Probability-proportional-to-size sampling explained

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

p _{i} to be selected to the sample when performing one draw. This p _{i} is proportional to some known quantity x _{i} so that p _{i} = (x _{i})/(\sum ^{N} _{i =1} x _{i}) .^{[1] [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] 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

p _{i} selection probability). If doing a without-replacement sampling, the schema can become more complex.

See also

• Bernoulli sampling
• Poisson distribution
• Poisson process
• Sampling design

Notes and References

1. Book: Model Assisted Survey Sampling . Carl-Erik Sarndal . Bengt Swensson . Jan Wretman . 978-0-387-97528-3 . 1992.
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.