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Add heterogenous conditional

remember that each person has a unit level treatment effect or unit level causal effect and that's the difference between their potential outcome if they have the treatment minus their potential outcome if they didn't have the treatment and the average treatment effect is just the average of all of these unit level causal effects. Take each person look at their unit causal effect and you take the average. Now we can also define the conditional average treatment effect as the average for just some of those people. For example the conditional average treatment for men where you just look at the average unit causal effects for men or the conditional average treatment effect for women or you just look at the average for women. So over here I have an example in this population there are four people person one, two, three and four . There are two men and two women. And here I've got their potential outcomes, so in this column their outcome if they got treated this guy would have a 40, this guy a 20, this four men a ten and this person a 30. Over here is The potential outcome if they don't get treated. 30, 20, 15, 30. So this first person if he is treated he gets forty. If he isn't treated he gets thirty. So his unit causal effect is ten, 40 minus 30. The second person gets 20 with treatment and 20 without treatment it actually doesn't matter whether they are treated or not they are the same outcome. So their unit causal effect is zero, 20 minus 20. Third person is ten with treatment 15 without so their unit causal effect is minus five 10 minus 15. And finally our last person has 30 with treatment and 30 without. So she also has a unit causal effect of zero. Now remember later on we are gonna deal with the problem that in data we never see both of these numbers for the same person because each person can only either get treated or not get treated. So one of these numbers will always be hidden for us depending on whether these people are treated or not. That's called the fundamental problem of causal inference that I discussed earlier. And that's gonna make it very difficult cuz we're not gonna be able to see the unit causal effects from the data, and we're gonna try to figure out how can we learn about these things when we have data that only shows us one of the numbers from these columns? But before we get to that, we just wanna understand the kinds of numbers we can learn if we observe both of these columns. So one of those numbers is the average treatment effect, which is just the average of all these unit causal effects. So in this example, the average treatment effect is equal to the sum of these 10 Plus zero, plus minus five, plus zero, divided by four. Cuz there are four people in the population that we're averaging over. So it's gonna equal ten minus five over four. Five over four, that's our average treatment effect. Now let's look at the conditional average treatment effect for men. [BLANK_AUDIO] this is the same thing as the average treatment effect but only looking at the men and the population. So we throw out women. So that's gonna ten plus zero, divided by two because we have two people that we're averaging over, and that's gonna be ten over two or five. So finally. let's look at the conditional average treatment effect for women [BLANK_AUDIO] and that's gonna be just like with men, we're gonna look at the average only just the women and threw out the men. So that's gonna be minus five, plus zero, divided by two, cuz there are two people. So that's gonna be minus five, over two. So we see the average treatment effect is five over four positive, so usually we conclude treatment must be good. We look at men, we see it's even bigger, so the treatment is very good for men, but we would say for women, the conditional average treatment effect is negative so we would say actually the treatment is bad for women so what we see here and also from unit causal effects is treatment effect hydrogenating. In this example the treatment affects different people differently. This person benefit , this person is hurt and both of these people it doesn't matter whether they are treated or not. Now when we go to the data and later on in these models we're gonna see how we learn this numbers from real data. Here I computed it but that's because I assumed I knew both columns and we're not gonna know both columns. So how do we learn this numbers and there are batch of other kinds of averages you can take and we will talk about later on for example there's something called late, a local treatment effect will talk about this later no pun intended. So those are Average Treatment Effects and conditional Average Treatment Effects. [MUSIC]