|The SURVEYSELECT Procedure
If you specify the option METHOD=PPS, PROC SURVEYSELECT selects
units with probability proportional to size and without replacement.
The selection probability for unit i in stratum h equals
nh Zhi . The procedure uses the Hanurav-Vijayan algorithm
for PPS selection without replacement. Hanurav (1967) introduced
this algorithm for the selection of two units per stratum, and
Vijayan (1968) generalized it for the selection of more than two units.
The algorithm enables computation of joint selection probabilities
and provides joint selection probability values that usually
ensure nonnegativiy and stability
of the Sen-Yates-Grundy variance estimator.
Refer to Fox (1989), Golmant (1990), and Watts (1991) for details.
Notation in the remainder of this section drops the stratum
subscript h for simplicity, but selection is still done
independently within strata if you specify a stratified design.
For a stratified design,
n now denotes the sample size for the current stratum,
N denotes the stratum population size, and Mi denotes the
size measure for unit i in the stratum.
If the design is not stratified, this notation applies
to the entire sampling frame.
According to the Hanurav-Vijayan algorithm, PROC SURVEYSELECT
first orders units within the stratum in ascending order by
size measure, so that
.Then the procedure selects the PPS sample of n observations
- The procedure randomly chooses one of the integers
1, 2, ... , n with probability , where
Zj = Mj / M, ,
and, by definition, ZN+1 = 1/n to ensure that
- If i is the integer selected in step 1, the
procedure includes the last (n-i) units of the stratum
in the sample, where the units are ordered by size measure
as described previously. The procedure then selects the
remaining i units according to steps 3 through 6 below.
- The procedure defines new normed size measures for
the remaining (N-n+i) stratum units that were not
selected in steps 1 and 2,
- The procedure selects the next unit from the first (N-n+1)
stratum units with probability proportional to aj(1), where
and Pk = Mk / ( Mk+1 + Mk+2 + ... + MN-n+i ) .
- If stratum unit j1 is the unit selected in step 4, then
the procedure selects the next unit from units j1+1 through
N-n+2 with probability proportional to aj(2,j1), where
- The procedure repeats step 5 until all n sample
units are selected.
If you request the JTPROBS option, PROC SURVEYSELECT computes
the joint selection probabilities for all pairs of selected units
in each stratum. The joint selection probability for units i and
j in the stratum equals
where Pk = Mk / ( Mk+1 + Mk+2 + ... + MN-n+r ) .
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.