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 ${\displaystyle p_{i}}$ to be selected to the sample when performing one draw. This ${\displaystyle p_{i}}$ is proportional to some known quantity ${\displaystyle x_{i}}$ so that ${\displaystyle p_{i}={\frac {x_{i}}{\sum _{i=1}^{N}x_{i}}}}$.[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 ${\displaystyle p_{i}}$ selection probability). If doing a without-replacement sampling, the schema can become more complex.[1]^: 93