We mentioned degrees of freedom in the last episode of this Basic Estimation module. So I want to say a bit more about that in this video. Degrees of freedom are associated with a variance estimator. And they are related to the stability of a variance estimator. So as the degrees of freedom increase, the precision of variance estimator increases, which just means the variance of the variance estimator decreases. Precision goes up, its variance goes down, that's the conventional jargon. Now there are rules of thumb, right, a rule of thumb that's used by all software packages. And what it is is you take the sum over strata of nh -1, where nh is number of primary sampling units selected in stratum h. So that's the total number of PSUs minus the total number of strata. Now if you use the degrees of freedom approximation, it's largely determined by the number of first stage units in the sample. So that may be fairly crude. The number of sample units within each PSU, notice, doesn't enter into that rule of thumb at all. So it's as if you don't get credit for drawing a big sample within various PSUs when you compute these degrees of freedom. Now, this rule is not always accurate, but it's easy to apply and that's why the software packages use it. So, how does this enter into confidence interval construction? A 1- alpha level confidence intervals for some quantity theta is computed this way. We take the estimator, theta hat and then we add and subtract a multiple of the standard error of that estimator. So right here the variance will be if theta hat is the symbol for the variance of theta hat. I take the square root of that and that's the estimated standard error. And then the multiplier I use is out of a t distribution. So, t of 1- alpha over t, or 2 per df is the upper percentile of a t distribution. So that 1- alpha / 2 is the area under the distribution is less than that value. And then the alpha over two is above it. And its 4 central t distribution that's got df degrees of freedom. Now, when is this valid? You need a large sample of first-stage units to get this working right. Essentially what's being said is if I were to graph the distribution of my estimator theta hat, it's going to be symmetric like this. It ideally is centered around the value theta, And then this 1- alpha / 2 percentile is up here somewhere. So in this area of the distribution up here, that constitutes or the amount of area in that is alpha / 2. So for example if you're doing a 95% confidence interval, you want 2% of the area in lower and upper symmetric interval. So, this would be the 97 and a half percentile, right there. On the distribution. That's the idea. And when does that happen? It happens when you got a large sample, first stage units. So, it's central limit theorem sort of result begins to take hold. So let's look at some easy examples. First think about a single stage sample with 150 schools. So there is no story there. I just subtract one to get my degrees of freedom rule of thumb, that's 149. The 97th and a half percentile of a t distribution with 149 degrees freedom is 1.976. Well, that's really, really close to the standard normal number 1.96. So very little difference, we've seen a normal when you get that many degrees of freedom. Now let's take another example. Let's suppose I've got a stratified single stage sample of establishments. I got three strata with 25, 45 and 75 establishments in the 3. So I subtract 1 from each of those numbers and add them up. So I get 24 plus 44 plus 74 is equal to 142 degrees of freedom. And if I look at the t value associated with that, well that's essentially the normal 1.96 [INAUDIBLE. So, 11 degrees of freedom. And that approximation looks just like the standard normal approximation. Now, suppose I've got a multistage sample with 10 strata, 2 PSUs per stratum, and I pick 50 households per stratum. And suppose I do that by simple room sampling without replacement but I select the PSU's probability proportional to size. Well, the rule-of-thumb, for degrees of freedom doesn't distinguish between PPS sampling and equal probability sampling. So what I do is I just take total number of PSUs, that's 20, minus number of strata, that's 10. And here's my rule-of-thumb degrees for freedom, 10. On the other hand, how many households did they have? I've got 10 times 2 times 50. That's 100 households. So my degrees of freedom number is 10, which is a whole lot less than 100, or 100 minus 1, or 99. And then if I look at the t value for 10 degrees of freedom, the upper 97 and a half percentile, I get 2.28. And that is quite a bit different from the 1.96 for standard normal. So the rule of thumb can lead to fairly big change in how you'd compute a confidence interval. Using this 2.28, will make a confidence interval that's wider then 1.96, reflecting that additional uncertainty due to the fact that your variance estimator is unstable.