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Excerpt from Sampling With Probability Proportional to Size From a Sorted List
Sampling with unequal probabilities of selecting a sample unit frequently can result in increased efficiency in estimating population totals. The technique requires that each unit in the population be characterized by a size variable that is correlated with the variable of real interest to be measured for that unit. If a list of the size variable can be obtained cheaply for all units in the population, then sampling with probability proportional to size (pps) according to the method developed by Hartley (1966) can be more efficient than pps sampling by other methods. How much more efficient depends on the character of the functional relation between size and the variable of real interest.
The distinguishing feature of Hartley's method is that the units of the population are sampled at uniform intervals of accumulated size from a random start in the list of units sorted into order by size. Hence, the sample that is drawn is more likely to contain both small and large units than would a pps sample drawn from a randomly ordered list. Hartley has provided an expression for the true variance of estimates based on this method of selecting the sample. However, the sample estimate of this variance that he suggests is based on the assumption that the ratio of the variable of interest to the size variable is a linear function of the size variable. In a subsequent part of this paper, the assumption concerning the linearity of this relation is relaxed to include polynomial functions of the size variable.
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