Sampford's PPS Method

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

P_hi = n_h [(M_hi)/(M_{h ·})] = n_h Z_hi

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

[(Z_hi)/(1-n_h Z_hi)]

and with replacement. If the same unit appears more than once in the sample of size n_h, then Sampford's algorithm rejects that sample and selects a new sample. The sample is accepted if it contains n_h distinct units.

The joint selection probability for units i and j in stratum h equals

$P_{h(ij)} = K_h \lambda_i \lambda_j \sum_{t=2}^{n_h} \biggl[ t - n_h (P_{hi} + P_{hj}) L_{n_h-t}(ij) \biggr] / n_h^{t-2}$

where

$\lambda_i = \frac{Z_{hi}}{1 - n_h Z_{hi}}$

$L_m = \sum_{S(m)} \lambda_{i_1} \lambda_{i_2} ... \lambda_{i_m}$

where S(m) denotes all possible samples of size m, for m = 1, 2, ... , N_h . The sum L_m(ij) is defined similarly to L_m but sums over all possible samples of size m that do not include units i and j, and

$K_h = ( \sum_{t=1}^{n_h} tL_{n_h-t}/n_h^t ) ^{-1}$

Sampford's method requires that the relative size Z_hi be less than 1/n_h for all units. Refer to Cochran (1977, pp. 262 -263) and Sampford (1967).

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