The SURVEYSELECT Procedure |
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
![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}](images/suseq20.gif)
where
![\lambda_i =
\frac{Z_{hi}}{1 - n_h Z_{hi}}](images/suseq21.gif)
![L_m =
\sum_{S(m)} \lambda_{i_1} \lambda_{i_2} ... \lambda_{i_m}](images/suseq22.gif)
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
![K_h =
( \sum_{t=1}^{n_h} tL_{n_h-t}/n_h^t ) ^{-1}](images/suseq23.gif)
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).
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.