Money, money, money. That's what we were talking about before, although it didn't seem that way. Sometimes people will say about this particular topic, it's not very practical. And I think they're getting bound up in formulas and things and they miss the point of why these things are being done. So here we are in lecture one of unit three, in which we're talking about saving money. We're talking about complex sampling now, cluster sampling. But a simple form of it, where the clusters are all equal in size, and they also have the property that when we sample them, we're going to take all the elements that are there on a given sample cluster. But we had looked at this particular problem within the context of our lecture, so we're still on lecture one here. But now what we're going to do is move back to, if you recall, our sampling distribution problem and see the consequences of dealing with this. And so what we're going to do is expand our example and talk a little bit about a different example and add some additional notation for a second and change the example. So we've been doing 18 blocks, right, with eight housing units each in our previous illustration and drew a sample and then looked at the different visual representations of what the sample would look like if we had drawn the elements directly, simple random sampling, versus drawing the elements only after having selected clusters. Let's try another Illustration, something that involves schools. To illustrate something that happens for the calculations here. And I borrow something from the work of a colleague. It's a slightly different of what's been done, but it has to do with sampling clusters that are equal in size. And suppose that we've got a population of 1000 clusters. Not 18, but 1000. So it's a bigger problem. And we've decided that what we can afford to do is draw a sample of ten of those and then go and collect all data about all the elements within them. And so I put this in the context of the following. Suppose that we're doing work for a school district, a geographical management area where there's a set of schools. And it's a fairly large school district, and they're interested in studying what's going on in elementary school classrooms, the early year classrooms, with respect to the immunization history of the children. They've had some concerns about outbreaks of childhood diseases among the student population, and they've decided they want to do a sample and go in and make sure they understand what fraction of their children are fully immunized. Now they could do this at the beginning of the school year and require the parents to bring the records, but they've tried doing that and it gets to be quite complicated. The parents can't find the records, they've got a lot of data to manage, and in this case, they've got a thousand classrooms. They have a list of those thousand classrooms, but they don't have a list of the children within them. So again, they face the problem. They could go and try to build the list of the children, but that would take a bit of time. And so they've decided to sample ten of the classrooms and then go and examine all of the children within those classrooms. Now, there are 24 children in each classroom. So in the population, there are 24,000 children in this grade level. But there are in the sample then 240, 1 in 100, one 100th of the population will be in the sample. One-tenth, no, one 100th of the classrooms. And then taking all of the students within. Now we're going to refer to the capital A here, that 1,000, as those classrooms as primary sampling units. Because in the previous example, we had blocks. Those were the primary sampling units. They're the sampling units at the first stage of selection. So the PSUs as they're going to be referred to. In their classrooms, as shown in the lower left illustration here. Now, when we've drawn the sample, we go and make our measurements. And in this particular case, the record shows the following. And I've organized this, or this illustration's organized by classroom. Here are the ten sample classrooms, the fractions of each classroom where children are fully immunized. Just put them in order. This isn't in the selection order, this is the order by the fraction from lowest to highest. 9 in 24, about one-third of the kids are immunized, to 21 in 24, where virtually all of them are immunized. Now, from our point of view, what we want to know is what's the overall fraction. We don't want to know by classroom. But this is what's coming about because of our randomized selection. We selected classrooms, this is what the classrooms looked like. To do our estimates, we don't have to worry about the classrooms. We can just add up the numerators there. So the 9, plus 11, plus 13, plus 15 and so on. And that sums up to 160 immunized children. And that's out of 240. So our rate, proportion immunized is 0.67, two-thirds. Now, that number that we've got is our sample estimate. We're going to use that to project to the population for dealing with immunizations in this particular case. This kind of cluster sampling, one stage if you will, or simple cluster sampling taking all. That estimate that we just computed is unbiased. The process is unbiased. Let me say that in a little different terms. The mean in this case is unbiased for the population mean. Because this particular process is unbiased from the mean. So, on average, if we did this again and again, do all possible samples of ten classrooms from the 1,000 and computed this proportionate in all cases, a mean, and averaged them across all possible samples, we would get the true population value. Now, that's also true of simple random sampling. Remember, for simple random sampling without replacement. Now we're going to build the whole list of the 24,000. Expensive operation for us. And compute the sample proportion. And that sample proportion we know, when we sample 240 at random, without replacement from the 24,000, will be unbiased. And we also know from that previous work that there is sampling variance that goes with it. And we can estimate the sampling variance from just the sample data. And as string formulas there, three expressions. The sampling variance, lowercase var of the proportion, is that 1-f s squared over n. I guess we should have put the lowercase n under the 1-f. We are familiar with that kind of an expression. And to the right of it, I wrote the version of that formula, the algebraic equivalence, that involves proportions. That will just be more convenient for us to use. That involves the p times 1 minus p divided by n minus 1 times 1 minus f. Now, again, these are, you recall, sort of two-step processes. We took the definition of the sampling variance, did some algebraic manipulation to get the 1 minus f capital S squared over n. And then said, but we don't know capital S squared, but we can estimate it from the data 1 minus f lowercase s squared over n. In the last step, by the way, lowercase s squared for proportions can be written as p times 1 minus p times n over n minus 1, leading to that last expression. That's the long story of what that represents. So we just put it in compact form so it didn't have to have multiple slides here. Now, what we have is not this. What we have is a simple random sample of 10 equal sized clusters from capital A equal 1000. And our randomization, as we noted, occurs at the cluster level. Now, the formula that we see there, the middle for the estimated sampling variance for simple random samples, is 240 random events. Here, we have 10. And so now, because the randomization occurs at that level, we're going to deal with the p's, the characteristics of the clusters. Well, what is that? For the child it's immunized or not immunized, a one or a zero. For the cluster it's the fraction of the children who are immunized or the proportion. So we have a P now sub alpha, alpha for each of the clusters. A, going with capital A, number of clusters in the population, lower case a, number of clusters in our sample. Alpha to index the clusters going from 1 to 10 in our particular sample. So those 10 fractions that we've got. Unlike in simple random sampling and cluster sampling, we're going to treat this sample as a sample of lowercase a from capital A. Lowercase a is the number of sample clusters, in our case, 10, from capital A, 1,000. It's a simple random sample. And when we do that, the sampling variance. We go back through all the things we talked about before for the sampling variance as we've seen. Here, this is the sample analog in which we have, expression involves the expression lowercase var of our proportion, as opposed to the capital VAR that we were looking at before. And we have three similar elements to what we had before for simple rand om sampling. A finite population correction, 1-f. A divisor by the number of random events in the sample. In this case, the sample size is 10, not 240, but 10 clusters. That's where the randomization, that's all the random numbers we used. And then that gets multiplied by an s squared. But in this case, that s squared is built around those cluster proportions that we just introduced in the last slide, p sub alphas. And their variability relative to their average. The overall proportion happens to be the average of those individual cluster proportions. So we compute that variability, dividing by a-1. There are a differences, square differences, there. And it's an s sub a squared that we're dealing with. And we've put it into our sampling variance formula. It's a bit of a mess. It's got our 1-f over a, and our s sub a squared expression. But when we want to move on to talk about comps intervals, standard errors, we just take the square root of that sample variance. The same thing that we've done before. Now, just to illustrate the calculations, to go back to the results that we had before, our s sub a squared, I put it out in front now, the 1 divided by 10 minus 1. That's 1 divided by the 10 random events minus 1. And then inside of the big square brackets, there's the fraction in the first stratum, versus the fraction overall. That's for the first of the clusters. The fraction in the second cluster compared to the fraction overall, in each case squared, and so on. And we get a number 0.02816. Not clear what that number means until we go two steps further with it. If we put that 1 minus s sub a squared then into our expression for 1 minus f, let's see, f is 0.01, right, 1% of the population is in the sample. So that's 0.99 times that s sub a squared, divided by 10. We get a number that is 0.002760, which is our sampling variance of the mean. But that's on the squared scale. On the original scale, when we take the square roots to get our standard error, we see that our standard error is 0.05, or 0.0525, but 0.05, now that relates directly to our estimate of the fraction of children who're immunized, that 0.67. This is the standard error on that estimate of 0.67. It's on the same scale now. We can envision it. We can even think about a confidence interval that goes with that. So that we can take 1.96 times that 0.05, add and subtract it from the 0.67. We'd have a confidence interval that goes from about, well, let's see, 2 times 0.05 is 0.1. This goes from about 0.57 to about 0.77. Now we can relate it to the actual estimate for our cluster sample. But there's an important adjustment that we need to make to this. We have only ten degrees of freedom here, ten random events. The 240 in the sample, that was a much larger number. When we did our first samples of size 20, it was a little bit larger number. Here we're going to introduce the idea of a different multiplier in these confidence intervals. We were using for example the 95% confidence interval the z-value, the value of 1.96 that I just cited. Which is appropriate for a type one error rate of 5%. Well, here because the degrees of freedom are so limited, that turns out, that 1.96 was only appropriate if we had large sample sizes. Now, how large is large is an interesting issue. For this particular purpose, it's at about 100 where things start to make the most sense. The 1.96 works just fine. But when you get down to smaller numbers, certainly 10, we've gotta make an adjustment here. Because what happens is that the s sub a squared is introducing so much variability because it's built around only ten square differences. It may sound like a lot to you, but from the point of view of the sampling distribution, it's problematic. That normal distribution doesn't work that well. And so they introduced the idea of a t-distribution. Now, I don't have time to go into where the t-distribution comes from. Actually, if you're interested in it, a little bit of homework. I suggest, and you might want to write this down, you look up a little bit of the history about the t-distribution, because it involves beer. It involves the Guinness brewery in Dublin, Ireland. And a man there named William Gosset who was faced with this kind of a problem. Small sample size. He was doing testing of beer of all things. And it led to the development of the t-distribution. So that's a separate issue here. But it has to do with small sample sizes. And we have small sample sizes here. So we're going to modify this. The confidence interval built on a standard error really does depend on the number of random events for its stability. And in this case, it's not a lowercase n, it's lowercase a. And so our degrees of freedom is really much smaller than our sample size. And as a result, we're going to use the t-distribution, because it takes into account that added variability, having fewer random events in the sample. So that t-distribution now has two things. The normal distribution, the Z, had just one. What's your type one error rate? is it 0.05? Okay here's the number, 1.96. Is it 0.1? Here it is, 1.64. So there's a specific number that goes with that regardless of the sample size. But for the t, we have to do, not only that error rate, but also the sample size. And that sample size is referred to as degrees of freedom in this particular case. It's one application of that term if you've heard it before. But for our purposes, we think of degrees of freedom as random events in our sample, in this case, lowercase a. So that that t is substituted, as you can see in our expression here in red, in the very first line, that t is introduced as the multiplier instead of the normal. And so we have our proportion minus t with 1 minus alpha over 2 and a minus 1 degrees of freedom times that standard error. That's the lower limit and the corresponding upper limit. It's the same proportion, it's the same standard error, but the multiplier, the t-distribution, is changed. And we end up with a t-value that is larger than the corresponding normal value, the Z value. So for 100 degrees of freedom, for example, if a were 100, that t-value's going to be around 1.98, 1.97, in that range. Not quite 1.96. So it inflates this a little bit. And it depends on how far you're willing to depart from what that underlying value is. That's why actually some people in their confidence intervals for 95% confidence intervals, will just use a multiplier of 2. They round that 1.96 up. They don't really worry that much about degrees of freedom. That's actually, when we talked about margin of error, what happens with the margin of error kind of calculation. Okay. So for our case, just to finish off the illustration and the calculation, the 95% confidence interval involves our 0.67. And it involves as well our t-value and our standard error. Oops, and I can see we have a typo here. This should be 0.05 as our standard error for both of these. It's right on the right-hand side, but wrong on the left-hand side. And when we do the calculation, the t-value here is not 1.96, but with 9 degrees of freedom, 10 minus 1, it's 2.26. So, about 12%, 13% larger than the normal value. So these things are going to be bumped out. And we can see now our confidence interval goes, same calculation, from 0.55 to 0.79. And that's a little wider confidence interval that also takes into account this added variability due to the fact that we have limited degrees of freedom, limited number of random events in our sample. Okay, this is the same thing we did for simple random sampling, right? We did a calculation of an estimate. Then we calculated a sampling variance for that estimate using our 1-f s squared over n. And then a standard error and a confidence interval. Same thing here for cluster sampling, but what we've done is change how we calculated that standard error. Okay, that's the basics for the cluster sampling. The type we're doing, the simple complex sampling, equal size clusters. How does it compare to what happened for simple random sampling? And so in our next lecture, what we're going to do Is a comparison, a contrast of what would happen if we had done a simple random sample to what we're doing with a cluster sample. And so join me again then for lecture two as we continue our consideration of complex or cluster sampling, and saving money. Thank you.
