Statistical Sampling Strategies for Survey of Soil Contamination - Dealing with Contaminated Sites: From Theory Towards Practical Application - page 187

4.3.1.1 Using Data from Outside the Block

If we can not afford a reasonable number of sampling locations per block so that the
quality of the design-based estimates of the block-means will be too poor, then either
a model-based sampling strategy can be considered (Section 4.3.2 ), or a design-
based sampling strategy with alternative design-based estimators that make use of
the data from outside the block to be estimated. The latter strategy may be con-
sidered if the total number of sampling locations is too small to obtain a reliable
variogram. The intention is that, by using data from outside the block, the precision
of the estimated block mean will increase. On the other hand, in general some bias
will be introduced.
Let us consider the situation where the study area can be divided into more or less
homogeneous sub-areas (e.g. map units) that cut across the blocks. The sub-areas
are used as strata in stratified simple random sampling. Note that we do not have
control of the numbers of sampling locations per block in this way. A first estimator
of the block-mean, referred to as the synthetic estimator, is

L

¯

y syn, u = a hu ¯

y s h , (4.32)

h = 1

where a hu is the surface area of stratum h in block u relative to the area of the
block ( a hu y s h is the sample mean for stratum h . In general the

synthetic estimator is not unbiased, but we hope that the reduction of the sampling
variance, due to the use of sample data from similar parts of the area outside the
block, outweighs the bias.
The second estimator, the regression estimator, tries to eliminate the bias of the
synthetic estimator

= A hu / A u ), and ¯

L L

1 A hu ¯

y s h ,

1
A u

¯

y pos, u =

a hu ¯ y s h + y s hu −¯ (4.33)

h = 1 h =

where y s hu are the sample means in stratum h , and in the intersection of

stratum h and block u , respectively, and A u and A hu are the estimated surface areas
of block u and of the intersection of stratum h and block u , respectively. Note that
even though we know both surface areas, the estimator uses their estimated areas.
This is because this gives higher precision. The areas are estimated by

y s h ¯ and ¯

n hu
n h

A hu =

A h , (4.34)

and

L

n hu
n h A h ,

A u =

(4.35)

h = 1