I'm looking at a published study that estimated the prevalence of a particular characteristic nationwide. The country is a large, sparsely populated country. The population is concentrated in three regions that have considerably different populations and are separated by large expanses that are uninhabited. The researchers used a multistage cluster sampling technique, and the first stage was to sample one of the three regions. To me this is intuitively unacceptable but I'm having difficulty describing the reasons. I would like to hear your opinion on this.
Amin -
You are correct. This makes no sense. It certainly will not do what it purports to do. One very different among three places cannot represent all three.
Let us now consider variance. The variance among the primary units is infinite. Thus the overall variance is infinte. See, for example, Cochran(1977), Sampling Techniques, 3rd ed, Wiley, page 278, where you see s1 will be infinite because n-1 is in the denominator.
There has to be some logic in a sampling plan. You are correct: This was "intuitively unacceptable," and unacceptable in fact. It seems to me that with just three regions, it makes more sense to sample each. Note that this would not necessarily mean stratified random sampling because the goal of stratified random sampling is to obtain a good overall number, and thus "optimal allocation" may not provide acceptably low standard errors for each stratum (or subpopulation). If you want that, you will likely need a larger overall sample size.
Cheers - Jim
PS -
"Basu's Elephant Fable" comes to mind here. You ran into a case where someone misused multistage sampling. In Basu's story, a "Circus Statistician" stops the circus owner from doing something "reasonable" (as I heard it described in the story, or probably from someone else's commentary on it), and then misuses unequal probability sampling, and makes a total disaster of estimating the elephants total weight (for transportation purposes, I think). Thus, Basu (an academician) says the Circus Statistician is fired, and then gets a job as an academician.
:-)
PSS -
Even if you are talking about a qualitative study - not my expertise - where you do not have numbers to show infinite variance above, it seems to me that the principle is still the same.
I think this demonstrates the difference between strictly Randomization Principle-based thinking and needing to do some model-based related thinking. Yes, if you choose a case at random, that is not biased in the sense of if there were repeated random sampling. But we only choose one case, and we know that they are different.
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