WEBVTT 1 00:00:49.940 --> 00:00:53.400 David Stephens, Prof: Okay, good morning, everybody. So… 2 00:00:53.690 --> 00:00:59.419 David Stephens, Prof: My name's, Dave Stevens, this is the module on Propensity Scores. 3 00:01:00.200 --> 00:01:09.600 David Stephens, Prof: So, I'm Dave Stevens from the Department of Mathematics and Statistics, McGill University in Montreal. So, I'm here in the Eastern Time Zone, it's 11.30 coming up here, 8.30 on the… 4 00:01:10.450 --> 00:01:11.649 David Stephens, Prof: on the West Coast. 5 00:01:11.910 --> 00:01:20.490 David Stephens, Prof: So yeah, this module, for today and tomorrow will, lead you through the construction of Propensity score logic. 6 00:01:20.880 --> 00:01:26.619 David Stephens, Prof: And give an introduction to the propensity score construction, and then… 7 00:01:27.150 --> 00:01:33.579 David Stephens, Prof: We'll look at some examples of how the Propensity score is used in… Causal adjustment. 8 00:01:33.990 --> 00:01:36.280 David Stephens, Prof: In epidemiological research. 9 00:01:36.760 --> 00:01:43.800 David Stephens, Prof: You can contact me on this email address here, and all of the materials, the module materials, are on this 10 00:01:44.430 --> 00:01:54.350 David Stephens, Prof: website, just a basic folder of the PDFs for you to download. It's also posted on the module website as well. 11 00:01:54.650 --> 00:02:01.060 David Stephens, Prof: If you've got any questions as we go through, you can put them in the chat, or… 12 00:02:01.340 --> 00:02:03.740 David Stephens, Prof: mail them to the Slack channel. 13 00:02:04.270 --> 00:02:06.560 David Stephens, Prof: Or contact me at this address afterwards. 14 00:02:06.960 --> 00:02:09.840 David Stephens, Prof: So, the… 15 00:02:12.700 --> 00:02:18.519 David Stephens, Prof: schedule I've also included there, this is roughly what we're going to try to achieve in the next couple of days. 16 00:02:18.640 --> 00:02:21.389 David Stephens, Prof: Today we'll be building the framework. 17 00:02:21.650 --> 00:02:26.599 David Stephens, Prof: And tomorrow, we'll be looking at the applications of the methods that we're going to develop. 18 00:02:26.960 --> 00:02:33.339 David Stephens, Prof: Including an analysis of a real data set from the NHANES study. 19 00:02:34.150 --> 00:02:44.880 David Stephens, Prof: So here I lay out in our how to do some of the analyses, and then there's also a simulation study to demonstrate some of the concepts that I'm going to try to get across as well. 20 00:02:47.160 --> 00:02:48.130 David Stephens, Prof: So… 21 00:02:52.420 --> 00:02:59.390 David Stephens, Prof: In this module, we're going to be talking a lot about Causal inference, and… 22 00:03:00.710 --> 00:03:06.860 David Stephens, Prof: what I try to argue here is that causal conclusions are really what we're aiming at in most 23 00:03:08.380 --> 00:03:12.250 David Stephens, Prof: Scientific investigations, most epidemiological investigations. 24 00:03:14.380 --> 00:03:24.779 David Stephens, Prof: And the key distinction that we usually make is between situations in which it's obvious how we can make causal conclusions, that is, let's say, experimental studies. 25 00:03:25.310 --> 00:03:35.639 David Stephens, Prof: And I draw the distinction between experimental studies and observational studies. The difference, of course, is that in the experimental setting, we are explicitly able 26 00:03:35.880 --> 00:03:40.010 David Stephens, Prof: To investigate the causal impact of a treatment or an exposure. 27 00:03:40.320 --> 00:03:45.179 David Stephens, Prof: Because it is the researcher that's assigning the treatment. 28 00:03:46.290 --> 00:03:52.060 David Stephens, Prof: So this gives us the capacity to compare, on an equal footing, groups that… 29 00:03:52.600 --> 00:03:55.289 David Stephens, Prof: Well, let's say treated and not treated. 30 00:03:56.350 --> 00:03:59.390 David Stephens, Prof: We can make the causal conclusions directly from that comparison. 31 00:04:02.020 --> 00:04:04.930 David Stephens, Prof: And that's in contrast from the observational case. 32 00:04:05.250 --> 00:04:09.070 David Stephens, Prof: Where… Treatment is not assigned. 33 00:04:09.680 --> 00:04:11.070 David Stephens, Prof: by the researcher. 34 00:04:11.720 --> 00:04:17.830 David Stephens, Prof: So, treated and untreated groups… Maybe, fundamentally, not comparable. 35 00:04:18.160 --> 00:04:19.370 David Stephens, Prof: Without adjustment. 36 00:04:21.110 --> 00:04:23.469 David Stephens, Prof: So, whereas in the experimental case. 37 00:04:24.290 --> 00:04:42.359 David Stephens, Prof: causal conclusions might be relatively straightforward to make. It might seem on the surface that causal conclusions from observational studies are much more complicated. So much of this focus of the module will be how exactly we can make causal conclusions, under some assumptions at least. 38 00:04:42.950 --> 00:04:44.550 David Stephens, Prof: From observational data. 39 00:04:46.620 --> 00:04:51.179 David Stephens, Prof: So this is, the five-part breakdown of the module. 40 00:04:51.510 --> 00:04:57.049 David Stephens, Prof: We'll get through, basically, the first two today, and then look at the last three tomorrow. 41 00:04:57.580 --> 00:05:06.680 David Stephens, Prof: So, I'm gonna spend some time building the framework, that is, talking about why we need to adjust using the propensity score. 42 00:05:07.460 --> 00:05:15.579 David Stephens, Prof: And then I'll show why the Propensity score in particular is the… is the tool that we can deploy fairly easily using standard methods. 43 00:05:16.180 --> 00:05:20.050 David Stephens, Prof: To make, causal conclusions from observational data possible. 44 00:05:22.030 --> 00:05:29.910 David Stephens, Prof: So what we'll see is that statistical tools that are commonly used in Propensity score adjustment are fairly standard. They are regression-type tools. 45 00:05:30.950 --> 00:05:35.529 David Stephens, Prof: It's just a particular combination of methods that, 46 00:05:35.960 --> 00:05:40.610 David Stephens, Prof: It leads us to be able to make these nice conclusions from the observational data. 47 00:05:41.430 --> 00:05:59.460 David Stephens, Prof: So we'll spend some time building the, sort of, fundamental framework based on binary treatments, so treated and untreated, say, and then look at some extensions of that to more complicated settings. For example, multi-value treatments, time-dependent treatments, courses of treatments, and so on. 48 00:05:59.800 --> 00:06:03.320 David Stephens, Prof: And then I'll also talk a little bit about some of the 49 00:06:04.060 --> 00:06:11.249 David Stephens, Prof: Current areas of research and propensity score work, including the use of machine learning-type methods. 50 00:06:11.590 --> 00:06:14.540 David Stephens, Prof: Maybe some Bayesian discussion as well. 51 00:06:15.700 --> 00:06:17.109 David Stephens, Prof: Okay, so that's the plan. 52 00:06:18.710 --> 00:06:24.160 David Stephens, Prof: I've broken it down into these five, parts, in a little more detail. 53 00:06:24.530 --> 00:06:29.740 David Stephens, Prof: Through these, sections, so these are the sections that you'll see appearing on the bottom of the slides. 54 00:06:30.030 --> 00:06:33.500 David Stephens, Prof: First part is about… Building the framework. 55 00:06:34.110 --> 00:06:41.080 David Stephens, Prof: And I'll talk about some fundamental statistical ideas, and some ways of thinking about why… 56 00:06:41.810 --> 00:06:45.030 David Stephens, Prof: Propensity score adjustment works in the way that it does. 57 00:06:46.150 --> 00:06:52.150 David Stephens, Prof: In particular, introduced this notion of confounding and how propensity scores can be used to 58 00:06:52.500 --> 00:06:55.360 David Stephens, Prof: Adjust for confounding in observational data. 59 00:06:58.640 --> 00:07:05.920 David Stephens, Prof: Second section is explicitly about, construction of the Propensity score, why we do it in the way we do it, why it works. 60 00:07:06.720 --> 00:07:12.779 David Stephens, Prof: We'll see that in the ordinary case of binary treatments, but then in more complicated cases as well. 61 00:07:12.940 --> 00:07:16.280 David Stephens, Prof: And then, in the second half of this, 62 00:07:16.500 --> 00:07:22.640 David Stephens, Prof: Section, which we'll really be looking at tomorrow in the first session tomorrow. 63 00:07:23.020 --> 00:07:29.900 David Stephens, Prof: We'll see the particular statistical methods that deploy the propensity score. 64 00:07:30.980 --> 00:07:35.420 David Stephens, Prof: Different types of regression, weighting, matching, stratification type approaches. 65 00:07:39.180 --> 00:07:45.229 David Stephens, Prof: Then we'll see… The… a bit of background on the statistical methodology. 66 00:07:46.310 --> 00:07:51.120 David Stephens, Prof: Using fairly standard ideas, again, maybe one of the key ones there is the use of, 67 00:07:52.020 --> 00:07:57.820 David Stephens, Prof: Sandwich-type estimation, robust estimation of variant… variability, 68 00:07:58.930 --> 00:08:07.499 David Stephens, Prof: And then talk a little bit about some considerations for how the propensity score models should be, constructed and deployed. 69 00:08:08.170 --> 00:08:12.229 David Stephens, Prof: And then we'll have this, extension on, sorry, this, 70 00:08:13.560 --> 00:08:19.220 David Stephens, Prof: example on the NHANES data, some sort of deployment of the methods in practice. 71 00:08:19.580 --> 00:08:26.709 David Stephens, Prof: And then, also an examination using a simulation study to… Try to re-emphasize there. 72 00:08:27.090 --> 00:08:30.330 David Stephens, Prof: The key properties of all of the methods that we're going to look at. 73 00:08:31.850 --> 00:08:34.309 David Stephens, Prof: Extensions to the longitudinal case. 74 00:08:36.240 --> 00:08:40.489 David Stephens, Prof: In a short section there, and then sum up with some, 75 00:08:41.270 --> 00:08:47.099 David Stephens, Prof: Practical challenges, current challenges, current areas of interest, and 76 00:08:47.540 --> 00:08:51.150 David Stephens, Prof: A little bit at the end about how the… the, 77 00:08:51.840 --> 00:08:53.960 David Stephens, Prof: Propensity score adjustment can be used in… 78 00:08:54.630 --> 00:08:59.269 David Stephens, Prof: Extended settings, like, the development of optimal treatment strategies. 79 00:09:01.840 --> 00:09:07.780 David Stephens, Prof: So, we're gonna… I'm gonna do, let's say, an hour and a half in the first, 80 00:09:08.410 --> 00:09:15.360 David Stephens, Prof: sub-session, and then we'll have a break for half an hour, so it'll be 1 o'clock my time, so roughly an hour and a half from now. 81 00:09:15.480 --> 00:09:16.890 David Stephens, Prof: And then, 82 00:09:17.150 --> 00:09:30.379 David Stephens, Prof: have a half an hour break, and then continue for the last hour and a half today. And in each of those, hour and a half sessions, I'm gonna do 45 minutes, and then have a short break for a comfort break, and allow me to have a… 83 00:09:30.660 --> 00:09:31.890 David Stephens, Prof: A drink of water. 84 00:09:32.870 --> 00:09:47.710 David Stephens, Prof: Okay, so… In the first part, we're gonna think about building the… the framework For Propensity score models. 85 00:09:48.470 --> 00:09:53.010 David Stephens, Prof: And… this will entail an inspection of some… 86 00:09:53.490 --> 00:09:58.390 David Stephens, Prof: Ideas from probability modelling, some ideas from Causal graphs. 87 00:09:58.690 --> 00:10:04.820 David Stephens, Prof: And then, towards the end of this section, we'll see the, 88 00:10:05.160 --> 00:10:11.220 David Stephens, Prof: The fundamental idea of confounding, and why confounding is important, and why the propensity score can be 89 00:10:11.360 --> 00:10:13.310 David Stephens, Prof: Used to remove it. 90 00:10:15.170 --> 00:10:20.260 David Stephens, Prof: Okay, so… Again, the central causal question. 91 00:10:20.720 --> 00:10:27.260 David Stephens, Prof: That we're going to study is, how do we, investigate the impact 92 00:10:27.800 --> 00:10:32.010 David Stephens, Prof: Of changing a treatment or an exposure on an outcome. 93 00:10:33.710 --> 00:10:41.859 David Stephens, Prof: So, in a health research setting, this might be looking at the benefit of having a new treatment or a new therapy compared to standard care. 94 00:10:42.000 --> 00:10:48.110 David Stephens, Prof: In economics applications, we might think about a training program or something, so if we have somebody 95 00:10:48.510 --> 00:10:51.550 David Stephens, Prof: Perform a training program, how does that benefit them? 96 00:10:51.780 --> 00:10:54.320 David Stephens, Prof: In terms of subsequent wages. 97 00:10:54.870 --> 00:10:59.450 David Stephens, Prof: One area I've worked in and used this kind of methods is in… 98 00:11:00.300 --> 00:11:05.680 David Stephens, Prof: Transportation planning, so if you want to, as a… 99 00:11:06.100 --> 00:11:16.709 David Stephens, Prof: civil engineer intervene to change a road network? What is the impact of that change on subsequent traffic outcomes? Traffic flow, traffic density, or so on? 100 00:11:17.610 --> 00:11:27.480 David Stephens, Prof: So, these are just some, common type examples. And… They're representative of typical scientific. 101 00:11:27.580 --> 00:11:36.150 David Stephens, Prof: investigation settings. Our interest is in finding out the true impact of Our intervention on the system. 102 00:11:36.280 --> 00:11:37.350 David Stephens, Prof: On the outcome. 103 00:11:39.730 --> 00:11:50.220 David Stephens, Prof: Now, the complicating factor is, in most settings, that we recognize that manipulating one variable and looking at the impact on the outcome. 104 00:11:50.510 --> 00:11:54.220 David Stephens, Prof: Is a rather simplistic view, because we recognize there are other 105 00:11:55.150 --> 00:11:59.260 David Stephens, Prof: Complicating factors, other predictors, other covariates that we can measure. 106 00:11:59.580 --> 00:12:01.469 David Stephens, Prof: That are part of the same system. 107 00:12:02.600 --> 00:12:10.300 David Stephens, Prof: And what we need to take care to do is to tease out the impact of our change 108 00:12:10.860 --> 00:12:16.989 David Stephens, Prof: our intervention… Our change of treatment level, let's say, on the outcome. 109 00:12:17.220 --> 00:12:19.810 David Stephens, Prof: As opposed to the contributions of the other factors. 110 00:12:23.910 --> 00:12:25.179 David Stephens, Prof: So, here's a… 111 00:12:25.580 --> 00:12:32.580 David Stephens, Prof: simple, made-up example. Let's say we're going to look at the impact of a diet, or nutrition on health. 112 00:12:32.750 --> 00:12:37.229 David Stephens, Prof: Let's say we're looking at, an exposure which is, 113 00:12:37.740 --> 00:12:42.930 David Stephens, Prof: dietary, level of, vitamin E, let's say. 114 00:12:43.200 --> 00:12:49.389 David Stephens, Prof: So we either have a normal exposure, a normal diet, or a low-intake diet. 115 00:12:49.890 --> 00:12:56.919 David Stephens, Prof: And then we want to relate this, in some way, to the incidence of cardiovascular disease in some time period. 116 00:12:58.420 --> 00:13:07.130 David Stephens, Prof: Okay, so our investigation is… Whether they're dietary habits, or dietary, 117 00:13:07.720 --> 00:13:13.219 David Stephens, Prof: Exposure has an effect on the incidence of cardiovascular disease. 118 00:13:13.360 --> 00:13:14.550 David Stephens, Prof: That's the setup. 119 00:13:15.140 --> 00:13:18.029 David Stephens, Prof: Here are some data that we might collect, made-up data. 120 00:13:19.500 --> 00:13:29.750 David Stephens, Prof: So we see normal and low as the two exposure groups. We see outcome in a fairly large cohort, cardiovascular and no cardiovascular disease in this five-year period. 121 00:13:30.090 --> 00:13:34.809 David Stephens, Prof: And our research question is, is there an association between these factors? 122 00:13:35.140 --> 00:13:41.029 David Stephens, Prof: Does a normal diet, Lead to a lower rate. 123 00:13:41.190 --> 00:13:46.260 David Stephens, Prof: Of cardiovascular disease, compared to… A low rate. 124 00:13:46.410 --> 00:13:53.009 David Stephens, Prof: A low dietary rate of… of, a low vitamin E diet. 125 00:13:53.740 --> 00:14:02.219 David Stephens, Prof: Okay? So, this is a simple statistical comparison, this is the kind of comparison that we teach in elementary statistics courses. 126 00:14:04.570 --> 00:14:08.750 David Stephens, Prof: But our question is a little bit richer than a simple two-group comparison. 127 00:14:10.260 --> 00:14:12.870 David Stephens, Prof: Our investigation really focuses on… 128 00:14:13.740 --> 00:14:22.250 David Stephens, Prof: Whether an intervention to change the diet from one level to the other would have the effect of changing the cardiovascular disease rate. 129 00:14:25.000 --> 00:14:28.070 David Stephens, Prof: Does it make it more likely, or indeed less likely? 130 00:14:29.110 --> 00:14:34.340 David Stephens, Prof: to… To lead to a cardiovascular disease outcome. 131 00:14:34.940 --> 00:14:38.949 David Stephens, Prof: If we have a normal diet compared to a low vitamin E diet. 132 00:14:40.170 --> 00:14:46.559 David Stephens, Prof: And… To answer that question, we really need to know more about the study design. 133 00:14:47.260 --> 00:14:51.450 David Stephens, Prof: Because we can't tell from this dataset whether… 134 00:14:52.140 --> 00:14:59.610 David Stephens, Prof: the data arose in an experimental fashion, or an observational fashion. That is, When we collected the data. 135 00:14:59.880 --> 00:15:04.309 David Stephens, Prof: Did we just collect exposure status and outcome status 5 years down the line? 136 00:15:04.660 --> 00:15:07.510 David Stephens, Prof: Or did we intervene to assign 137 00:15:08.230 --> 00:15:14.459 David Stephens, Prof: one diet category, or the other diet category. Now, in this case, the observational setting seems more realistic. 138 00:15:15.340 --> 00:15:24.219 David Stephens, Prof: We've just gone out, collected information on the diet, exposure, and the outcome, and looked at the relationship between them. 139 00:15:24.710 --> 00:15:28.290 David Stephens, Prof: So, we can certainly do that calculation in this case using… 140 00:15:28.460 --> 00:15:33.239 David Stephens, Prof: Too good comparison of binomial proportions, or odds ratio, or whatever. 141 00:15:34.130 --> 00:15:34.880 David Stephens, Prof: Okay. 142 00:15:35.180 --> 00:15:39.760 David Stephens, Prof: But that doesn't answer the question necessarily about the effect of an intervention. 143 00:15:40.040 --> 00:15:45.199 David Stephens, Prof: Because, of course, what we recognize in this setting is that there are other variables that might affect 144 00:15:46.090 --> 00:15:50.120 David Stephens, Prof: A cardiovascular disease outcome, which also might be related to exposure. 145 00:15:51.850 --> 00:15:58.599 David Stephens, Prof: The usual one, of course, we think of in this kind of setting is the, socioeconomic status. 146 00:16:01.070 --> 00:16:09.899 David Stephens, Prof: So, what we're gonna have to do is to build a framework that really explicitly allows us to think about the effect of an intervention, first of all. 147 00:16:10.400 --> 00:16:15.639 David Stephens, Prof: And… Second of all, to think about… 148 00:16:15.810 --> 00:16:19.330 David Stephens, Prof: How we do the comparison in an appropriate way. 149 00:16:19.960 --> 00:16:26.920 David Stephens, Prof: To recognize the fact that we're isolating in our target inference. 150 00:16:28.200 --> 00:16:35.129 David Stephens, Prof: The intervened upon variable, whilst acknowledging the presence of other variables that might be related to the outcome. 151 00:16:37.310 --> 00:16:41.960 David Stephens, Prof: So, we're gonna start building a mathematical framework to allow us to do that. 152 00:16:45.020 --> 00:16:48.409 David Stephens, Prof: I'm gonna start introducing some mathematical notation in a minute. 153 00:16:49.230 --> 00:16:54.289 David Stephens, Prof: Really, we're thinking about, in general terms, the effect of an outcome 154 00:16:55.050 --> 00:17:02.390 David Stephens, Prof: The effect on an outcome of changes in the value of an exposure or treatment. The outcome could be a binary outcome? Yes, no? 155 00:17:02.910 --> 00:17:09.320 David Stephens, Prof: CBD or no CBD. It could be, multi-category integer-valued. 156 00:17:09.800 --> 00:17:13.470 David Stephens, Prof: It could be continuous value, it could be a survival time, something like that. 157 00:17:13.920 --> 00:17:16.200 David Stephens, Prof: The exposure could also be binary. 158 00:17:17.040 --> 00:17:18.500 David Stephens, Prof: Treated, not treated. 159 00:17:18.920 --> 00:17:23.909 David Stephens, Prof: Or integer value, treated at level 1, treated at level 2, could be categorically. 160 00:17:24.349 --> 00:17:26.060 David Stephens, Prof: valued. 161 00:17:27.470 --> 00:17:30.619 David Stephens, Prof: Treatment Type 1, Treatment Type 2, Treatment Type 3. 162 00:17:31.570 --> 00:17:37.249 David Stephens, Prof: Could be continuous valued, even. Could be an exposure aggregate over time, say. 163 00:17:39.340 --> 00:17:48.439 David Stephens, Prof: We could also be carrying out this study in a cross-sectional way, single-type, that's the one we'll focus on most of all. But it could also be longitudinal, and we'll see 164 00:17:48.690 --> 00:17:51.290 David Stephens, Prof: Some extensions to that case later on. 165 00:17:54.090 --> 00:18:02.790 David Stephens, Prof: But what's critical in what we're thinking about is that we're studying the effect of an intervention. So the intervention by the researcher to change the exposure status. 166 00:18:04.030 --> 00:18:09.829 David Stephens, Prof: Now… The critical thing that we'll see is that 167 00:18:10.580 --> 00:18:20.369 David Stephens, Prof: We want to make that conclusion about the effect of an intervention, even if the data that we're analyzing do not come from a study where there was an actual intervention. 168 00:18:22.140 --> 00:18:27.190 David Stephens, Prof: Perhaps individuals chose their own exposure status, their own treatment status. 169 00:18:28.550 --> 00:18:32.349 David Stephens, Prof: So studying the impact of an intervention in that setting is necessarily more complicated. 170 00:18:34.680 --> 00:18:37.159 David Stephens, Prof: So the notation that I'll adopt, so… 171 00:18:37.710 --> 00:18:44.599 David Stephens, Prof: some mathematical notation is necessary here, so it certainly, I think, eases understanding what it does. 172 00:18:44.710 --> 00:18:48.130 David Stephens, Prof: In most cases, he's understanding about how the methods work. 173 00:18:48.620 --> 00:18:50.750 David Stephens, Prof: So Y is going to be the outcome? 174 00:18:50.960 --> 00:18:53.650 David Stephens, Prof: ZAD is going to be the exposure or treatment. 175 00:18:53.870 --> 00:18:58.810 David Stephens, Prof: X is going to be the collection of other predictors or covariates that we might have access to. 176 00:18:59.390 --> 00:19:07.670 David Stephens, Prof: our… Interest, then, is to investigate the impact of changing Z on Y. 177 00:19:08.920 --> 00:19:11.899 David Stephens, Prof: in the presence of these other variables, X. 178 00:19:13.670 --> 00:19:18.410 David Stephens, Prof: In a cross-sectional study, Y and Z will be single-valued, scalar-valued. 179 00:19:19.260 --> 00:19:25.429 David Stephens, Prof: In the longitudinal case, we might have a series of intermediate outcomes and a series of treatments. 180 00:19:26.920 --> 00:19:31.869 David Stephens, Prof: X is typically a vector-valued quantity, so we might have 10 other predictors. 181 00:19:32.940 --> 00:19:34.700 David Stephens, Prof: 50, 100, or whatever. 182 00:19:37.180 --> 00:19:40.279 David Stephens, Prof: But our focus is on the impact of Z on Y. 183 00:19:41.280 --> 00:19:46.920 David Stephens, Prof: I just indexes individuals, and we'll be collecting a dataset of size N. 184 00:19:47.090 --> 00:19:49.270 David Stephens, Prof: To make our conclusions. 185 00:19:50.730 --> 00:19:51.420 David Stephens, Prof: Okay? 186 00:19:52.460 --> 00:20:03.669 David Stephens, Prof: So… One tool that I'll deploy quite often in the module is the causal graph representation of 187 00:20:04.320 --> 00:20:08.129 David Stephens, Prof: A statistical setup, or an experimental setup. 188 00:20:08.360 --> 00:20:16.140 David Stephens, Prof: So I… there was a module, earlier on in the week that I gave about causal DAGs, causal graphs. 189 00:20:16.340 --> 00:20:33.120 David Stephens, Prof: And I've included some of the details on this link. If you click on this in the, click on this link in the bottom here in the PDF, you'll be taken to the appendix, which is at the end of this set of slides, which gives you a basic run-through of the ideas. 190 00:20:33.840 --> 00:20:37.260 David Stephens, Prof: The fundamental idea is that we have a… 191 00:20:37.560 --> 00:20:44.910 David Stephens, Prof: collection of variables X, which are our… Characteristics of the study individuals? 192 00:20:45.840 --> 00:20:47.319 David Stephens, Prof: We have a treatment. 193 00:20:47.720 --> 00:20:48.960 David Stephens, Prof: And we have an outcome. 194 00:20:49.960 --> 00:20:53.190 David Stephens, Prof: Right? And our belief about the way that the… 195 00:20:54.070 --> 00:20:57.360 David Stephens, Prof: Data-related is encapsulated through this graph. 196 00:20:57.710 --> 00:21:00.799 David Stephens, Prof: We read this in the following way. We say that… 197 00:21:01.230 --> 00:21:05.109 David Stephens, Prof: Individuals in the population have characteristics X, 198 00:21:05.340 --> 00:21:08.569 David Stephens, Prof: Which, in general, in an observational setting. 199 00:21:08.810 --> 00:21:11.980 David Stephens, Prof: Lead them to choose a particular treatment or exposure. 200 00:21:14.560 --> 00:21:22.450 David Stephens, Prof: Once we've observed those characteristics and the treatment. Those two things get together to cause the outcome. 201 00:21:24.280 --> 00:21:34.439 David Stephens, Prof: Okay, that's what the… that's what this DAG basically means. This is the DAG which represents the most general observational setting that we can think of, so there may be multiple variables here. 202 00:21:35.610 --> 00:21:38.790 David Stephens, Prof: Typically a single variable here, so the… 203 00:21:40.270 --> 00:21:47.089 David Stephens, Prof: Popular, characteristics of the individual lead to a choice of treatment, and then the characteristics and the treatment together. 204 00:21:47.280 --> 00:21:49.119 David Stephens, Prof: Give rise to the outcome. 205 00:21:51.030 --> 00:21:53.269 David Stephens, Prof: This kind of graph is very useful. 206 00:21:53.780 --> 00:21:58.420 David Stephens, Prof: It's gonna be useful in the construction of the Propensity score logic. 207 00:22:00.720 --> 00:22:09.770 David Stephens, Prof: Once you've explained… What's problematic about this DAG for… about this graph for, 208 00:22:10.910 --> 00:22:13.030 David Stephens, Prof: For the kind of inference that we want to do. 209 00:22:13.270 --> 00:22:16.269 David Stephens, Prof: This is called the Directed Acyclic Graph, or DAG. 210 00:22:16.460 --> 00:22:19.950 David Stephens, Prof: Because it's a graph, Which has got no cycles in it. 211 00:22:20.260 --> 00:22:24.590 David Stephens, Prof: Okay, you can find out more about the… The general terminology here. 212 00:22:24.690 --> 00:22:29.029 David Stephens, Prof: But that's the basic idea. This is our model for how the data arrives. 213 00:22:30.640 --> 00:22:40.000 David Stephens, Prof: And in any practical situation, Right at the start of the exercise, when carrying out an epidemiological Study. 214 00:22:40.550 --> 00:22:44.489 David Stephens, Prof: The first step, really, in planning of the study 215 00:22:45.250 --> 00:22:49.580 David Stephens, Prof: should include something like a consideration of a DAG like this. 216 00:22:49.800 --> 00:22:54.270 David Stephens, Prof: So, how do we think the data arise. 217 00:22:57.640 --> 00:23:02.950 David Stephens, Prof: Now, that's not something a statistician can do, they need subject matter expertise. 218 00:23:03.970 --> 00:23:14.510 David Stephens, Prof: So the idea is that the researcher and the statistician get together to come up with a DAG like this. And the structure of this DAG is pretty important in the kind of adjustments that we need to do. 219 00:23:14.760 --> 00:23:18.589 David Stephens, Prof: Because it's going to be an analysis of this kind of DAG that tells us 220 00:23:19.230 --> 00:23:22.169 David Stephens, Prof: Why we need to adjust using the propensity score. 221 00:23:22.280 --> 00:23:23.959 David Stephens, Prof: And how we should do it. 222 00:23:28.470 --> 00:23:34.210 David Stephens, Prof: So in that DAG, we think of X as being what's called a direct cause of Y and Z. 223 00:23:34.790 --> 00:23:37.190 David Stephens, Prof: Z is a direct cause of Y. 224 00:23:37.330 --> 00:23:41.629 David Stephens, Prof: But it's also a mediator of the indirect cause of Z on Y. 225 00:23:44.850 --> 00:23:48.800 David Stephens, Prof: The variables there are nodes, And the paths… 226 00:23:48.990 --> 00:23:51.479 David Stephens, Prof: That appear are just a series of arrows. 227 00:23:52.890 --> 00:23:54.350 David Stephens, Prof: So, in this case. 228 00:23:57.460 --> 00:24:01.710 David Stephens, Prof: We have a direct path from Z to Y, and an indirect path from Z to Y. 229 00:24:02.450 --> 00:24:06.340 David Stephens, Prof: Okay, now… Our interest is gonna be… 230 00:24:06.760 --> 00:24:10.439 David Stephens, Prof: In studying the effect of the treatment on the outcome. 231 00:24:10.590 --> 00:24:13.819 David Stephens, Prof: So our main interest is in this path. 232 00:24:14.980 --> 00:24:21.320 David Stephens, Prof: But we recognize also that there is this path, that connects Z to Y. 233 00:24:21.460 --> 00:24:25.409 David Stephens, Prof: If we ignore the direction of the arrows, the so-called undirected path. 234 00:24:26.590 --> 00:24:30.509 David Stephens, Prof: There's a connection between Z and Y that goes through X. 235 00:24:30.980 --> 00:24:32.280 David Stephens, Prof: And it's that. 236 00:24:34.420 --> 00:24:43.250 David Stephens, Prof: additional path, That we need to take care of in our… Propensity score adjustment. 237 00:24:44.880 --> 00:24:56.000 David Stephens, Prof: Because you can think of this as saying, well, we're interested in the direct cause of… Sorry, the direct impact of Z on Y, that's our interventional notion, that's what we want to really study. 238 00:24:56.110 --> 00:25:01.599 David Stephens, Prof: But we recognize that in our data generating model, there is another path 239 00:25:01.810 --> 00:25:04.969 David Stephens, Prof: Via which the effect of Z and Y could leak out. 240 00:25:05.120 --> 00:25:06.020 David Stephens, Prof: Think of it like that. 241 00:25:07.020 --> 00:25:11.089 David Stephens, Prof: So our task, really, will be to handle 242 00:25:11.440 --> 00:25:13.700 David Stephens, Prof: In some way, the presence of that path. 243 00:25:15.910 --> 00:25:23.460 David Stephens, Prof: Okay, so… In that kind of, Dag? 244 00:25:23.610 --> 00:25:30.809 David Stephens, Prof: We have directed paths, so here we have the treatment, going to X1, going to X2, going to Y, 245 00:25:31.080 --> 00:25:34.629 David Stephens, Prof: That's called a directed path, because all the arrows point in the same direction. 246 00:25:35.830 --> 00:25:38.530 David Stephens, Prof: An undirected path is a similar path. 247 00:25:38.660 --> 00:25:41.720 David Stephens, Prof: But where the arrows don't point in the same direction. 248 00:25:42.330 --> 00:25:50.709 David Stephens, Prof: So, there are… in both cases, there's a path from Z to Y. In this case, all the arrows point in the same direction. It's called directed. 249 00:25:51.210 --> 00:25:56.959 David Stephens, Prof: And in this case, the arrows don't point in the same direction, so that it's called undirected. 250 00:25:58.010 --> 00:26:03.959 David Stephens, Prof: Our causal analysis, Our, a study of the effective intervention. 251 00:26:04.690 --> 00:26:09.509 David Stephens, Prof: of on Z on the outcome. We'll be interested in directed paths. 252 00:26:10.220 --> 00:26:15.490 David Stephens, Prof: Undirected paths of this nature will be problematic to us. We have to find some way of handling them. 253 00:26:21.720 --> 00:26:22.680 David Stephens, Prof: Now. 254 00:26:23.020 --> 00:26:30.800 David Stephens, Prof: The other small bit of terminology that I'll introduce is the idea of a collider. So a collider is a node. 255 00:26:30.960 --> 00:26:32.550 David Stephens, Prof: A variable in the graph. 256 00:26:32.720 --> 00:26:35.770 David Stephens, Prof: That has arrows pointing into it only. 257 00:26:36.480 --> 00:26:38.970 David Stephens, Prof: So here there is a path from Z to Y. 258 00:26:40.530 --> 00:26:46.199 David Stephens, Prof: But, the path is undirected, because the arrows don't all point in the same direction. 259 00:26:47.060 --> 00:26:49.890 David Stephens, Prof: And the arrows incoming meter X. 260 00:26:51.850 --> 00:26:57.279 David Stephens, Prof: So, Whereas these two paths are so-called open paths. 261 00:26:57.770 --> 00:27:02.409 David Stephens, Prof: This one is a… Closed path, or blocked path. 262 00:27:02.820 --> 00:27:04.940 David Stephens, Prof: Because there is a collider at X. 263 00:27:09.270 --> 00:27:14.200 David Stephens, Prof: Our interest, ultimately, Will be to… 264 00:27:14.910 --> 00:27:21.779 David Stephens, Prof: Study the causal effect of treatment by looking at the open, directed paths between the treatment and the outcome. 265 00:27:22.170 --> 00:27:23.409 David Stephens, Prof: As it says here. 266 00:27:26.090 --> 00:27:28.999 David Stephens, Prof: The problem we have is a problem of confounding. 267 00:27:30.870 --> 00:27:35.250 David Stephens, Prof: And confounding occurs if there are open, undirected paths. 268 00:27:35.520 --> 00:27:37.150 David Stephens, Prof: Between the treatment and the outcome. 269 00:27:39.740 --> 00:27:44.560 David Stephens, Prof: So the statistical task, when we come to do statistical modeling, to make… 270 00:27:46.300 --> 00:27:56.300 David Stephens, Prof: Causal conclusions from observational data, our statistical task will be to block the open, undirected paths between the two variables. 271 00:27:58.760 --> 00:28:11.159 David Stephens, Prof: Now, it turns out that we do this in… can do this in several different ways. The typical route involves conditioning on variables, so in particular, in a regression setting, we use conditioning both 272 00:28:12.120 --> 00:28:17.130 David Stephens, Prof: Through including variables in the regression model, as we'll see as we go through. 273 00:28:18.090 --> 00:28:26.649 David Stephens, Prof: But we don't have to do it only through regression, we could do it through other methods like stratification and matching. So we'll study all of this in detail as we get to the… 274 00:28:26.790 --> 00:28:29.759 David Stephens, Prof: To the Propensity school construction, and… 275 00:28:29.870 --> 00:28:33.789 David Stephens, Prof: The statistical tools that use the propensity score construction. 276 00:28:33.960 --> 00:28:36.029 David Stephens, Prof: I'm just trying to give the basic idea. 277 00:28:36.630 --> 00:28:43.739 David Stephens, Prof: So again, the starting point is the DAG. It's our picture of the data-generating model. Once we've got the DAG, 278 00:28:44.120 --> 00:28:45.870 David Stephens, Prof: We can analyze it. 279 00:28:47.640 --> 00:28:52.410 David Stephens, Prof: Identify the open, undirected paths between the treatment and the outcome. 280 00:28:53.700 --> 00:28:54.859 David Stephens, Prof: And block them. 281 00:28:55.180 --> 00:28:56.940 David Stephens, Prof: Using conditioning. 282 00:28:57.990 --> 00:29:00.289 David Stephens, Prof: That's, as we'll see, that's there. 283 00:29:00.460 --> 00:29:02.839 David Stephens, Prof: That's the sort of high-level version of the… 284 00:29:03.520 --> 00:29:07.180 David Stephens, Prof: Logic. We'll see the details as we go through the module. 285 00:29:11.190 --> 00:29:22.310 David Stephens, Prof: So, why do we think of causal modeling in this way? Well, again, a piece of jargon that I'll introduce, a piece of terminology that I'll introduce, is this idea of a structural model. 286 00:29:23.080 --> 00:29:28.269 David Stephens, Prof: And as we'll see, the structural models can tie in directly to regression models. 287 00:29:31.170 --> 00:29:41.369 David Stephens, Prof: So, the DAG, one good way to think about it is that it really encapsulates functional relationships between variables. 288 00:29:42.390 --> 00:29:50.470 David Stephens, Prof: So in that confounding DAG that I saw before, it's basically saying that the treatment is some function of X, 289 00:29:50.630 --> 00:29:52.940 David Stephens, Prof: Plus some randomness. 290 00:29:53.650 --> 00:30:01.130 David Stephens, Prof: On the outcome, It's some function of the predictor's And the treatment. 291 00:30:01.320 --> 00:30:02.820 David Stephens, Prof: Plus some randomness. 292 00:30:05.150 --> 00:30:05.940 David Stephens, Prof: Okay? 293 00:30:06.160 --> 00:30:10.959 David Stephens, Prof: So… This is a concrete mathematical representation. 294 00:30:11.290 --> 00:30:13.929 David Stephens, Prof: Of the relationship between the variables of interest. 295 00:30:17.970 --> 00:30:22.749 David Stephens, Prof: And this kind of representation is important for the reasons that I'll explain in a second. 296 00:30:23.220 --> 00:30:27.580 David Stephens, Prof: To do with our interpretation of what cause and effect is. 297 00:30:29.310 --> 00:30:37.739 David Stephens, Prof: Now, this is a mathematical framework for talking about cause and effect. It doesn't necessarily exactly correspond to a mechanistic 298 00:30:37.830 --> 00:30:50.109 David Stephens, Prof: cause and effect relationships. So we're not necessarily thinking about here about molecular mechanisms that lead from one thing to another. It's a mathematical representation. Now, the mathematical representation 299 00:30:50.850 --> 00:30:59.110 David Stephens, Prof: Should try to reflect the true physical situation, but again, it's a… it's a… it's really only a mathematical representation ever. 300 00:31:00.430 --> 00:31:04.900 David Stephens, Prof: Okay? So what these equations tell us that… 301 00:31:05.040 --> 00:31:09.309 David Stephens, Prof: Well, once we've got X, we plug them into the function gz. 302 00:31:10.670 --> 00:31:15.909 David Stephens, Prof: Add in some randomness, and that gives us the… the treatment. 303 00:31:16.460 --> 00:31:18.410 David Stephens, Prof: In the observational setting, this is. 304 00:31:18.630 --> 00:31:26.050 David Stephens, Prof: And for Y, once we've got X and Z, plus some randomness, we plug those into G, and that gives us the response. 305 00:31:28.820 --> 00:31:32.289 David Stephens, Prof: Okay? So the idea is that, 306 00:31:33.400 --> 00:31:42.759 David Stephens, Prof: So this is a… maybe it'll become more apparent when I put up some equations in a second. This is how variables relate to each other in our mathematical model. 307 00:31:45.090 --> 00:31:46.410 David Stephens, Prof: So, for example. 308 00:31:50.320 --> 00:31:56.150 David Stephens, Prof: In our observational study, imagine people coming in off the street with their characteristics X. 309 00:31:56.990 --> 00:32:01.270 David Stephens, Prof: So you can make measurements on them, like blade age, whatever. 310 00:32:02.560 --> 00:32:07.060 David Stephens, Prof: What this first model says is that while the treatment is just… 311 00:32:07.610 --> 00:32:11.130 David Stephens, Prof: That, in the univariate case, let's say. 312 00:32:11.330 --> 00:32:14.289 David Stephens, Prof: Is the treatment plus some random error. 313 00:32:15.400 --> 00:32:18.540 David Stephens, Prof: Sorry, the treatment is the X plus some random error. 314 00:32:22.100 --> 00:32:25.089 David Stephens, Prof: And then for the outcome. 315 00:32:25.330 --> 00:32:33.520 David Stephens, Prof: Well, this function on the right-hand side tells us how we combine X and Z plus a random error to get Y. 316 00:32:34.670 --> 00:32:52.970 David Stephens, Prof: Okay, so this second model here looks very much like a regression model, right? You'll have seen this in regression courses. It just says, well, our belief in how Y arises is that you take X and you take Z, and you plug them into this formula, and then you add on a bit of randomness. 317 00:32:53.510 --> 00:32:54.990 David Stephens, Prof: And that gives you Y. 318 00:32:56.920 --> 00:32:57.650 David Stephens, Prof: Okay? 319 00:32:57.850 --> 00:33:07.810 David Stephens, Prof: So the… so this basic first point is that We can think of Causal structures. 320 00:33:08.650 --> 00:33:12.129 David Stephens, Prof: Through things that look like regression models. 321 00:33:13.250 --> 00:33:19.190 David Stephens, Prof: These are, in fact, structural models, but they're… structural regression models. 322 00:33:20.590 --> 00:33:22.729 David Stephens, Prof: Okay, so that's fairly simple, hopefully. 323 00:33:23.490 --> 00:33:28.650 David Stephens, Prof: I'm gonna use it, though, to illustrate the… causal calculation. 324 00:33:31.910 --> 00:33:34.570 David Stephens, Prof: Because our goal is to understand 325 00:33:35.250 --> 00:33:40.310 David Stephens, Prof: the effect of Z on Y in an unconfounded setting. That is, when 326 00:33:40.510 --> 00:33:46.610 David Stephens, Prof: We regard Z as being something that's set by us, by manipulation, by intervention. 327 00:33:48.070 --> 00:33:56.999 David Stephens, Prof: And where X doesn't influence Z. So if you think in a randomized study setting, Randomized study says. 328 00:33:57.410 --> 00:33:59.619 David Stephens, Prof: Take a whole bunch of study participants. 329 00:34:00.110 --> 00:34:10.000 David Stephens, Prof: Randomly allocate treatment, Or no treatment to each individual. And then do the subsequent analysis of the outcome. 330 00:34:11.300 --> 00:34:18.380 David Stephens, Prof: In that setting, The person's characteristics, Do not influence their treatment. 331 00:34:18.800 --> 00:34:22.200 David Stephens, Prof: The only thing that changes the treatment is a coin flip. 332 00:34:22.719 --> 00:34:25.120 David Stephens, Prof: Or a deterministic allocation, even. 333 00:34:26.400 --> 00:34:27.070 David Stephens, Prof: Okay? 334 00:34:27.550 --> 00:34:31.920 David Stephens, Prof: So, in this case, we should write the DAG that looks like this. 335 00:34:32.060 --> 00:34:32.780 David Stephens, Prof: Right? 336 00:34:33.190 --> 00:34:38.509 David Stephens, Prof: So we still accept that X and Z are causes of Y, 337 00:34:39.010 --> 00:34:43.480 David Stephens, Prof: Right? Manipulating X and Z will lead to different Ys. 338 00:34:44.719 --> 00:34:48.990 David Stephens, Prof: But in this… Hypothetical, experimental setting. 339 00:34:49.489 --> 00:34:51.940 David Stephens, Prof: We now know that X 340 00:34:52.190 --> 00:34:58.619 David Stephens, Prof: doesn't influence ZAD. The only thing that influences ZAD, the only thing that influences the treatment, is us as the researcher. 341 00:35:00.830 --> 00:35:03.840 David Stephens, Prof: So… The point here is that. 342 00:35:04.010 --> 00:35:07.170 David Stephens, Prof: When we think about causal conclusions. 343 00:35:07.300 --> 00:35:11.179 David Stephens, Prof: We are thinking about conclusions from a DAG like this. 344 00:35:13.070 --> 00:35:24.779 David Stephens, Prof: We want to say, what is the effect of changing Z on Y? We can analyze that by only looking down this arrow, basically. There's no possibility of the effect of Z leaking out through some other route. 345 00:35:26.830 --> 00:35:29.750 David Stephens, Prof: So this is the DAG on which we're focused. 346 00:35:30.300 --> 00:35:36.049 David Stephens, Prof: Where there is no open path between Z and Y apart from the direct path. 347 00:35:38.770 --> 00:35:49.349 David Stephens, Prof: So, when we think about an experimental study, this is exactly the DAG that we're working with, right? People may come in with their own characteristics into the study, that's fine. We know those characteristics affect the outcome. 348 00:35:49.970 --> 00:36:02.479 David Stephens, Prof: But we also know that those characteristics don't affect the treatment level. The only thing that affects the treatment level is our intervention. Now, we can modify this a little to account for stratification by… 349 00:36:04.550 --> 00:36:18.289 David Stephens, Prof: sex, gender, or age, if we like, we could do that, or other factors. And then the data gets a little bit more complicated. But the key thing is that the thing that we regard as confounders are acknowledged explicitly in the analysis. 350 00:36:18.630 --> 00:36:21.539 David Stephens, Prof: And in the simplest version of the experimental study. 351 00:36:22.290 --> 00:36:25.629 David Stephens, Prof: Those characteristics, at the individual level, don't affect treatment. 352 00:36:27.740 --> 00:36:29.210 David Stephens, Prof: Okay, so… 353 00:36:31.750 --> 00:36:38.160 David Stephens, Prof: When we think about comparisons, then, in the structural model, that regression model that I wrote up previously. 354 00:36:38.950 --> 00:36:43.269 David Stephens, Prof: Let's imagine the treatment level boom fixed to some value, little red z. 355 00:36:43.400 --> 00:36:55.159 David Stephens, Prof: Right, so this is a numerical value. Z is the random variable representing the treatment that the person receives, but let's say we're fixing that to be some particular value, so it could be treated or untreated, say. 356 00:36:56.080 --> 00:37:01.139 David Stephens, Prof: Then, the structural model tells us how we expect Y to behave. 357 00:37:02.170 --> 00:37:08.879 David Stephens, Prof: Right? It still obeys the same function that we saw before, but now, instead of having a random Z, 358 00:37:09.310 --> 00:37:12.269 David Stephens, Prof: we've got a fixed numerical value of Z. 359 00:37:14.430 --> 00:37:19.879 David Stephens, Prof: So, the notation sort of hides it, but little red z is just a number. 360 00:37:21.530 --> 00:37:24.500 David Stephens, Prof: Could be 0 or 1, say, for the binary case. 361 00:37:24.680 --> 00:37:27.069 David Stephens, Prof: So we plug Z equals 0 in. 362 00:37:28.250 --> 00:37:31.709 David Stephens, Prof: To get the response under no treatment? 363 00:37:32.230 --> 00:37:38.209 David Stephens, Prof: And we plug the value little red z equal to 1 in if we want to account for treat… if we want to say that that person is treated. 364 00:37:40.190 --> 00:37:43.490 David Stephens, Prof: And that structural interpretation really tells us 365 00:37:43.940 --> 00:37:46.620 David Stephens, Prof: About what the causal effect is. 366 00:37:47.460 --> 00:37:49.770 David Stephens, Prof: So if we believe this to be the model. 367 00:37:50.240 --> 00:38:00.669 David Stephens, Prof: then we can think about looking at differences in the Y that we observe when changing that little red Z from 0 to 1. 368 00:38:01.380 --> 00:38:06.609 David Stephens, Prof: That, in that form, again, this is no longer a random quantity, Z isn't. 369 00:38:06.730 --> 00:38:13.889 David Stephens, Prof: the random variable cost. But what we're saying is, in the intervention, we're fixing it, To be some value. 370 00:38:14.470 --> 00:38:18.679 David Stephens, Prof: So imagine sticking Z equal to 0, and Z equal to 1 in this formula. 371 00:38:19.730 --> 00:38:30.409 David Stephens, Prof: Okay, that gives us the two cases of when the person is non-treated for Z equals 1, treated… sorry, non-treated for Z equals 0, treated for Z equals 1. 372 00:38:32.650 --> 00:38:38.740 David Stephens, Prof: Okay, so… A good way to think about, then, causal… 373 00:38:38.840 --> 00:38:49.819 David Stephens, Prof: impact is through the kind of structural model, and the construction of what are called counterfactual or potential outcomes. So, a counterfactual or potential outcome is… 374 00:38:50.480 --> 00:38:54.759 David Stephens, Prof: Typically denoted in this way, with a random variable and then brackets. 375 00:38:55.780 --> 00:39:00.230 David Stephens, Prof: With the argument little red z in there. That's the hypothetical outcome. 376 00:39:00.360 --> 00:39:06.189 David Stephens, Prof: for the ith individual, if we were to intervene to set exposure equal to little red z. 377 00:39:08.170 --> 00:39:13.859 David Stephens, Prof: So, that is called a counterfactual or potential outcome. It is what we would observe 378 00:39:14.190 --> 00:39:17.120 David Stephens, Prof: If the person were treated, or not treated. 379 00:39:17.690 --> 00:39:19.010 David Stephens, Prof: Two different outcomes. 380 00:39:20.060 --> 00:39:24.629 David Stephens, Prof: So it's like we're saying that, well, the person comes in with their characteristics. 381 00:39:24.840 --> 00:39:28.199 David Stephens, Prof: We recognized at that point that if they were to be treated. 382 00:39:28.420 --> 00:39:33.779 David Stephens, Prof: They would have a certain response, whereas if they were not to be treated, then they would have a different response. 383 00:39:36.470 --> 00:39:39.330 David Stephens, Prof: So that's the so-called counterfactual potential outcome. 384 00:39:39.900 --> 00:39:44.849 David Stephens, Prof: So, for example, if exposure is binary, then the pair of potential outcomes 385 00:39:44.960 --> 00:39:51.159 David Stephens, Prof: Y bracket 0 and Y brackets 1 are the two counterfactuals of interest. 386 00:39:51.660 --> 00:39:58.090 David Stephens, Prof: They're the outcomes that would result if that individual was subject to no treatment or was not exposed. 387 00:39:58.420 --> 00:40:01.819 David Stephens, Prof: Or exposed, respectively, those two cases. 388 00:40:03.910 --> 00:40:06.790 David Stephens, Prof: So these are hypothetical quantities. 389 00:40:08.340 --> 00:40:14.610 David Stephens, Prof: What we would get if the person was not treated, or what we would get if the person was treated. 390 00:40:16.750 --> 00:40:19.209 David Stephens, Prof: And we can relate them to the observed 391 00:40:20.430 --> 00:40:29.779 David Stephens, Prof: Outcome, the actual response, the actual outcome that we get through the simple, combination of those two, potential outcomes. 392 00:40:30.280 --> 00:40:37.470 David Stephens, Prof: Taking into account which… treatment they've actually received. And what this formula simply says is. 393 00:40:38.400 --> 00:40:44.190 David Stephens, Prof: If they receive treatment, So ZI is equal to 1. 394 00:40:44.450 --> 00:40:48.280 David Stephens, Prof: Then we observe… That counterfactual outcome. 395 00:40:49.170 --> 00:40:53.750 David Stephens, Prof: Whereas if the treatment is equal to 0, This term kicks in. 396 00:40:54.140 --> 00:40:56.800 David Stephens, Prof: And we get to observe Yi0. 397 00:41:00.420 --> 00:41:05.710 David Stephens, Prof: So ZI records the actual treatment, or actual exposure level. 398 00:41:07.190 --> 00:41:15.009 David Stephens, Prof: YI is the actual observed outcome, and we can phrase it in terms of the two potential outcomes using this formula. 399 00:41:16.790 --> 00:41:18.220 David Stephens, Prof: That's the basic idea. 400 00:41:19.930 --> 00:41:25.629 David Stephens, Prof: Okay, so… I think it's useful to display that on a… 401 00:41:26.130 --> 00:41:28.989 David Stephens, Prof: DAG, so that's what I've tried to do on this slide. 402 00:41:30.440 --> 00:41:34.980 David Stephens, Prof: It makes clear that In the data generating mechanism. 403 00:41:35.820 --> 00:41:41.939 David Stephens, Prof: It's the characteristics themselves, X, the covariates, the predictors. 404 00:41:42.620 --> 00:41:45.360 David Stephens, Prof: That give rise to these potential outcomes. 405 00:41:47.040 --> 00:41:47.720 David Stephens, Prof: Right. 406 00:41:48.000 --> 00:41:55.020 David Stephens, Prof: And that these potential outcomes don't depend in any way on the treatment Received. 407 00:41:55.590 --> 00:42:00.929 David Stephens, Prof: These are the things that we keep back, that we store in our mind. 408 00:42:02.760 --> 00:42:06.499 David Stephens, Prof: To realize the actual outcome once we've observed the treatment. 409 00:42:07.710 --> 00:42:10.309 David Stephens, Prof: So this is a modification of the previous graph. 410 00:42:10.920 --> 00:42:18.480 David Stephens, Prof: X is a cause of these two potential outcomes. The only thing that we need to know in order to write these down is X. 411 00:42:19.760 --> 00:42:21.670 David Stephens, Prof: X also causes Z. 412 00:42:22.820 --> 00:42:24.430 David Stephens, Prof: In the observational setting. 413 00:42:25.070 --> 00:42:31.500 David Stephens, Prof: And once we've observed Z and have access to the two counterfactual outcomes. 414 00:42:31.800 --> 00:42:39.300 David Stephens, Prof: we can use that to get the observed outcome. So this… Collection says that… this formula. 415 00:42:40.050 --> 00:42:41.900 David Stephens, Prof: tells us how to generate Y. 416 00:42:46.150 --> 00:42:53.620 David Stephens, Prof: If we know the value of X, then we know the values, or at least the distributions of Y0 and Y1, 417 00:42:53.740 --> 00:42:57.640 David Stephens, Prof: We don't need to know anything about the actual treatment in order to get that. 418 00:42:58.120 --> 00:43:03.620 David Stephens, Prof: But once we've observed the actual treatment, and the collection of potential outcomes. 419 00:43:04.440 --> 00:43:06.269 David Stephens, Prof: We get to observe the real outcome. 420 00:43:08.200 --> 00:43:08.940 David Stephens, Prof: Okay. 421 00:43:10.820 --> 00:43:14.449 David Stephens, Prof: So I think… Personally, this DAG is useful. 422 00:43:15.000 --> 00:43:20.929 David Stephens, Prof: for… Explaining some of the key assumptions that'll come up later on. 423 00:43:21.420 --> 00:43:25.750 David Stephens, Prof: What I've tried to do here is explain how the 424 00:43:26.450 --> 00:43:30.959 David Stephens, Prof: The real data and the hypothetical data, the counterfactual outcomes, arise. 425 00:43:31.440 --> 00:43:32.969 David Stephens, Prof: In a sensible way. 426 00:43:33.280 --> 00:43:42.219 David Stephens, Prof: And we'll see this DAG being supported by subsequent mathematical assumptions, assumptions that you see commonly in the construction of the propensity score idea. 427 00:43:42.400 --> 00:43:47.129 David Stephens, Prof: Basically, focus on the idea of… what the jargon term is, ignorability. 428 00:43:50.960 --> 00:43:53.030 David Stephens, Prof: But the ideas here are quite subtle. 429 00:43:53.470 --> 00:43:58.879 David Stephens, Prof: In the sense that… We know… I mean, we have a… 430 00:43:59.430 --> 00:44:03.140 David Stephens, Prof: common sense idea that the actual outcome depends on Z. 431 00:44:05.120 --> 00:44:11.480 David Stephens, Prof: In the observational setting, the treatment affects the outcome in most cases. That's our focus of our interest, anyway. 432 00:44:11.630 --> 00:44:17.789 David Stephens, Prof: The subtlety here is that… The treatment doesn't affect the counterfactual outcomes. 433 00:44:20.980 --> 00:44:22.520 David Stephens, Prof: Anyway, we'll come back to that. 434 00:44:22.990 --> 00:44:35.209 David Stephens, Prof: Now, that was the binary case. We can consider constructions of the collection of potential outcomes if we have multiple levels of treatment, so we can have a potential outcome for every possible level of treatment. 435 00:44:37.560 --> 00:44:51.629 David Stephens, Prof: And then, again, use the same sort of idea to say, well, this indicator here just picks off the counterfactual outcome corresponding to the actual observed treatment. Again, don't worry too much about the mathematical construction here. It's basically saying. 436 00:44:52.200 --> 00:45:04.030 David Stephens, Prof: The same thing as before, in that we have this array of potential outcomes, and the actual outcome that we get to observe is the one of those potential outcomes corresponding to 437 00:45:04.730 --> 00:45:06.330 David Stephens, Prof: The observed level of treatment. 438 00:45:07.640 --> 00:45:09.260 David Stephens, Prof: Okay, that's all that's saying. 439 00:45:09.710 --> 00:45:19.100 David Stephens, Prof: So, in the multi-treatment level case, we can get that. We can also extend this to continuous value to have a continuous family of 440 00:45:19.460 --> 00:45:27.260 David Stephens, Prof: Potential outcomes as the, Hypothetical levels of treatment vary across some set. 441 00:45:27.580 --> 00:45:34.210 David Stephens, Prof: Okay, so it's all completely general. Most of the formulation that we'll see will be framed in terms of, 442 00:45:34.700 --> 00:45:40.210 David Stephens, Prof: Of the binary case. But this is just to show that 443 00:45:40.560 --> 00:45:46.360 David Stephens, Prof: There's nothing really special about the binary case, we can… Go beyond that very easily. 444 00:45:48.000 --> 00:45:49.399 David Stephens, Prof: Okay, so… 445 00:45:53.140 --> 00:45:58.040 David Stephens, Prof: So, our interest in counterfactual outcomes 446 00:45:58.270 --> 00:46:05.459 David Stephens, Prof: Counterfactual modeling is really there so that we can frame and talk about causal effects. 447 00:46:08.530 --> 00:46:15.750 David Stephens, Prof: Our interest in the causal setting is that experimental DAG, where there was no connection between X and Z. 448 00:46:16.740 --> 00:46:20.280 David Stephens, Prof: However, complication is that our data 449 00:46:22.370 --> 00:46:31.569 David Stephens, Prof: are not generated from that model. Our observational data are generated from a model in which there is an arrow between X and Z. 450 00:46:34.360 --> 00:46:42.060 David Stephens, Prof: So, the central problem of causal inference is that we want to make inference in this experimental-like setting. 451 00:46:42.620 --> 00:46:45.440 David Stephens, Prof: But we only have data from the observational setting. 452 00:46:48.160 --> 00:47:01.240 David Stephens, Prof: The other thing to note now is that, I've talked about these counterfactual outcomes. It's rare that we would get to observe both counterfactual outcomes under no treatment and treatment in the same study. 453 00:47:02.490 --> 00:47:07.880 David Stephens, Prof: There are sort of exceptions to that, if you allow for crossover trials, and… 454 00:47:08.210 --> 00:47:11.900 David Stephens, Prof: washout, and so on. So there's, you know, there are… there are cases that 455 00:47:12.130 --> 00:47:15.260 David Stephens, Prof: Make it somewhat more possible, but it's, 456 00:47:15.390 --> 00:47:17.959 David Stephens, Prof: More complicated, kind of setup. 457 00:47:20.590 --> 00:47:25.209 David Stephens, Prof: But typically, we're not going to be able to observe both of these things in the same study, and that… 458 00:47:25.340 --> 00:47:30.360 David Stephens, Prof: Is one of the fundamental limitations of 459 00:47:32.600 --> 00:47:45.969 David Stephens, Prof: one of the main obstacles to thinking about causal conclusions from observational data, that we don't get to observe both of these things. As we'll see, the propensity score-like constructions 460 00:47:47.280 --> 00:47:51.820 David Stephens, Prof: Allow us to, make inference where it would otherwise not be possible. 461 00:47:53.340 --> 00:47:59.979 David Stephens, Prof: So, for example, in that hypothetical study that I had before, we can't observe both the CBDA outcome 462 00:48:00.440 --> 00:48:08.749 David Stephens, Prof: Under the assumption that the subject did, and simultaneously did not have a low vitamin E diet. That's a comparison that we'd be interested in doing. 463 00:48:08.970 --> 00:48:11.240 David Stephens, Prof: But we can't do it based on our data. 464 00:48:11.760 --> 00:48:15.380 David Stephens, Prof: And I've called this here the first fundamental challenge of causal inference. 465 00:48:16.030 --> 00:48:20.300 David Stephens, Prof: The comparison that we want to do can't be done because we don't have the data to do it. 466 00:48:27.110 --> 00:48:29.430 David Stephens, Prof: So when we think about causal estimation. 467 00:48:29.680 --> 00:48:35.730 David Stephens, Prof: We have to start thinking about the S- demand concern, that is the target parameter that we're after. 468 00:48:36.230 --> 00:48:37.460 David Stephens, Prof: And… 469 00:48:39.120 --> 00:48:44.779 David Stephens, Prof: you can see in the way that I've set it up that the central causal question relates to comparing 470 00:48:45.440 --> 00:48:46.940 David Stephens, Prof: potential outcomes. 471 00:48:47.320 --> 00:48:54.520 David Stephens, Prof: In particular, the difference, let's say, between the outcome that we would observe If the person were treated. 472 00:48:55.240 --> 00:49:01.129 David Stephens, Prof: Compared to… The outcome that we would observe if they were not treated. 473 00:49:02.130 --> 00:49:08.039 David Stephens, Prof: So, we might think about differences in potential outcomes. It's not the only thing we could look at, but it's the simplest. 474 00:49:08.750 --> 00:49:13.900 David Stephens, Prof: So, causal outcomes… Oh, sorry, the causal effect is the difference in outcomes 475 00:49:14.170 --> 00:49:20.090 David Stephens, Prof: That we would observe in an intervention to change, From untreated to treated, say. 476 00:49:21.690 --> 00:49:27.750 David Stephens, Prof: Now, this is a statistical rather than necessarily mechanistic definition of causality. 477 00:49:29.250 --> 00:49:36.729 David Stephens, Prof: But I would argue that this is the role of the statistician and the epidemiologist in this setting. We're not necessarily looking at 478 00:49:37.190 --> 00:49:45.629 David Stephens, Prof: biological or mechanistic causes. There are some settings in which we can do that in principle, but in general, we can't… we'd find that difficult to do. 479 00:49:45.770 --> 00:49:52.210 David Stephens, Prof: So it's, the kind of conclusion that we would typically make from a statistical model. 480 00:49:52.750 --> 00:50:02.190 David Stephens, Prof: In the same way that we would in regression, by picking off a coefficient, let's say an estimated coefficient, and saying, well, the effect of this variable is captured by that coefficient. 481 00:50:03.970 --> 00:50:13.539 David Stephens, Prof: So, it is somewhat of a limitation to think of this in a statistical rather than mechanistic way, but it's the usual compromise we have to make in statistical analysis. 482 00:50:19.150 --> 00:50:25.099 David Stephens, Prof: So we can think about binary exposures, And look at additive difference. 483 00:50:26.170 --> 00:50:36.010 David Stephens, Prof: So, if we've got treated and untreated individual, then we can think of the causal effect of treatment as being, let's say, the difference in the random variables that we would have got 484 00:50:36.380 --> 00:50:40.110 David Stephens, Prof: If they were not treated, Compared to the fact that they were treated. 485 00:50:41.010 --> 00:50:49.560 David Stephens, Prof: Again, a hypothetical quantity phrased in terms of potential outcomes, but a way of measuring the true effect, the impact of the intervention 486 00:50:49.910 --> 00:50:51.260 David Stephens, Prof: To change treatment level. 487 00:50:51.900 --> 00:50:54.069 David Stephens, Prof: We could do it on a multiplicative scale. 488 00:50:54.530 --> 00:51:03.309 David Stephens, Prof: For a continuous… so, the ratio. We could do it, on a continuous scale, we might think about a dose-response function for some… 489 00:51:05.360 --> 00:51:09.539 David Stephens, Prof: Between, different levels of treatment and the response. 490 00:51:16.610 --> 00:51:17.270 David Stephens, Prof: Okay? 491 00:51:17.510 --> 00:51:22.689 David Stephens, Prof: So I think I'll take, because my throat is going a little bit here, I'll take a 5-minute break now. 492 00:51:23.040 --> 00:51:29.089 David Stephens, Prof: We'll start again at 25 minutes past, so 25 past 12 on my time. 493 00:51:29.530 --> 00:51:35.989 David Stephens, Prof: And, pick up again with this discussion to think more about how we measure 494 00:51:36.810 --> 00:51:38.900 David Stephens, Prof: Causal effects in different settings. 495 00:55:49.350 --> 00:55:54.190 David Stephens, Prof: Okay, and so we'll pick up for another 35 minutes, and then break for a longer break. 496 00:55:54.360 --> 00:56:00.510 David Stephens, Prof: So, we're thinking about… how to… 497 00:56:00.900 --> 00:56:05.329 David Stephens, Prof: measure the causal effect of a treatment Z on an outcome. 498 00:56:05.760 --> 00:56:12.060 David Stephens, Prof: In the previous discussion here, this is really the, benefit of using the counterfactual 499 00:56:12.170 --> 00:56:20.999 David Stephens, Prof: framework, the potential outcome framework. We can measure the effect of treatment by thinking about the different things that would happen to an individual. 500 00:56:22.120 --> 00:56:24.530 David Stephens, Prof: comparing… If they were. 501 00:56:25.510 --> 00:56:26.780 David Stephens, Prof: Not treated. 502 00:56:28.390 --> 00:56:30.049 David Stephens, Prof: To what would happen if they were treated. 503 00:56:31.570 --> 00:56:37.410 David Stephens, Prof: So that would ha- that's at the individual level. Of course, for statistical purposes. 504 00:56:38.830 --> 00:56:43.109 David Stephens, Prof: It's hard to think about individual-level contrasts, and what we typically do 505 00:56:43.430 --> 00:56:48.190 David Stephens, Prof: Is aggregate those to think about population-level impacts. 506 00:56:49.140 --> 00:56:54.789 David Stephens, Prof: So… We might look at a so-called expected or average potential outcome. 507 00:56:55.090 --> 00:57:00.729 David Stephens, Prof: As being the quantity of interest, so across a population, what would be the… 508 00:57:00.890 --> 00:57:04.360 David Stephens, Prof: Expected response to a particular treatment, let's say. 509 00:57:04.820 --> 00:57:09.389 David Stephens, Prof: So that's the average potential outcome across the population for 510 00:57:09.520 --> 00:57:11.590 David Stephens, Prof: Treatment at level little red z. 511 00:57:13.580 --> 00:57:21.069 David Stephens, Prof: We can make that more specific. We can think about subgroup-specific equivalent quantities. 512 00:57:21.250 --> 00:57:26.670 David Stephens, Prof: Or we could look at conditional expected quantities of the same nature, so if 513 00:57:27.130 --> 00:57:31.650 David Stephens, Prof: for a certain subpopulation identified by characteristic X, 514 00:57:31.910 --> 00:57:35.360 David Stephens, Prof: then we could look at the expected outcome, the expected 515 00:57:35.590 --> 00:57:39.559 David Stephens, Prof: For different treatment levels, as Little Red Z varies. 516 00:57:39.820 --> 00:57:48.829 David Stephens, Prof: Right? We're mainly going to focus on this one, but all of the methods really extend to look at more… slightly more general, or slightly different settings. 517 00:57:53.450 --> 00:57:55.529 David Stephens, Prof: So, for example, the… 518 00:57:55.940 --> 00:58:03.440 David Stephens, Prof: For the binary treatment case, we can think about the so-called average treatment effect, or ATE. 519 00:58:03.700 --> 00:58:07.269 David Stephens, Prof: Which is the expected difference across the population. 520 00:58:07.420 --> 00:58:10.359 David Stephens, Prof: Between the outcome, if the person were treated. 521 00:58:11.030 --> 00:58:13.259 David Stephens, Prof: And the outcome if the person were not treated. 522 00:58:14.180 --> 00:58:21.960 David Stephens, Prof: So that's the simple difference between expected values under Treatment, and no treatment. 523 00:58:22.440 --> 00:58:24.910 David Stephens, Prof: That's our causal measure of interest. 524 00:58:25.340 --> 00:58:29.860 David Stephens, Prof: I'm not talking yet about how we would think about estimating these things. 525 00:58:29.990 --> 00:58:33.070 David Stephens, Prof: I'm just defining the parameter of interest. 526 00:58:34.290 --> 00:58:42.769 David Stephens, Prof: And as we'll see, estimation in some settings is straightforward, but estimation in other settings is more complicated, that's why we need the Propensity score. 527 00:58:43.500 --> 00:58:49.400 David Stephens, Prof: Okay? So it's the… here it's the expected outcome if everybody were treated. 528 00:58:49.920 --> 00:58:53.880 David Stephens, Prof: Here, it's the expected outcome if everybody were untreated. 529 00:58:55.170 --> 00:58:55.920 David Stephens, Prof: Okay? 530 00:58:56.410 --> 00:59:05.780 David Stephens, Prof: If the outcome were binary, we can look at the same kind of thing. We could look at, odds, or odds ratios. All of these are valid. 531 00:59:06.210 --> 00:59:10.059 David Stephens, Prof: Causal estimands that we might want to look at in different settings. 532 00:59:12.740 --> 00:59:20.439 David Stephens, Prof: Okay, so that's the… this is identifying the target parameter, okay? In our causal setting, this is what we want to estimate. 533 00:59:24.240 --> 00:59:37.469 David Stephens, Prof: It's not the only thing we might be interested in. We might also be interested in the average treatment effect on the treated, or ATT, which is the same expected counterfactual difference, but now conditional on being observed to receive treatment. 534 00:59:38.100 --> 00:59:44.740 David Stephens, Prof: So this is a subset of the entire observational population. 535 00:59:45.910 --> 00:59:49.789 David Stephens, Prof: We're only looking in those individuals that were treated in the study. 536 00:59:50.440 --> 00:59:52.340 David Stephens, Prof: Okay? Now… 537 00:59:52.510 --> 01:00:04.820 David Stephens, Prof: There are specific methods, causal methods, that have been defined to look at this kind of thing. The same principles, the same regression principles, the same weighting principles apply to this kind of quantity. 538 01:00:05.540 --> 01:00:15.109 David Stephens, Prof: I'm mainly going to focus on the ATE rather than the ATT. There are some settings where the ATT is more directly available. 539 01:00:15.360 --> 01:00:19.260 David Stephens, Prof: And I'll mention those as we go through some of the methods, tomorrow. 540 01:00:20.110 --> 01:00:24.979 David Stephens, Prof: A setting that we might be interested in, 541 01:00:25.380 --> 01:00:31.530 David Stephens, Prof: looking at the ATT rather than the ATE, though, is a situation in which the, 542 01:00:32.420 --> 01:00:37.120 David Stephens, Prof: The… it's the, the treated population 543 01:00:39.040 --> 01:00:47.020 David Stephens, Prof: specifically that we're interested in, or subpopulation. So, this is a study from, published in JAMA Paediatrics from… 544 01:00:47.390 --> 01:00:49.710 David Stephens, Prof: Quebec? 545 01:00:51.460 --> 01:00:56.809 David Stephens, Prof: And, it looks at, the, effect on, 546 01:00:57.270 --> 01:01:07.960 David Stephens, Prof: The outcome of, so, which is autism spectrum disorder, in a cohort of mothers that… 547 01:01:08.150 --> 01:01:11.930 David Stephens, Prof: May or may not have used antidepressants during pregnancy. 548 01:01:14.160 --> 01:01:23.129 David Stephens, Prof: So… The idea here is that antidepressants quite widely prescribed for a variety of mental health concerns. 549 01:01:23.550 --> 01:01:28.049 David Stephens, Prof: Pregnant women may be reluctant to embark on a course during pregnancy, though. 550 01:01:28.660 --> 01:01:32.090 David Stephens, Prof: We want to investigate in a population of… 551 01:01:32.560 --> 01:01:39.249 David Stephens, Prof: users and potential users of antidepressants, the incidence of ASD in early childhood. 552 01:01:39.400 --> 01:01:45.340 David Stephens, Prof: To assess the possibility of… A causal influence of antidepressants on this… on this incidence. 553 01:01:46.940 --> 01:01:54.370 David Stephens, Prof: So we might have an outcome which is a recording of a particular ASD diagnosis. 554 01:01:55.350 --> 01:02:00.950 David Stephens, Prof: Exposure would be antidepressant use during pregnancy, second or third trimester. 555 01:02:01.730 --> 01:02:11.819 David Stephens, Prof: What we're interested in doing Is looking at, in the causal world, The difference in incidence rates. 556 01:02:11.930 --> 01:02:18.600 David Stephens, Prof: Between those individuals that Actually used antidepressants. 557 01:02:20.710 --> 01:02:26.059 David Stephens, Prof: The difference in the outcome that would have been observed if they were to take it. 558 01:02:26.710 --> 01:02:31.060 David Stephens, Prof: Compared to… What would have happened had they not taken it? 559 01:02:33.130 --> 01:02:38.459 David Stephens, Prof: And the argument here is that this is a more imp… A more relevant 560 01:02:38.930 --> 01:02:43.930 David Stephens, Prof: causal S demand, because the overall 561 01:02:44.330 --> 01:02:48.130 David Stephens, Prof: Average treatment effect in the entire population is not of interest. 562 01:02:49.550 --> 01:02:53.200 David Stephens, Prof: Well, the ordinary ATE, the difference in expected. 563 01:02:53.740 --> 01:02:56.410 David Stephens, Prof: Outcomes that we would have observed 564 01:02:56.980 --> 01:03:02.789 David Stephens, Prof: under antidepressant use, and under no antidepressant use. That quantity in itself 565 01:03:03.020 --> 01:03:09.570 David Stephens, Prof: is not of interest. We're more interested in looking at the effect on those individuals who actually took 566 01:03:10.310 --> 01:03:11.590 David Stephens, Prof: the antidepressants. 567 01:03:11.920 --> 01:03:14.440 David Stephens, Prof: So this quantity is the ATT. 568 01:03:15.710 --> 01:03:19.090 David Stephens, Prof: Okay, and this paper… goes into… 569 01:03:19.430 --> 01:03:24.369 David Stephens, Prof: discuss… goes into discussion of that issue in this particular cohort. It's very interesting. 570 01:03:24.880 --> 01:03:30.300 David Stephens, Prof: paper. There are other similar kinds of… Studies that have been ongoing. 571 01:03:32.090 --> 01:03:46.509 David Stephens, Prof: So all this to say that we'll mainly focus on the ATE, but other causal quantities may be of interest. The propensity score methods that I'm going to explain really can be applied in any of these causal lesson amounts. 572 01:03:49.720 --> 01:03:53.039 David Stephens, Prof: So… If we think about, 573 01:03:55.040 --> 01:03:57.790 David Stephens, Prof: Now, statistical estimation of those quantities. 574 01:03:59.790 --> 01:04:05.330 David Stephens, Prof: And we think about the usual setup of statistical estimation of a population average. 575 01:04:05.620 --> 01:04:08.339 David Stephens, Prof: Then our go-to method is to use a sample mean. 576 01:04:09.690 --> 01:04:14.520 David Stephens, Prof: That is, if we have a random sample from a population, and we want to estimate the population mean. 577 01:04:16.590 --> 01:04:20.410 David Stephens, Prof: Our go-to estimator, Is the sample mean. 578 01:04:22.180 --> 01:04:23.890 David Stephens, Prof: probably uncontroversial. 579 01:04:24.890 --> 01:04:26.780 David Stephens, Prof: Here's the definition of it here. 580 01:04:28.430 --> 01:04:40.479 David Stephens, Prof: And we're… this idea of population averages being estimated by sample averages is something that is universally agreed upon in 581 01:04:41.010 --> 01:04:43.530 David Stephens, Prof: Statistics and epidemiology, for good reason. 582 01:04:43.890 --> 01:04:49.519 David Stephens, Prof: Sample mean is a good estimator of things when we're looking at population averages. 583 01:04:54.460 --> 01:05:02.269 David Stephens, Prof: Now, In the causal setting, we're interested in estimating the average potential outcome. 584 01:05:05.330 --> 01:05:07.629 David Stephens, Prof: Okay, so in the simplest situation. 585 01:05:07.970 --> 01:05:14.249 David Stephens, Prof: that we're gonna think about first. Let's imagine that, everybody was given exposure a little red Z. 586 01:05:17.210 --> 01:05:26.349 David Stephens, Prof: In that case… Our estimate would be the… Population mean… For that treatment level. 587 01:05:26.790 --> 01:05:31.310 David Stephens, Prof: Which is the expected value of the potential outcomes for that treatment level. 588 01:05:33.940 --> 01:05:35.860 David Stephens, Prof: Okay? So, in this case. 589 01:05:36.350 --> 01:05:42.869 David Stephens, Prof: we're really back in the simple sample average world, right? Everybody received the same tube into level Z, 590 01:05:43.440 --> 01:05:46.240 David Stephens, Prof: So we can estimate the expected outcome. 591 01:05:46.380 --> 01:05:48.249 David Stephens, Prof: for that treatment level Z. 592 01:05:48.480 --> 01:05:50.330 David Stephens, Prof: By just taking the sample mean. 593 01:05:54.410 --> 01:05:56.620 David Stephens, Prof: Okay? Here is the relevant DAG. 594 01:05:57.850 --> 01:06:00.330 David Stephens, Prof: X is causing the potential outcomes. 595 01:06:00.480 --> 01:06:04.350 David Stephens, Prof: But we are intervening to set the treatment level to be Z. 596 01:06:05.340 --> 01:06:07.619 David Stephens, Prof: There's no arrow from X to Z. 597 01:06:11.000 --> 01:06:14.229 David Stephens, Prof: We're setting this value, so that's where the square box is here. 598 01:06:14.360 --> 01:06:16.860 David Stephens, Prof: We're setting that value… 599 01:06:17.670 --> 01:06:27.040 David Stephens, Prof: That leads to this potential outcome only. We only pick off one, whether little red z is 1 or 0. Doesn't matter, we get that outcome. So this… 600 01:06:27.110 --> 01:06:38.249 David Stephens, Prof: intervention here is just picking one of these, whichever the appropriate one is for little red z equals 0, we get this one. Little red z equals 1, we get this one, that's what we see here. 601 01:06:39.930 --> 01:06:41.810 David Stephens, Prof: That's the idea of an intervention. 602 01:06:45.300 --> 01:06:50.240 David Stephens, Prof: Now… Why does the sample mean? 603 01:06:50.690 --> 01:06:52.250 David Stephens, Prof: Workers are good. 604 01:06:53.140 --> 01:06:55.430 David Stephens, Prof: Estimator of the population mean. 605 01:06:55.920 --> 01:06:56.790 David Stephens, Prof: Well… 606 01:06:57.080 --> 01:07:05.269 David Stephens, Prof: This is one of the things we learn about in elementary courses in statistics. What we learn about is that the population average 607 01:07:05.900 --> 01:07:22.469 David Stephens, Prof: which we can write in a mathematical way like this. It's just the average of the integral y times the density Y dY. Basically, what this says is that we average the collection of possible values weighted by the probability that they are observed. 608 01:07:27.090 --> 01:07:33.480 David Stephens, Prof: So, the mathematical definition of the S demand… It's through this integral form. 609 01:07:35.740 --> 01:07:36.710 David Stephens, Prof: And… 610 01:07:39.480 --> 01:07:51.780 David Stephens, Prof: What we'll see in a short while is that sample averaging basically just replaces this theoretical distribution with the empirical distribution of our observed data. 611 01:07:53.870 --> 01:08:00.820 David Stephens, Prof: Alright, so… There's another way that we can write that expectation in the… 612 01:08:00.980 --> 01:08:07.419 David Stephens, Prof: Previous calculation. And that's to recognize that in our data-generating situation. 613 01:08:08.150 --> 01:08:10.640 David Stephens, Prof: There are also these other quantities, X. 614 01:08:11.410 --> 01:08:21.199 David Stephens, Prof: So, average apop… average across the population calculations are really also averaging across all X values, right? 615 01:08:22.390 --> 01:08:26.229 David Stephens, Prof: So we can also write this expectation as a double integral. 616 01:08:26.399 --> 01:08:30.880 David Stephens, Prof: Over the distribution, under the hypothetical intervention. 617 01:08:31.670 --> 01:08:35.679 David Stephens, Prof: over the possible values of Y and the possible values of X. 618 01:08:36.080 --> 01:08:41.520 David Stephens, Prof: So, if you studied this mathematically, you'll know that this calculation 619 01:08:41.939 --> 01:08:44.459 David Stephens, Prof: And this calculation gives the same answer. 620 01:08:45.670 --> 01:08:49.009 David Stephens, Prof: They are exactly the same quantity. 621 01:08:50.359 --> 01:08:58.140 David Stephens, Prof: So, why do we bother writing it in this way? Well, it recognizes the impact of Y… of, sorry, of X on Y. 622 01:08:58.840 --> 01:09:09.129 David Stephens, Prof: Right? This joint distribution here tells us how the hypothetical Y and X vary together. 623 01:09:09.830 --> 01:09:17.479 David Stephens, Prof: It tells us about the relationship between X and the counterfactuals. Counterfactual outcomes, the potential outcomes. 624 01:09:18.399 --> 01:09:21.410 David Stephens, Prof: And in fact, that joint distribution 625 01:09:22.160 --> 01:09:29.910 David Stephens, Prof: according to our DAG, factorizes in this way. Again, what this means is that people come in off the street and they bring their exes with them. 626 01:09:30.920 --> 01:09:38.279 David Stephens, Prof: Once we've seen their X value, that leads to a generation of the counterfactual or potential Y that we would observe. 627 01:09:39.410 --> 01:09:42.269 David Stephens, Prof: If we had a given treatment level. 628 01:09:45.080 --> 01:09:48.700 David Stephens, Prof: So, it's getting a bit mathematical now, I appreciate that. 629 01:09:48.800 --> 01:09:55.439 David Stephens, Prof: We won't see too much of this as we go through. What I'm trying to make the point is here that this population average 630 01:09:55.640 --> 01:09:58.709 David Stephens, Prof: is an average over the Y and X distribution. 631 01:10:01.830 --> 01:10:09.250 David Stephens, Prof: Alright, so… To make it even more… complicated. 632 01:10:09.880 --> 01:10:14.940 David Stephens, Prof: but actually to yield the same answer. We can further extend the… 633 01:10:15.620 --> 01:10:21.580 David Stephens, Prof: expectation. So this is, again, the same quantity, this is a target estimate, the expected outcome. 634 01:10:23.260 --> 01:10:32.429 David Stephens, Prof: When the treatment is set to be level Z, We can take that Z intervention. 635 01:10:32.770 --> 01:10:37.110 David Stephens, Prof: And introduce that into the expectation in exactly the same way. 636 01:10:38.490 --> 01:10:44.029 David Stephens, Prof: So we now write it as a triple integral over all of the variables concerned. 637 01:10:44.710 --> 01:10:57.900 David Stephens, Prof: It's just that we make sure that we recognize the intervention by including this term here, which basically just says, only consider the case where the treatment was equal to 638 01:10:58.750 --> 01:10:59.930 David Stephens, Prof: The target treatment. 639 01:11:02.570 --> 01:11:03.270 David Stephens, Prof: Okay. 640 01:11:03.970 --> 01:11:14.340 David Stephens, Prof: Now, what this, expectation does, it says, well, it's a, it's an average over the population. The population, is really, 641 01:11:14.770 --> 01:11:17.520 David Stephens, Prof: A 3-variable quantity. 642 01:11:17.900 --> 01:11:18.820 David Stephens, Prof: X. 643 01:11:19.450 --> 01:11:21.580 David Stephens, Prof: Z and Y. 644 01:11:22.580 --> 01:11:31.190 David Stephens, Prof: According to our DAG, X generates the counterfactuals, Z… Is intervened upon separately? 645 01:11:34.200 --> 01:11:41.540 David Stephens, Prof: So there's one and only one value that the treatment takes in this hypothetical quantity, and it's identified with a little red Z. 646 01:11:44.350 --> 01:11:48.499 David Stephens, Prof: So, at the end of the day, the story is that this 647 01:11:48.930 --> 01:11:52.259 David Stephens, Prof: Target quantity can be written as this triple integral. 648 01:11:53.460 --> 01:11:56.699 David Stephens, Prof: So it's an average across the population quantity. 649 01:11:57.000 --> 01:12:01.899 David Stephens, Prof: But where we pretended that we've intervened to set the treatment equal to little red z. 650 01:12:03.330 --> 01:12:08.859 David Stephens, Prof: Now, this calculation will come up later on, when we think about one particular of the methods. 651 01:12:09.820 --> 01:12:13.319 David Stephens, Prof: Which is, inverse probability weighting. 652 01:12:14.430 --> 01:12:22.269 David Stephens, Prof: So if you don't quite catch what's going on here at the moment, don't worry about it too much. We'll revisit it later on, and I'll show the importance of recognizing this. 653 01:12:23.320 --> 01:12:24.740 David Stephens, Prof: calculation. 654 01:12:28.460 --> 01:12:37.960 David Stephens, Prof: Now, In that, hypothetical situation where everybody received the train… the same treatment said by intervention. 655 01:12:38.980 --> 01:12:42.769 David Stephens, Prof: Okay, we've just seen that data generating model, that's fine. 656 01:12:43.280 --> 01:12:51.669 David Stephens, Prof: How do we estimate the relevant average potential outcome for that hypothetical setting? Remember, the hypothetical setting says. 657 01:12:53.090 --> 01:12:55.269 David Stephens, Prof: Everybody brought their own axes. 658 01:12:55.790 --> 01:13:01.259 David Stephens, Prof: We intervene to set the value of Z, and then we observe the outcome. 659 01:13:02.640 --> 01:13:04.829 David Stephens, Prof: So how do we estimate this quantity? 660 01:13:05.460 --> 01:13:11.890 David Stephens, Prof: Which is the, expected value under that hypothetical intervention. 661 01:13:12.370 --> 01:13:13.649 David Stephens, Prof: It's very easy. 662 01:13:13.790 --> 01:13:16.049 David Stephens, Prof: We just take the sample mean of the outcomes. 663 01:13:16.560 --> 01:13:18.600 David Stephens, Prof: Everybody received the same treatment. 664 01:13:19.780 --> 01:13:21.699 David Stephens, Prof: So everybody's on the same footing. 665 01:13:22.080 --> 01:13:24.439 David Stephens, Prof: We want to average across the population. 666 01:13:24.600 --> 01:13:31.460 David Stephens, Prof: Average across the characteristics to look at the response, the impact of that intervention. But it's very easy, we just take the sample mean. 667 01:13:31.630 --> 01:13:33.250 David Stephens, Prof: There's nothing else to be done. 668 01:13:36.200 --> 01:13:37.950 David Stephens, Prof: So really, this is just a… 669 01:13:38.330 --> 01:13:46.390 David Stephens, Prof: sort of glorified version of that previous calculation. If everybody received the same treatment, then they're all on the same footing. And we could just take the sample mean. 670 01:13:46.650 --> 01:13:51.879 David Stephens, Prof: To estimate their res- the population response, Under that treatment. 671 01:13:53.730 --> 01:13:59.700 David Stephens, Prof: Now, of course, this is a very limited setting, right? Because typically what we want to do is to compare treatments. 672 01:14:00.070 --> 01:14:06.770 David Stephens, Prof: So this kind of design would be useless for doing that, right? We've only got one assigned treatment level. 673 01:14:07.540 --> 01:14:15.790 David Stephens, Prof: So we can estimate the mean under that treatment, but we don't have any other information about the other treatments. So clearly, this on its own is not good enough for what we want to achieve. 674 01:14:16.220 --> 01:14:21.180 David Stephens, Prof: But it lays down the basic elements of the calculation that we're going to do. 675 01:14:26.310 --> 01:14:32.450 David Stephens, Prof: If we didn't want to focus on the… the population… Average. 676 01:14:32.780 --> 01:14:42.659 David Stephens, Prof: We can easily extend that to look at functions of that. So if we wanted an odds, we could look at, in the binary outcome case, we could look at the empirical odds. 677 01:14:43.290 --> 01:14:48.310 David Stephens, Prof: That would be… An easy calculation to do. 678 01:14:49.900 --> 01:14:55.789 David Stephens, Prof: So having estimated this by the sample mean, we can just use transformations to get other quantities. 679 01:14:57.800 --> 01:14:58.790 David Stephens, Prof: Alright. 680 01:15:00.660 --> 01:15:06.340 David Stephens, Prof: And… Just as a, sort of, an aside here, 681 01:15:09.270 --> 01:15:14.910 David Stephens, Prof: I think it can be useful to think about the calculation that I've just laid down from a… 682 01:15:15.080 --> 01:15:19.010 David Stephens, Prof: From a slightly different perspective, which is the so-called Monte Carlo perspective. 683 01:15:19.610 --> 01:15:24.509 David Stephens, Prof: This is what's going to inspire the inverse probability weighting approach that we see later on. 684 01:15:24.900 --> 01:15:29.259 David Stephens, Prof: So basically what this is saying is that, so Monte Carlo is a way for doing integration. 685 01:15:29.840 --> 01:15:30.920 David Stephens, Prof: Mathematically. 686 01:15:31.220 --> 01:15:33.360 David Stephens, Prof: Using numerical approximations. 687 01:15:33.550 --> 01:15:37.560 David Stephens, Prof: And it says, If we want to estimate this expectation. 688 01:15:38.170 --> 01:15:42.679 David Stephens, Prof: Written out in this way, so this is the mathematical definition of the expectation of G. 689 01:15:43.320 --> 01:15:51.730 David Stephens, Prof: Instead of doing this integral analytically, using clever tools from mathematics, what we can do is to use sampling-based methods. 690 01:15:52.110 --> 01:16:01.470 David Stephens, Prof: Which is basically to say, we generate a sample from the Y distribution, And then we average The G's. 691 01:16:02.020 --> 01:16:03.980 David Stephens, Prof: Across the sample that we've obtained. 692 01:16:04.120 --> 01:16:08.220 David Stephens, Prof: So the idea is we use a random number generator to generate the Y's. 693 01:16:09.990 --> 01:16:19.640 David Stephens, Prof: and then estimate the expectation of interest just by taking the sample average of the G's, evaluated those Ys. That's the basic Monte Carlo calculation. 694 01:16:20.720 --> 01:16:28.120 David Stephens, Prof: And if the Monte Carlo sample is large enough, then this thing approximates this thing well. That's the idea. That's Monte Carlo. 695 01:16:31.200 --> 01:16:38.500 David Stephens, Prof: I'm… One thing I want to point out is that This is just Frequentist statistics. 696 01:16:39.320 --> 01:16:48.459 David Stephens, Prof: So we often think about Monte Carlo as being a sort of numerical integration method idea, nothing to do with statistics. 697 01:16:48.750 --> 01:16:54.750 David Stephens, Prof: But actually, The calculation of a sample mean to approximate a population mean. 698 01:16:54.860 --> 01:16:57.190 David Stephens, Prof: Is applying exactly the same principles. 699 01:16:57.640 --> 01:17:04.559 David Stephens, Prof: We collect a data sample of size N, we don't generate it on the computer anymore, we go out into the real world and collect data. 700 01:17:05.140 --> 01:17:09.960 David Stephens, Prof: And then we use those data to form estimates based on the sample, in particular the… 701 01:17:10.080 --> 01:17:12.939 David Stephens, Prof: Moment-based estimators work exactly like that. 702 01:17:14.300 --> 01:17:16.209 David Stephens, Prof: And if the sample is large enough. 703 01:17:16.420 --> 01:17:18.719 David Stephens, Prof: The estimate will be close to the true value. 704 01:17:18.970 --> 01:17:21.640 David Stephens, Prof: That is the logic of frequented statistics. 705 01:17:23.510 --> 01:17:30.199 David Stephens, Prof: So you can think of frequentis statistics, if you've got a certain mindset, as being really just a Monte Carlo-type method. 706 01:17:31.540 --> 01:17:37.129 David Stephens, Prof: It's just that we don't generate data from a computer, we go out and collect data. 707 01:17:40.600 --> 01:17:45.330 David Stephens, Prof: Why that's important is because there's another method called importance sampling. 708 01:17:45.610 --> 01:17:48.680 David Stephens, Prof: Which is based on forming weighted averages. 709 01:17:49.580 --> 01:17:57.649 David Stephens, Prof: instead of sampling from the distribution FY, we generate from a different distribution, FY star. 710 01:17:57.980 --> 01:18:06.040 David Stephens, Prof: and then do a re-weighted calculation in order to estimate the quantity of interest. Our target is still the original expectation. 711 01:18:06.310 --> 01:18:10.229 David Stephens, Prof: But rather than sampling from F, we sample from F star. 712 01:18:10.730 --> 01:18:12.689 David Stephens, Prof: And that can recover the same quantity. 713 01:18:14.160 --> 01:18:16.109 David Stephens, Prof: So again, I put a little, 714 01:18:16.280 --> 01:18:20.690 David Stephens, Prof: recap of Monte Carlo methods at the end in the appendix, you can look at that if you want to, we'll… 715 01:18:20.910 --> 01:18:23.250 David Stephens, Prof: Touch on elements of this as we go through. 716 01:18:25.810 --> 01:18:32.760 David Stephens, Prof: But, the idea, basically, is that we're gonna do a form of standardized estimation. 717 01:18:33.440 --> 01:18:42.649 David Stephens, Prof: Right, so, like, standardized mortality rates and whatever, by thinking not about, estimation in the target population. 718 01:18:43.210 --> 01:18:48.659 David Stephens, Prof: But how to port estimates from the non… from a non-target population. 719 01:18:48.850 --> 01:18:51.220 David Stephens, Prof: Into estimates for the target population. 720 01:18:51.810 --> 01:18:54.950 David Stephens, Prof: What we'll see is that we'll be able to do that through re-weighting. 721 01:18:57.510 --> 01:19:05.549 David Stephens, Prof: And… In the causal setting, this is important, because remember I said one of the fundamental points is that 722 01:19:07.110 --> 01:19:11.409 David Stephens, Prof: Our target of inference, Is in the experimental world. 723 01:19:11.730 --> 01:19:16.439 David Stephens, Prof: Where we intervene to change the treatment from one level to another. 724 01:19:16.540 --> 01:19:18.109 David Stephens, Prof: And look at the impact on outcome. 725 01:19:21.200 --> 01:19:23.100 David Stephens, Prof: That's the target of inference. 726 01:19:23.700 --> 01:19:28.300 David Stephens, Prof: But in observational data, In the observational data setting. 727 01:19:28.550 --> 01:19:33.110 David Stephens, Prof: Our data are not generated in that way. They're generated from a different mechanism. 728 01:19:34.380 --> 01:19:37.929 David Stephens, Prof: So how can we use those differently generated data 729 01:19:38.520 --> 01:19:46.949 David Stephens, Prof: to estimate the things that we're really interested in, which really arise from a different probability model altogether. We have to port the data. 730 01:19:47.290 --> 01:19:49.110 David Stephens, Prof: From one model to the other. 731 01:19:51.330 --> 01:19:56.680 David Stephens, Prof: And it's going to turn out that important sampling or re-weighting is one way that we can do that. 732 01:20:00.430 --> 01:20:03.300 David Stephens, Prof: Okay, so the final little bit of, 733 01:20:04.520 --> 01:20:12.660 David Stephens, Prof: mathematical notation that I'm going to use, is here to help me talk about, conditional and marginal measures. 734 01:20:13.370 --> 01:20:15.810 David Stephens, Prof: So, the calculation that we just did. 735 01:20:17.060 --> 01:20:23.240 David Stephens, Prof: set up the S demand as being this, expectation here. 736 01:20:23.560 --> 01:20:28.819 David Stephens, Prof: And it's arrived at by averaging over the X distribution. 737 01:20:31.290 --> 01:20:35.850 David Stephens, Prof: So again, don't worry too much about the mathematical notation, that is the basic principle. 738 01:20:36.340 --> 01:20:43.439 David Stephens, Prof: The expected quantity of interest is an expectation over the distribution of characteristics, basically. 739 01:20:44.610 --> 01:20:48.240 David Stephens, Prof: So this formula is sometimes known as the G-computation formula. 740 01:20:50.630 --> 01:20:57.349 David Stephens, Prof: And, it arises because of the assumption that X and Z are independent. 741 01:21:01.810 --> 01:21:05.730 David Stephens, Prof: what we're really doing is assuming that X and Z are independent. 742 01:21:05.920 --> 01:21:14.080 David Stephens, Prof: as we would need to do for the experimental setting, remember there was no arrow going from X into Z. 743 01:21:14.700 --> 01:21:18.290 David Stephens, Prof: X is not a cause of Z, they're unassociated, right? 744 01:21:18.860 --> 01:21:20.679 David Stephens, Prof: We're choosing Zed. 745 01:21:21.880 --> 01:21:23.680 David Stephens, Prof: It's nothing to do with X. 746 01:21:25.810 --> 01:21:35.090 David Stephens, Prof: So… This is the calculation that we'd have to do, and it's basically arrived at from this… conditional distribution. 747 01:21:35.280 --> 01:21:38.670 David Stephens, Prof: Saying, what is the distribution of Y? 748 01:21:38.980 --> 01:21:41.540 David Stephens, Prof: Forgivens that, under independence. 749 01:21:46.490 --> 01:22:01.690 David Stephens, Prof: The one thing I want to point out here is that marginal measures are not the same as conditional measures. So this is not really a causal logic or a causal idea, it's just a model, a model specification issue, in that… 750 01:22:01.790 --> 01:22:04.279 David Stephens, Prof: If we build a, 751 01:22:04.820 --> 01:22:10.550 David Stephens, Prof: conditional model for Y given X and Z. Imagine a regression model for Y given X and Z. 752 01:22:13.130 --> 01:22:16.940 David Stephens, Prof: Then that conditional model Has parameters in it. 753 01:22:17.820 --> 01:22:24.690 David Stephens, Prof: That don't necessarily correspond to the same parameters in the marginal model. 754 01:22:27.220 --> 01:22:31.410 David Stephens, Prof: So, the interpretation of parameters in the conditional model 755 01:22:31.990 --> 01:22:35.840 David Stephens, Prof: let's say coefficients for individual Z terms. 756 01:22:38.940 --> 01:22:44.139 David Stephens, Prof: Those parameters don't have the same interpretation as in the corresponding marginal model. 757 01:22:47.980 --> 01:22:53.650 David Stephens, Prof: Okay, so that will be important, because at some points later on, we will be making the distinction between 758 01:22:54.100 --> 01:22:56.559 David Stephens, Prof: conditional and marginal. 759 01:22:57.490 --> 01:22:59.810 David Stephens, Prof: calculations. 760 01:23:00.410 --> 01:23:06.740 David Stephens, Prof: And it's important to remember this, issue of non-collapsibility. 761 01:23:07.160 --> 01:23:15.499 David Stephens, Prof: So there are in… so imagine doing a logistic regression for an outcome given characteristics, X and Z. 762 01:23:16.350 --> 01:23:20.199 David Stephens, Prof: the coefficient of Z in that conditional model 763 01:23:20.630 --> 01:23:22.929 David Stephens, Prof: Will not be the coefficient of… 764 01:23:23.290 --> 01:23:29.040 David Stephens, Prof: the, equivalent impact of Z in the marginal model, where we've integrated that over X. 765 01:23:32.180 --> 01:23:48.470 David Stephens, Prof: Yeah, so there's a nice question in the chat there. Is the assumption of independence between X and Z valid, when from the DAG we know that they're not? By definition, X is a joint cause of Z and Y? Yes, that's a very good point. So in our observational data. 766 01:23:50.460 --> 01:23:52.009 David Stephens, Prof: Z and X. 767 01:23:52.340 --> 01:23:54.550 David Stephens, Prof: Cannot be assumed to be independent. 768 01:23:55.450 --> 01:24:03.229 David Stephens, Prof: However, The target of our inference is the world where X and Z are independent. 769 01:24:04.840 --> 01:24:11.830 David Stephens, Prof: So our hypothetical quantity of interest arises in the model where X and Z are independent. 770 01:24:12.140 --> 01:24:19.769 David Stephens, Prof: But you're absolutely right, in the real-world data that we're going to collect in the observational setting, we cannot assume X and Z are independent. 771 01:24:20.120 --> 01:24:26.179 David Stephens, Prof: So part of the causal question is, how do we use those apparently, 772 01:24:26.700 --> 01:24:35.229 David Stephens, Prof: Confounded data, or lack of independent data, to make inferences in a different situation where we assume independence. 773 01:24:38.780 --> 01:24:40.669 David Stephens, Prof: Hopefully that answers the question. 774 01:24:44.840 --> 01:24:45.550 David Stephens, Prof: Good. 775 01:24:49.450 --> 01:24:52.140 David Stephens, Prof: Okay, so… in the… 776 01:24:57.620 --> 01:25:01.929 David Stephens, Prof: Must be getting time for a… close for a break, I'm losing my voice, but anyway. 777 01:25:02.660 --> 01:25:12.680 David Stephens, Prof: To set up the afternoon's work, I'm gonna start thinking about the nature of this comparison that we're interested in. Indeed, building on the question that just came up. 778 01:25:12.940 --> 01:25:16.020 David Stephens, Prof: Which is a very important question. It's a central question in the… 779 01:25:16.220 --> 01:25:18.299 David Stephens, Prof: In the construction of these methods. 780 01:25:23.140 --> 01:25:31.780 David Stephens, Prof: So we can use… Sample… ordinary sample means to estimate average potential outcomes in the special situation where everybody 781 01:25:33.150 --> 01:25:36.200 David Stephens, Prof: Receives the same treatment by intervention. 782 01:25:37.790 --> 01:25:41.590 David Stephens, Prof: So they come in off the street, they bring their exes with them. 783 01:25:42.370 --> 01:25:46.300 David Stephens, Prof: We give them all the same treatment, and then measure the outcome. 784 01:25:47.710 --> 01:25:52.809 David Stephens, Prof: So in that setting, It's not even really a causal… 785 01:25:52.940 --> 01:25:55.790 David Stephens, Prof: Question, then we're just estimating the response 786 01:25:56.140 --> 01:25:59.130 David Stephens, Prof: On average, across the population, subject to that treatment. 787 01:26:00.960 --> 01:26:06.690 David Stephens, Prof: But as I mentioned before, this is not a very useful setting, because really we're interested in comparing treatments. 788 01:26:07.770 --> 01:26:09.930 David Stephens, Prof: So how do we set up the situation? 789 01:26:10.130 --> 01:26:14.180 David Stephens, Prof: Or set up the, set up the calculation in order to handle that case. 790 01:26:15.640 --> 01:26:23.219 David Stephens, Prof: Well, the usual, Resolution is through the randomized controlled trial kind of design. 791 01:26:23.900 --> 01:26:27.559 David Stephens, Prof: So we base our conclusions on randomization. 792 01:26:29.420 --> 01:26:31.910 David Stephens, Prof: Alright, that is, we randomize treatment. 793 01:26:32.440 --> 01:26:37.169 David Stephens, Prof: Again, we don't allow for the impact of X on the treatment assignment. 794 01:26:37.360 --> 01:26:44.090 David Stephens, Prof: But what we do is, we collect, let's say, a sample of 2N individuals, 795 01:26:44.200 --> 01:26:49.760 David Stephens, Prof: We might randomly pick N of them, and assign Treatment to those. 796 01:26:49.960 --> 01:26:52.589 David Stephens, Prof: Whereas for the remainder, we leave them untreated. 797 01:26:54.510 --> 01:27:01.309 David Stephens, Prof: Now, this is not the usual way we'd carried out a randomized trial. We don't collect our data first, we do it sequentially, and I'll come back to that in a second. 798 01:27:01.470 --> 01:27:05.949 David Stephens, Prof: But this is one simple mechanism, right? We collect a data sample of size 2N, 799 01:27:06.840 --> 01:27:11.129 David Stephens, Prof: We randomly pick n of those individuals and place them in the treated group. 800 01:27:11.450 --> 01:27:15.049 David Stephens, Prof: We leave the remaining N untreated. 801 01:27:15.890 --> 01:27:18.150 David Stephens, Prof: Now, in this case, we still have… 802 01:27:19.220 --> 01:27:22.719 David Stephens, Prof: No connection between the X's and the treatment level. 803 01:27:23.640 --> 01:27:26.050 David Stephens, Prof: They're still, independent. 804 01:27:28.480 --> 01:27:33.979 David Stephens, Prof: How do we go about estimating the ATE in this case? Well, it's a simple extension of the previous idea. 805 01:27:35.030 --> 01:27:40.590 David Stephens, Prof: Because we know that, in this Randomized selection of individuals. 806 01:27:40.720 --> 01:27:41.600 David Stephens, Prof: Case. 807 01:27:43.810 --> 01:27:46.879 David Stephens, Prof: The first group, The treated group. 808 01:27:47.150 --> 01:27:55.579 David Stephens, Prof: Are still fundamentally comparable with the second group, In every respect, Apart from their treatment assignment. 809 01:27:56.310 --> 01:27:58.519 David Stephens, Prof: They're directly comparable groups. 810 01:27:59.430 --> 01:28:03.590 David Stephens, Prof: So we can just estimate the mean in the first group, using the sample mean. 811 01:28:03.710 --> 01:28:08.969 David Stephens, Prof: And estimate the mean in the second group, using the sample mean from those data, and look at the difference. 812 01:28:11.370 --> 01:28:15.840 David Stephens, Prof: And we can get away with doing this because we recognize, due to the… 813 01:28:16.150 --> 01:28:23.390 David Stephens, Prof: design that we've used, that the two halves of the original sample are exchangeable with each other. They're really 814 01:28:25.960 --> 01:28:31.810 David Stephens, Prof: fundamentally exchangeable in terms of their X characteristics. The only difference 815 01:28:32.560 --> 01:28:34.689 David Stephens, Prof: Is due to their exposure assignment. 816 01:28:34.940 --> 01:28:36.289 David Stephens, Prof: The treatment assignment. 817 01:28:38.890 --> 01:28:48.950 David Stephens, Prof: Now, we recognize that in this setup, there's no impact of X on the treatment assignment. We've just randomly selected, so everybody came in, the two N individuals came in. 818 01:28:49.330 --> 01:28:51.509 David Stephens, Prof: We just split them up at random. 819 01:28:52.150 --> 01:28:53.870 David Stephens, Prof: We've got no reason to believe. 820 01:28:54.060 --> 01:28:55.910 David Stephens, Prof: They're any different from each other? 821 01:28:56.640 --> 01:28:58.300 David Stephens, Prof: On their X basis. 822 01:28:59.160 --> 01:29:02.650 David Stephens, Prof: The first half got the treatment, the second half got no treatment. 823 01:29:03.310 --> 01:29:10.819 David Stephens, Prof: We can use sample mean estimation to estimate the average response in the treated group, and in the untreated group, and then look at the difference. 824 01:29:11.110 --> 01:29:12.060 David Stephens, Prof: Very simple. 825 01:29:14.960 --> 01:29:19.400 David Stephens, Prof: So, in this randomized setup, where… 826 01:29:19.670 --> 01:29:24.570 David Stephens, Prof: X doesn't affect treatment. X and Z are independent. 827 01:29:24.750 --> 01:29:26.270 David Stephens, Prof: We can use this method. 828 01:29:30.790 --> 01:29:34.339 David Stephens, Prof: The more common way that Randomized studies, 829 01:29:34.660 --> 01:29:41.290 David Stephens, Prof: employed, not to do it exactly like that. Typically, what happens is that we collect 830 01:29:42.180 --> 01:29:49.389 David Stephens, Prof: or recruit N individuals, and we randomly assign treatment at the individual level. 831 01:29:49.810 --> 01:29:53.560 David Stephens, Prof: Right? So, somebody signs up, they're recruited. 832 01:29:54.540 --> 01:29:57.009 David Stephens, Prof: And then we basically flip a coin. 833 01:29:57.540 --> 01:30:03.879 David Stephens, Prof: or a bias coin with probability P to say whether they get treatment or not. There are more sophisticated ways to do it. 834 01:30:04.090 --> 01:30:09.739 David Stephens, Prof: Block randomization, and so on, stratified by… could be stratified by, 835 01:30:09.970 --> 01:30:13.300 David Stephens, Prof: Center, or by, by characteristics. 836 01:30:14.190 --> 01:30:20.340 David Stephens, Prof: of the… Individuals say. But the basic mechanism is the same. We end up flipping a coin somehow. 837 01:30:21.630 --> 01:30:29.289 David Stephens, Prof: Okay? Now, in this simple case where we have n individuals and we are doing a random assignment sequentially with probability P, 838 01:30:29.720 --> 01:30:38.459 David Stephens, Prof: In the final sample, the number actually treated is some number n1, say. That's a realization of a random variable which is binomial. 839 01:30:40.060 --> 01:30:44.270 David Stephens, Prof: Alright, so… And individuals each have the chance of 840 01:30:44.810 --> 01:30:57.809 David Stephens, Prof: being allocated treatment with probability P. Therefore, the number that would end up being treated in the entire sample is a random variable with probability N… with a… which is going to be denoted N1, which is binomial NP, standard. 841 01:30:58.460 --> 01:30:59.800 David Stephens, Prof: Probability logic. 842 01:31:01.110 --> 01:31:16.210 David Stephens, Prof: Now, that number treated, or indeed the number not treated, can be written basically as this sum. Again, the indicator there is just a counting mechanism, so it counts the number of individuals that were treated, and it counts the number of individuals that were not treated. 843 01:31:21.480 --> 01:31:26.780 David Stephens, Prof: And, if we're in that situation, then not much changes. 844 01:31:27.230 --> 01:31:32.850 David Stephens, Prof: The only thing… This, sort of complicated… more complicated. 845 01:31:33.120 --> 01:31:34.670 David Stephens, Prof: notation. 846 01:31:34.910 --> 01:31:51.730 David Stephens, Prof: really says is that, well, if we're going to do it that way, we have to remember to, when we take the sample means, divide by the correct number of individuals. So, if, there are N0 individuals that were not treated, then we have to remember to divide by N0. 847 01:31:52.750 --> 01:31:59.080 David Stephens, Prof: If there are N1 individuals that were treated, we have to remember to divide by N1. That's all this basically says. 848 01:32:01.130 --> 01:32:04.570 David Stephens, Prof: And this formula essentially just says, okay, well. 849 01:32:04.740 --> 01:32:12.870 David Stephens, Prof: Take the mean of the treatment group, take the mean of the no treatment group, And then… 850 01:32:13.810 --> 01:32:16.390 David Stephens, Prof: We can get at the ATE by, again, looking at the difference. 851 01:32:16.920 --> 01:32:23.910 David Stephens, Prof: So again, this is not a problem. It's really just correct accounting, correct sample size, Counting and so on. 852 01:32:24.460 --> 01:32:26.419 David Stephens, Prof: So that's all pretty straightforward. 853 01:32:32.900 --> 01:32:44.179 David Stephens, Prof: The other thing that we can do, though, is to say, well, instead of regarding these as random variables, we can just put in the expected sample sizes. That's something that's done sometimes. 854 01:32:44.570 --> 01:32:54.539 David Stephens, Prof: And to use an alternative estimator of the expected response under no treatment or treatment, which looks at the calculation in this way. 855 01:32:57.900 --> 01:32:58.650 David Stephens, Prof: Okay? 856 01:32:59.040 --> 01:33:06.799 David Stephens, Prof: So instead of dividing by N0, we divide by N1 minus P, which is the expected sample size. 857 01:33:08.500 --> 01:33:17.150 David Stephens, Prof: Here, it's the expected sample size for the treated group. So you might think that's a little bit of a weird thing to do, but it turns out that these estimators are still… 858 01:33:17.300 --> 01:33:22.350 David Stephens, Prof: Really perfectly good, and they're easier to study than the previous estimators. 859 01:33:24.880 --> 01:33:34.119 David Stephens, Prof: Okay? And you might think also, well, okay, so N1 is random, N0 is random, so maybe it's better just to plug in the expectation, because that takes out some randomness. 860 01:33:38.130 --> 01:33:46.209 David Stephens, Prof: Okay? It turns out that, actually, that's not the right thing to do. The right thing to do is this. Turns out that the sample mean is always the best estimator. 861 01:33:46.340 --> 01:33:56.759 David Stephens, Prof: Not some modified version of it. Now, why we might look at these things is because it turns out to be easier to study their statistical properties when we generalize to more complicated settings. 862 01:33:58.820 --> 01:34:00.990 David Stephens, Prof: Alright, so, 863 01:34:07.050 --> 01:34:15.030 David Stephens, Prof: What we'll study in the latter half of today's module is what happens when we're not in this situation. 864 01:34:15.260 --> 01:34:19.889 David Stephens, Prof: So we've used, hypothetically simple situations to set up the problem. 865 01:34:20.030 --> 01:34:28.669 David Stephens, Prof: and said, well, if we were lucky enough to have an experimental study, then our causal problems would be over. We could just easily 866 01:34:28.910 --> 01:34:34.750 David Stephens, Prof: Estimate the causal quantities just by taking sample means. That's what the logic of the last 15 minutes has been. 867 01:34:35.290 --> 01:34:37.409 David Stephens, Prof: But, of course, the reality is that 868 01:34:37.950 --> 01:34:41.279 David Stephens, Prof: Most… in most cases, we're in the observational world. 869 01:34:41.630 --> 01:34:51.520 David Stephens, Prof: Where… the benefits of randomization, the benefits of breaking the connection between X and Z are no longer present. 870 01:34:53.850 --> 01:34:57.339 David Stephens, Prof: So, in general, as pointed out in the question in the chat there. 871 01:34:57.590 --> 01:35:01.960 David Stephens, Prof: In general, we can't make the assumption that Z and X are independent. 872 01:35:03.060 --> 01:35:09.829 David Stephens, Prof: So what we'll see in the session after the… after the break is… Why that's a problem. 873 01:35:10.470 --> 01:35:15.269 David Stephens, Prof: And we'll start to see how we can resolve it using propensity score type ideas. 874 01:35:15.950 --> 01:35:23.979 David Stephens, Prof: So, I'll stop there for the… for the first half session. We'll pick up again on the half hour. So, for me here, it's at, 875 01:35:24.600 --> 01:35:31.140 David Stephens, Prof: 1.30, so on the west coast it's, I believe, 10.30. 876 01:35:33.240 --> 01:35:44.159 David Stephens, Prof: And then, we'll push through to the end of the day, and by the end of the sec- by the end of this, second sub-session, we'll have seen the construction of the Propensity score 877 01:35:44.640 --> 01:35:48.590 David Stephens, Prof: how… It's constructed, and why it works. 878 01:35:48.970 --> 01:35:51.479 David Stephens, Prof: Okay, so we'll start again at, 879 01:35:51.780 --> 01:35:54.330 David Stephens, Prof: On the half hour. So, 1.30 my time.