The SURVEYSELECT Procedure

Murthy’s PPS Method

Murthy’s method (METHOD=PPS_MURTHY ) selects two units from each stratum, with probability proportional to size and without replacement. The selection probability for unit i in stratum h is

\[ P_{hi} = Z_{hi} \bigl ( 1 + K_ h - (Z_{hi}/(1-Z_{hi}) \bigr ) \]

where $Z_{hi} = M_{hi} / M_{h \cdot }$ and

\[ K_ h = \sum _{j=1}^{N_ h} \bigl ( Z_{hj} / ( 1 - Z_{hj} ) \bigr ) \]

Murthy’s algorithm first selects a unit with probability $Z_{hi}$. Then a second unit is selected from the remaining units with probability $Z_{hj}/(1-Z_{hi})$, where unit i is the first unit selected. The joint selection probability for units i and j in stratum h is

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

For more information, see Cochran (1977, pp. 263–265) and Murthy (1957).