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 is n Subscript h Baseline upper M Subscript h i Baseline slash upper M Subscript h dot Baseline equals n Subscript h Baseline 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 n Subscript h.)

Sampford’s method first selects a unit from stratum h with probability upper Z Subscript h i. Then subsequent units are selected with probability proportional to

lamda Subscript h i Baseline equals upper Z Subscript h i Baseline slash left-parenthesis 1 minus n Subscript h Baseline upper Z Subscript h i Baseline right-parenthesis

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

If you specify 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 stratum h is

upper P Subscript h left-parenthesis i j right-parenthesis Baseline equals upper K Subscript h Baseline lamda Subscript h i Baseline lamda Subscript h j Baseline sigma-summation Underscript t equals 2 Overscript n Subscript h Baseline Endscripts left-parenthesis left-bracket t minus n Subscript h Baseline left-parenthesis upper Z Subscript h i Baseline plus upper Z Subscript h j Baseline right-parenthesis right-bracket upper L Subscript h comma left-parenthesis n Sub Subscript h Subscript minus t right-parenthesis Baseline left-parenthesis ModifyingAbove i j With bar right-parenthesis right-parenthesis slash n Subscript h Superscript t minus 2

where

upper K Subscript h Baseline equals 1 slash sigma-summation Underscript t equals 1 Overscript n Subscript h Baseline Endscripts left-parenthesis t upper L Subscript h comma left-parenthesis n Sub Subscript h Subscript minus t right-parenthesis Baseline slash n Subscript h Superscript t Baseline right-parenthesis
upper L Subscript h comma m Baseline equals sigma-summation Underscript upper S Subscript h Baseline left-parenthesis m right-parenthesis Endscripts lamda Subscript h i 1 Baseline lamda Subscript h i 2 Baseline midline-horizontal-ellipsis lamda Subscript h i Sub Subscript m

and upper S Subscript h Baseline left-parenthesis m right-parenthesis denotes all possible samples of size m, for m equals 1 comma 2 comma ellipsis comma upper N Subscript h Baseline . The sum upper L Subscript h comma m Baseline left-parenthesis ModifyingAbove i j With bar right-parenthesis is defined similarly to upper L Subscript h comma m but sums over all possible samples of size m that do not include units i and j. For more information, see Cochran (1977, pp. 262–263) and Sampford (1967).

Last updated: December 09, 2022