The SURVEYSELECT Procedure

PPS Sampling with Replacement

If you specify the METHOD=PPS_WR option, PROC SURVEYSELECT selects units with probability proportional to size and with replacement. The procedure makes $n_ h$ independent random selections from the stratum of $N_ h$ units, selecting with probability $Z_{hi} = M_{hi}/M_{h \cdot }$ . Because units are selected with replacement, a unit can be selected for the sample more than once. The expected number of hits (selections) for unit i in stratum h equals $n_ h Z_{hi}$ . If you specify the JTPROBS option, PROC SURVEYSELECT computes the joint expected number of hits for all pairs of selected units in each stratum. The joint expected number of hits for units i and j in stratum h equals

$\begin{equation*} P_{h(ij)} = \begin{cases} n_ h (n_ h - 1) Z_{hi} Z_{hj} & \mr {for} \hspace{.1in} j \neq i \\ n_ h (n_ h - 1) Z_{hi} Z_{hi} / 2 & \mr {for} \hspace{.1in} j = i \\ \end{cases}\end{equation*}$