Geoscience Reference
In-Depth Information
As an example, suppose that 90 uniformly spaced sample units are
arranged in a rectangular grid over an intertidal study area, and that it is nec-
essary to estimate the average barnacle density. A visual assessment is made
of the density on the first three units, which are then on that basis ordered
from the one with the lowest density to the one with the highest density. The
density is then determined accurately for the highest ranked unit. The next
three units are then visually ranked in the same way, and the density is then
determined accurately for the unit with the middle of the three ranks. Next,
sample units 7, 8, and 9 are ranked and the density determined accurately for
the unit with the lowest rank. The process of visually ranking sets of three
units and measuring first the highest-ranking unit, then the middle-ranking
unit, and finally the lowest ranking unit is then repeated using units 10 to
18, units 19 to 27, and so on. After the completion of this procedure on all 90
units, a ranked set sample of size 30 is available based on the accurate esti-
mation of density. This sample is not as good as would have been obtained
by measuring all 90 units accurately, but it should have considerably better
precision than a standard sample of size 30.
Ratio estimation is often used to allow for the varying sizes of sample
units where the value of a variable of interest
Y
is assumed to be approxi-
mately proportional to the size of the sample unit that it is recorded in
U
, so
that
Y
≈
RU
where
R
is a constant. For example, the number of animals might
be counted in patches of vegetation of different sizes, with the assumption
that the expected number of animals in a patch is proportional to the area
of the patch. Another possibility with ratio estimation is that the size of an
animal or plant population needs to be estimated, and that it is expensive to
accurately determine the abundance in a sample unit
Y
. However, there is
some other variable
U
that is not expensive to measure that is approximately
proportional to the abundance. For example, this second variable might just
be an estimate based on a quick visual survey of the sample unit. The tech-
nique used involves making the visual survey of either all or a large sample
of the population of sample units and making the accurate determination
of abundance on a small number of these units. The idea then is to adjust
the estimated abundance based on the small sample using the relationship
between the accurate measurements of abundance and the values of the
ine
xp
e
n
sive-to-measure variable. This is done by first estimating the ratio
Ryu
ˆ
/
using the sample units for which both
Y
and
U
are known and then
multiplying this by the mean value of
U
based on either all units in the popu-
lation of
r
the mean of a large sample of
U
values to give the estimated mean
of
Y
as
ˆ
ratio mean
.
Ratio estimation assumes that the ratio of the variable of interest
Y
to the
subsidiary variable
U
is approximately constant for the sample units in the
population. A less-restrictive assumption is that
Y
and
U
are approximately
related by a linear equation of the form
Y
= α + β
U
, in which case regression
estimation can be used. As for ratio estimation, a sample of units is selected
and values of
Y
and
U
obtained. Ordinary linear regression is then used
y U
