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ST 432 Unequal Probability Sampling (Chapter 6)
This is an important topic for later in the semester. Unequal probability sampling
is very important to many topics including cluster sampling and two-stage sampling,
network sampling, and sampling of animals where detection is an issue. I emphasize that
these notes do not replace careful reading of text as they are too brief for that.
Why do we use Unequal Probability Sampling?
1. Unequal Importance of Sampling Units .
I am reminded of George Orwell’s famous quote from “Animal Farm”.
“ All animals are created equal but some animals are more equal than others” (☺).
Sampling units are often not of equal importance.
Example-If we were surveying the state then regions may not be equally important. In a
survey on coastal development the surveyor may want to give more weight to the Coast
vs the Piedmont and the Mountains.
2. Unequal Sizes of Sampling Units
Some units may be bigger and therefore should be sampled at a higher rate.
Example 1- If the sampling unit is a family then you might want sample proportional to
the size of the family.
Example 2- Hospital survey where individual hospital is the unit. Here it might make
sense to sample proportional to the number of beds in the hospital. The No. of beds is a
measure of size of the hospital.
Example 3- Business survey where an individual business is the sampling unit. Sample
proportional to some know measure of size (no of employees, gross sales etc).
Sampling With or Without Replacement.
With replacement- Hansen Hurwitz Theorem
In this case we assume that we are sampling with replacement from a population
{y1, y2, …,yN}and we define pi as the probability of drawing unit i on each draw. We then
have a sample {y1, y2, …,yn}. It is reasonably simple to show that
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