdoor paths. 497 00:50:37.020 --> 00:50:39.900 David Stephens, Prof: And all open backdoor paths are biasing paths. 498 00:50:41.200 --> 00:50:46.770 David Stephens, Prof: So our task, really, will be to block all these open Backdoor paths. 499 00:50:50.920 --> 00:50:59.090 David Stephens, Prof: Right? To obtain an unbiased estimate of the effect of Z and Y, all backdoor paths between Z and Y must be blocked. 500 00:51:10.400 --> 00:51:18.999 David Stephens, Prof: And then the jargon term that we use is the backdoor criterion, so the S… the set S satisfies the backdoor criterion. If, after conditioning upon it. 501 00:51:20.180 --> 00:51:23.820 David Stephens, Prof: There's no open backdoor path from Z to Y. 502 00:51:26.510 --> 00:51:30.760 David Stephens, Prof: Also, it rules out the case of having a descendant of the treatment, as we've just seen. 503 00:51:31.480 --> 00:51:45.659 David Stephens, Prof: So, our task in a statistical, or indeed, maybe in a preliminary phase of doing an analysis of our data set is to identify this set S, this set of variables that we need to condition upon. 504 00:51:46.170 --> 00:51:52.039 David Stephens, Prof: So that we can get an unbiased estimate of the effect of the treatment on the outcome. 505 00:51:53.280 --> 00:51:55.350 David Stephens, Prof: We have to avoid all the pitfalls. 506 00:51:55.800 --> 00:51:59.570 David Stephens, Prof: In particular, conditioning on… colliders. 507 00:52:00.330 --> 00:52:02.579 David Stephens, Prof: Conditioning on descendants of treatment. 508 00:52:08.770 --> 00:52:15.719 David Stephens, Prof: So finally, a confounding path between Z and Y is a biasing path that ends in an arrow, ends with an arrow into Y. 509 00:52:20.920 --> 00:52:25.899 David Stephens, Prof: And variables on a confounding path are termed confounders, so this is the classic setup. 510 00:52:27.040 --> 00:52:28.209 David Stephens, Prof: Where we have… 511 00:52:28.360 --> 00:52:34.740 David Stephens, Prof: X being a confound of the relationship, the backdoor path goes from Z to X to Y. 512 00:52:35.070 --> 00:52:38.850 David Stephens, Prof: It's a backdoor path, it's a… confounding path. 513 00:52:40.140 --> 00:52:42.520 David Stephens, Prof: Because of that arrow, and that arrow. 514 00:52:44.060 --> 00:52:49.049 David Stephens, Prof: So this is the generic… Picture of confounding that we always go back to. 515 00:52:49.970 --> 00:52:55.880 David Stephens, Prof: When we're… when thinking about confounding, when thinking about all of these… Arguments. 516 00:52:56.110 --> 00:52:59.209 David Stephens, Prof: This simple 3-node DAG is where to start. 517 00:52:59.420 --> 00:53:04.060 David Stephens, Prof: Right? We are interested in the relationship between Z and Y, 518 00:53:04.350 --> 00:53:11.919 David Stephens, Prof: We acknowledge there's a third variable, X, where the relationship between ZX and Y is encapsulated in this graph. 519 00:53:12.360 --> 00:53:15.800 David Stephens, Prof: We are interested in the direct effect down this arrow. 520 00:53:16.510 --> 00:53:17.889 David Stephens, Prof: We acknowledge. 521 00:53:18.090 --> 00:53:25.660 David Stephens, Prof: that there is a potential for bias caused by the open, undirected path between Z and Y that goes through X. 522 00:53:26.060 --> 00:53:27.260 David Stephens, Prof: our task. 523 00:53:27.510 --> 00:53:31.219 David Stephens, Prof: is to block the path in X. We do that by conditioning. 524 00:53:32.560 --> 00:53:35.790 David Stephens, Prof: And in all the causal problems associated with DAGs. 525 00:53:36.860 --> 00:53:52.309 David Stephens, Prof: that basic set of principles will apply. It'll become more and more complicated as the graph gets more complicated, but the principles are always the same. We need to identify this set of variables S, such that when we condition upon that set S, 526 00:53:53.910 --> 00:53:56.910 David Stephens, Prof: The open backdoor paths become blocked. 527 00:54:04.780 --> 00:54:08.719 David Stephens, Prof: So, in this DAG, We've got, 528 00:54:09.700 --> 00:54:18.790 David Stephens, Prof: again, the relationship between Z and Y. Now, in this case, we've got… 529 00:54:19.700 --> 00:54:32.980 David Stephens, Prof: more complicated situation. So, Z… And why are they related? Well, there is a… path… that goes ZX1WY. 530 00:54:33.250 --> 00:54:38.510 David Stephens, Prof: and a path that goes ZX1, W, X2, Y. 531 00:54:41.800 --> 00:54:46.950 David Stephens, Prof: The first of those… is an open… Undirected path. 532 00:54:47.130 --> 00:54:48.370 David Stephens, Prof: Before we condition. 533 00:54:52.810 --> 00:54:59.030 David Stephens, Prof: The second path, ZX1, W, X2. 534 00:54:59.240 --> 00:55:00.250 David Stephens, Prof: Why? 535 00:55:02.410 --> 00:55:03.610 David Stephens, Prof: is not open. 536 00:55:03.860 --> 00:55:08.479 David Stephens, Prof: Because on that path, W acts as a collider. 537 00:55:12.980 --> 00:55:17.149 David Stephens, Prof: So you can see straight away that there's a bit of a quandary that we're in now, because… 538 00:55:19.620 --> 00:55:22.559 David Stephens, Prof: let's say we were gonna act upon W, 539 00:55:23.020 --> 00:55:25.279 David Stephens, Prof: To block the first open path. 540 00:55:26.420 --> 00:55:32.530 David Stephens, Prof: Then, if we conditioned on W there, that would for sure block the path at W. 541 00:55:34.000 --> 00:55:36.720 David Stephens, Prof: But it would open the other path. 542 00:55:37.700 --> 00:55:41.879 David Stephens, Prof: Because on that second path, W is a collider. 543 00:55:45.190 --> 00:55:48.970 David Stephens, Prof: So what that tells us is that, conditioning on W alone. 544 00:55:49.100 --> 00:55:50.660 David Stephens, Prof: Is not the way to go. 545 00:56:00.140 --> 00:56:05.830 David Stephens, Prof: If we're going to go down the route of conditioning on W, we also need to condition on X2. 546 00:56:06.770 --> 00:56:11.990 David Stephens, Prof: Conditioning on W and X2 blocks all of the open paths. 547 00:56:12.270 --> 00:56:24.340 David Stephens, Prof: And doesn't unblock… yeah, so, the originally closed path… Z… X1, W, X2, Y, 548 00:56:25.500 --> 00:56:30.719 David Stephens, Prof: becomes open when I condition on the collider W, 549 00:56:31.420 --> 00:56:34.480 David Stephens, Prof: So I have to close it again by conditioning on X2. 550 00:56:36.490 --> 00:56:43.129 David Stephens, Prof: So, if I condition on W and X2, then… X and Z become desceparated. 551 00:56:44.470 --> 00:56:49.239 David Stephens, Prof: Okay? Of course, the other thing we could do here is just a condition on X1, and that would block everything as well. 552 00:56:52.570 --> 00:56:57.030 David Stephens, Prof: Okay, so… Not just, identifies. 553 00:56:57.200 --> 00:57:00.660 David Stephens, Prof: one of the kinds of things that we have to consider, right? So… 554 00:57:00.830 --> 00:57:15.149 David Stephens, Prof: There are multiple paths, maybe involving the same variable, so it's not quite as simple as saying condition or not condition, because conditioning might have one effect, whilst at the same time having another effect on the different path. 555 00:57:19.780 --> 00:57:28.429 David Stephens, Prof: If the… so, the difference here is that we've now removed this arrow, and we're back to this example of the… from Rothman. 556 00:57:33.160 --> 00:57:37.250 David Stephens, Prof: In this case, it's clear that conditioning on W is a bad idea. 557 00:57:38.260 --> 00:57:47.400 David Stephens, Prof: Right? Z and Y are unconditionally Uncorrelated, unconditionally unassociated. 558 00:57:48.520 --> 00:57:50.690 David Stephens, Prof: Because of the collider at W. 559 00:57:52.510 --> 00:57:56.950 David Stephens, Prof: If I condition on W, Then that path becomes open. 560 00:57:58.880 --> 00:58:03.609 David Stephens, Prof: And there appears to be an association. What that means is, essentially, if I… 561 00:58:04.410 --> 00:58:08.010 David Stephens, Prof: were to regress Y on Z, 562 00:58:08.230 --> 00:58:12.390 David Stephens, Prof: I would get the correct answer that Z and Y are unassociated. 563 00:58:13.100 --> 00:58:15.660 David Stephens, Prof: Whereas if I throw W into the model. 564 00:58:15.780 --> 00:58:22.269 David Stephens, Prof: then I get the spurious result that shows me that actually Y and Z are correlated. 565 00:58:24.340 --> 00:58:39.289 David Stephens, Prof: So in this case, conditioning on W doesn't do it. I would have to condition on X1 and X2 if I'm going to condition on anything. But in this case, Z and Y are unconditionally, uncorrelated anyway. They are de-separated, because there are no open undirected paths between them. 566 00:58:44.390 --> 00:58:47.399 David Stephens, Prof: If I simply change the direction of the arrow here… 567 00:58:47.650 --> 00:58:54.020 David Stephens, Prof: then W is no longer a collider. I still have a confounding path between the variables. 568 00:58:55.330 --> 00:58:58.050 David Stephens, Prof: So… 569 00:59:01.770 --> 00:59:05.429 David Stephens, Prof: In reality, There is no… 570 00:59:05.730 --> 00:59:11.770 David Stephens, Prof: effect of Z and Y. There's no arrow between them. Z is not a cause of Y. 571 00:59:12.540 --> 00:59:16.450 David Stephens, Prof: But if I just simply regress Y on Z, it would appear 572 00:59:17.190 --> 00:59:18.929 David Stephens, Prof: that Z is a cause of Y. 573 00:59:19.460 --> 00:59:23.390 David Stephens, Prof: Because of this spurious backdoor and confounding path. 574 00:59:25.290 --> 00:59:29.450 David Stephens, Prof: So in this case, I do have to condition on one of these variables in order to remove that. 575 00:59:37.690 --> 00:59:48.140 David Stephens, Prof: Okay, so I think about… maybe I'll stop there for a couple of minutes, and then before we finish in the last half hour, and go on to some of the examples in the knitter sheet, we'll talk a bit more about the… 576 00:59:48.930 --> 00:59:51.359 David Stephens, Prof: The nature of, 577 00:59:52.480 --> 00:59:59.119 David Stephens, Prof: how we measure the impact of the treatment variable on the outcome variable. So I'll pick up again at, 578 00:59:59.390 --> 01:00:01.089 David Stephens, Prof: Let's say 34. 579 01:00:02.250 --> 01:00:03.699 David Stephens, Prof: 34 minutes passed. 580 01:04:28.130 --> 01:04:31.980 David Stephens, Prof: Okay then, so for the final section, we'll, 581 01:04:32.230 --> 01:04:45.390 David Stephens, Prof: first of all, tidy up this idea of measuring the effect of the treatment on the outcome. So, we've got, in this complicated setting where we've got multiple variables, we've been able to do sufficient conditioning that allow us to 582 01:04:45.600 --> 01:04:52.180 David Stephens, Prof: restrict… The, open paths that remain. 583 01:04:52.570 --> 01:04:56.639 David Stephens, Prof: between Z and Y to be only those that are directed. 584 01:04:59.080 --> 01:05:08.370 David Stephens, Prof: So our conditioning has allowed us to block the open, undirected paths in the original DAG by conditioning on those variables. 585 01:05:08.640 --> 01:05:11.390 David Stephens, Prof: So we're left with only the directed paths. 586 01:05:11.710 --> 01:05:16.569 David Stephens, Prof: And our Causal effect is going to be felt through those directed paths. 587 01:05:17.720 --> 01:05:27.450 David Stephens, Prof: And those paths might contain two mechanisms for… Capturing how… Zed. 588 01:05:28.010 --> 01:05:31.090 David Stephens, Prof: influences Y. There could be a direct effect. 589 01:05:32.690 --> 01:05:37.649 David Stephens, Prof: Or there could be an indirect effect. So the direct effect is felt down those arrows 590 01:05:38.090 --> 01:05:42.270 David Stephens, Prof: which essentially go directly from Z to Y. 591 01:05:43.620 --> 01:05:48.189 David Stephens, Prof: Whereas the indirect effect passes through this set of third variables, X. 592 01:05:50.640 --> 01:05:54.980 David Stephens, Prof: And we think of those as intermediate or mediator variables. 593 01:05:55.950 --> 01:06:04.489 David Stephens, Prof: So if you imagine all the, after conditioning the open, directed paths between the variables, there will either be direct arrows from 594 01:06:05.330 --> 01:06:06.610 David Stephens, Prof: Z to Y? 595 01:06:06.730 --> 01:06:12.310 David Stephens, Prof: or… paths with directed arrows that go through X. 596 01:06:15.660 --> 01:06:20.710 David Stephens, Prof: So we can think about the causal effect in both of… through both of these mechanisms. 597 01:06:21.880 --> 01:06:28.430 David Stephens, Prof: So, in this first DAG here, we've got… Z influencing X, influencing Y. 598 01:06:28.960 --> 01:06:35.749 David Stephens, Prof: So we can think of this as being really an indirect effect of Z on Y that is action through X. 599 01:06:36.490 --> 01:06:39.970 David Stephens, Prof: In this DAG, we've got two open underrated paths. 600 01:06:41.210 --> 01:06:43.509 David Stephens, Prof: Sorry, open directed paths. 601 01:06:44.020 --> 01:06:48.269 David Stephens, Prof: There's the direct path, and the indirect path that goes through X. 602 01:06:49.130 --> 01:06:54.799 David Stephens, Prof: So, legitimate causal effects can be measured through both of these paths. 603 01:07:01.500 --> 01:07:09.230 David Stephens, Prof: In this case, in this slightly more complicated DAG, we've got a direct effect felt for Z on Y through the direct path. 604 01:07:09.720 --> 01:07:13.680 David Stephens, Prof: But there are other variables here, X and U say. 605 01:07:15.150 --> 01:07:18.469 David Stephens, Prof: So there are two paths, overall, from Z to Y. 606 01:07:20.190 --> 01:07:25.230 David Stephens, Prof: There's the direct path, and the indirect path that goes to X and U. 607 01:07:26.020 --> 01:07:28.010 David Stephens, Prof: But X is a collider. 608 01:07:28.150 --> 01:07:29.850 David Stephens, Prof: So that path is blocked. 609 01:07:31.190 --> 01:07:38.350 David Stephens, Prof: So here, the only effect of Z on Y is the direct effect felt down this D arrow. 610 01:07:38.800 --> 01:07:41.330 David Stephens, Prof: There is no other open path that we have to worry about. 611 01:07:44.500 --> 01:07:50.490 David Stephens, Prof: And as we saw before, If we were… if we were in this situation, in reality. 612 01:07:50.870 --> 01:07:56.789 David Stephens, Prof: We had Z on Y. Regressing Y on Z would give us the correct measure of the association. 613 01:07:57.580 --> 01:08:00.439 David Stephens, Prof: If we condition on X, however. 614 01:08:00.810 --> 01:08:03.660 David Stephens, Prof: Conditioning on X in the regression model. 615 01:08:03.940 --> 01:08:08.820 David Stephens, Prof: Would open the undirected path through you, and lead to bias. 616 01:08:15.330 --> 01:08:18.390 David Stephens, Prof: That brings us on to the idea of unmeasured confounding. 617 01:08:18.560 --> 01:08:28.699 David Stephens, Prof: Which is, I guess a real practical problem that we have to address when studying any dataset, even if we're not going down the Causal route. 618 01:08:28.950 --> 01:08:33.950 David Stephens, Prof: Because the presence of unmeasured confounding can really mess things up in terms of our results. 619 01:08:34.880 --> 01:08:37.840 David Stephens, Prof: So suppose that there is a further variable U, 620 01:08:38.010 --> 01:08:41.390 David Stephens, Prof: Which is a confounder, but it's not measured in the observed data. 621 01:08:42.189 --> 01:08:45.609 David Stephens, Prof: That is, we want to study the relationship between Z and Y. 622 01:08:45.830 --> 01:08:55.120 David Stephens, Prof: We recognize the measured confound as X, but we don't recognize the unmeasured confound as U. So if this situation were the data generating mechanism. 623 01:08:56.319 --> 01:09:02.259 David Stephens, Prof: We could account for the presence of X in this open, undirected path by conditioning. 624 01:09:03.870 --> 01:09:06.210 David Stephens, Prof: we throw X into the regression model. 625 01:09:06.890 --> 01:09:13.409 David Stephens, Prof: However, if U is unmeasured, then there's nothing we can do about U. This confounding path is still there. 626 01:09:14.370 --> 01:09:16.359 David Stephens, Prof: Z U Y. 627 01:09:16.800 --> 01:09:28.430 David Stephens, Prof: And because we don't have access to those variables, there's nothing we can do. So all of… basically all of causal inference analysis is based on the assumption that there are no unmeasured confounders. 628 01:09:28.779 --> 01:09:34.689 David Stephens, Prof: That is, all the confounders in the system are measured. 629 01:09:34.990 --> 01:09:36.449 David Stephens, Prof: In the data collection. 630 01:09:37.790 --> 01:09:39.670 David Stephens, Prof: So, 631 01:09:40.160 --> 01:09:49.919 David Stephens, Prof: It could be argued that's a strong assumption, that's a limiting assumption. Of course, it's the assumption that we make in any statistical analysis, it just happens to be a bit more formalized in the causal setting. 632 01:09:52.390 --> 01:10:11.189 David Stephens, Prof: And it shows, again, that there's no substitute in any of this for, subject matter expertise. So a subject matter expert should be able to look at the problem and say, well, I'm sure we've measured all the confounders of interest here. There are no unmeasured confounders, our causal analysis can proceed. 633 01:10:15.650 --> 01:10:22.890 David Stephens, Prof: One other point about the unmeasured confounding, though, is that, It can lead… 634 01:10:23.280 --> 01:10:31.640 David Stephens, Prof: To problems in settings that, are slightly more complicated, if we do our analysis carelessly. 635 01:10:33.180 --> 01:10:37.660 David Stephens, Prof: So, in this situation, this is the underlying data-generating model. 636 01:10:37.820 --> 01:10:41.040 David Stephens, Prof: I've got Z… And why? 637 01:10:41.420 --> 01:10:43.910 David Stephens, Prof: And a third variable, X, that I do measure. 638 01:10:44.930 --> 01:10:50.440 David Stephens, Prof: Now, if I've got Z and X and Y formulated without the red lines. 639 01:10:50.700 --> 01:10:57.459 David Stephens, Prof: Then that DAG… well, it's not even a DAG, really. There are 3 independent nodes. They're unconnected. 640 01:10:58.050 --> 01:11:02.189 David Stephens, Prof: So, certainly, Z has no association with Y. 641 01:11:05.770 --> 01:11:12.590 David Stephens, Prof: If I just had the three variables, Z, X, and Y, if I regressed Y on Z, 642 01:11:12.810 --> 01:11:17.700 David Stephens, Prof: If I regressed Y and Z, then I would get a null result. 643 01:11:17.860 --> 01:11:19.620 David Stephens, Prof: And that would be the correct answer. 644 01:11:20.310 --> 01:11:24.989 David Stephens, Prof: However, in my analysis, if I condition on X, 645 01:11:26.690 --> 01:11:29.719 David Stephens, Prof: And, in fact, these unmeasured confounders are there. 646 01:11:30.570 --> 01:11:41.890 David Stephens, Prof: I create problems for myself, because conditioning on X in the regression model opens this true confounding path. 647 01:11:44.980 --> 01:11:49.290 David Stephens, Prof: So, this mechanism is called… M-bias. 648 01:11:54.740 --> 01:11:56.609 David Stephens, Prof: Because of the shape M here. 649 01:11:57.300 --> 01:11:58.390 David Stephens, Prof: And… 650 01:12:03.790 --> 01:12:11.339 David Stephens, Prof: It shows the… so if you… if you really believed… whoops, if you really believed this… whoops… this structure. 651 01:12:14.750 --> 01:12:15.740 David Stephens, Prof: then… 652 01:12:19.430 --> 01:12:23.159 David Stephens, Prof: So, the black nodes here. You would, certainly not… 653 01:12:23.270 --> 01:12:36.149 David Stephens, Prof: need to condition on X. But you might think to yourself in the background, oh, hang on, maybe it's just safer to condition on X. Well, this example shows you why that unnecessary step can cause problems. It can induce bias. 654 01:12:36.250 --> 01:12:39.800 David Stephens, Prof: Whereas the original analysis did not induce bias. 655 01:12:41.770 --> 01:12:46.899 David Stephens, Prof: Okay? Conditioning on X spuriously opened the undirected 656 01:12:48.890 --> 01:12:51.470 David Stephens, Prof: open path through U1 and U2. 657 01:12:55.230 --> 01:12:59.800 David Stephens, Prof: So it tells you, again, to be judicious in your… modeling choices. 658 01:13:04.020 --> 01:13:06.600 David Stephens, Prof: So, just finally then, 659 01:13:09.240 --> 01:13:18.440 David Stephens, Prof: to recap where we… where we are. What is the… what are the implications for all of these… all of these graphical ideas on statistical modeling? 660 01:13:19.170 --> 01:13:23.339 David Stephens, Prof: Well, again, remember that the first step in any of our, 661 01:13:24.160 --> 01:13:32.049 David Stephens, Prof: analysis framework is to sit down and draw the DAG that we think represents the connections between our variables. 662 01:13:33.940 --> 01:13:38.349 David Stephens, Prof: Our subject matter expertise will tell us how to do that. Once we've done that. 663 01:13:39.410 --> 01:13:51.339 David Stephens, Prof: Our next task is to construct a statistical model that will allow us to assess the true causal effect connecting of the treatment variable on the outcome variable. 664 01:13:53.120 --> 01:13:56.569 David Stephens, Prof: So our task is really to choose those variables. 665 01:13:57.390 --> 01:14:03.080 David Stephens, Prof: to throw into the regression model that will block the Blyasing paths. 666 01:14:03.640 --> 01:14:08.029 David Stephens, Prof: Because only that will allow us to get an unbiased assessment of the causal effect. 667 01:14:09.510 --> 01:14:14.060 David Stephens, Prof: In the classic confounding situation, it's clear we've got a condition on X. 668 01:14:16.180 --> 01:14:17.150 David Stephens, Prof: So… 669 01:14:19.850 --> 01:14:26.630 David Stephens, Prof: First step, collect our data. Second step, draw the DAG that we think connects the variables in our analysis. 670 01:14:29.550 --> 01:14:33.099 David Stephens, Prof: on the basis of the DAG, choose our statistical model. 671 01:14:33.410 --> 01:14:37.930 David Stephens, Prof: that correctly blocks the problematic paths in the DAG. 672 01:14:40.250 --> 01:14:42.070 David Stephens, Prof: Usually we do that by conditioning. 673 01:14:42.420 --> 01:14:45.470 David Stephens, Prof: Conditioning on those variables, throwing those variables into the model. 674 01:14:49.920 --> 01:15:03.969 David Stephens, Prof: Now, if we believe that all of the variables that we observe are normally distributed, a typical kind of simplifying assumption, then regression is exactly the right thing to do, because regressing Y on X and Z threat, say, through a model like this. 675 01:15:04.230 --> 01:15:14.090 David Stephens, Prof: is the exactly correct way to account for the dependence on X. 676 01:15:15.420 --> 01:15:29.520 David Stephens, Prof: So outside of the normal case, and outside of linear regression, it's not quite so exact, but, often a good approximation. However, if we genuinely believe that YX, and Z were all normally distributed, then linear regression 677 01:15:30.060 --> 01:15:37.639 David Stephens, Prof: But the appropriate terms in the regression model would be exactly the right way to adjust for the confounding variables. 678 01:15:38.670 --> 01:15:42.270 David Stephens, Prof: Okay, so in this case, 679 01:15:44.450 --> 01:15:49.449 David Stephens, Prof: So in the general confounding case, this model would allow us to uncover 680 01:15:50.080 --> 01:15:54.310 David Stephens, Prof: the effect of Z on Y by estimating psi. 681 01:15:59.910 --> 01:16:02.440 David Stephens, Prof: That point raises, though, a… 682 01:16:07.830 --> 01:16:12.770 David Stephens, Prof: I think an important and sometimes underappreciated issue, which is that 683 01:16:13.390 --> 01:16:19.010 David Stephens, Prof: Whereas this is the correct thing to do if the data are normally distributed, jointly normally distributed. 684 01:16:19.250 --> 01:16:21.900 David Stephens, Prof: And fitting this model is the correct thing to do. 685 01:16:22.460 --> 01:16:27.670 David Stephens, Prof: More generally, conditioning 686 01:16:28.410 --> 01:16:35.339 David Stephens, Prof: can't quite be achieved by throwing a term into a regression model. It only works exactly in the normal case. 687 01:16:35.560 --> 01:16:39.650 David Stephens, Prof: And therefore works well in an approximate case in many settings. 688 01:16:40.620 --> 01:16:44.839 David Stephens, Prof: But, more generally, you have to think about 689 01:16:45.140 --> 01:16:50.390 David Stephens, Prof: correct specification of this, what is essentially a parametric model, right? 690 01:16:50.530 --> 01:16:56.359 David Stephens, Prof: So, if the data were normally distributed, fitting this would be exactly the right thing to do. You'd correct 691 01:16:57.160 --> 01:16:58.919 David Stephens, Prof: In exactly the right way. 692 01:16:59.600 --> 01:17:05.729 David Stephens, Prof: for the confounding variable X by this form of conditioning, and we'd get the right answer for psi. 693 01:17:07.260 --> 01:17:09.410 David Stephens, Prof: Outside of the normal case, though. 694 01:17:10.620 --> 01:17:19.540 David Stephens, Prof: That kind of procedure can only ever be approximate, and we are then contingent on having correct parametric specifications. 695 01:17:19.810 --> 01:17:25.409 David Stephens, Prof: Which is really why we, in the, in the econometrics literature. 696 01:17:26.010 --> 01:17:32.640 David Stephens, Prof: rather than the biostatistics or public health or epidemiological literature. They basically focus on this kind of model. 697 01:17:32.990 --> 01:17:39.520 David Stephens, Prof: And assume that, the normality assumption is approximately correct. 698 01:17:40.360 --> 01:17:50.170 David Stephens, Prof: Whereas in epidemiology, public health, biostatistics, there's been a bit more work done on what constitutes the correct form of conditioning, at least in an asymptotic setting. 699 01:17:52.570 --> 01:17:57.449 David Stephens, Prof: Alright, so, before we move on to the few examples, just a recap of what we've seen. 700 01:18:00.500 --> 01:18:13.739 David Stephens, Prof: So in an epidemiological research problem, where we're studying the effect of a treatment on an act, and we identify those variables Z and X, we acknowledge the presence of other… sorry, Z and Y, and we acknowledge the presence of those other variables X. 701 01:18:14.170 --> 01:18:17.440 David Stephens, Prof: We form the DAG connecting all of the variables. 702 01:18:18.460 --> 01:18:21.490 David Stephens, Prof: And then to study the effect of the, 703 01:18:21.770 --> 01:18:24.770 David Stephens, Prof: Causal effect of Z on Y, 704 01:18:25.260 --> 01:18:31.600 David Stephens, Prof: We identify in our DAG the open, directed paths between the variables. 705 01:18:33.120 --> 01:18:44.060 David Stephens, Prof: the open, directed paths from Z to Y. Our Causal effect is felt through those paths. Now, those paths could be direct, or they could have indirect elements, but 706 01:18:44.250 --> 01:18:49.370 David Stephens, Prof: The analysis of those paths alone will give us the correct measure of association. 707 01:18:51.670 --> 01:18:59.689 David Stephens, Prof: The problematic paths are the open, undirected paths between the variables, and those are the paths that we have to block. 708 01:19:01.020 --> 01:19:08.120 David Stephens, Prof: Typically, we do that by… Conditioning, typically by throwing variables into a regression model, although 709 01:19:08.280 --> 01:19:12.030 David Stephens, Prof: That's not always guaranteed to work exactly in finite sample. 710 01:19:13.100 --> 01:19:24.389 David Stephens, Prof: But those are the paths that we have to identify. So the DAG and the Causal graph helps us to identify which variables we need to condition upon in order to block the paths. 711 01:19:26.780 --> 01:19:37.689 David Stephens, Prof: The other thing to say is that the collider node, if you ever see that on a path connecting the two variables, in some sense that's helpful to you, because it's blocking an otherwise problematic path. 712 01:19:38.940 --> 01:19:45.670 David Stephens, Prof: The only thing we have to remember, though, is that conditioning on a collider, so throwing a variable into a model. 713 01:19:46.490 --> 01:19:50.160 David Stephens, Prof: When, in fact, it's a collider on a relevant path. 714 01:19:50.370 --> 01:19:56.339 David Stephens, Prof: Is the wrong thing to do, and it can give us, Incorrect answers. 715 01:19:57.620 --> 01:20:03.070 David Stephens, Prof: Okay, so those are the ends of the slides. I'll just quickly, for the last… 10 minutes or so. 716 01:20:03.280 --> 01:20:08.560 David Stephens, Prof: Go over to the… Examples. 717 01:20:08.740 --> 01:20:10.000 David Stephens, Prof: Right? So… 718 01:20:10.160 --> 01:20:17.980 David Stephens, Prof: you can take this code, and it's just the code in R, simple code in R, I've laid out in a little detail here with comments exactly what's going on. 719 01:20:18.230 --> 01:20:28.420 David Stephens, Prof: You can take these, bits of code, and manipulate the numbers, and test different cases, and so on. 720 01:20:28.520 --> 01:20:38.899 David Stephens, Prof: In order to make the concepts that I've described more concrete. So the first example is the distinction between conditional independence and dependence. 721 01:20:39.220 --> 01:20:44.830 David Stephens, Prof: What I've done here is to set it up such that Z and Y… 722 01:20:45.230 --> 01:20:55.630 David Stephens, Prof: ZY and X are all continuous variables, but I'm specifying that conditional on X, Z and Y are negatively correlated. 723 01:20:56.990 --> 01:20:58.870 David Stephens, Prof: That's what the specification means. 724 01:21:01.800 --> 01:21:08.770 David Stephens, Prof: Whereas… In the joint distribution, They're positively correlated. 725 01:21:09.130 --> 01:21:12.820 David Stephens, Prof: So, if I simulate from this model, it's carefully set up to do that. 726 01:21:14.560 --> 01:21:22.630 David Stephens, Prof: I see the strong positive correlation between all of the variables, even though conditionally they're negatively correlated Z and Y. 727 01:21:24.140 --> 01:21:28.000 David Stephens, Prof: So this is a sort of classic confounding case mimicking. 728 01:21:28.640 --> 01:21:36.520 David Stephens, Prof: There's the positive correlation between all the variables. These are point clouds, 10,000 points simulated from the system to show that they're all positively correlated. 729 01:21:39.060 --> 01:21:47.490 David Stephens, Prof: if I… Then look at the, the ZY relationship in particular. 730 01:21:47.920 --> 01:21:59.199 David Stephens, Prof: unconditionally on X, we see a positive correlation. That tells us that if I were to regress Y on Z, then I would get a positive coefficient. 731 01:22:04.700 --> 01:22:09.189 David Stephens, Prof: So, to illustrate the… The reverse of that relationship. 732 01:22:09.570 --> 01:22:15.549 David Stephens, Prof: After conditioning, here I look at subsets of the data conditioning on X in different ranges. 733 01:22:15.690 --> 01:22:17.710 David Stephens, Prof: And I do 3 different plots. 734 01:22:18.180 --> 01:22:26.840 David Stephens, Prof: So, for the first range, X lies between 4.5 and 5.5, then 9.5, 10.5, 14.5, 15.5. 735 01:22:27.100 --> 01:22:32.310 David Stephens, Prof: What I observe when I do the different point clouds for those three subsets is that I have negative correlation. 736 01:22:32.690 --> 01:22:35.180 David Stephens, Prof: That's how I deliberately simulated the data. 737 01:22:36.680 --> 01:22:41.670 David Stephens, Prof: So, then if I do a regression, of… of the… 738 01:22:41.790 --> 01:22:46.570 David Stephens, Prof: of Y on Z in the three subsets, I get a strongly negative coefficient. 739 01:22:48.270 --> 01:22:55.379 David Stephens, Prof: Whereas if I regress Y on Z overall, I get a strongly positive coefficient. 740 01:22:56.670 --> 01:23:05.840 David Stephens, Prof: So this simulation illustrates that point about the difference between marginal dependence, and… conditional dependence. 741 01:23:08.060 --> 01:23:13.840 David Stephens, Prof: There are the three individual subsets, conditioning on the three different values of X. 742 01:23:15.480 --> 01:23:22.920 David Stephens, Prof: So in this analysis, in order to get… to understand the correct relationship between Y and Z, I need to condition on X. 743 01:23:27.590 --> 01:23:34.150 David Stephens, Prof: So now if, that's where there was a general companion relationship. In this case, I've got… 744 01:23:34.550 --> 01:23:40.779 David Stephens, Prof: a simpler relationship, where X causes Y, X causes Z, they're conditionally independent. 745 01:23:42.020 --> 01:23:45.700 David Stephens, Prof: Here I get a positive association, marginally. 746 01:23:46.340 --> 01:23:58.240 David Stephens, Prof: But if I do the conditioning again, that's the Z versus Y relationship, so that's unchanged from before. However, now if I do conditioning, I see that the variables are not associated anymore. 747 01:23:59.390 --> 01:24:00.070 David Stephens, Prof: Right. 748 01:24:00.300 --> 01:24:06.160 David Stephens, Prof: In the three subsets, conditioning on X, Z and Y are uncorrelated. 749 01:24:07.310 --> 01:24:10.219 David Stephens, Prof: Okay? And again, that's evinced through… 750 01:24:10.560 --> 01:24:18.240 David Stephens, Prof: regressing Y on Z in each of the three subsets. Here I get essentially a zero coefficient each time. 751 01:24:19.560 --> 01:24:24.750 David Stephens, Prof: Whereas, marginally, again, I get this strong positive association. 752 01:24:27.040 --> 01:24:33.530 David Stephens, Prof: So this simple simulation shows me the importance of doing the right type of conditional analysis. 753 01:24:36.730 --> 01:24:37.980 David Stephens, Prof: There's the plot again. 754 01:24:39.390 --> 01:24:49.029 David Stephens, Prof: Okay? Then I've got a little illustration of colliders doing the same kind of thing, so X and Y being independent in the original simulation. 755 01:24:49.740 --> 01:24:54.320 David Stephens, Prof: If I condition on Z… They become negatively correlated. 756 01:24:55.870 --> 01:24:58.679 David Stephens, Prof: That's the basic collider concept. 757 01:25:00.130 --> 01:25:03.659 David Stephens, Prof: Then I've got some… some illustrations of de-separation. 758 01:25:04.070 --> 01:25:14.099 David Stephens, Prof: Again, showing that if we do the analysis of this DAG correctly, find out what the de-separating variables are between X2 and X3, 759 01:25:14.710 --> 01:25:22.579 David Stephens, Prof: then I can mimic that in the simulation by doing appropriate conditioning subsetting of the data. 760 01:25:22.910 --> 01:25:23.580 David Stephens, Prof: Okay? 761 01:25:23.930 --> 01:25:25.180 David Stephens, Prof: So… 762 01:25:26.390 --> 01:25:34.059 David Stephens, Prof: to show de-separation, all I need to do is to say, well, to show de-separation of X2 and X3 state. 763 01:25:35.240 --> 01:25:36.490 David Stephens, Prof: by X1. 764 01:25:36.930 --> 01:25:43.439 David Stephens, Prof: I show that, when I regress X2 on X3, I get a significantly non-zero coefficient. 765 01:25:44.670 --> 01:25:51.000 David Stephens, Prof: Whereas if I… Condition on X1, Then the coefficient becomes zero. 766 01:25:55.320 --> 01:26:01.749 David Stephens, Prof: Okay? So conditioning on X1 induces de-separation between X2 and X3. 767 01:26:04.900 --> 01:26:09.520 David Stephens, Prof: Contrary to that, if I condition on X1 and X4, Then… 768 01:26:09.830 --> 01:26:16.080 David Stephens, Prof: I don't get this de-separation idea, and this is what the regression models here show. So again, you can play around with these… 769 01:26:16.200 --> 01:26:20.909 David Stephens, Prof: examples to… Explore the different possibilities. 770 01:26:22.370 --> 01:26:24.359 David Stephens, Prof: Some more on de-separation. 771 01:26:26.230 --> 01:26:32.139 David Stephens, Prof: And then… these more complicated DAGs that I was talking about here, so again. 772 01:26:32.450 --> 01:26:37.489 David Stephens, Prof: You can take this, take this data, take this code, simulate your own data. 773 01:26:38.090 --> 01:26:40.630 David Stephens, Prof: Attach variables and so on, however you like. 774 01:26:43.380 --> 01:26:50.980 David Stephens, Prof: Two more examples that show that. And in all of these cases, because I'm generating from a multivariate normal model. 775 01:26:51.130 --> 01:26:55.519 David Stephens, Prof: The regression through LM is always going to give me a correct picture. 776 01:26:58.390 --> 01:27:01.669 David Stephens, Prof: And then the simulation study shows the different results there. 777 01:27:02.460 --> 01:27:11.119 David Stephens, Prof: So you can look through that. These are a bit more complicated, but it's really identifying the same sorts of issues as we've identified in the slides. 778 01:27:11.970 --> 01:27:16.569 David Stephens, Prof: And then… oh yeah, another, another desceparation example. 779 01:27:17.950 --> 01:27:27.290 David Stephens, Prof: Bit more complicated. And then finally, these two, examples, I think are also taken from the Rothman textbook. 780 01:27:27.450 --> 01:27:33.609 David Stephens, Prof: But… loaded into the R package, which is called GGDAG, 781 01:27:33.980 --> 01:27:39.649 David Stephens, Prof: Which is, for drawing DAGs, in a nice way. 782 01:27:41.640 --> 01:27:57.439 David Stephens, Prof: So, here we've got this setup, so I'll talk about this one a bit more. We want to, here we've got smoking status, glioma, hospitalization, broken bone, and tendency for reckless behavior. We sit down and we believe that this is the DAG connecting our variables. 783 01:28:00.780 --> 01:28:03.069 David Stephens, Prof: So we see here that, 784 01:28:03.770 --> 01:28:06.529 David Stephens, Prof: We're gonna look at the relationship between G and S. 785 01:28:07.620 --> 01:28:16.109 David Stephens, Prof: So G and S are… Unconditionally uncorrelated, because in this path. 786 01:28:16.550 --> 01:28:18.590 David Stephens, Prof: There is a collider at H. 787 01:28:22.620 --> 01:28:24.289 David Stephens, Prof: H is hospitalisation. 788 01:28:24.910 --> 01:28:29.569 David Stephens, Prof: So, in practice, what that… what this might mean is that, although this is the… 789 01:28:30.210 --> 01:28:34.080 David Stephens, Prof: DAG structure, and actually, glioma and smoking status. 790 01:28:34.430 --> 01:28:37.910 David Stephens, Prof: That's a null relationship. 791 01:28:38.230 --> 01:28:41.240 David Stephens, Prof: If we look at only people who are hospitalized. 792 01:28:41.830 --> 01:28:51.509 David Stephens, Prof: then that's gonna condition on the DAG variable H, And that's gonna open the… path between G and S. 793 01:28:52.360 --> 01:28:58.750 David Stephens, Prof: So, conditioning on H, renders G and S dependent. 794 01:29:00.160 --> 01:29:04.940 David Stephens, Prof: So, if I do a… A study of… 795 01:29:06.230 --> 01:29:10.119 David Stephens, Prof: Smoking… the relationship between smoking status and glioma. 796 01:29:12.060 --> 01:29:14.979 David Stephens, Prof: But I only look in hospitalized patients. 797 01:29:16.000 --> 01:29:19.489 David Stephens, Prof: then I'll pick up a spurious positive association. 798 01:29:24.430 --> 01:29:26.099 David Stephens, Prof: This simulation shows. 799 01:29:26.380 --> 01:29:28.130 David Stephens, Prof: Puts the numerics onto that. 800 01:29:29.770 --> 01:29:30.440 David Stephens, Prof: Okay? 801 01:29:30.750 --> 01:29:35.370 David Stephens, Prof: So this shows us that… so a careful analysis of this DAG, 802 01:29:36.260 --> 01:29:43.260 David Stephens, Prof: Tells us what we should get from our statistical analysis, and what statistical analysis we should not embark upon. 803 01:29:43.900 --> 01:29:44.610 David Stephens, Prof: Right. 804 01:29:45.270 --> 01:29:49.350 David Stephens, Prof: Or it tells us the danger of only looking in a particular cohort. 805 01:29:52.000 --> 01:29:54.330 David Stephens, Prof: The second example is that, 806 01:29:54.430 --> 01:30:01.759 David Stephens, Prof: CD4 and HIV drug example from before. Here is the DAG that was proposed for this analysis. 807 01:30:02.520 --> 01:30:07.890 David Stephens, Prof: So this is a, a, 808 01:30:10.460 --> 01:30:13.590 David Stephens, Prof: Conditioning on a descendant of the treatment. 809 01:30:14.820 --> 01:30:23.040 David Stephens, Prof: Issue, and descendant of a collider issue. So, D is the assignment of the new drug. That causes… 810 01:30:23.580 --> 01:30:25.750 David Stephens, Prof: Symptoms, relief? 811 01:30:27.090 --> 01:30:31.439 David Stephens, Prof: But those symptoms also… Lead to inclusion in the follow-up. 812 01:30:34.170 --> 01:30:39.940 David Stephens, Prof: The relationship between D and C, the outcome of interest. 813 01:30:40.060 --> 01:30:45.649 David Stephens, Prof: is a null one, because D and C are… 814 01:30:46.050 --> 01:30:49.859 David Stephens, Prof: unconditionally uncorrelated due to the collider S. 815 01:30:51.040 --> 01:31:00.930 David Stephens, Prof: So, in fact, in this setup, there is no relationship, no null… there is a null association between D and C, 816 01:31:01.840 --> 01:31:10.420 David Stephens, Prof: However, If we condition on you, i.e. those people followed up, those people staying in the trial, U… 817 01:31:10.890 --> 01:31:13.840 David Stephens, Prof: is a descendant of the Collider. 818 01:31:14.080 --> 01:31:22.739 David Stephens, Prof: So conditioning on U opens the path for S, and leads to a spurious association being detected between D and C. 819 01:31:28.260 --> 01:31:32.010 David Stephens, Prof: There's the numerical, the numerical. 820 01:31:33.220 --> 01:31:34.860 David Stephens, Prof: Evidence for this. 821 01:31:35.200 --> 01:31:41.319 David Stephens, Prof: Okay, again, arrived at through regression models. Actually, not even regression models here, just proportions in data. 822 01:31:42.870 --> 01:31:46.740 David Stephens, Prof: Finally, an M-bias simulation to show the M-bias concept. 823 01:31:49.080 --> 01:32:00.760 David Stephens, Prof: Okay, so, I'll stop there for the module. So, a lot of material. If you've got any questions on that immediately, I'll stick around… I can stick around for a few minutes. If you've got any, 824 01:32:01.560 --> 01:32:04.130 David Stephens, Prof: Questions that arise after… 825 01:32:04.250 --> 01:32:09.669 David Stephens, Prof: consideration of the notes and so on. You can always email me directly or post to the… 826 01:32:09.980 --> 01:32:11.820 David Stephens, Prof: to the Slack channel. 827 01:32:12.320 --> 01:32:28.779 David Stephens, Prof: The sessions have been recorded. It takes usually a couple of hours for the system to process it at the McGill Zoom end, so, they should be posted on the Slack channel, hopefully by the end of today. 828 01:32:29.390 --> 01:32:30.910 David Stephens, Prof: Okay, thank you very much.