The SURVEYSELECT Procedure

### Sampford's PPS Method

Sampford's method (METHOD=PPS_SAMPFORD) is an extension of Brewer's method that selects more than two units from each stratum, with probability proportional to size and without replacement. The selection probability for unit i in stratum h equals
Phi = nh   [(Mhi)/(Mh ·)] = nh   Zhi

Sampford's method first selects a unit from stratum h with probability Zhi . Then subsequent units are selected with probability proportional to

[(Zhi)/(1-nh   Zhi)]
and with replacement. If the same unit appears more than once in the sample of size nh, then Sampford's algorithm rejects that sample and selects a new sample. The sample is accepted if it contains nh distinct units.

The joint selection probability for units i and j in stratum h equals where  where S(m) denotes all possible samples of size m, for m = 1, 2, ... , Nh . The sum Lm(ij) is defined similarly to Lm but sums over all possible samples of size m that do not include units i and j, and Sampford's method requires that the relative size Zhi be less than 1/nh for all units. Refer to Cochran (1977, pp. 262 -263) and Sampford (1967).