Probability-proportional-to-size sampling

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 [math]\displaystyle{ p_i }[/math] to be selected to the sample when performing one draw. This [math]\displaystyle{ p_i }[/math] is proportional to some known quantity [math]\displaystyle{ x_i }[/math] so that [math]\displaystyle{ p_i = \frac{x_i}{\sum_{i=1}^N x_i} }[/math].

One of the cases this occurs in, as developed by Hanson and Hurwitz in 1943, 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. 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 [math]\displaystyle{ p_i }[/math] selection probability). If doing a without-replacement sampling, the schema can become more complex.

See also

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