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 equals $2M_{hi}/M_{h \cdot } = 2 Z_{hi}$. (Because selection probabilities cannot exceed 1, the relative size for each unit, $Z_{hi}$, must not exceed $1/2$.)

Brewer’s algorithm first selects a unit with probability

\[  \frac{Z_{hi} (1-Z_{hi})}{D_ h (1-2Z_{hi})}  \]


\[  D_ h = \sum _{i=1}^{N_ h} \frac{Z_{hi} (1-Z_{hi})}{1-2Z_{hi}}  \]

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

\[  \frac{Z_{hj}}{1-Z_{hi}}  \]

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

\[  P_{h(ij)} = \frac{2 Z_{hi} Z_{hj}}{D_ h} \left( \frac{1 - Z_{hi} - Z_{hj}}{(1-2Z_{hi}) (1-2Z_{hj})} \right)  \]

See Cochran (1977, pp. 261–263) and Brewer (1963) for details. Brewer’s method yields the same selection probabilities and joint selection probabilities as Durbin’s method. See Cochran (1977) and Durbin (1967) for details.