9597

• UCREFRP Catalog Number: 9597
• Author: U.S. Environmental Protection Agency.
• Title: Biological Field and Laboratory Methods for Measuring the Quality of Surface Waters and Effluents.
• USFW Year: 1973.
• USFW - Doc Type: Cincinnati, OH.
• Copyright Material: NO

BIOLOGICAL METHODS

Expected values are always computed based
upon the null hypothesis. The computation for
X2 is

X2 = (892 - 919)2 + (946 - 919)2 - 1 59 *
919 . n.s.

*n.s. = not significant

There is one degr:ee of freedom for this test.
Since computed X2 is not greater than tabulated
X 2 (3.84), the null hypothesis is not rejected.
This test, of course, applies equally well to data
that has not been pooled, i.e., where the values
are from two unpooled categories.
The information contained in each of the
collections is partially obliterated by pooling. If
the identity of the collections is maintained, two
types of test may be made: a test of the null
hypothesis for each collection separately; and a
test of interaction, i.e., whether the ratio
depends upon the lake from which the sample
was obtained (Table 5).

TABLE 5. FISH SEX DATA FROM 3 LAKES

Lake No Males No Females Total x2
1 346* (354)t 362 (354) 708 .36 n.s.
2 302 (288) 274 (288) 576 1.30 n.s.
3 244 (277) 310 (277) 554 7.88
P = .005
Total 892 (919) 946 (919) 1838 1.59 n.s.

*Observed values.
t Expected, or hypothesized values.

With the use of the same null hypothesis, the
following results are obtained.
The individual X2,s were computed in the
same manner as equation (34), in separate tests
of the hypothesis for each lake. Note that the
first two are not significant whereas the third is
significant. This points to probable ecological
differences among lakes, a possibility that would
not have been discerned by pooling the data.
The test for interaction (dependence) is made
by summing the individual X2 's and subtracting
the X 2 obtained using totals, i.e.,
x2 (interactions) = Df (individuals) - X2 (total)
= .36 + 1.30 + 7.88 - 1.59 = 7.95
The degrees of freedom for the interaction X2
are the number of individual X 2 's minus one; in
this case, two. This interaction X 2 is significant
(P > .025), which indicates that the SI;;X ratio is
indeed dependent upon the lake.

4

Another X2 test may be illustrated by the
following example. Suppose that comparable
techniques were used to collect from four
streams. With the use of three species common
to all streams, it is desired to test the hypothesis
that the three species occur in the same ratio
regardless of stream, i.e., that their ratio is
independent of stream (Table 6).

TABLE 6. OCCURRENCE OF THREE
SPECIES OF FISH

Stream

Number of organisms
Species 1 Species 2 Species 3
24* (21.7)t 12 (12.5) 30 (31.7)
15 (18.5) 14 (10.6) 27 (26.9)
28 (27.4) 15 (15.7) 40 (39.9)
20 (19.4) 9 (11.2) 30 (28.4)
87 50 127

66
56
83
59
264

Frequency

1
2
3
4
Total
Expected
ratio

87/264

50/264 127/264

*Observed values.
t Expected, or hypothesized

To discuss the table above, OJ j = the observa-
tion for the iU! stream and the jU! species.
Hence, 023 is the observation for stream two
and species three, or 27. A similar indexing
scheme applies to the expected values, Ej j. For
the totals, a subscript replaced by a dot (.)
symbolizes that summation has occurred for the
observations indicated by that subscript. Hence,
0.2 is the total for species two (50); 03. is the
total for stream three (83); and 0.. is the grand
total (264).
Computations of expected values make use of
the null hypothesis that the ratios are the same
regardless of stream. The best estimate of this

ratio for any species is~, the ratio of the sum

for species j to the total of all species. This ratio
multiplied by the total for stream i gives the
expected number of organisms of species j in
stream i:

4

(o.j)
Ej . = ~ (Oi.>
J v...

(35)

For example,

E12 = (~:~) (01)

- 2Q. (66)
- 264
= 12.5

4

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