Spatial Sampling Designs - Sampling Spatial Units for Agricultural Surveys - page 168

Fig. 7.4 Sample selected with CUBE method and first-order inclusion probabilities proportional
to size (balanced π ps samples)

> framebal < - framepop[bal2 ¼¼ 1,]
> points(framebal$xc,framebal$yc, pch ¼ 1, cex ¼ 2)

Note that the cube method is also implemented into the package Balanced-
Sampling through the command cube.
The difficulties that arise when varying π k (see the previous R output) are much
greater than when we have constant probabilities. The selected sample may not be
able to produce HT estimates close to the known totals of the auxiliary variables.
There have been several studies on the properties of the cube method. Particu-
larly the fast version, which can easily be applied to real life surveys because it can
deal with frames that have a very high number of records. Amongst others,
balanced sampling has been investigated in terms of variance estimation (Deville
and Till´ 2005 ), extending the constraints to sub-populations (Chauvet 2009 ),
allocating the sample size (Till´ and Favre 2005 ), and optimal selection probabil-
ities when dealing with multivariate auxiliaries (Chauvet et al. 2011 ).
Following Till´ ( 2011 ), we can summarize the main features of balanced
sampling as follows:

1. It increases the accuracy of the HT estimator, because its variance depends only
on the regression residuals of the variable of interest by the balancing variables.