Statistical Sampling Strategies for Survey of Soil Contamination - Dealing with Contaminated Sites: From Theory Towards Practical Application - page 182

b

,

y reg = ¯

¯

− ¯

y π + z ¯ z π (4.24)

where

¯

y is the spatial mean of the soil contaminant y estimated from the measure-

ments of y in the probability sample only;

π

¯

z is the estimated mean of the ancillary variable;

π

¯ z is the true mean of the ancillary variable; and

b is the estimated regression coefficient (slope) for the ancillary variable.

¯ ¯

The estimators y and z are the design-specific estimators for the mean,

π π

presented in Section 4.2.2.
As with spatial means, in estimating the slope the sampling design must be taken
into account. For SI, the slope can be estimated by the least squares estimator

i = 1 ( y i − ¯

y SI )( z i − ¯ z SI )

b = . (4.25)

i = 1 ( y i − ¯

y SI ) 2

For STSI, Eq. ( 4.25 ) is used to estimate the slopes per stratum. If the number of
sampling locations selected in a stratum is small, e.g. n h < 10 to 20 (depending on

the number of regression coefficients), this stratum must be combined with others
to obtain valid estimates of the sampling variance. For a combination of strata, the
regression coefficients can be estimated by the weighted means of the coefficients
per stratum, using the relative areas as weights (Eq. 4.7 ). For SY, the slope can be
simply estimated by Eq. ( 4.25 ).
An interesting application of the regression estimator is the use of legacy data
not selected by probability sampling in unbiased estimation of spatial means (Brus
and de Gruijter 2003 ). For instance, one may have legacy data on a soil contam-
inant, possibly preferentially sampled at contaminated or uncontaminated sites. In
the method proposed by Brus and de Gruijter ( 2003 ) a relatively small probabil-
ity sample is added to the non-probability sample. The concentrations measured at
the legacy sampling locations are interpolated to these new sampling locations. The
interpolated variable is then used as a covariate in the regression estimator.

4.2.4 Composite Sampling

Laboratory costs can be saved by taking composite samples, i.e. by bulking the
soil aliquots taken at individual sampling locations, and mixing them thoroughly
(Boswell et al. 1996 ; Elder et al. 1980 ; Rohde 1976 ). To profit fully from com-
positing it is necessary to mix and homogenize the subsamples (increments) that are
pooled together to form a composite sample, to the extent that the spatial variation
between the increments is eliminated. In practice this can be hard to achieve, so that
it can become efficient to repeat the subsampling of the composite for laboratory
analysis.