The SURVEYSELECT Procedure

Brewer’s PPS Method

Brewer’s method (METHOD=PPS_BREWER) selects two units from each stratum, with probability proportional to size and without replacement. The selection probability for unit i in stratum h is 2 upper M Subscript h i Baseline slash upper M Subscript h dot Baseline equals 2 upper Z Subscript h i. (Because selection probabilities cannot exceed 1, the relative size for each unit, upper Z Subscript h i, must not exceed 1 slash 2.)

Brewer’s algorithm first selects a unit with probability

StartFraction upper Z Subscript h i Baseline left-parenthesis 1 minus upper Z Subscript h i Baseline right-parenthesis Over upper D Subscript h Baseline left-parenthesis 1 minus 2 upper Z Subscript h i Baseline right-parenthesis EndFraction

where

upper D Subscript h Baseline equals sigma-summation Underscript i equals 1 Overscript upper N Subscript h Baseline Endscripts StartFraction upper Z Subscript h i Baseline left-parenthesis 1 minus upper Z Subscript h i Baseline right-parenthesis Over 1 minus 2 upper Z Subscript h i Baseline EndFraction

Then a second unit is selected from the remaining units with probability

StartFraction upper Z Subscript h j Baseline Over 1 minus upper Z Subscript h i Baseline EndFraction

where unit i is the first unit selected. The joint selection probability for units i and j in stratum h is

upper P Subscript h left-parenthesis i j right-parenthesis Baseline equals StartFraction 2 upper Z Subscript h i Baseline upper Z Subscript h j Baseline Over upper D Subscript h Baseline EndFraction left-parenthesis StartFraction 1 minus upper Z Subscript h i Baseline minus upper Z Subscript h j Baseline Over left-parenthesis 1 minus 2 upper Z Subscript h i Baseline right-parenthesis left-parenthesis 1 minus 2 upper Z Subscript h j Baseline right-parenthesis EndFraction right-parenthesis

For more information, see Cochran (1977, pp. 261–263) and Brewer (1963). Brewer’s method yields the same selection probabilities and joint selection probabilities as Durbin’s method (Cochran 1977; Durbin 1967).

Last updated: March 08, 2022