Probability-proportional-to-size sampling

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In survey methodology, probability-proportional-to-size (pps) sampling is a sampling process where each element of the population (of size N) has some (independent) chance to be selected to the sample when performing one draw. This is proportional to some known quantity so that . [1] :97 [2]

One of the cases this occurs in, as developed by Hanson and Hurwitz in 1943, [3] is when we have several clusters of units, each with a different (known upfront) number of units, then each cluster can be selected with a probability that is proportional to the number of units inside it. [4] :250 So, for example, if we have 3 clusters with 10, 20 and 30 units each, then the chance of selecting the first cluster will be 1/6, the second would be 1/3, and the third cluster will be 1/2.

The pps sampling results in a fixed sample size n (as opposed to Poisson sampling which is similar but results in a random sample size with expectancy of n). When selecting items with replacement the selection procedure is to just draw one item at a time (like getting n draws from a multinomial distribution with N elements, each with their own selection probability). If doing a without-replacement sampling, the schema can become more complex. [1] :93

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References

  1. 1 2 Carl-Erik Sarndal; Bengt Swensson; Jan Wretman (1992). Model Assisted Survey Sampling. ISBN   978-0-387-97528-3.
  2. Skinner, Chris J. "Probability proportional to size (PPS) sampling." Wiley StatsRef: Statistics Reference Online (2014): 1-5. (link)
  3. Hansen, Morris H., and William N. Hurwitz. "On the theory of sampling from finite populations." The Annals of Mathematical Statistics 14.4 (1943): 333-362.
  4. Cochran, W. G. (1977). Sampling Techniques (3rd ed.). Nashville, TN: John Wiley & Sons. ISBN   978-0-471-16240-7