Welcome back as we continue our lecture four on designing two stage samples. And going on to consider two additional topics. One, design effects and their effect on sample size. And two, projecting standard errors and confidence intervals together for new designs. Now design effects, when projected, also can be used to help us to determine sample size in cluster sampling. So there's not just their value, their utility, in calculating standard errors, Projected standard errors, but also, it can have an important impact on our design by dealing with sample size. So, cluster sampling, as we know, increases variances by a factor that is that design effect. Our variances are going up by the factor 1+(b-1)roh and we now know how to project that, at least building on past information And choosing a value of b. And so, that design effect is a comparison to simple random sampling. Here's what we can do. We can offset this increase by increasing the sample size. That is, we're going to have larger variances because of this design effect compared to simple random sampling. And we can offset that by taking the cluster sample and inflating its sample size. That is what we're going to do is start with a simple random sample sample size and inflate it by a design effect. Now again, keep in mind that we are saving substantial money by doing cluster sampling. Either because our list, assembly cost are much smaller. We're only assembling lists of elements, for example, within selected clusters. Or, and/or, our travel costs are substantially reduced because we're only going to a sample of the clusters. And not to all of them, possibly in a simple random sample that spreads our sample out across all of our clusters. Or even a much larger number than we might do in cluster sampling. So, suppose that we start where we did before for simple random sampling. And say that we have decided in our design process that we're going after this proportion. We think it's around 0.4, and we want to have a 95% confidence interval, us and the client. The client and our team have negotiated around this and decided that a 95% confidence interval from 0.37 to 0.43 would be satisfactory. Now, that confidence interval, as we talked about before, translates into a margin of error, a plus or minus 0.03. So, we have taken the estimate, the 0.4 and added 0.03 and subtracted 0.03, that margin of error. And we know that margin of error is comprised of both a multiplier that we've derived from a theoretical distribution, a normal or a t-distribution and a standard error. If what we're doing is 95% confidence intervals and we've been using that normal distribution for that multiplier effect in our uncertainty statement, then we can take 1.96 or 2 and use that to work backwards from the margin of error to the standard error that's being implied. That is, in this particular case, 0.03 divided by 1.96. Well, 0.03 divided by 2, just to keep it round numbers that we can readily do in our heads. It says that we need a standard error 0.015 for this particular 95% confidence interval. That means that if we're going to calculate a simple random sample now, we now know what that sampling variance should be, it's going to be 0.015 squared. All we need to do then is to couple that with our element variance, S squared. So, if you recall, for a simple random sample size calculation, assuming that our population size is very large, we're not going to worry about adjusting it here. We didn't do that before because we thought that we had a very large population. So, we simply would take an element variance, S squared, and divide it by a projected or desired sampling variance, v sub d, but in our cases that v sub d is going to be that standard error squared. So, you'll see here in the numerator for our S squared we're going to use p times 1 minus p, roughly and approximately. It's satisfactory for these kinds of calculations. 0.4 times 0.6, and that's going to be divided by 0.015 squared. It gives us a sample size then, simple random sample, necessary sample size of 1066.67. Now, let's not worry about rounding at this point up or down, lets just keep that number with the decimals and add to this a consideration that deals with the cluster sampling. We know that if we were to draw a cluster sample of this size, 1066, our design effects would be such that our confidence interval limits would not be from 0.37 to 0.43, but wider. Because we would have an inflation in the variance. So, how are we going to bring those confidence intervals back into alignment? We're going to increase the sample size by a factor of the design effect, that width widening factor. Now, it's the design effect, not necessarily the square root of the design effect. So, mechanically then, if the cluster sample was projected to have a design effect of 2.18, 2.1795, just to keep that number there, the sample size for the cluster sample would be our simple random sampling, sample size 1066.67, times that design effect. And now, I've rounded the sample size, 2,325 is our sample size. All right, so we've got a way now to use the design effect to help us in understanding what we need to do in our cluster sample to counteract the effects of cluster sampling and obtain a sample size. We can take this use of design effects and this projection process one step further. Now, we've got the sample size, we can look and see what a 95% confidence interval would look like for our design under an alternative set of assumptions. Now, in this particular case, what I'm going to do is, if I were to use what we just did and work backwards, we're going to end up with the same confidence intervals, the ones that go from 0.37 to 0.43. But here, let's consider design B. Remember design alternative B? Where we had 60 clusters and 20 elements in each, a sample size of 1,200, a projected design effect of 1.575, 1.58. And what we would do to get a confidence interval from this kind of thing, would be to use that design effect as an adjustment on our simple random sampling variance. We all ready knew that was 0.0002. And we all ready calculated a varience then under this projection system of 0.000315. The 95% confidence interval then builds on this by taking 0.4 and adding and subtracting around 0.4, our margin of error to well, 1.96 times the square root of that, 0.000315. That, remember now, is the simple random sampling variance inflated to account for cluster sampling. And we see our confidence interval width goes from 0.365 to 0.435, very similar to what we had before. But remember, this is a different design effect. It's going to be a different outcome here than what we had before. All right, there we have it. So we've seen how to project standard errors and variances. We've seen how to project confidence intervals from our design. We've seen how to use projected design effects to calculate a sample size, a simple random sample size adjusted for clustering. Those are the basic features of the design process. There are others that are involved, but that's as far as we're going to be able to go in this particular course. And what we're going to do now is turn to a topic, also in the realm of cluster sampling, we want to relax this constraint that the cluster sizes all have to be the same size. We're going to deal with unequal size clusters in our next lecture before we wrap up things by talking about subsample size as well. So, join us in the next lecture, lecture five in unit three where we will deal with the real world. Where we have unequal size clusters. Thank you.