View source for Probability-proportional-to-size sampling

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*[[Pps sampling]]
*[[Probability proportional to size sampling]]
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In [[survey methodology]], '''probability-proportional-to-size (pps) sampling''' is a [[sampling (statistics)|sampling]] process where each element of the [[statistical population|population]] (of size ''N'') has some ([[statistical independence|independent]]) chance <math>p_i</math> to be selected to the sample when performing one draw. This <math>p_i</math> is proportional to some known quantity <math>x_i</math> so that <math>p_i = \frac{x_i}{\sum_{i=1}^N x_i}</math>.<ref name="sarndal1992">{{cite book
 |title = Model Assisted Survey Sampling
 |author=Carl-Erik Sarndal |author2=Bengt Swensson |author3=Jan Wretman
 |isbn= 978-0-387-97528-3
 |year = 1992
}}</ref>{{rp|97}}<ref>Skinner, Chris J. "Probability proportional to size (PPS) sampling." Wiley StatsRef: Statistics Reference Online (2014): 1-5. ([https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118445112.stat03346.pub2 link])</ref>

One of the cases this occurs in, as developed by Hanson and Hurwitz in 1943,<ref>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.</ref> 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.<ref name = "Cochran1977">Cochran, W. G. (1977). Sampling Techniques (3rd ed.). Nashville, TN: John Wiley & Sons. {{ISBN|978-0-471-16240-7}}</ref>{{rp|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&nbsp;''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 <math>p_i</math> selection probability). If doing a without-replacement sampling, the schema can become more complex.<ref name="sarndal1992" />{{rp|93}}

==See also==
*[[Bernoulli sampling]]
*[[Poisson distribution]]
*[[Poisson process]]
*[[Sampling design]]

==References==
{{Reflist}}

[[Category:Sampling techniques]]

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