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

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X 2 is computed as
_ 2: (0 ij- Eij)2
X2 -.. E" - 2.69(n.s.)
I] IJ

F or this type of hypothesis, there are (rows - 1)
(colums - 1) degrees of freedom, in this case
(3) (2) = 6

In the example, X2 is nonsignificant. Thus, there
is no evidence that the ratios among the organ-
isms are different for different streams.

Tests of two types of hypotheses by X2 have
been illustrated. The first type of hypothesis was
one where there was a theoretical ratio, i.e., the
ratio of males to females is 1: 1. The second type
of hypothesis was one where equal ratios were
hypothesized, but the values of the ratios
themselves were computed from the data. To
draw the proper inference, it is important to
make a distinction between these two types of
hypotheses. Because the ratios are derived from
the data in the later case, a better fit to these
ratios (smaller X2) is expected. This is compen-
sated for by loss of degrees of freedom. Thus,
smaller computed X2's may be judged signifi-
cant than would be in the case where the ratios
are hypothesized independently of the data.

5.3 F-test

The F distribution is used for testing equality
of variance. Values of F are found in books of
mathematical and statistical tables as well as in
most statistics texts. Computation of the F
statistic involves the ratio of two variances, each
with associated degrees of freedom. Both of
these are used to enter the table. At any entry of
the F tables for (nl - 1) and (n2 - 1) degrees of
freedom, there are usually two or more entries.
These entries are for various levels of probability
of rejection of the null hypothesis when in fact
it is true.
The simplest F may be computed by forming
the ratio of two variances. The null hypothesis is
~ : a12 = a2 2. The F statistic is
Sl2
F=-
S22

(36)

.

where S12 is computed from nl observations
and S22 from n2' For simple variances, the
degrees of freedom, f, will be f1 = n1 - I and

BIOMETRICS - F-TEST

f2 = n2 - 1. The table is entered at the chosen
probability level, a, and if F exceeds the tabu-
lated value, it is said that there is a 1 - a
probability that al 2 exceeds a2 2.

5.4 Analysis of Variance

Two simple but potentially useful examples
of the analysis of variance are presented to
illustrate the use of this technique. The analysis
of variance is a powerful and general technique
applicable to data from virtually any experimen-
tal or field study. There are restrictions, however,
in the use of the technique. Experimental errors
are assumed to be normally (or approximately
normally) distributed about a mean of zero and
have a common variance; they are also assumed
to be independent (i.e., there should be no cor-
relations among responses that are unaccounted
for by the identifiable factors of the study or by
the model). The effects tested must be assumed
to be linearly additive. In practice these assump-
tions are rarely completely fulfilled, but the
analysis of variance can be used unless signifi-
cant departures from normality, or correlations
among adjacent observations, or other types of
measurement bias are suspected. It would be
prudent, however, to check with a statistician
regarding any uncertainties about the appli-
cability of the test before issuing final reports or
publications.

5.4.1 Randomized design

The analysis of variance for completely
randomized designs provides a technique often
useful in field studies. This test is commonly
used for data derived from highly-controlled
laboratory or field experiments where treat-
ments are applied randomly to all experimental
units, and the interest lies in whether or not the
treatments significantly affected the response of
the experimental units. This case may be of use
in water quality studies, but in these studies the
treatments are the conditions found, or are
classifications based upon ecological criteria.
Here the desire is to detect any differences in
some type of measurement that might exist in
conjunction with the field situation or the
classifications or criteria.
For example, suppose it is desired to test
whether the biomass of organisms attaching to

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