Geoscience Reference
In-Depth Information
The standard deviation of
p
is
−
(
)
2
∑
p
n
2
(
803 5
. )
∑
2
2
2
…
p
(. )
78 5
+
(
835
.)
−
10
s
p
=
=
=
10 002778
.
=
3 163
.
(
n
−
1
)
9
And the standard error for
p
(ignoring the fpc) is
2
s
n
n
N
10 002778
10
.
=
p
s
=
1
−
=
1 000
.
p
Note that
n
and
N
in these equations refer to the number of clusters, not to the number of individuals.
The 95% confidence interval, computed by the procedure for continuous variables is
pt
±
(
)( ,
s
wherehas (
t
n
−
1
)
=
9degrees of freedom
005
p
80.35
±
(
2.262)(1.000)
=
78.1 to 82.
6
7.13.1 t
ransFormations
The above method of computing confidence limits assumes that the individual percentages follow
something close to a normal distribution with homogeneous variance (i.e., same variance regard-
less of the size of the percent). If the clusters are small (say, less than 100 individuals per cluster) or
some of the percentages are greater than 80 or less than 20, the assumptions may not be valid and
the computed confidence limits will be unreliable. In such cases, it may be desirable to compute the
transformation
y
= arcsin
percent
and to analyze the transformed variable.
7.14
CHI-SQUARE TESTS
7.14.1 t
est
oF
i
ndependenCe
Individuals are often classified according to two (or more) distinct systems. A tree can be classified
as to species and at the same time according to whether it is infected or not infected with some
disease. A milacre plot can be classified as to whether or not it is stocked with adequate reproduc-
tion and whether it is shaded or not shaded. Given such a cross-classification, it may be desirable
to know whether the classification of an individual according to one system is independent of its
classification by the other system. In the species-infection classification, for example, independence
of species and infection would be interpreted to mean that there is no difference in infection rate
among species (i.e., infection rate does not depend on species).
The hypothesis that two or more systems of classification are independent can be tested by chi-
square. The procedure can be illustrated by a test of three termite repellents. A batch of 1500
wooden stakes was divided at random into three groups of 500 each, and each group received
a different termite-repellent treatment. The treated stakes were driven into the ground, with the
treatment at any particular stake location being selected at random. Two years later the stakes were
examined for termites. The number of stakes in each classification is shown in the following 2×3
(two rows by three columns) contingency table:
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