WEBVTT 1 00:00:05.240 --> 00:00:11.060 David Stephens, Prof: Okay, good morning. So we'll start with, second day of this module. 2 00:00:11.660 --> 00:00:14.649 David Stephens, Prof: And today we'll see the… 3 00:00:14.760 --> 00:00:24.390 David Stephens, Prof: Use of propensity Scores using statistical methods, various statistical methods for performing causal estimation. 4 00:00:25.450 --> 00:00:27.009 David Stephens, Prof: We saw yesterday. 5 00:00:27.820 --> 00:00:30.619 David Stephens, Prof: The need for the propensity score, the need to… 6 00:00:32.689 --> 00:00:37.560 David Stephens, Prof: block problematic paths in the underlying DAG for the data generation. 7 00:00:37.680 --> 00:00:38.560 David Stephens, Prof: mechanism. 8 00:00:39.640 --> 00:00:44.650 David Stephens, Prof: We saw that the Propensity score is one mechanism for… blocking. 9 00:00:45.000 --> 00:00:46.759 David Stephens, Prof: problematic paths. 10 00:00:47.620 --> 00:00:50.179 David Stephens, Prof: And in such a way that… 11 00:00:50.860 --> 00:00:54.080 David Stephens, Prof: Conditional on the Propensity score, we had this 12 00:00:54.820 --> 00:00:59.820 David Stephens, Prof: Balance concept, so that conditional on the Propensity score. 13 00:01:01.590 --> 00:01:03.789 David Stephens, Prof: The two groups that we want to compare. 14 00:01:03.900 --> 00:01:11.310 David Stephens, Prof: Let's say, treated and untreated individuals, are now directly comparable, irrespective of the fact that, well. 15 00:01:14.350 --> 00:01:18.909 David Stephens, Prof: Despite the fact that The data generating mechanism. 16 00:01:19.750 --> 00:01:23.380 David Stephens, Prof: It has a confounding structure to it, so… 17 00:01:23.900 --> 00:01:33.300 David Stephens, Prof: the initial labeling by treatment group leads to two subgroups that are not directly comparable. What we saw was that conditional on the propensity score. 18 00:01:33.660 --> 00:01:35.990 David Stephens, Prof: We could, then… 19 00:01:36.760 --> 00:01:43.330 David Stephens, Prof: Perform that comparison directly, because conditional on the Propensity score, the groups became comparable again, due to this balance mechanism. 20 00:01:44.290 --> 00:01:48.249 David Stephens, Prof: So, it looked a little bit yesterday out there. 21 00:01:49.670 --> 00:01:53.040 David Stephens, Prof: The, the concept of matching and stratification. 22 00:01:53.740 --> 00:01:57.410 David Stephens, Prof: And… I try to give a logic for that. 23 00:01:58.570 --> 00:02:02.890 David Stephens, Prof: Based on, this balancing logic. 24 00:02:03.440 --> 00:02:06.190 David Stephens, Prof: What we're gonna see today, though, is that there are other… 25 00:02:06.790 --> 00:02:12.369 David Stephens, Prof: Statistical methods that can be used that deploy the Propensity score in a particular way. 26 00:02:14.210 --> 00:02:27.619 David Stephens, Prof: That lean on the same balancing ideas, but are perhaps more, statistically understandable, and in fact, can have better statistical performance when performing causal estimation. 27 00:02:28.840 --> 00:02:33.329 David Stephens, Prof: So the… so… and then later on, we'll see the, the use of… 28 00:02:33.470 --> 00:02:38.870 David Stephens, Prof: This, these methods in practice in some, in some examples and some simulations. 29 00:02:39.580 --> 00:02:44.220 David Stephens, Prof: Okay, so the first method that I'm going to talk about is propensity score regression. 30 00:02:44.900 --> 00:02:55.070 David Stephens, Prof: And the idea here is that we're gonna use the propensity score as… Simply, an additional regression term. 31 00:02:55.950 --> 00:03:02.960 David Stephens, Prof: That is, we're going to include the propensity score in a regression model as a predictor in its own right. 32 00:03:03.940 --> 00:03:09.399 David Stephens, Prof: And the logic there is that, well, performing conditioning on the propensity score 33 00:03:09.480 --> 00:03:28.240 David Stephens, Prof: If we have a jointly normal structure, the correct thing to do is to include that propensity score in as an additional term. That will perform exactly the right type of conditioning that blocks all of the necessary… sorry, the open undirected paths that are problematic in our causal analysis. 34 00:03:29.880 --> 00:03:35.529 David Stephens, Prof: So, Regression models are fairly straightforward, we know how to fit them in R. 35 00:03:35.670 --> 00:03:42.020 David Stephens, Prof: We just use the LM function. In all standard packages, fitting a regression model is pretty straightforward. 36 00:03:43.110 --> 00:03:47.229 David Stephens, Prof: What we'll see is that it can be effective, but there are some limitations on 37 00:03:47.690 --> 00:03:51.870 David Stephens, Prof: The analysis that we can carry out using regression methods. 38 00:03:56.840 --> 00:04:06.389 David Stephens, Prof: So the idea of including the propensity score in a regression is really the same logic that we used previously. 39 00:04:07.130 --> 00:04:09.530 David Stephens, Prof: when we had conditioning on X, 40 00:04:10.260 --> 00:04:14.810 David Stephens, Prof: As being the mechanism for blocking The undirected path. 41 00:04:15.090 --> 00:04:29.229 David Stephens, Prof: Here we see the DAG with the Propensity score included. It sits on the path between X and Z. It's a function of the X's, deterministic function of the X's, if you remember. In the binary case, it's the probability that X equals… 42 00:04:29.900 --> 00:04:32.460 David Stephens, Prof: sorry, Z equals 1 given… 43 00:04:32.740 --> 00:04:35.179 David Stephens, Prof: The covariate values, or predictive values. 44 00:04:35.430 --> 00:04:42.970 David Stephens, Prof: In the continuous case, it's some sufficient summary of the… Xz relationship. 45 00:04:43.850 --> 00:04:49.219 David Stephens, Prof: That allows you to characterize the distribution of Z in terms of this function of X. 46 00:04:49.550 --> 00:04:52.099 David Stephens, Prof: So it sits on that path between X and Z. 47 00:04:52.420 --> 00:05:03.589 David Stephens, Prof: the idea of regressing on the propensity score is simply that we include that EX term or BX term in the more general balancing case in a regression model. 48 00:05:03.840 --> 00:05:04.910 David Stephens, Prof: for Y. 49 00:05:05.140 --> 00:05:06.489 David Stephens, Prof: Given X and Z. 50 00:05:13.210 --> 00:05:19.460 David Stephens, Prof: So… the… there's the logic. The idea is that we… 51 00:05:19.720 --> 00:05:25.949 David Stephens, Prof: condition on X EX, and so add the E-axis included in there to break the confounding. 52 00:05:26.340 --> 00:05:30.190 David Stephens, Prof: We still have X as a predictor of Y. 53 00:05:30.820 --> 00:05:43.989 David Stephens, Prof: In the original model, if I go back to that previous graph, we still see that there is this path from X to Y. So we know that X is a predictor of Y in its own right. We include the propensity score, though, to break the confounding structure. 54 00:05:44.820 --> 00:05:49.290 David Stephens, Prof: So that means that we regress Y on X, and EX. 55 00:05:49.440 --> 00:05:50.530 David Stephens, Prof: and Z. 56 00:05:51.690 --> 00:05:57.060 David Stephens, Prof: Okay, we have a separate coefficient for the X values, and the EX values, and the Z values. 57 00:05:57.630 --> 00:06:03.180 David Stephens, Prof: So you can think of this as just basically a transformed Version of the predictors. 58 00:06:03.790 --> 00:06:10.169 David Stephens, Prof: Single, one-dimensional summary of the predictors that is there to block this path. 59 00:06:11.490 --> 00:06:12.679 David Stephens, Prof: Okay? By conditioning. 60 00:06:13.560 --> 00:06:17.169 David Stephens, Prof: So we build that model, and it's an ordinary regression model. 61 00:06:18.080 --> 00:06:23.350 David Stephens, Prof: So we have a coefficient for the X here, it's a coefficient for EX, and a coefficient for Z. 62 00:06:25.230 --> 00:06:30.130 David Stephens, Prof: And the idea is that, This is a more robust approach. 63 00:06:30.360 --> 00:06:34.799 David Stephens, Prof: because of this balancing effect that EX has, this, this, 64 00:06:35.100 --> 00:06:38.120 David Stephens, Prof: balancing effect, the fact that X and Z are 65 00:06:38.290 --> 00:06:45.809 David Stephens, Prof: conditionally independent given EX. This is a more robust kind of model to use than a simple regression model. 66 00:06:48.250 --> 00:06:48.940 David Stephens, Prof: Okay? 67 00:06:49.560 --> 00:06:56.919 David Stephens, Prof: So, the original regression models that we first started thinking about just had X and Z in them. 68 00:06:59.230 --> 00:07:01.840 David Stephens, Prof: That was prone to possible misspecification. 69 00:07:02.300 --> 00:07:04.640 David Stephens, Prof: So the idea is that we include this 70 00:07:04.840 --> 00:07:09.589 David Stephens, Prof: EX to break the dependence between Z and X. 71 00:07:10.080 --> 00:07:15.310 David Stephens, Prof: And therefore allow… correct estimation of the coefficient of Z, which is R. 72 00:07:15.890 --> 00:07:17.529 David Stephens, Prof: Target, in most cases. 73 00:07:21.590 --> 00:07:27.739 David Stephens, Prof: So, for example, we might have a model that looks like this. So in this model, I've only got dependence on the treatment. 74 00:07:28.090 --> 00:07:36.690 David Stephens, Prof: and the propensity score. There are coefficients, psi 0 and phi 0, attached to those things. This is the model we build and fit. 75 00:07:37.510 --> 00:07:38.670 David Stephens, Prof: And then… 76 00:07:39.260 --> 00:07:53.590 David Stephens, Prof: if Y is a continuous valued outcome response variable, then we just use ordinary least squares to estimate this model. We use LM to estimate B to 0, psi 0, and phi 0. 77 00:07:53.730 --> 00:07:55.309 David Stephens, Prof: So it's fairly simple. 78 00:07:55.540 --> 00:07:59.790 David Stephens, Prof: And regression is a powerful kind of technique. 79 00:08:02.100 --> 00:08:06.739 David Stephens, Prof: Okay? And the idea here is that this is a simpler model to fit. 80 00:08:07.070 --> 00:08:20.800 David Stephens, Prof: than the model that has arbitrary functions of X that appear on the right-hand side here. Remember that regression is a perfectly good causal method if you can guarantee correct specification of the regression function. 81 00:08:21.700 --> 00:08:31.969 David Stephens, Prof: But we've gone into this exercise of looking at propensity scores with the notion that that model is itself… the original regression model is itself very difficult to fit. 82 00:08:32.140 --> 00:08:35.980 David Stephens, Prof: Here we've got… A model for the Propensity score. 83 00:08:37.179 --> 00:08:40.890 David Stephens, Prof: presumed correctly specified, and a simple model for Z. 84 00:08:42.450 --> 00:08:47.730 David Stephens, Prof: So, in this model, the parameter psi is the 85 00:08:47.850 --> 00:08:52.570 David Stephens, Prof: average treatment effect parameter. That captures the impact of Z on the outcome. 86 00:08:54.070 --> 00:08:56.080 David Stephens, Prof: So, in the simplest possible case. 87 00:08:56.330 --> 00:09:02.900 David Stephens, Prof: we could fit this model, and under some conditions which I'll continue to talk about, it would give 88 00:09:03.390 --> 00:09:06.689 David Stephens, Prof: Consistent, unbiased estimation of the 89 00:09:07.270 --> 00:09:09.749 David Stephens, Prof: I've assumed effect parameter size 0. 90 00:09:12.220 --> 00:09:17.449 David Stephens, Prof: Now, we could extend this model by including further terms of x of our choice. 91 00:09:18.110 --> 00:09:18.920 David Stephens, Prof: Okay? 92 00:09:19.400 --> 00:09:25.800 David Stephens, Prof: At the moment, we've just got a simple intercept term, beta 0, but we could also include 93 00:09:26.020 --> 00:09:34.290 David Stephens, Prof: other functions of X, polynomial functions, linear functions of X, and so on. In order to improve the fit of this model. 94 00:09:34.580 --> 00:09:40.689 David Stephens, Prof: With the idea that we're improving the representation of this arrow. 95 00:09:50.970 --> 00:09:57.000 David Stephens, Prof: Let's think about the properties of a model such as this, or a slightly general version, which 96 00:09:57.510 --> 00:10:04.470 David Stephens, Prof: Builds a regression model In terms of XZ, and a balancing score, which could be the Propensity score. 97 00:10:06.090 --> 00:10:11.020 David Stephens, Prof: Well, if the true conditional mean model turns out to be the one that we've chosen. 98 00:10:12.820 --> 00:10:25.399 David Stephens, Prof: then we can still get consistent estimation of the average potential outcome at NEZ by the same mechanism as before. What we really do is just fit this model. 99 00:10:25.520 --> 00:10:35.189 David Stephens, Prof: And then plug in the fitted values of the from the model, Into the sample average that 100 00:10:35.370 --> 00:10:37.580 David Stephens, Prof: reflect this quantity. That is… 101 00:10:38.570 --> 00:10:48.749 David Stephens, Prof: We fit the balancing model, here I've got it represented in terms of Z and X, it could be just a function of X as well, could be just the propensity score for the binary treatment case. 102 00:10:50.150 --> 00:10:52.869 David Stephens, Prof: We build this model by regressing Z on X. 103 00:10:53.400 --> 00:10:57.550 David Stephens, Prof: Then we fit this model, incorporating those fitted values. 104 00:10:58.380 --> 00:11:11.830 David Stephens, Prof: And then, for each Z of interest, we estimate the average potential outcome by taking this sample average. So, again, it's predictions from the fitted model that lead us to this estimate of the average potential outcome. 105 00:11:14.230 --> 00:11:16.019 David Stephens, Prof: So it's a fairly simple mechanism. 106 00:11:18.690 --> 00:11:29.679 David Stephens, Prof: Typically, we also have to fit the propensity score, or balancing model, and in the binary exposure case, we would do that through, let's say, a logistic regression. 107 00:11:30.080 --> 00:11:37.260 David Stephens, Prof: We could do a logistic regression of Z on X, the predictors, 108 00:11:37.370 --> 00:11:43.650 David Stephens, Prof: That would require us to fit a parametric model, typically. We estimate the parameters, alpha, in that parametric model. 109 00:11:45.280 --> 00:11:47.220 David Stephens, Prof: We get the fitted values. 110 00:11:48.480 --> 00:11:52.930 David Stephens, Prof: And we use those in the simple form of the propensity score regression. 111 00:11:55.260 --> 00:11:55.940 David Stephens, Prof: Okay. 112 00:11:57.790 --> 00:12:10.140 David Stephens, Prof: We can then use this model with these fitted values, these are sort of derived predictors, if you like, fit those in the model here, estimate B to 0, psi 0, and phi 0, using least squares. 113 00:12:10.330 --> 00:12:16.259 David Stephens, Prof: Psi 0 is the estimate of the average treatment effect in this model. 114 00:12:21.520 --> 00:12:25.429 David Stephens, Prof: So when we do this calculation of the resulting estimator. 115 00:12:25.850 --> 00:12:28.460 David Stephens, Prof: Well, in this case, the estimate, because I've got the, 116 00:12:28.780 --> 00:12:36.470 David Stephens, Prof: little values in. We have to remember that the values that go in here are the… 117 00:12:37.720 --> 00:12:42.760 David Stephens, Prof: Propensity score values fitted under the model, and not the, 118 00:12:43.620 --> 00:12:49.059 David Stephens, Prof: And we've worked this out, the targeted Z value, not the observed Z values. 119 00:12:49.520 --> 00:12:50.240 David Stephens, Prof: Right. 120 00:12:50.430 --> 00:12:57.310 David Stephens, Prof: Because… The idea is that we obtain the fit of this model by using the observed data. 121 00:13:00.810 --> 00:13:04.129 David Stephens, Prof: But then evaluate the mean function at the target level. 122 00:13:04.950 --> 00:13:14.249 David Stephens, Prof: So it's like extrapolating from the model for a different value, Z, compared to the observed Zs that we obtained… that we used to obtain the fit. 123 00:13:16.080 --> 00:13:28.969 David Stephens, Prof: So, Propensity score regression is a really pretty simple mechanism. We basically have to fit two regression functions. We fit the logistic regression for the treatment model, that captures the relationship between Z and X, 124 00:13:30.910 --> 00:13:33.000 David Stephens, Prof: We get the fitted values from that model. 125 00:13:34.140 --> 00:13:41.880 David Stephens, Prof: Conditional on E, Z and X are… Independent by construction. 126 00:13:42.350 --> 00:13:44.970 David Stephens, Prof: So what that means is that we can have 127 00:13:45.260 --> 00:13:51.349 David Stephens, Prof: correct estimation of the Psi Zero parameter in this model by dint of that independence. 128 00:13:52.590 --> 00:13:56.120 David Stephens, Prof: Okay? Now, this may not be the best model to fit. 129 00:13:56.500 --> 00:14:03.320 David Stephens, Prof: But under the stated conditions of correct specification of the Propensity model. 130 00:14:04.490 --> 00:14:07.599 David Stephens, Prof: We will get consistent estimation of psi 0. 131 00:14:09.910 --> 00:14:14.379 David Stephens, Prof: There's another assumption In this model, which is, 132 00:14:15.560 --> 00:14:26.759 David Stephens, Prof: Maybe a restrictive assumption, which is that we have to assume that the contribution of the treatment to the outcome is captured by this linear term. 133 00:14:27.620 --> 00:14:33.830 David Stephens, Prof: That is, it's a linear additive effect of of the treatment. 134 00:14:36.390 --> 00:14:45.259 David Stephens, Prof: We'll be able… we'll see later on that we'll be able to relax that using other types of method, but it's a… it's a constraint of this propensity score regression type approach. 135 00:14:47.660 --> 00:14:53.119 David Stephens, Prof: So, if we believe this captures the effect of treatment, then this model 136 00:14:53.840 --> 00:15:00.770 David Stephens, Prof: We'll correctly estimate Psi, we'll correctly estimate the… The treatment effect. 137 00:15:05.130 --> 00:15:08.770 David Stephens, Prof: So, to summarize that, we've fitted these two regression models. 138 00:15:12.290 --> 00:15:14.989 David Stephens, Prof: We've got to fit the E or B model. 139 00:15:15.340 --> 00:15:17.520 David Stephens, Prof: And then we've got to fit the Mu model. 140 00:15:18.100 --> 00:15:21.329 David Stephens, Prof: The Propensity model and the outcome model. 141 00:15:23.020 --> 00:15:33.290 David Stephens, Prof: We need to have correct… the Propensity score model correctly specified, propensity or balancing model correctly specified, and we need to have the outcome mean model to be correctly specified. 142 00:15:38.170 --> 00:15:40.820 David Stephens, Prof: And just to illustrate that point a little further. 143 00:15:41.400 --> 00:15:46.029 David Stephens, Prof: Suppose that we have the true data-generating model looked like this. 144 00:15:47.150 --> 00:15:53.360 David Stephens, Prof: So, there's some function of the axis, Which is the treatment-free part. 145 00:15:54.200 --> 00:15:58.930 David Stephens, Prof: And then… Some function of the axis which captures the treatment effect. 146 00:15:59.800 --> 00:16:07.940 David Stephens, Prof: So mu1X here is a… Potentially complicated function of the… of the X variables. 147 00:16:08.850 --> 00:16:15.489 David Stephens, Prof: This captures the contribution of the treatment overall, the… sometimes termed the blip due to treatment. 148 00:16:16.460 --> 00:16:22.000 David Stephens, Prof: Mu zero is the baseline model, which corresponds to the treatment-free. 149 00:16:22.610 --> 00:16:23.630 David Stephens, Prof: response. 150 00:16:23.880 --> 00:16:26.159 David Stephens, Prof: Right, here we're dealing with a binary case. 151 00:16:26.270 --> 00:16:32.379 David Stephens, Prof: So all mean models of this nature can be written in this way. There's a contribution which depends on 152 00:16:33.270 --> 00:16:35.990 David Stephens, Prof: Sorry, contribution which depends on Z. 153 00:16:37.000 --> 00:16:41.219 David Stephens, Prof: And a contribution that doesn't depend on Z. We can always factorise it in this way. 154 00:16:42.280 --> 00:16:43.920 David Stephens, Prof: Or break it down in this way. 155 00:16:45.370 --> 00:16:49.530 David Stephens, Prof: So let's suppose that this is the model that generates the data. 156 00:16:51.230 --> 00:16:54.509 David Stephens, Prof: We don't know this. Instead, we fit this model. 157 00:16:56.120 --> 00:17:02.660 David Stephens, Prof: So the model is… a different choice for muzero. 158 00:17:04.730 --> 00:17:12.440 David Stephens, Prof: But we have a correct specification, matching the data generating mechanism, For the treatment effect. 159 00:17:14.750 --> 00:17:15.410 David Stephens, Prof: Okay? 160 00:17:15.550 --> 00:17:21.780 David Stephens, Prof: So the Propensity score regression model we have to fit looks like this. It's this arbitrary choice of the treatment. Free part… 161 00:17:22.270 --> 00:17:28.129 David Stephens, Prof: And it's Z times the correct specification of the treatment effect component. 162 00:17:29.200 --> 00:17:33.280 David Stephens, Prof: plus A, Times the same term. 163 00:17:35.500 --> 00:17:38.109 David Stephens, Prof: So, for this model to give the right answer. 164 00:17:38.500 --> 00:17:44.910 David Stephens, Prof: We need correct specification of the propensity model, And correct specification of mu1. 165 00:17:46.090 --> 00:17:51.909 David Stephens, Prof: Okay? In the sim… in the earlier example, mu1 of X was just the constant value 1. 166 00:17:52.160 --> 00:17:55.289 David Stephens, Prof: Okay, but in general, you could have treatment effects that are… 167 00:17:55.580 --> 00:18:00.309 David Stephens, Prof: dependent on X, X squared as well, so you'd have to include that in the Mu1 model. 168 00:18:01.830 --> 00:18:09.109 David Stephens, Prof: So if this is the true data-generating model, but instead we fit this model, where this can be misspecified. 169 00:18:10.970 --> 00:18:14.919 David Stephens, Prof: Then, we can still get correct estimation of the average treatment effect. 170 00:18:17.250 --> 00:18:25.260 David Stephens, Prof: Basically, what happens is that This treatment-free part, Can be misestimated. 171 00:18:25.670 --> 00:18:29.389 David Stephens, Prof: But when we come to take the difference, that term cancels out. 172 00:18:30.720 --> 00:18:35.290 David Stephens, Prof: So, provided that the mu1 part is correctly specified. 173 00:18:35.520 --> 00:18:38.229 David Stephens, Prof: We correctly capture the treatment-free effect. 174 00:18:38.380 --> 00:18:40.699 David Stephens, Prof: Sorry, sorry, the treatment effect. 175 00:18:41.390 --> 00:18:46.920 David Stephens, Prof: So, irrespective of the specification of the treatment-free model, we still get the right answer. 176 00:18:48.640 --> 00:18:58.380 David Stephens, Prof: So that is a little bit better than where we started off, using ordinary simple regression. 177 00:18:59.210 --> 00:19:06.879 David Stephens, Prof: Because there we needed complete correct specification. We couldn't get away with misspecification of this part. 178 00:19:10.880 --> 00:19:11.650 David Stephens, Prof: Okay? 179 00:19:14.650 --> 00:19:21.289 David Stephens, Prof: So how does this get modified if we have a slightly more complicated treatment effect? 180 00:19:21.540 --> 00:19:22.640 David Stephens, Prof: contribution. 181 00:19:24.350 --> 00:19:34.310 David Stephens, Prof: So let's suppose the treatment had an effect on the outcome captured through this first model here. 182 00:19:36.520 --> 00:19:46.159 David Stephens, Prof: So the simple model says that we have an intercept, and then a term which is Z multiplied by this linear term in X. 183 00:19:46.530 --> 00:19:52.340 David Stephens, Prof: It's psi 0 plus psi 1 times X. So this says that there, in a sense, there is an interaction 184 00:19:52.630 --> 00:19:59.690 David Stephens, Prof: between the treatment and the first predictor X1, say, In predicting their response. 185 00:20:00.020 --> 00:20:03.389 David Stephens, Prof: Okay, so if the data generating model were this. 186 00:20:04.990 --> 00:20:10.530 David Stephens, Prof: then what we would have to do, in fact, would be to fit this propensity score regression model. 187 00:20:11.690 --> 00:20:14.549 David Stephens, Prof: We have the correct specification of mu1, 188 00:20:14.760 --> 00:20:19.639 David Stephens, Prof: And then a term involving the propensity score, which, in a sense, matches it. 189 00:20:21.990 --> 00:20:25.569 David Stephens, Prof: Every term in Z has a corresponding term in E. 190 00:20:26.860 --> 00:20:29.980 David Stephens, Prof: And remember that in the binary case. 191 00:20:30.340 --> 00:20:33.079 David Stephens, Prof: this E of X, this propensity score function. 192 00:20:33.410 --> 00:20:39.740 David Stephens, Prof: Evaluated here is, indeed, the conditional expectation of Z given X. 193 00:20:40.510 --> 00:20:45.879 David Stephens, Prof: In the binary case, it's also the probability that Z equals 1, but that's also the expectation. 194 00:20:46.020 --> 00:20:56.529 David Stephens, Prof: So what we have here is an arbitrary intercept term, a… function which… Captures the treatment effect. 195 00:20:56.680 --> 00:21:00.470 David Stephens, Prof: And then a matching term, Which… 196 00:21:03.670 --> 00:21:07.049 David Stephens, Prof: Replaces the treatment by the propensity score. 197 00:21:07.300 --> 00:21:08.150 David Stephens, Prof: Value. 198 00:21:09.930 --> 00:21:10.730 David Stephens, Prof: No. 199 00:21:11.130 --> 00:21:24.649 David Stephens, Prof: this works. The logic for why it works is given in another appendix at the end, so I've included some of the details of that in the appendix. 200 00:21:25.090 --> 00:21:35.630 David Stephens, Prof: Essentially, Including a term like this ensures that all the problematic, biasing, confounding paths are blocked. 201 00:21:36.550 --> 00:21:46.739 David Stephens, Prof: So it turns out that this is exactly the right kind of conditioning to do in order to allow us to unbiasedly estimate Psi 0 and psi 1. 202 00:21:51.820 --> 00:21:55.379 David Stephens, Prof: So that's an important thing to remember when using propensity score regression. 203 00:21:56.510 --> 00:22:02.730 David Stephens, Prof: If you have a more complicated treatment effect model, Then you need to correctly 204 00:22:02.980 --> 00:22:05.629 David Stephens, Prof: Include terms involving the propensity score. 205 00:22:09.760 --> 00:22:14.249 David Stephens, Prof: And this, Propensity score regression is really, 206 00:22:15.670 --> 00:22:23.750 David Stephens, Prof: a slightly different version of a method that's well known in the causal literature called G-estimation. 207 00:22:25.300 --> 00:22:32.649 David Stephens, Prof: And… I've included a few bits of logic about gestimation here. Essentially. 208 00:22:33.470 --> 00:22:36.070 David Stephens, Prof: If we have the same setup as before. 209 00:22:36.580 --> 00:22:43.110 David Stephens, Prof: The true daisy-generating mean model being of this nature, so now a linear model. 210 00:22:45.010 --> 00:22:59.120 David Stephens, Prof: Involving some true betas and the covariates, plus Z times X2 times psi, so these are all vectors now, so there's a vector of coefficients, so this is just a compact way to write this. 211 00:23:00.440 --> 00:23:08.300 David Stephens, Prof: Right? So we fit this… so this is the data-generating model, but instead we fit this model, Okay? 212 00:23:08.540 --> 00:23:13.479 David Stephens, Prof: So… We have a misspecified treatment-free part. 213 00:23:14.220 --> 00:23:22.169 David Stephens, Prof: a… Term that matches the treatment effect part, and then a matching term in the propensity score. 214 00:23:24.230 --> 00:23:28.510 David Stephens, Prof: Okay? So this part is misspecified. 215 00:23:29.310 --> 00:23:31.320 David Stephens, Prof: In this gestation approach. 216 00:23:32.090 --> 00:23:38.120 David Stephens, Prof: What essentially we have is that if we fit this model using ordinary least squares. 217 00:23:38.670 --> 00:23:47.819 David Stephens, Prof: We have these estimating equations, so this is one way to think about how we perform ordinary least squares estimation, right? It's through the estimating 218 00:23:47.920 --> 00:23:49.279 David Stephens, Prof: equation approach. 219 00:23:49.650 --> 00:23:57.379 David Stephens, Prof: Basically we end up solving this score-like equation for beta, psi, and phi. 220 00:23:58.010 --> 00:23:58.720 David Stephens, Prof: Okay? 221 00:23:59.100 --> 00:24:04.419 David Stephens, Prof: Our interest will still be in the size. Here, there again, a vector of parameters. 222 00:24:05.020 --> 00:24:15.230 David Stephens, Prof: So we solve this system of equations. We have a system for beta, a system for psi, and a system for phi, encapsulated in this 223 00:24:16.150 --> 00:24:17.960 David Stephens, Prof: Column vector of terms here. 224 00:24:19.130 --> 00:24:23.839 David Stephens, Prof: So this is the usual setup for statistical estimation that we use in regression. 225 00:24:24.200 --> 00:24:27.959 David Stephens, Prof: Okay? And if I were to fit this model using OLS, 226 00:24:28.910 --> 00:24:40.230 David Stephens, Prof: This is the set of equations I would have. Now, it turns out that in this linear case, this linear model, linear regression case, you can solve this analytically, which is why we have nice, 227 00:24:41.600 --> 00:24:47.950 David Stephens, Prof: easy-to-apply functions in our matrix, and back to functions in our… Alright. 228 00:24:48.390 --> 00:24:51.000 David Stephens, Prof: So, OLS of the, in this model. 229 00:24:51.590 --> 00:24:55.640 David Stephens, Prof: OLS estimation of these parameters involves solving these equations. 230 00:24:58.520 --> 00:25:03.500 David Stephens, Prof: Propensity score regression works… 231 00:25:04.970 --> 00:25:13.350 David Stephens, Prof: By studying properties of this, or… we can see that it works by studying properties of this estimating function system. 232 00:25:16.380 --> 00:25:25.539 David Stephens, Prof: So, what we'll see is that the… if the mean model is misspecified, but the Propensity score model is correctly specified, and this residual-like term 233 00:25:26.110 --> 00:25:27.990 David Stephens, Prof: Doesn't depend on Z. 234 00:25:29.450 --> 00:25:35.030 David Stephens, Prof: So that means that the effect of Z is correctly captured through this term, that's the same assumption as before. 235 00:25:35.370 --> 00:25:47.849 David Stephens, Prof: then… The solutions to this equation… Particularly for Persai, Are the consistent estimating solutions. 236 00:25:50.090 --> 00:25:51.550 David Stephens, Prof: Okay? Now… 237 00:25:54.770 --> 00:25:58.630 David Stephens, Prof: An interesting way to review that system. 238 00:25:59.060 --> 00:26:04.849 David Stephens, Prof: Here, so this is just ordinary least squares, which gives us the right answer under these assumptions. 239 00:26:12.250 --> 00:26:17.770 David Stephens, Prof: An interesting way to review those estimating equations, though, is to simply subtract 240 00:26:18.350 --> 00:26:21.840 David Stephens, Prof: Replace the third equation by a difference 241 00:26:22.250 --> 00:26:32.709 David Stephens, Prof: Between the original second and third equations. So basically what I'm doing is coming back to here and subtracting the second… sorry, subtracting the third equations from the second. 242 00:26:34.480 --> 00:26:37.320 David Stephens, Prof: If I do that, I end up with this system. 243 00:26:37.950 --> 00:26:44.879 David Stephens, Prof: Now, as you all know, In this, linear system of equations. 244 00:26:45.390 --> 00:26:48.939 David Stephens, Prof: I can do that because of the zero on the right-hand side. 245 00:26:49.350 --> 00:26:53.019 David Stephens, Prof: Subtracting one set of equations from another doesn't change the solution. 246 00:26:55.540 --> 00:26:58.550 David Stephens, Prof: Okay? So that's a little algebraic trick. 247 00:26:59.510 --> 00:27:10.359 David Stephens, Prof: Which shows us that… As we'll see in a second, if the main model That is, this specification. 248 00:27:11.460 --> 00:27:15.550 David Stephens, Prof: Or the Propensity score model is correctly specified. 249 00:27:18.230 --> 00:27:21.109 David Stephens, Prof: Then we get correct estimation of the size. 250 00:27:21.980 --> 00:27:25.929 David Stephens, Prof: Provided that this term correctly captures the treatment effect. 251 00:27:29.320 --> 00:27:30.030 David Stephens, Prof: Okay? 252 00:27:33.270 --> 00:27:38.749 David Stephens, Prof: So we're in a slightly better situation than the original regression setting that I spoke about yesterday. 253 00:27:39.220 --> 00:27:42.019 David Stephens, Prof: Because we're allowed a little bit of misspecification. 254 00:27:42.930 --> 00:27:45.610 David Stephens, Prof: We can misspecify this term here. 255 00:27:47.750 --> 00:27:50.930 David Stephens, Prof: Provided we correctly specify the propensity score. 256 00:27:52.100 --> 00:27:52.780 David Stephens, Prof: model. 257 00:27:53.180 --> 00:27:53.870 David Stephens, Prof: Right. 258 00:27:54.130 --> 00:27:55.440 David Stephens, Prof: So, if… 259 00:27:56.720 --> 00:28:08.910 David Stephens, Prof: the treatment effect model is correctly specified, then we can get away with misspecifying the model, provided one of the component models is correctly specified. Either the mean model overall, that is. 260 00:28:09.350 --> 00:28:12.829 David Stephens, Prof: The model captured here. That is, basically. 261 00:28:13.840 --> 00:28:18.469 David Stephens, Prof: This model is correctly specified, so i.e. this matches this. 262 00:28:20.230 --> 00:28:24.260 David Stephens, Prof: Or… The Propensity score model is correctly specified. 263 00:28:25.110 --> 00:28:27.400 David Stephens, Prof: So this is known as double robustness. 264 00:28:28.370 --> 00:28:33.080 David Stephens, Prof: It's a fairly elementary form of double robustness, has to be said, because… 265 00:28:33.370 --> 00:28:41.779 David Stephens, Prof: The first part of the double robustness assumption, that it, or the first, option, this mean model correct specification. 266 00:28:42.590 --> 00:28:46.380 David Stephens, Prof: is… Perhaps unrealistic. 267 00:28:46.830 --> 00:28:52.839 David Stephens, Prof: What we have to believe for that double robust assumption is that this model is correctly specified. 268 00:28:57.790 --> 00:29:01.189 David Stephens, Prof: And that means that the mean model depends on the propensity score. 269 00:29:03.120 --> 00:29:10.760 David Stephens, Prof: So, it's a bit unrealistic, perhaps, to assume that that is correctly specified. So really, this, although it is a doubly robust procedure. 270 00:29:13.730 --> 00:29:17.789 David Stephens, Prof: The first element of double robustness, mean model correct specification. 271 00:29:18.010 --> 00:29:33.850 David Stephens, Prof: is perhaps unrealistic. So we're really relying on this in the… for the second correct specification. So, rather than having the mean model correctly specified, we must correctly specify the propensity score model, the probability that Z equals 1 given XA, 272 00:29:34.820 --> 00:29:41.790 David Stephens, Prof: And then, provided the treatment effect model is correctly specified, we end up getting the right answer for the average treatment effect. 273 00:29:47.880 --> 00:29:49.739 David Stephens, Prof: Now, if, 274 00:29:57.160 --> 00:30:02.080 David Stephens, Prof: If we think about a simplified version of the model, Which basically… 275 00:30:02.900 --> 00:30:07.560 David Stephens, Prof: Incorporates the Propensity score term into this first term. 276 00:30:07.940 --> 00:30:11.909 David Stephens, Prof: Right? Because if we think about it, This function here. 277 00:30:12.160 --> 00:30:15.230 David Stephens, Prof: could just be viewed as some function of X. 278 00:30:17.080 --> 00:30:23.759 David Stephens, Prof: with a set of coefficients. So, what we could think about doing is combining this term with this term. 279 00:30:25.270 --> 00:30:26.749 David Stephens, Prof: And that yields this. 280 00:30:29.920 --> 00:30:36.259 David Stephens, Prof: So, again, we have a double robust situation, provided that now that this model is correctly specified. 281 00:30:39.160 --> 00:30:42.100 David Stephens, Prof: Or the Propensity score model is correctly specified. 282 00:30:42.510 --> 00:30:46.750 David Stephens, Prof: Assuming that the treatment effect mechanism is correctly specified. 283 00:30:49.220 --> 00:30:49.910 David Stephens, Prof: Okay. 284 00:30:51.270 --> 00:30:57.260 David Stephens, Prof: Now, we… so we still have double robustness, but again, the… the first… Double robust. 285 00:30:57.370 --> 00:31:01.079 David Stephens, Prof: Assumption that this model is correctly specified is a bit unrealistic. 286 00:31:01.470 --> 00:31:06.780 David Stephens, Prof: So what I've done here is rely on the inclusion of this term. 287 00:31:07.110 --> 00:31:11.870 David Stephens, Prof: In order for me to get consistent estimation of the size. 288 00:31:11.970 --> 00:31:14.579 David Stephens, Prof: We've modified the estimating equation. 289 00:31:15.240 --> 00:31:17.750 David Stephens, Prof: In a similar way to the way we had it here. 290 00:31:19.400 --> 00:31:22.799 David Stephens, Prof: But we've thought about combining those two terms. 291 00:31:23.940 --> 00:31:28.070 David Stephens, Prof: So it's a slightly different model, The key thing 292 00:31:28.220 --> 00:31:33.280 David Stephens, Prof: that gives me the consistent estimation is the inclusion of this term here. Now. 293 00:31:33.650 --> 00:31:36.269 David Stephens, Prof: As you can see, this is not… 294 00:31:36.760 --> 00:31:45.620 David Stephens, Prof: a system that arises from ordinary least squares, right? If you had an ordinary least squares, that would be basically this term 295 00:31:46.100 --> 00:31:56.689 David Stephens, Prof: squared, and summed, to give the objective function. So when I take the partial derivatives, I would have a term involving beta, and a term involving psi. I would not get this term. 296 00:31:57.870 --> 00:32:00.480 David Stephens, Prof: So this is a special form of estimation. 297 00:32:00.770 --> 00:32:03.229 David Stephens, Prof: Not quite ordinary least squares. 298 00:32:03.550 --> 00:32:06.150 David Stephens, Prof: And it's become known as g-estimation. 299 00:32:07.530 --> 00:32:10.560 David Stephens, Prof: So it's a modified form of estimating equation. 300 00:32:14.010 --> 00:32:18.080 David Stephens, Prof: And, if I make this even simpler and say, well, this, 301 00:32:18.370 --> 00:32:23.710 David Stephens, Prof: let's have no term here at all, I just have a… have this term missing altogether. 302 00:32:23.890 --> 00:32:29.629 David Stephens, Prof: then the system Reduces to this, single equation system. 303 00:32:33.600 --> 00:32:45.960 David Stephens, Prof: So this is very explicitly singly robust, in that it will give me the correct answer for Psi, provided that the propensity score model is correctly specified. 304 00:32:51.110 --> 00:32:51.900 David Stephens, Prof: No. 305 00:32:52.240 --> 00:32:58.979 David Stephens, Prof: In the simplest form of this model, so this is the basic G-estimating equation, where we allow treatment 306 00:32:59.610 --> 00:33:01.110 David Stephens, Prof: predictor interaction. 307 00:33:01.350 --> 00:33:02.730 David Stephens, Prof: In the treatment effect. 308 00:33:03.180 --> 00:33:14.380 David Stephens, Prof: But if I make an even simpler version of that, to say, well, okay, let's go back to that original, simpler specification, and say that the effect of treatment is captured entirely by a simple blip 309 00:33:14.630 --> 00:33:16.940 David Stephens, Prof: Due to the treatment. That is. 310 00:33:17.740 --> 00:33:25.960 David Stephens, Prof: If Z is 0 in the binary case, we get 0 contribution here, and if Z is 1, we get psi 0. 311 00:33:26.750 --> 00:33:31.640 David Stephens, Prof: So, if that model correctly captures The treatment effect. 312 00:33:32.750 --> 00:33:35.240 David Stephens, Prof: I… it's basically a change in intercept. 313 00:33:35.600 --> 00:33:41.809 David Stephens, Prof: Then this estimating equation will consistently estimate for me psi 0. 314 00:33:41.950 --> 00:33:46.849 David Stephens, Prof: Provided… The Propensity score model is correctly specified. 315 00:33:48.430 --> 00:33:54.040 David Stephens, Prof: So this is the simplest form of G-estimation, in a sense, the simplest form of propensity score regression. 316 00:33:56.580 --> 00:34:07.160 David Stephens, Prof: The good thing about that equation here is that… so this is the estimating equation, we're solving this for psi 0, it's the only parameter now that's unknown. 317 00:34:08.449 --> 00:34:11.180 David Stephens, Prof: We can solve that explicitly, and it gives this form. 318 00:34:13.320 --> 00:34:17.880 David Stephens, Prof: Okay, just by rearranging the terms, we end up with this form of G estimator. 319 00:34:18.909 --> 00:34:20.559 David Stephens, Prof: In this case, the G-estimate. 320 00:34:21.750 --> 00:34:23.850 David Stephens, Prof: Here's the corresponding estimator here. 321 00:34:24.429 --> 00:34:29.839 David Stephens, Prof: So, this is really pretty simple to estimate. All we have to do is to fit the 322 00:34:30.219 --> 00:34:39.250 David Stephens, Prof: let's say logistic regression for the propensity score term by regressing Z on X, And then include that term. 323 00:34:39.989 --> 00:34:42.090 David Stephens, Prof: In this, difference here. 324 00:34:42.429 --> 00:34:46.370 David Stephens, Prof: And… Basically, look at the ratio of this to this. 325 00:34:47.610 --> 00:34:48.929 David Stephens, Prof: So it's fairly simple. 326 00:34:49.699 --> 00:34:53.669 David Stephens, Prof: This estimator will be consistent. 327 00:34:54.130 --> 00:35:00.280 David Stephens, Prof: For the right answer, for the right average treatment effect, provided the propensity score model is correctly specified. 328 00:35:01.390 --> 00:35:06.610 David Stephens, Prof: Under the assumption that the treatment effect is captured by the simple Psi 0 is that term. 329 00:35:08.750 --> 00:35:09.430 David Stephens, Prof: Okay. 330 00:35:12.350 --> 00:35:21.000 David Stephens, Prof: We can turn that into a more complicated estimator by recognizing the appeal to double robustness. 331 00:35:21.320 --> 00:35:22.600 David Stephens, Prof: Essentially. 332 00:35:22.890 --> 00:35:31.429 David Stephens, Prof: what we can do is to modify the presence of Y here by making this a residual of Y on some fitted model. 333 00:35:33.180 --> 00:35:35.940 David Stephens, Prof: So the idea here is that we have a… 334 00:35:36.500 --> 00:35:43.980 David Stephens, Prof: an Xi beta term that partially explains the variation in Y. 335 00:35:45.500 --> 00:35:49.059 David Stephens, Prof: But the key thing is that this variation is independent of the treatment. 336 00:35:50.620 --> 00:35:55.719 David Stephens, Prof: So again, looking at a residual here based on some fit that doesn't involve the treatment. 337 00:35:59.690 --> 00:36:05.379 David Stephens, Prof: So then we've got two models to fit again. We've got this treatment-free component. 338 00:36:05.690 --> 00:36:13.740 David Stephens, Prof: To estimate the betas. And then we've got the Propensity score, if we don't know the Propensity score, we've got that model to fit as well. 339 00:36:13.900 --> 00:36:16.110 David Stephens, Prof: And then we can form this estimator. 340 00:36:22.020 --> 00:36:25.619 David Stephens, Prof: So, G-estimation, and Propensity score regression. 341 00:36:25.980 --> 00:36:35.970 David Stephens, Prof: Incorporate the propensity score into the analysis in exactly the right way, under some assumptions, to get correct estimation of the average treatment effect. 342 00:36:38.740 --> 00:36:44.330 David Stephens, Prof: Here, I focused on the linear model with binary treatment. We'll see some generalizations of that later on. 343 00:36:45.320 --> 00:36:49.949 David Stephens, Prof: But essentially, the ideas carry across to more complicated settings. 344 00:36:53.380 --> 00:36:59.959 David Stephens, Prof: Now… The next set of methods that I'm going to look at relax at least one of the assumptions. 345 00:37:00.850 --> 00:37:06.730 David Stephens, Prof: Which is, in particular, the assumption that we have to correctly specify the treatment effect model. 346 00:37:10.470 --> 00:37:15.680 David Stephens, Prof: So we had a… a setup that… meant that… 347 00:37:27.850 --> 00:37:32.500 David Stephens, Prof: The treatment effect was captured by that Psi 0, Z term. We had to assume that. 348 00:37:32.610 --> 00:37:36.860 David Stephens, Prof: Now, in many practical situations, that's not an unreasonable assumption. 349 00:37:37.640 --> 00:37:42.550 David Stephens, Prof: Right, we can get consistent estimation of Psi 0, the treatment effect parameter. 350 00:37:42.950 --> 00:37:46.870 David Stephens, Prof: Provided the actual dependence on treatment looks like this. 351 00:37:47.280 --> 00:37:51.509 David Stephens, Prof: The contribution of the treatment is this additive term in Z. 352 00:37:53.150 --> 00:37:56.030 David Stephens, Prof: Psi 0 is just the coefficient in that regression model. 353 00:38:04.630 --> 00:38:11.569 David Stephens, Prof: then subsequent… assumptions… Lead us to be able to consistently estimate that parameter. 354 00:38:11.960 --> 00:38:15.040 David Stephens, Prof: Under single or doubly robust. 355 00:38:16.120 --> 00:38:16.910 David Stephens, Prof: Settings. 356 00:38:21.320 --> 00:38:28.119 David Stephens, Prof: Let's say we didn't want to make that assumption, though. Let's say we didn't want to have to assume the form of the treatment effect. 357 00:38:29.800 --> 00:38:30.930 David Stephens, Prof: contribution. 358 00:38:31.510 --> 00:38:44.050 David Stephens, Prof: We don't want to assume it's psi 0 times Z, or psi 0 times Z plus psi 1 times Z times X1, or whatever. Let's say we didn't want to assume that form is correctly specified. 359 00:38:44.200 --> 00:38:52.770 David Stephens, Prof: And we want to have… A complete… to be completely, ambivalent to what that specification actually is. 360 00:38:54.340 --> 00:38:57.860 David Stephens, Prof: Then the g-estimation and Propensity score methods won't work for us. 361 00:38:58.470 --> 00:39:00.150 David Stephens, Prof: We have to find another method. 362 00:39:00.640 --> 00:39:07.129 David Stephens, Prof: And the other method that we can think about using is based on weighting or re-weighting. 363 00:39:08.650 --> 00:39:11.909 David Stephens, Prof: And the re-weighting, as we'll see, also depends on the propensity score. 364 00:39:14.160 --> 00:39:15.170 David Stephens, Prof: So… 365 00:39:15.690 --> 00:39:35.110 David Stephens, Prof: In order to justify this approach, I'm going to go back to this integral form of the definition of the average student effect that we saw yesterday. Basically, this is the average potential outcome, and it's rewritten in terms of this double integral. So again, this is a theoretical quantity for the moment, we'll see how to estimate it in a short while. 366 00:39:35.150 --> 00:39:40.749 David Stephens, Prof: This is the definition of the average potential outcome. This is our… basically the form of our S demand. 367 00:39:45.010 --> 00:39:52.670 David Stephens, Prof: As I mentioned briefly yesterday, we can rewrite that estimation, sorry, that estimate, in terms of a… 368 00:39:54.460 --> 00:39:57.839 David Stephens, Prof: Integral over all of the available variables. 369 00:39:58.490 --> 00:40:01.500 David Stephens, Prof: And the key thing about the form of the… 370 00:40:01.960 --> 00:40:10.510 David Stephens, Prof: Quantity being estimated is that it represents The hypothetical data-generating mechanism 371 00:40:11.630 --> 00:40:17.029 David Stephens, Prof: In the experimental world that we know doesn't match our observational world. 372 00:40:18.320 --> 00:40:27.390 David Stephens, Prof: So, the average Look, average potential outcome defined here is the calculation that we are aiming at. 373 00:40:30.210 --> 00:40:37.109 David Stephens, Prof: It's an expectation over a distribution, which is the… we can think of it as the experimental distribution. 374 00:40:37.540 --> 00:40:40.559 David Stephens, Prof: Whereas our data actually arrives from a different distribution. 375 00:40:42.070 --> 00:40:44.400 David Stephens, Prof: If we were to have data from this. 376 00:40:46.590 --> 00:40:51.019 David Stephens, Prof: from this world. If we were carrying out an experimental study. 377 00:40:51.330 --> 00:40:54.470 David Stephens, Prof: Then, it turns out we do have a sample from this. 378 00:40:54.640 --> 00:41:01.610 David Stephens, Prof: And so estimating using the standard sample average approach is gonna work for us. 379 00:41:03.350 --> 00:41:09.939 David Stephens, Prof: And again, I'll appeal back to the Monte Carlo idea, or the usual use of frequentist statistical thinking. 380 00:41:10.200 --> 00:41:20.419 David Stephens, Prof: Our estimator is gonna be an empirical approximation to this interval. That is, we draw a sample from this distribution. 381 00:41:21.400 --> 00:41:24.659 David Stephens, Prof: I intervened to fix the treatment level. 382 00:41:24.870 --> 00:41:29.969 David Stephens, Prof: And then… worry about the joint distribution of Y and X. 383 00:41:30.710 --> 00:41:31.490 David Stephens, Prof: Awesome. 384 00:41:31.600 --> 00:41:35.530 David Stephens, Prof: Only through our drawn distribution. 385 00:41:35.670 --> 00:41:41.659 David Stephens, Prof: So we draw a sample from that, which is basically, we carry out our experimental study. 386 00:41:41.950 --> 00:41:45.440 David Stephens, Prof: And then we take the sample average of the resulting 387 00:41:46.270 --> 00:41:57.840 David Stephens, Prof: outcomes for those individuals, right? We intervene to set Truman to level Z, and therefore, this estimator will be a consistent estimator of the right thing. 388 00:42:00.020 --> 00:42:05.840 David Stephens, Prof: Right? So if we were carrying out an experimental study, this method would work fine, as we talked about yesterday. 389 00:42:06.020 --> 00:42:09.309 David Stephens, Prof: But, of course, the problem is, we don't have an experimental study. 390 00:42:10.730 --> 00:42:12.499 David Stephens, Prof: We have an observational study. 391 00:42:15.410 --> 00:42:31.579 David Stephens, Prof: So, the two estimators that I wrote down yesterday as 1 and 2 basically look like this. So, again, this quantity here, it's a complicated version of the previous formula, essentially. All it says is, take the sample average in the treatment groups. 392 00:42:33.180 --> 00:42:40.080 David Stephens, Prof: The treatment groups are defined by the little red Zs, so little red Z equal to 0 means untreated. 393 00:42:40.320 --> 00:42:43.140 David Stephens, Prof: Little red z equal to 1 means treated. 394 00:42:44.770 --> 00:42:55.120 David Stephens, Prof: So when Little Red Z is 0, we have the sample average for the untreated individuals. When it's 1, we have the sample average for the treated individuals. 395 00:42:55.540 --> 00:43:00.620 David Stephens, Prof: That's the obvious estimator, but what I commented on yesterday is that an equivalent 396 00:43:02.150 --> 00:43:09.080 David Stephens, Prof: Conceptually estimator, uses a replacement of the actual 397 00:43:09.620 --> 00:43:13.040 David Stephens, Prof: Sample size by the expected sample size. 398 00:43:13.690 --> 00:43:15.520 David Stephens, Prof: In the study, right? 399 00:43:15.690 --> 00:43:22.469 David Stephens, Prof: So again, we're imagining randomization, randomization to treatment, treatment, Or non-treatment, according to some. 400 00:43:22.770 --> 00:43:26.260 David Stephens, Prof: Model, probability model, could be flipping a coin. 401 00:43:26.420 --> 00:43:28.909 David Stephens, Prof: with probability P of being treated, or whatever. 402 00:43:29.260 --> 00:43:32.870 David Stephens, Prof: This number here is the expected sample size. 403 00:43:32.990 --> 00:43:38.860 David Stephens, Prof: The expected number of individuals who received treatment, Little Red said. 404 00:43:40.640 --> 00:43:48.490 David Stephens, Prof: Okay, so these two estimators work fairly well for the experimental study, but again, we don't have an experimental study, so how do we change them? 405 00:43:50.200 --> 00:43:56.899 David Stephens, Prof: Well… The key is… Essentially a reweighting argument. 406 00:43:57.340 --> 00:44:05.150 David Stephens, Prof: So you can think of this as, in a sense, a standardization calculation in, sort of, standard epidemiological terms. 407 00:44:05.260 --> 00:44:11.409 David Stephens, Prof: Or, I think of it as an important sampling argument, or a change of measure argument. 408 00:44:11.860 --> 00:44:18.790 David Stephens, Prof: Which exploits the fact that we can write down the expectation of interest, In two different ways. 409 00:44:20.820 --> 00:44:25.410 David Stephens, Prof: Our target is this expectation, which I can write in this way. 410 00:44:26.020 --> 00:44:34.779 David Stephens, Prof: It's the expectation of the outcome, given treatment at a level Z, Under the experimental measure. 411 00:44:34.990 --> 00:44:38.740 David Stephens, Prof: Where Z and X are, are independent. 412 00:44:40.410 --> 00:44:51.209 David Stephens, Prof: So this just encapsulates all those ideas that we wrote down yesterday, saying that our target S demand is in the world where Z and X are independent. 413 00:44:52.470 --> 00:44:53.150 David Stephens, Prof: Right. 414 00:44:53.550 --> 00:44:59.500 David Stephens, Prof: We're intervening to set the treatment level, that's our target causal quantity. 415 00:45:02.190 --> 00:45:05.229 David Stephens, Prof: And the key thing is that we do that independently of X. 416 00:45:06.560 --> 00:45:12.039 David Stephens, Prof: Right? So this is the hypothetical experimental world in which we want to make inference. 417 00:45:13.460 --> 00:45:18.809 David Stephens, Prof: But we recognize that actually our data don't come from that world. So we'll see how to handle that in a second. 418 00:45:19.140 --> 00:45:26.370 David Stephens, Prof: Now, this expectation here, this conditional expectation here, which gives me the thing I'm after, can be rewritten in this way. 419 00:45:29.460 --> 00:45:42.159 David Stephens, Prof: And, really, you can think of this as the theoretical version of the empirical calculation that we have here. This one says… this one says, take the sample average by 420 00:45:43.200 --> 00:45:47.360 David Stephens, Prof: Summing up all the Y's that received the right level of treatment. 421 00:45:47.750 --> 00:45:50.759 David Stephens, Prof: In the numerator, and in the denominator. 422 00:45:51.050 --> 00:45:53.610 David Stephens, Prof: Counting the number of people that received that treatment. 423 00:45:53.920 --> 00:45:56.029 David Stephens, Prof: That's all that this notation means. 424 00:45:56.950 --> 00:46:00.689 David Stephens, Prof: So the corresponding theoretical quantity we're right in this way. 425 00:46:01.100 --> 00:46:08.870 David Stephens, Prof: So all we've done, essentially, is to replace the Data-driven calculation here. 426 00:46:09.390 --> 00:46:12.319 David Stephens, Prof: By the theoretically driven calculation here. 427 00:46:15.740 --> 00:46:18.879 David Stephens, Prof: So this mathematical machinery… 428 00:46:19.170 --> 00:46:27.639 David Stephens, Prof: you can kind of take with a pinch of salt if you like. It's not too critical, but for me, it helps to understand where this idea of re-weighting comes from. 429 00:46:29.730 --> 00:46:36.930 David Stephens, Prof: Alright, so, this is the hypothetical calculation that we want to do, this is the estimate, and this is the way we would get at it. 430 00:46:39.490 --> 00:46:44.859 David Stephens, Prof: Unfortunately, we don't have data that allow us to estimate the quantities here. 431 00:46:45.360 --> 00:46:46.600 David Stephens, Prof: In the right way. 432 00:46:47.240 --> 00:46:53.780 David Stephens, Prof: Right? This calculation works in the experimental world if we did have a randomized study. 433 00:46:54.740 --> 00:46:55.889 David Stephens, Prof: But we don't. 434 00:46:56.880 --> 00:47:02.119 David Stephens, Prof: So, how do we estimate these quantities when we have data from the observational world? 435 00:47:03.980 --> 00:47:13.750 David Stephens, Prof: Well, I'm going to denote the observational measure in this way. And of course, the only difference is that now we have a dependence between X and Z. Remember back to the DAGs that we saw yesterday? 436 00:47:14.910 --> 00:47:20.800 David Stephens, Prof: The curly O indicates… The observational world, meaning that 437 00:47:20.940 --> 00:47:27.320 David Stephens, Prof: Z and X are dependent. In particular, we can think of people coming in with their characteristics X, 438 00:47:28.100 --> 00:47:32.300 David Stephens, Prof: And those characteristics determining the treatment that they receive. 439 00:47:33.540 --> 00:47:37.490 David Stephens, Prof: Having both access to Z and X, we can then generate the outcome. 440 00:47:41.250 --> 00:47:48.379 David Stephens, Prof: So this model… This observational model, Of course, reflect the… 441 00:47:51.690 --> 00:47:56.210 David Stephens, Prof: DAG that we saw… Yesterday. 442 00:48:04.020 --> 00:48:06.370 David Stephens, Prof: Connecting the three variables in this way. 443 00:48:09.800 --> 00:48:20.330 David Stephens, Prof: Generate the X's first, generate the Z's condition on the X's, and then generate the Y's condition on the X's and Zs. So this is how the data are arising. 444 00:48:21.860 --> 00:48:29.509 David Stephens, Prof: And, without further correction, Those data are no good for estimating these quantities. 445 00:48:30.620 --> 00:48:31.889 David Stephens, Prof: So what do we do? 446 00:48:34.120 --> 00:48:41.940 David Stephens, Prof: Well, again, to remind you of the difference between the two models, the target world is here. 447 00:48:44.650 --> 00:48:46.530 David Stephens, Prof: That's the experimental world. 448 00:48:51.840 --> 00:48:54.090 David Stephens, Prof: But our data live over here. 449 00:48:56.870 --> 00:49:02.929 David Stephens, Prof: So the difference between the two setups is contained in these two equations. This is the data-generating model 450 00:49:03.390 --> 00:49:05.729 David Stephens, Prof: Under the assumed experimental world. 451 00:49:06.090 --> 00:49:09.840 David Stephens, Prof: And this is the data-generated model under the observational world. 452 00:49:10.100 --> 00:49:16.390 David Stephens, Prof: And what we notice is that the two things are exactly the same, apart from this contribution concerning Zed. 453 00:49:22.770 --> 00:49:27.810 David Stephens, Prof: This has been the setup from the start. The only difference between the experimental world and the observational world 454 00:49:27.930 --> 00:49:34.319 David Stephens, Prof: is that this arrow from Z to X, which is present in the observational world. 455 00:49:34.590 --> 00:49:36.179 David Stephens, Prof: sorry, from X to Z. 456 00:49:36.290 --> 00:49:41.250 David Stephens, Prof: This arrow that's present here is not present in the experimental world. 457 00:49:42.280 --> 00:49:45.810 David Stephens, Prof: What we want is over here, but we've only got data over here. 458 00:49:48.030 --> 00:50:01.440 David Stephens, Prof: So… The way that we… Enable ourselves to use the observed data, To make conclusions about this. 459 00:50:02.070 --> 00:50:06.050 David Stephens, Prof: hypothetical experimental world is to use re-weighting. 460 00:50:07.680 --> 00:50:18.459 David Stephens, Prof: We can rewrite the quantity of interest, In terms of this ratio, Expectations over the observational. 461 00:50:19.430 --> 00:50:22.900 David Stephens, Prof: Measure, probability distribution, data generating model. 462 00:50:23.890 --> 00:50:34.380 David Stephens, Prof: of the same quantity in the numerator and denominator, but in both cases, multiplied by W, where W is the ratio of the previous two densities. 463 00:50:36.130 --> 00:50:41.660 David Stephens, Prof: So this is, in effect, essentially a standardization calculation. 464 00:50:41.890 --> 00:50:49.909 David Stephens, Prof: In the same way that you would take data from one population, some rates in one population, and extrapolate them to another population by 465 00:50:50.150 --> 00:50:54.629 David Stephens, Prof: using appropriate reweighting. The idea is somewhat similar to that. 466 00:50:55.470 --> 00:51:03.549 David Stephens, Prof: It turns out that this ratio of weights here, this ratio of densities here, is exactly the right weight to use to turn 467 00:51:04.280 --> 00:51:06.680 David Stephens, Prof: That ratio of expectations… 468 00:51:10.280 --> 00:51:13.270 David Stephens, Prof: Into this one, without changing the response. 469 00:51:14.220 --> 00:51:16.230 David Stephens, Prof: Sorry, without changing the S demand. 470 00:51:20.580 --> 00:51:23.880 David Stephens, Prof: So all we have to do is weighted estimation. 471 00:51:24.770 --> 00:51:26.729 David Stephens, Prof: Weighted sample averages. 472 00:51:27.680 --> 00:51:37.300 David Stephens, Prof: this logic shows us that that will still give us the right answer. So I think of this as a… basically a Monte Carlo-type person, as a… an important sampling argument. 473 00:51:39.230 --> 00:51:43.709 David Stephens, Prof: Right, the weights represent… Important sampling weights. 474 00:51:43.940 --> 00:51:52.550 David Stephens, Prof: Which allow me to turn samples from the… Observed population. 475 00:51:54.280 --> 00:51:58.580 David Stephens, Prof: Into samples from the Hypothetical experimental situation. 476 00:52:00.470 --> 00:52:01.420 David Stephens, Prof: population. 477 00:52:03.400 --> 00:52:07.880 David Stephens, Prof: So the right weight is this, and it's the ratio of those two densities. 478 00:52:11.580 --> 00:52:15.749 David Stephens, Prof: Okay, that is exactly the right weight to include to turn 479 00:52:16.040 --> 00:52:26.650 David Stephens, Prof: The experimental expectation into the… sorry, the observational experiment… the observational expectation into the experimental expectation. 480 00:52:27.480 --> 00:52:31.869 David Stephens, Prof: Now, what we notice from this is that there are terms that cancel. 481 00:52:32.900 --> 00:52:37.329 David Stephens, Prof: If we break those numerator and denominator terms down into the three parts. 482 00:52:37.710 --> 00:52:41.769 David Stephens, Prof: We see that the first term is just 1, 483 00:52:42.090 --> 00:52:44.100 David Stephens, Prof: And the third term is just 1. 484 00:52:45.430 --> 00:52:52.840 David Stephens, Prof: The first term is 1, due to the similarity between the two DAGs in the relationship between Z 485 00:52:53.280 --> 00:52:54.809 David Stephens, Prof: X and Y. 486 00:52:54.990 --> 00:53:00.049 David Stephens, Prof: Right, the only difference between these two worlds is that this arrow is removed. 487 00:53:01.120 --> 00:53:07.160 David Stephens, Prof: The relationship between XZ and Y is the same in both models. 488 00:53:08.900 --> 00:53:12.060 David Stephens, Prof: So… that ratio is 1. 489 00:53:13.420 --> 00:53:17.519 David Stephens, Prof: Similarly, the ratio in the third term is also 1. 490 00:53:17.770 --> 00:53:26.420 David Stephens, Prof: Because we're making conclusions in the same population of individuals. The X characteristics in this population 491 00:53:26.570 --> 00:53:31.960 David Stephens, Prof: Are the same for any individual, basically, the same… come from the same distribution. 492 00:53:34.420 --> 00:53:41.319 David Stephens, Prof: So, if the first term is 1, and the third term is 1, the only term that you have to think about is this second term. 493 00:53:44.390 --> 00:53:50.330 David Stephens, Prof: So that second term, It's precisely the right term. 494 00:53:51.380 --> 00:54:00.649 David Stephens, Prof: To change the observational sample, into a pseudo… experimental sample. 495 00:54:01.860 --> 00:54:11.390 David Stephens, Prof: All we need to do is to re-weight by this term, In order to get… pseudosamples. 496 00:54:11.530 --> 00:54:15.140 David Stephens, Prof: From a world where confounding is removed. 497 00:54:18.610 --> 00:54:25.930 David Stephens, Prof: Now, the term in the numerator. 498 00:54:26.610 --> 00:54:29.370 David Stephens, Prof: That's the… 499 00:54:29.600 --> 00:54:45.019 David Stephens, Prof: probability of being treated, let's say, in the experimental world. So that's, in some sense, a fixed quantity. So the only thing we really have to worry about is what's in the denominator. So that's the prob… that's the probability distribution of Z given X. 500 00:54:45.620 --> 00:54:49.900 David Stephens, Prof: And of course, that is precisely what we had to model 501 00:54:50.040 --> 00:54:53.270 David Stephens, Prof: When using the propensity… when constructing the propensity score. 502 00:54:54.920 --> 00:55:02.540 David Stephens, Prof: Okay, so again, we see the role of this conditional model for Z given X in correcting The confounding. 503 00:55:03.510 --> 00:55:09.470 David Stephens, Prof: And if we go back to the DAGs, we can see why that term is there, right? So the… this arrow here… 504 00:55:10.090 --> 00:55:19.380 David Stephens, Prof: captures… basically the impact of Z on X. It captures the structural relationship between the treatment And, predictors. 505 00:55:20.070 --> 00:55:28.590 David Stephens, Prof: So it's only natural that The difference between these worlds depends critically on what that arrow represents. 506 00:55:28.780 --> 00:55:34.050 David Stephens, Prof: Right? And the arrow basically represents the conditional distribution of Z given X. 507 00:55:37.370 --> 00:55:39.250 David Stephens, Prof: That's the logic I use, anyway. 508 00:55:40.120 --> 00:55:43.980 David Stephens, Prof: Alright, so, this term accounts for the imbalance. 509 00:55:45.210 --> 00:55:47.189 David Stephens, Prof: That is present due to confounding. 510 00:55:47.800 --> 00:55:52.660 David Stephens, Prof: It accounts for the imbalance between the, 511 00:55:52.950 --> 00:55:58.450 David Stephens, Prof: or the lack of balance between the X distributions in the treated and untreated groups. 512 00:56:03.250 --> 00:56:08.079 David Stephens, Prof: So then, implementing this, For a re-weighted estimator. 513 00:56:08.200 --> 00:56:13.350 David Stephens, Prof: is pretty straightforward. All we need to do is to take a weighted average. 514 00:56:13.610 --> 00:56:19.820 David Stephens, Prof: With weights determined by 1 over… The fitted Propensity score model. 515 00:56:23.060 --> 00:56:30.289 David Stephens, Prof: Or we do a standardized weight version, which basically makes sure that the weights that are attached to the Y's 516 00:56:30.440 --> 00:56:31.490 David Stephens, Prof: Sum to 1. 517 00:56:32.980 --> 00:56:39.459 David Stephens, Prof: So, in this version, We've just taken a simple average of the re-weighted Ys. 518 00:56:39.840 --> 00:56:42.259 David Stephens, Prof: But the weight's been determined by… 519 00:56:42.660 --> 00:56:45.779 David Stephens, Prof: Essentially, 1 over the propensity score type term. 520 00:56:46.670 --> 00:56:52.869 David Stephens, Prof: In the second version, We divide by the sum of the weights. 521 00:56:53.260 --> 00:56:57.020 David Stephens, Prof: To make sure that the weights that attach to the Y is all sum to 1. 522 00:56:58.060 --> 00:57:15.330 David Stephens, Prof: We don't have that in the first term, we have it in the second term. So again, we're presented with two estimators in the same way that we were for the experimental study, and the logic for the two is the same, basically. We have a version here that, 523 00:57:17.010 --> 00:57:21.060 David Stephens, Prof: Essentially accounts for the empirical 524 00:57:21.620 --> 00:57:28.120 David Stephens, Prof: sample size, whereas this one assumes that the sample size is N always, or at least N times F. 525 00:57:33.610 --> 00:57:39.880 David Stephens, Prof: So now we've got these two inverse weighting estimators that Provided… the… 526 00:57:40.040 --> 00:57:50.070 David Stephens, Prof: Propensity model is correctly specified, will give us unbiased estimation of the average potential outcome. 527 00:57:51.800 --> 00:57:59.480 David Stephens, Prof: So it's a very different method to using regression directly, it's just a reweighting method, but it can still give us the correct answer. 528 00:58:00.640 --> 00:58:19.760 David Stephens, Prof: Alright, so I think I'll take a few minutes break there, before we move on to, a bit more of a justification, or I'll start off with a reminder of the logic of inverse weighting, and then show some extensions that still use the weighting idea. So now we've seen a number of different methods for use of the propensity score. 529 00:58:20.950 --> 00:58:27.590 David Stephens, Prof: Matching, stratification, regression-based, and… Inverse weighting. 530 00:58:28.440 --> 00:58:30.559 David Stephens, Prof: Okay, and they can all give us the correct… 531 00:58:30.830 --> 00:58:41.959 David Stephens, Prof: causal conclusions. Okay, so I'll take a break now until 33, and then we'll pick up in the last half hour in the first session, talking a bit more about connecting these methods. 532 01:02:53.860 --> 01:03:01.810 David Stephens, Prof: Okay, to pick up for the last half hour then, we'll, look in a bit more detail at this inverse weighting method and some extensions. 533 01:03:02.210 --> 01:03:05.570 David Stephens, Prof: So, again, inverse weighting. 534 01:03:06.220 --> 01:03:17.390 David Stephens, Prof: There's a simple mechanism, For constructing a pseudosample, From this hypothetical experimental world. 535 01:03:17.880 --> 01:03:21.900 David Stephens, Prof: When originally, the data were from the observational world. 536 01:03:22.010 --> 01:03:27.879 David Stephens, Prof: In the experimental world, after re-weighting, We have no confounding. 537 01:03:30.210 --> 01:03:37.769 David Stephens, Prof: That is… It's removed the imbalances between the exposure groups due to the ZX relationship. 538 01:03:41.510 --> 01:03:46.800 David Stephens, Prof: After re-waiting, The pseudopulation, the pseudo-sample. 539 01:03:47.370 --> 01:03:53.650 David Stephens, Prof: should be balanced, and that allows us to do a direct comparison. That's the key role in 540 01:03:53.770 --> 01:03:56.530 David Stephens, Prof: In the calculation of the re-weighting. 541 01:03:56.690 --> 01:03:57.470 David Stephens, Prof: component. 542 01:04:00.030 --> 01:04:13.509 David Stephens, Prof: Again, that term in the denominator, and if we take the binary treatment case, we'll see that that is basically the exposure model, the Z given X model, and in the binary case, this reduces to the simple Bernoulli model. 543 01:04:13.830 --> 01:04:24.829 David Stephens, Prof: Right, so it's the propensity score to the Z, 1 minus the Propensity score to the 1 minus Z, so we get this as the weight if Z is 1, and this as the weight if Z is 0. 544 01:04:27.150 --> 01:04:32.359 David Stephens, Prof: So again, the Propensity score model, the Z given X model, is playing a crucial role. 545 01:04:33.540 --> 01:04:39.789 David Stephens, Prof: There's one critical assumption, though, which is that when we form that re-weighting estimate. 546 01:04:42.420 --> 01:04:47.560 David Stephens, Prof: At one point, we were dividing by the probability model for Z given X. 547 01:04:48.220 --> 01:04:53.420 David Stephens, Prof: So it's, of course, clear that we need this thing to be non-zero. 548 01:04:54.050 --> 01:04:57.450 David Stephens, Prof: It's a probability, or probability density, therefore it's… 549 01:04:57.730 --> 01:05:00.900 David Stephens, Prof: Not gonna be negative, but it could be zero. 550 01:05:02.370 --> 01:05:12.280 David Stephens, Prof: So… this assumption… Is necessary for… This calculation to work. 551 01:05:18.040 --> 01:05:24.300 David Stephens, Prof: We need the term in the denominator here, the terms in the denominator here to be non-zero. If they're zero. 552 01:05:24.500 --> 01:05:25.760 David Stephens, Prof: We got messed up. 553 01:05:25.940 --> 01:05:28.440 David Stephens, Prof: Right, we get infinite values. 554 01:05:29.820 --> 01:05:36.339 David Stephens, Prof: So let's think about this assumption. This assumption says that For any observed treatment level. 555 01:05:37.570 --> 01:05:42.519 David Stephens, Prof: I need a positive probability of observing that for any given X. 556 01:05:44.760 --> 01:05:48.589 David Stephens, Prof: So that term, that sometimes is called the positivity assumption. 557 01:05:51.330 --> 01:05:58.549 David Stephens, Prof: It basically means that We must be able to attain any treatment level for any 558 01:05:59.170 --> 01:06:02.999 David Stephens, Prof: Vector of predictors. Any vector of characteristics. 559 01:06:03.720 --> 01:06:06.520 David Stephens, Prof: In the observational… in the observational study. 560 01:06:08.110 --> 01:06:13.100 David Stephens, Prof: What that means, of course, in practice, is that for anybody in our population. 561 01:06:13.630 --> 01:06:23.600 David Stephens, Prof: each of them must be able to obtain any of the target treatment levels. So, in the binary case, any individual must be able to be treated. 562 01:06:24.230 --> 01:06:25.670 David Stephens, Prof: Or not treated. 563 01:06:27.260 --> 01:06:31.139 David Stephens, Prof: It can't be that one of those options is excluded. 564 01:06:34.270 --> 01:06:38.879 David Stephens, Prof: That is, there's not a deterministic relationship between Z and X. 565 01:06:41.910 --> 01:06:46.900 David Stephens, Prof: And that… Also leads to this, sort of, second… 566 01:06:47.830 --> 01:06:54.170 David Stephens, Prof: Piece of terminology, which is that this is the… Experimental Treatment Assignment Assumption. 567 01:06:56.190 --> 01:07:02.140 David Stephens, Prof: That is, it is possible for any individual to receive any of the treatments. 568 01:07:06.630 --> 01:07:13.799 David Stephens, Prof: Now, actually, this assumption was necessary in the de-estimation setting as well, it was just hidden a little bit in the… 569 01:07:14.190 --> 01:07:16.429 David Stephens, Prof: In the… in the mechanics. 570 01:07:17.400 --> 01:07:26.800 David Stephens, Prof: But anyway, this positivity assumption turns out to be critical in all of these causal estimation comparisons. What it does effectively is ensure that 571 01:07:27.610 --> 01:07:34.309 David Stephens, Prof: Treated and untreated individuals with the same X value are always going to be available. 572 01:07:35.310 --> 01:07:38.319 David Stephens, Prof: Right? So the causal comparison makes sense. 573 01:07:38.640 --> 01:07:43.809 David Stephens, Prof: If there were some exes for which you could never be treated, or never be not treated. 574 01:07:44.010 --> 01:07:48.679 David Stephens, Prof: Then, the causal comparison would make no sense for those individuals. 575 01:07:50.530 --> 01:07:54.779 David Stephens, Prof: So this just mathematically incorporates that logic. 576 01:07:58.020 --> 01:08:02.719 David Stephens, Prof: Okay, and as we'll see later on, this is something that we need to check for in practice. 577 01:08:02.910 --> 01:08:05.490 David Stephens, Prof: In order to implement their causal methods. 578 01:08:07.350 --> 01:08:16.110 David Stephens, Prof: Alright, so, an extension of the inverse probability weighting idea is to use what's called augmentation. 579 01:08:17.590 --> 01:08:27.180 David Stephens, Prof: So we can think of augmentation as a modification of the inverse probability weighting method that allows for a little bit of regression modeling. 580 01:08:28.779 --> 01:08:31.219 David Stephens, Prof: And the way I think of it is the following. 581 01:08:31.740 --> 01:08:35.320 David Stephens, Prof: We can always break down this expectation 582 01:08:35.510 --> 01:08:40.569 David Stephens, Prof: which gives us the average potential outcome in this way. This is an identity. 583 01:08:41.590 --> 01:08:46.410 David Stephens, Prof: All I've done is basically subtract, and then add back in mu. 584 01:08:47.000 --> 01:08:52.009 David Stephens, Prof: Where mu is the conditional expectation of Z given Y given X and Z. 585 01:08:53.029 --> 01:08:53.870 David Stephens, Prof: Okay? 586 01:08:54.240 --> 01:08:57.719 David Stephens, Prof: So, once we've seen that decomposition. 587 01:08:59.010 --> 01:09:05.340 David Stephens, Prof: what I need to do is to think about estimating this part, Using it as probability weighting. 588 01:09:05.689 --> 01:09:08.119 David Stephens, Prof: So that leads to this term. 589 01:09:08.729 --> 01:09:10.860 David Stephens, Prof: In the subsequent estimator. 590 01:09:12.029 --> 01:09:14.369 David Stephens, Prof: And then estimate this term. 591 01:09:14.490 --> 01:09:17.020 David Stephens, Prof: Which is essentially a regression. 592 01:09:17.800 --> 01:09:18.930 David Stephens, Prof: Estimator. 593 01:09:19.990 --> 01:09:29.670 David Stephens, Prof: So, I have now an alternative estimator, really an extended or augmented estimator, an augmented version of the IPW estimator. 594 01:09:32.899 --> 01:09:35.940 David Stephens, Prof: Which incorporates this term. 595 01:09:36.880 --> 01:09:42.180 David Stephens, Prof: And this term… In combination to give the result. 596 01:09:42.760 --> 01:09:43.479 David Stephens, Prof: Okay? 597 01:09:44.830 --> 01:09:50.040 David Stephens, Prof: So, in this case, we're taking the residual from the fit. 598 01:09:51.010 --> 01:09:55.689 David Stephens, Prof: Y given mu, so the idea is that we're proposing mu. 599 01:09:56.120 --> 01:10:07.120 David Stephens, Prof: Fitting a regression of Y on X and Z to get estimates of that mu, taking the residuals from that fit, and then average them using reweighting. 600 01:10:08.240 --> 01:10:11.780 David Stephens, Prof: And then, adding to that, average… 601 01:10:12.340 --> 01:10:19.120 David Stephens, Prof: in the first term, this contribution of the fitted mu terms for the target Z. 602 01:10:21.820 --> 01:10:29.629 David Stephens, Prof: So, this is an extension of the ordinary IPW estimator by the inclusion of this mu term. 603 01:10:30.250 --> 01:10:36.189 David Stephens, Prof: So I've written it here as if mu is the correct data-generating conditional expectation. 604 01:10:36.610 --> 01:10:44.420 David Stephens, Prof: In practice, of course, we won't know what that is, so we're going to have to substitute a proxy model. 605 01:10:44.980 --> 01:10:48.499 David Stephens, Prof: Right, so we'll have an M rather than the true mu. 606 01:10:49.110 --> 01:10:51.200 David Stephens, Prof: But we can still form this estimator. 607 01:10:51.640 --> 01:11:04.790 David Stephens, Prof: Right? So this is called the augmented IPW, or AIPW estimator. It requires me now to fit two regression models again. It requires me to understand the conditional relationship between Z and X, as in the propensity model. 608 01:11:05.210 --> 01:11:06.140 David Stephens, Prof: Treatment. 609 01:11:06.260 --> 01:11:07.600 David Stephens, Prof: Give an X model. 610 01:11:08.500 --> 01:11:11.320 David Stephens, Prof: And also, it requires me to fit this new model. 611 01:11:15.110 --> 01:11:18.860 David Stephens, Prof: It turns out that if both of those models are correctly specified. 612 01:11:19.080 --> 01:11:26.780 David Stephens, Prof: then we can guarantee that the variance of the AIPW estimator is smaller than the variance of the IPW estimator. 613 01:11:27.280 --> 01:11:30.440 David Stephens, Prof: That is, by including 614 01:11:30.550 --> 01:11:37.699 David Stephens, Prof: This regression, this little bit of extra regression modeling, allows me to reduce the estimator variance. 615 01:11:39.190 --> 01:11:46.639 David Stephens, Prof: Okay? It also provides a doubly robust result, in the sense that if the mean model 616 01:11:46.930 --> 01:11:49.990 David Stephens, Prof: Or the Propensity model is correctly specified. 617 01:11:50.300 --> 01:11:52.099 David Stephens, Prof: Then we have consistency. 618 01:11:53.500 --> 01:11:59.570 David Stephens, Prof: That is, we can model, correctly model the Z given X model, and get the mu model wrong. 619 01:11:59.860 --> 01:12:03.130 David Stephens, Prof: Or vice versa, and still get the correct answer. 620 01:12:06.200 --> 01:12:10.530 David Stephens, Prof: Correct, correct estimation, consistent estimation of the average treatment effect. 621 01:12:20.290 --> 01:12:24.110 David Stephens, Prof: To see why it works, basically, we, 622 01:12:24.800 --> 01:12:43.480 David Stephens, Prof: propose maybe misspecified… potentially misspecified models, and imagine the estimator formed from those misspecified models. So, mu is my chosen model for… sorry, M is my chosen model for mu, F is my chosen model for the propensity score, the estimator that I would form looks like this. 623 01:12:48.160 --> 01:12:51.120 David Stephens, Prof: It turns out that the bias of the estimator 624 01:12:51.840 --> 01:12:55.629 David Stephens, Prof: Of the average potential outcome turns out to be this. 625 01:12:56.210 --> 01:12:59.889 David Stephens, Prof: So… Reflect the difference in the… 626 01:13:01.150 --> 01:13:08.730 David Stephens, Prof: Propensity model, the difference in the mu model, Appropriately re-weighted by the… misspecified model. 627 01:13:11.660 --> 01:13:17.109 David Stephens, Prof: Now, this quantity is 0 if either that term or that term is 0. 628 01:13:18.900 --> 01:13:23.819 David Stephens, Prof: What you can see here, though, is that if they're… if both of those models are wrong. 629 01:13:24.720 --> 01:13:28.840 David Stephens, Prof: then you don't get consistent estimation. This bias is non-zero. 630 01:13:31.040 --> 01:13:36.260 David Stephens, Prof: And so, it's still a practical limitation. You still have a mathematical assumption of 631 01:13:36.880 --> 01:13:40.659 David Stephens, Prof: Correct specification of this, at least one of these models being correct. 632 01:13:42.910 --> 01:13:45.590 David Stephens, Prof: To recover consistent estimation. 633 01:13:47.220 --> 01:13:56.459 David Stephens, Prof: Now, this is a bit of a… if we think back to G-estimation, there we had a specific assumption which required that the treatment effect component 634 01:13:57.340 --> 01:14:00.549 David Stephens, Prof: Z times psi 0, say. That… 635 01:14:00.680 --> 01:14:05.759 David Stephens, Prof: Components of the model had to be correctly specified under all circumstances. 636 01:14:06.910 --> 01:14:11.099 David Stephens, Prof: Right? We could not get away using those regression methods and making that assumption. 637 01:14:12.260 --> 01:14:18.180 David Stephens, Prof: In the inverse probability weighting and AIPW setting, we don't need to have that assumption. 638 01:14:21.760 --> 01:14:30.149 David Stephens, Prof: I can… Mispecify the mu, as M, So this is not the… 639 01:14:30.810 --> 01:14:34.670 David Stephens, Prof: This is my proposed model, this is not necessarily the true model. 640 01:14:34.830 --> 01:14:41.159 David Stephens, Prof: capturing the conditional expectation of Y given X and Z. It's my proposed model. I can get that completely wrong. 641 01:14:42.230 --> 01:14:44.970 David Stephens, Prof: Provided I get the propensity model correct. 642 01:14:49.800 --> 01:14:58.270 David Stephens, Prof: So, what this suggests on the surface, and why this is a popular method, perhaps the most popular method for doing causal estimation, is that 643 01:14:58.610 --> 01:15:06.169 David Stephens, Prof: I don't have… A correct specification component of the outcome bean model at all. 644 01:15:06.280 --> 01:15:06.970 David Stephens, Prof: Right. 645 01:15:08.610 --> 01:15:17.880 David Stephens, Prof: So you might wonder, well, why do we ever think about gestimation in the first place? Why, and Propensity score regression, why do we ever bother thinking about that? 646 01:15:18.100 --> 01:15:26.700 David Stephens, Prof: when it comes with this additional assumption. Well, first of all, that additional assumption of correct specification of the treatment effect part. 647 01:15:27.550 --> 01:15:29.719 David Stephens, Prof: Could be thought of as not too strong an assumption. 648 01:15:31.870 --> 01:15:35.030 David Stephens, Prof: The second reason why we think about G-estimation 649 01:15:35.150 --> 01:15:41.380 David Stephens, Prof: sometimes above these inverse probability weighting methods is that if I go back to this variance result. 650 01:15:41.500 --> 01:15:44.990 David Stephens, Prof: Then here we see that the variance of the IPW estimator 651 01:15:45.210 --> 01:15:48.990 David Stephens, Prof: When augmented, it's lower than the variance of the 652 01:15:49.200 --> 01:15:55.889 David Stephens, Prof: IPW estimator, under correct specification, it turns out that if we do a similar comparison for the G estimator. 653 01:15:56.580 --> 01:16:02.959 David Stephens, Prof: the G estimator, Has an even lower variance under its correct specification assumptions. 654 01:16:04.230 --> 01:16:13.590 David Stephens, Prof: So… And essentially what they're saying is that, inverse weighting methods Can have worse variance performance. 655 01:16:14.770 --> 01:16:18.310 David Stephens, Prof: You can have a more variable estimator arising from 656 01:16:18.640 --> 01:16:22.909 David Stephens, Prof: the, AIPW form down from the G-E estimator form. 657 01:16:23.090 --> 01:16:29.550 David Stephens, Prof: So if the G-estimation assumptions are correct, that you do correctly capture the treatment effect model. 658 01:16:30.310 --> 01:16:32.470 David Stephens, Prof: Then you will get a lower variance. 659 01:16:35.110 --> 01:16:37.060 David Stephens, Prof: Okay, so it's preferable in that way. 660 01:16:42.980 --> 01:16:48.910 David Stephens, Prof: Yeah, so that's basically summarizing the… The misspecification argument there. 661 01:16:52.740 --> 01:17:00.170 David Stephens, Prof: We need parametric specifications for these models, that's inevitably true. Parameters in the propensity model, parameters in the outcome model. 662 01:17:00.330 --> 01:17:06.049 David Stephens, Prof: So we are again forced to do two… estimations? Estimation of the… 663 01:17:06.450 --> 01:17:11.030 David Stephens, Prof: Treatment model, the treatment assignment model, and of the outcome model. 664 01:17:11.670 --> 01:17:14.619 David Stephens, Prof: And then plug those into the previous estimator form. 665 01:17:17.940 --> 01:17:25.260 David Stephens, Prof: This little note says that it is, perhaps surprisingly. 666 01:17:28.440 --> 01:17:39.270 David Stephens, Prof: advantageous to do estimation of the propensity score model, even if you know the treatment assignment probabilities, or treatment assignment densities. It turns out that 667 01:17:39.520 --> 01:17:44.080 David Stephens, Prof: Oh, there's an adaptation effect. 668 01:17:44.760 --> 01:17:49.559 David Stephens, Prof: Which, means that if we estimate the propensity score. 669 01:17:51.290 --> 01:17:57.839 David Stephens, Prof: Non-parametrically or parametrically, then that can do a better job than using the known values. 670 01:17:59.580 --> 01:18:06.979 David Stephens, Prof: Okay, so that's, it's basically the inference adapts to the data, so that's something special about this model. 671 01:18:07.400 --> 01:18:08.359 David Stephens, Prof: Turns out. 672 01:18:10.640 --> 01:18:11.390 David Stephens, Prof: Okay. 673 01:18:15.210 --> 01:18:28.379 David Stephens, Prof: Finally, one little point about augmentation is that we can also regard it as a purely regression method, and I'll describe how we can do that in 674 01:18:28.590 --> 01:18:30.710 David Stephens, Prof: The following couple of slides. 675 01:18:31.090 --> 01:18:31.750 David Stephens, Prof: Good. 676 01:18:31.960 --> 01:18:39.810 David Stephens, Prof: So, we had a previous estimating equation, based on the W-robust AIPW estimator. 677 01:18:40.110 --> 01:18:46.199 David Stephens, Prof: Which looked like this. So basically, we're estimating this quantity, and we're solving this equation. 678 01:18:46.860 --> 01:18:49.580 David Stephens, Prof: To estimate mu, the average potential outcome. 679 01:18:49.960 --> 01:18:51.309 David Stephens, Prof: Right? That's the idea. 680 01:18:52.480 --> 01:19:02.229 David Stephens, Prof: Okay? And the way that we've thought about forming this is to fit the two models, and then combine this term with this term, and then that gives the… 681 01:19:02.860 --> 01:19:07.670 David Stephens, Prof: Gives the, Resulting estimate. Basically, the… 682 01:19:08.420 --> 01:19:13.509 David Stephens, Prof: We rearrange this estimating form, and that gives us an expression for the estimator. 683 01:19:14.830 --> 01:19:21.169 David Stephens, Prof: Okay, so… It turns out that you can justify that estimating equation 684 01:19:21.330 --> 01:19:37.609 David Stephens, Prof: purely using a regression perspective. That is, it turns out that this estimating equation is the estimating equation you would get using ordinary least squares if you chose this model. 685 01:19:40.370 --> 01:19:42.039 David Stephens, Prof: As the regression model. 686 01:19:46.020 --> 01:19:51.820 David Stephens, Prof: So, basically what that means is that you can do the, 687 01:19:53.950 --> 01:20:02.330 David Stephens, Prof: you can implement a procedure which fits a regression model based on this. So this is some proposed M0, so some proposed model 688 01:20:02.580 --> 01:20:04.080 David Stephens, Prof: Playing the role of Mew. 689 01:20:04.210 --> 01:20:07.409 David Stephens, Prof: And then we augment that model with this type of term. 690 01:20:08.090 --> 01:20:13.160 David Stephens, Prof: So this is a coefficient times the The appropriate weight. 691 01:20:15.490 --> 01:20:16.180 David Stephens, Prof: Okay? 692 01:20:16.380 --> 01:20:23.179 David Stephens, Prof: So, the appropriate weight here is… it's 1 when the right target level is met. 693 01:20:23.740 --> 01:20:26.889 David Stephens, Prof: Divided by the probability of receiving that. 694 01:20:28.300 --> 01:20:29.310 David Stephens, Prof: treatment level. 695 01:20:30.420 --> 01:20:33.439 David Stephens, Prof: This quantity is sometimes called the clever covariant. 696 01:20:35.580 --> 01:20:40.000 David Stephens, Prof: So the idea is simple, you include terms like this. 697 01:20:40.670 --> 01:20:45.140 David Stephens, Prof: In a regression model, and just use ordinary least squares, you just use ln. 698 01:20:46.360 --> 01:20:53.279 David Stephens, Prof: So this is an alternative view of augmentation, sometimes it's called augmented outcome regression. 699 01:20:55.510 --> 01:20:59.230 David Stephens, Prof: We fit the models, and then we take the averaged 700 01:20:59.630 --> 01:21:05.380 David Stephens, Prof: Output from those models, the average predictions from those models to estimate the average potential outcome. 701 01:21:05.540 --> 01:21:07.569 David Stephens, Prof: To lead us to the average treatment effect. 702 01:21:09.100 --> 01:21:11.459 David Stephens, Prof: Okay, that's what these equations basically say. 703 01:21:12.060 --> 01:21:12.770 David Stephens, Prof: Okay? 704 01:21:12.920 --> 01:21:19.830 David Stephens, Prof: So, we fit this model, we estimate any parameters in here, we estimate Phi? 705 01:21:20.400 --> 01:21:25.269 David Stephens, Prof: And then we plug those into this average prediction in order to get 706 01:21:25.580 --> 01:21:27.759 David Stephens, Prof: An estimate of the average potential outcome. 707 01:21:33.590 --> 01:21:35.690 David Stephens, Prof: This turns out to be equivalent. 708 01:21:36.310 --> 01:21:39.839 David Stephens, Prof: In large sample to the, average. 709 01:21:40.110 --> 01:21:50.199 David Stephens, Prof: to the augmented IPW estimator. Here's some of the logic given here. I won't… in the interest of time, I want to get to one more point before I get to the half-time break. 710 01:21:50.370 --> 01:21:58.560 David Stephens, Prof: This is the logic here. And, it's just to show that we can, implement the regression model using 711 01:21:58.670 --> 01:22:05.960 David Stephens, Prof: A simpler version, if Our target is the average treatment effect. 712 01:22:07.270 --> 01:22:07.980 David Stephens, Prof: Right. 713 01:22:08.940 --> 01:22:20.210 David Stephens, Prof: ordinary AIPW estimation, and this form of augmented regression, is really estimating the Average potential outcome. 714 01:22:24.770 --> 01:22:26.289 David Stephens, Prof: This quantity here. 715 01:22:29.030 --> 01:22:33.789 David Stephens, Prof: Whereas the average treatment effect is the difference between terms like that. 716 01:22:34.670 --> 01:22:39.070 David Stephens, Prof: So, if we're only interested in estimating this quantity. 717 01:22:39.430 --> 01:22:47.440 David Stephens, Prof: Rather than the two individual terms, so we can estimate the difference, and not the first term and the second term. 718 01:22:48.960 --> 01:22:54.049 David Stephens, Prof: It turns out that we can use the clever covariate that looks like this, or like this. 719 01:22:57.660 --> 01:23:03.539 David Stephens, Prof: Okay? And that performs a form of augmented IPW estimation. 720 01:23:11.530 --> 01:23:17.800 David Stephens, Prof: All of those arguments that we just looked at applied in the binary treatment case. 721 01:23:17.980 --> 01:23:20.939 David Stephens, Prof: Just quickly to finish now, I want to point out that 722 01:23:22.410 --> 01:23:29.759 David Stephens, Prof: All of the same ideas apply if we have continuous treatments. And I think for a long time in the literature, this was not recognized. 723 01:23:31.140 --> 01:23:40.489 David Stephens, Prof: But actually, the gestation approaches, the inverse probability weighting approaches, can all apply to the continuous treatment case as well. 724 01:23:41.150 --> 01:23:47.840 David Stephens, Prof: Instead of worrying about the, propensity score model, so now we've got a continuous treatment. 725 01:23:48.450 --> 01:24:00.590 David Stephens, Prof: we instead think about modeling the conditional expectation of that treatment given Z, and that gives us a balancing score. Remember that the idea is that this is a suitable summary statistic. 726 01:24:00.730 --> 01:24:05.809 David Stephens, Prof: On which we can condition in order to block the confounding paths. 727 01:24:06.670 --> 01:24:08.710 David Stephens, Prof: So we can either use, 728 01:24:09.150 --> 01:24:17.200 David Stephens, Prof: this B in the replacement for the propensity score, or for inverse probability weighting, we could use the whole of this model. 729 01:24:19.550 --> 01:24:22.299 David Stephens, Prof: So, what changes for continuous treatments? 730 01:24:23.770 --> 01:24:29.080 David Stephens, Prof: Not much. In G estimation, Literally nothing changes. 731 01:24:29.800 --> 01:24:35.670 David Stephens, Prof: We replace the propensity model by this conditional expectation, and that gives all the same results as before. 732 01:24:36.390 --> 01:24:42.500 David Stephens, Prof: Right? So G-estimation, or Propensity score regression, in the continuous treatment case. 733 01:24:42.710 --> 01:24:44.949 David Stephens, Prof: Proceeds in exactly the same way. 734 01:24:48.370 --> 01:25:01.230 David Stephens, Prof: The only subtlety is that… We need to think carefully about handling functions of Z. 735 01:25:02.190 --> 01:25:04.510 David Stephens, Prof: That are not the identity function. 736 01:25:05.510 --> 01:25:07.639 David Stephens, Prof: Let me give you an idea of what I mean by that. 737 01:25:08.260 --> 01:25:10.240 David Stephens, Prof: If we had, in fact, that the 738 01:25:10.640 --> 01:25:15.550 David Stephens, Prof: data-generated model looked like this, and it was, in fact, dependent on Z squared. 739 01:25:17.970 --> 01:25:23.509 David Stephens, Prof: Or, in fact, we had a model that depended on Z squared and some interactions. 740 01:25:24.090 --> 01:25:26.060 David Stephens, Prof: Okay? Then… 741 01:25:26.620 --> 01:25:34.470 David Stephens, Prof: What we notice that this case can arise in the continuous treatment case, whereas it couldn't arise in the binary case. 742 01:25:34.650 --> 01:25:38.430 David Stephens, Prof: Because in the binary case, Z squared is always equal to Z. 743 01:25:39.360 --> 01:25:44.410 David Stephens, Prof: Right? So… There's no possibility of including these more complicated terms. 744 01:25:46.430 --> 01:25:47.120 David Stephens, Prof: Alright? 745 01:25:47.280 --> 01:25:50.399 David Stephens, Prof: So, in the continuous case, or indeed the multi… 746 01:25:50.520 --> 01:25:53.919 David Stephens, Prof: level treatment case. We do have to worry about things like this. 747 01:25:55.270 --> 01:25:56.350 David Stephens, Prof: So… 748 01:25:56.670 --> 01:26:03.380 David Stephens, Prof: We saw before that in the ordinary binary case, the propensity score could be used to block the appropriate pass. 749 01:26:03.670 --> 01:26:04.969 David Stephens, Prof: by conditioning. 750 01:26:06.640 --> 01:26:08.379 David Stephens, Prof: Does anything change here? 751 01:26:12.530 --> 01:26:20.510 David Stephens, Prof: Well… What might change is that the… We need… 752 01:26:21.120 --> 01:26:25.190 David Stephens, Prof: to build some separate blocking models. So, for example. 753 01:26:26.060 --> 01:26:35.739 David Stephens, Prof: Remembering the logic of the propensity score is that we're going to include conditioning on that term to block appropriate paths. Maybe it's the case that we need 754 01:26:35.990 --> 01:26:39.329 David Stephens, Prof: Different blocking or balancing models. 755 01:26:39.980 --> 01:26:42.439 David Stephens, Prof: for Z and Z squared separately. 756 01:26:42.890 --> 01:26:51.529 David Stephens, Prof: So in general, it turns out that we have to construct terms like this. So how do we do that? Well, we regress Z on X and Z squared on X. 757 01:26:51.780 --> 01:26:53.589 David Stephens, Prof: And that gives us these functions. 758 01:26:53.720 --> 01:26:57.060 David Stephens, Prof: So, in practice, that's still pretty straightforward. 759 01:26:58.830 --> 01:26:59.590 David Stephens, Prof: Okay? 760 01:26:59.820 --> 01:27:06.650 David Stephens, Prof: But this extension to the continuous treatment case means that we do have to handle terms like this. 761 01:27:08.160 --> 01:27:13.460 David Stephens, Prof: And these extended propensity ideas are necessary, in some cases, to do that. 762 01:27:20.550 --> 01:27:33.240 David Stephens, Prof: Now… For models without interaction, It turns out that blocking using the first of these is sufficient. 763 01:27:34.120 --> 01:27:41.509 David Stephens, Prof: So if, in fact, the data-generating model looked like this, so we have an effect of Z and effect of Z squared. 764 01:27:42.050 --> 01:27:46.350 David Stephens, Prof: Then this propensity model, would be sufficient. 765 01:27:49.630 --> 01:27:57.319 David Stephens, Prof: And the reason is… That… There is the balancing score blocking the appropriate path. 766 01:27:59.690 --> 01:28:03.870 David Stephens, Prof: In the case where we have Z squared also affecting Y, 767 01:28:04.350 --> 01:28:10.149 David Stephens, Prof: then the confounding path is blocked by the same B1, right? This says that Z 768 01:28:10.400 --> 01:28:13.779 David Stephens, Prof: causes Z squared, and Z squareds influence Y. 769 01:28:14.520 --> 01:28:16.919 David Stephens, Prof: But if we block this path here. 770 01:28:17.270 --> 01:28:20.210 David Stephens, Prof: then that's still the only path through X. 771 01:28:20.930 --> 01:28:24.519 David Stephens, Prof: So, conditioning on B1 is completely sufficient. 772 01:28:28.630 --> 01:28:34.689 David Stephens, Prof: this tag… This blocking argument tells us precisely what we need to do. 773 01:28:34.980 --> 01:28:39.110 David Stephens, Prof: In order to correctly… Get the average treatment effect. 774 01:28:40.870 --> 01:28:44.960 David Stephens, Prof: Whereas if we have an interaction, the situation changes. 775 01:28:48.350 --> 01:28:53.620 David Stephens, Prof: We need to fit this extended model, Which allows me to blog. 776 01:28:53.890 --> 01:28:59.630 David Stephens, Prof: the interaction term with X, and the interaction term with, 777 01:29:02.950 --> 01:29:04.979 David Stephens, Prof: Between z squared and X. 778 01:29:08.290 --> 01:29:23.390 David Stephens, Prof: So that's a subtle point, and this is where I'll break off for the lunch break to give you some time to think about this. The argument still comes down to blocking the right paths, right? And I didn't draw the DAG, but if we have an XZ interaction. 779 01:29:24.650 --> 01:29:30.969 David Stephens, Prof: Then, what we find is that the… there's an additional path that needs blocking. 780 01:29:32.460 --> 01:29:35.829 David Stephens, Prof: If there's a XZ squared interaction. 781 01:29:36.080 --> 01:29:38.969 David Stephens, Prof: Then we also need to find a way to block that path, too. 782 01:29:40.830 --> 01:29:43.229 David Stephens, Prof: So this is a subtle point that comes up. 783 01:29:43.660 --> 01:29:50.189 David Stephens, Prof: In the context of continuous treatments that we don't have to handle in the binary treatment case. 784 01:29:51.420 --> 01:30:02.090 David Stephens, Prof: We can use the same methods, based on the propensity score, But… We need to ad… Carefully. 785 01:30:02.280 --> 01:30:03.260 David Stephens, Prof: applied them. 786 01:30:08.950 --> 01:30:14.250 David Stephens, Prof: Okay, so I think I'll, how many slides have I got on this? Okay, yeah, so there are some… 787 01:30:14.400 --> 01:30:19.720 David Stephens, Prof: points to doing the risk probability weighting here. So I'll take a break there until 1.30. 788 01:30:19.950 --> 01:30:26.429 David Stephens, Prof: And when we pick up at 1.30, I'll come back to this idea of, 789 01:30:27.080 --> 01:30:30.220 David Stephens, Prof: 1.30, or on the half hour, 1.30 my time. 790 01:30:31.430 --> 01:30:41.339 David Stephens, Prof: come back to this idea of deploying G-estimation and inverse probability weighting in the continuous treatment case, and study what we need to do for 791 01:30:41.770 --> 01:30:43.340 David Stephens, Prof: Inverse probability weighting. 792 01:30:44.650 --> 01:30:48.570 David Stephens, Prof: That'll take, maybe 15-20 minutes. And then… 793 01:30:48.830 --> 01:30:56.739 David Stephens, Prof: Soon after that, we'll move on to considering some applications of these methods to real data and simulated data. 794 01:30:56.950 --> 01:30:58.899 David Stephens, Prof: Okay, so I'll break off now for half an hour.