So, last time, we introduced potential outcomes notation which is the key thing. Then, we used that notation to define unit in average treatment effects, and we briefly discussed unbiased estimation of average treatment effects. So, this time, I want to go on and talk about randomized experiments. As these are the bridge to observational studies, we need to understand these. So, that's what we're going to do in this module. Okay. So, the first thing we're going to do in the first lesson, we'll talk about some randomized experiments. In the second lesson, I'm going to introduce you to randomization-based inference. You might not be familiar with it, and in particular, how to test the null hypothesis of no effect. Then in lesson three, we're going to go on and talk about estimation of the sample average treatment effect. So, let's start with the randomized experiments. The key to understanding the differences between these randomized experiments are the rules by which subjects are assigned to either a treatment group and control group. Now, in order to set this up properly, we're going to require some notation. So, let me introduce that first. The units i equals one through n, you should be familiar with that. Then I have this row vector which consists of Zi which is the treatment assignment of unit i. You'll notice that's a random variable, and Xi which is a column vector of covariates more pre-treatment characteristics associated with unit i. Then I have Yi zero and Yi one which are i's potential outcomes. I'm going to group all the assignments into a column vector z and then I'm going to let x, y zero, y one as follows denote the corresponding matrix with rows corresponding to the units. Now, as in module one, we're going to be interested in the relationship between treatment assignment, potential outcomes and covariates. So, equation one writes the entire treatment assignment vector given x, y zero, and y one, given all the x's and y zeros, and ones for all the n units. Then, Z sits in some set, which, since you can be treated or not treated, which is a Cartesian product of 01 to the nth. So, z sits in that set omega. Of course, it may be for some of these rules. You don't see all of that. So, that's why omega is a subset and it can be a proper subset. We are going to be interested primarily in assignment rules that have the form below. Let me go through and talk about each one of those things. Okay. So, the first equation on the left of course is the probability for that particular assignment and the set omega. The next equality says that we can write it in this way which we call sort of individualistic assignment, which consists of the idea that, as you can see, that the assignment of unit i to either the treatment or control group, now depends only on i's covariates in potential outcomes. So, you'll see that differs in that equation than from the one we started with. Because the y zero and y one and x place respectively with Xi, Yi zero, and Yi one. So, we're saying that, and then we're also saying we can write the entire assignment rule as the product of these individual assignments, suitably normalized by a constant k. Then, when we go to the second line, you'll notice now that the probability that Zi equals Xi which did depend on Xi, Yi zero, and Yi one, now only depends on Xi. So, now, we're making the assumption that treatment assignment is unconfounded. Now, we'll write that to form the last thing, where that's just Bernoulli, e to the Xi to the Zi, but the important point, that's just re-writing it. But the important point is that, we're going to require the probability that z is one is strictly between zero and one for all the units. In other words, each unit can be potentially exposed to either treatment or its absence. So, I've just sort of gone through what those things mean. When equation two holds equivalently conditions one, two, and three, we say following the terminology introduced by Rosenbaum and Rubin in their seminal article in 1983, a treatment assignment is strongly ignorable given covariates. This E is the so-called propensity score. It's a probability being treated given the covariates. Right now, it's not going to be so important but it's going to be super important later on. Let's consider some simple experiments. First of all, the Bernoulli experiment. You're just tossing a coin with probability of heads equal to Lambda, and you're doing it independently for each subject i. Omega consists of all possibilities and k is equal to one before, and each unit has the same known probability of assignment Lambda. So, propensity scores is same. For all units, it's Lambda and it doesn't really depend on x. So, that's a very sort of simple starting point. Now, another thing that we're going to be very interested in, is so-called Completely randomized experiment. Here, the investigator decides beforehand that n one of the n units will be treated, and of course, n minus n one equals n naught will be on treated. So, omega's now can consist of all possible assignments in which n one of the subjects are treated. Now, in this completely randomized experiment, we will furthermore going to say all the possible assignments are equally likely. Then of course, you can see that the e of Xi is n one over n for all i, and again, doesn't depend on covariates. Now, we come to an extension of the completely randomized experiment called the randomized block experiment. Now, the n units are grouped into strata or blocks on the basis of the covariates, the Xi's. Within each block, we're going to conduct a completely randomized experiment. So, we just need a little bit of notation. Let f be the function that's mapping in the covariates into the stratum s and there are capital S of these guys. Then, within each stratum, instead of n, now we have n sub s units and n sub s one that are assigned to the treatment group. Now, we're going to have to change the notation a little bit. So, Zsi prime, which is the i-prime unit within stratum s is one if assigned to treatment zero otherwise. Now, for stratum s, we're going to have a column vector z sub s, all assignments in that stratum, and we're going to have omega sub s which is all possible assignments in stratum s with n sub s one subjects treated. Now, omega will be the Cartesian product of all these sets, and the probability of any of these vector Z1 and Z1 and Zs given all of the x's is just the product of all the within stratum which are equally likely. Then, the propensity score is n sub s one over n sub s. Now, that differs from the completely randomized experiment and that the propensity score now depends upon the covariates. Okay. Another kind of randomized experiment that I'd like to introduce, is a paired randomized experiment. In that case, it's randomized block experiment, where within each stratum, there are two subjects. One of which is assigned to the treatment group and one of which is assigned to the control group. This typically arises when an investigator groups units into pairs matched on the basis of their closeness on covariates, then assigns each member of the pair to receive treatment with probability one half. So, those of you who remember your elementary statistics course and the t-tests for match pairs, will find this pretty familiar going.
