7116 ‎(2)‎

• UCREFRP Catalog Number: 7116
• Author: Baxter, G. and J. Simon
• Title: Editor
• USFW Year: Series
• USFW - Doc Type: 1970
• Copyright Material: YES

Y = ( f ~j)/n = 127.04

^
V(~) = 2867.1

n
^
L V(~J = 21.08

^
Y = 37, 127.04 = 4700

^ ^ 37 37
V(Y) = - (37 - 7) 2867.1 + ~ 21.08 =
7 7
= 454640 + 111 = 454751

SE(Y) = )454751 = 674

The coefficient of variation C = 674/4700 = 0.14

^ ^
Note that the last term in V(Y) is extremely
small (0.02% of the total variance) and can safely
be ignored. This will probably be the case in many
cases when p is of 'normal' magnitude and 3
removals applied. If 2 removals are carried out in
the case above, the last term is still negligible
because the p is large.
If the sampling sections are of unequal size,
formulas (21)-(22) will still hold. Ifhowever, they
^ ^
differ greatly, V(Y) in eq. (22) will be large and
thus the sampling variance large. In this case, a
better method is to include the size (e.g. area) mj
in the calculation. If so, and if still using simple
random sampling within strata, this leads to a
ratio estimation of the population size. We call this
method SRS ratio estimation or SRS with
auxiliary variable since mi is also included. In
addition to the size mj of the sections in the
sample, the total size (area) M ofthe stratum must
also be known. With the notation above, the total
fish population in the stratum is estimated as

n n

Y = M L ~ J L mj

(23)

,

and the population density (e.g. per 100 m2) as

Ann

Y = L ~ J L mj

(23')

31

As the ratio estimation is biased (in the order of
^
a factor l/n), the precision of Y might be
^
expressed as the Mean Square Error of (Y),
^ ^
MSE(Y), rather than the variance V(Y). The
MSE(Y) is larger than V(Y) for biased esti-
mates since it also includes the (squared) bias. In
^
practice, the MSE(Y) for the ratio estima}e
above can be compared with the variance V(Y)
of the foregoing method (eq. 22). An estimate of
^
MSE(Y) is

MSE(Y) = N(N - n) f mi(Yi -)I) + N f V(Yi)
n(n - 1) n
(24)

The MSE for the population density Y is

^..::.. ^ ^
MSE(Y) = MSE(Y)/M

(24')

Example 6. The data from example 5 are used
(although the sample size n = 7 is too small for
this method, see below), together with data on the
area ffi. of these sections. We assume that the total
I
stratum area is 10000 m2. The result is shown
on the next page.

U sing these figures we find, from (23),
Y = 10000 889.3/1874 = 4745, and from (23'),

~ 2
mean density Y = 889.3/1874 = 0.4745 per m .

^ ^ 37 . 30 37
From (24), MSE(Y) = - 11853 + - 21.08
7,6 7

= 313369

JMSE(Y) = 560, which can be compared with
SE(Y) = 674 in example 5.
As in the previous method, the ratio estimators
can be added in the case of more than one
stratum.

Methods based on proportional probability sampling
(PPS)
If areas of unequal size are chosen, another way
of selecting a sample of such areas is to use
sampling probabilities proportional to the size