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

Sampford’s method first selects a unit from stratum h with probability $Z_{hi}$. Then subsequent units are selected with probability proportional to

\[  \lambda _{hi} = Z_{hi} ~  / ~  (1-n_ h ~  Z_{hi})  \]

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

\[  P_{h(ij)} = K_ h ~  \lambda _{hi} ~  \lambda _{hj} ~  \sum _{t=2}^{n_ h} \Bigl ( ~  \left[ t - n_ h ~  (Z_{hi} + Z_{hj}) \right] ~  L_{h,(n_\mi {h}-t)}(\bar{ij}) \Bigr ) ~  / ~  n_ h^{t-2}  \]


\[  K_ h = 1 ~  / ~  \sum _{t=1}^{n_\mi {h}} \left( t~ L_{h,(n_\mi {h}-t)} ~  / ~  n_ h^{t} \right)  \]
\[  L_{h,m} = \sum _{S_ h(m)} \lambda _{h i_1} ~  \lambda _{h i_2} ~  \cdots ~  \lambda _{h i_ m}  \]

and $S_ h(m)$ denotes all possible samples of size m, for $m = 1, 2, \ldots , N_ h$ . The sum $L_{h,m}(\bar{ij})$ is defined similarly to $L_{h,m}$ but sums over all possible samples of size m that do not include units i and j. See Cochran (1977, pp. 262–263) and Sampford (1967) for details.