Biological field and laboratory methods for measuring the quality of ...

performed by computing the ratio, (mean square
for streams)/(mean square for slides), in this
151.065
case, 29.055 =5.20.
When the calculated F value (5.20) is compared
with the F values in the table (tabular F
values) where df = 2 for the numerator and df =
8 for the denominator, we find that the calculated
F exceeds the value of the tabular F for
probability .05. Thus, the experiment indicates a
high probability (greater than 0.95) of there
being a difference in biomass attached to the
slides, a difference attributable to differences in
streams.
Note that this analysis presumes good biological
procedure and obviously cannot discriminate
differences in streams from differences arising,
for example, from the slides having been placed
in a riffle in one stream and a pool in the next.
In general, the form of any analysis of variance
derives from a model describing an observation
in the experiment. In the example, the model,
although not stated explicitly, assumed only two
factors affecting a biomass measurement ­
streams and slides within streams. If the model
had included other factors, a more complicated
analysis of variance would have resulted.
5.4.2 Factorial design
Another application of a simple analysis of
variance may be made where the factors are
arranged factorially. Suppose a field study where
the effect of a suspected toxic effluent upon the
fish fauna of a river was in question (Tables 9
and 10). Five samples were taken about onequarter
mile upstream and five, one-quarter mile
downstream in August of the summer before the
plant began operation, and the sampling scheme
was repeated in August of the summer after
operations began.
Standard statistical terminology refers to each
of the combinations PI T I , P2 TI , PI T 2, and
P 2 T2 as treatments or treatment combinations.
Of use in the analysis is a table of treatment
totals.
In planning for this field study, a null and
alternate hypothesis should have been formed.
In fact, whether stated explicitly or not, the null
hypothesis was:
Ho: The toxic effluent has no effect upon
the weight of fish caught
17
BIOMETRICS - ANALYSIS OF VARIANCE
This hypothesis is not stated in statistical terms
and, therefore, only implicitly tells us what test
to make. Let us look further at the analysis
before attempting to state a null hypothesis in
statistical terms.
In this study two factors are identifiable:
times and positions. A study could have been
done on each of the two factors separately, i.e.,
an attempt could have been made to distinguish
whether there was a difference associated with
times, assuming all other factors insignificant,
and likewise with the positions. The example,
used here, however, includes both factors
simultaneously. Data are given for times and for
positions but with the complication that we
cannot assume that one is insignificant when
studying the other. For the purpose of this
study, whether there is a significant difference
with times or on the other hand with positions,
are questions that are of little interest. Of
interest to this study is whether the upstreamdownstream
difference varies with times. This
type ofcontrast is termed a positions-times interaction.
Thus, our null hypothesis is, in statistical
TABLE 9. POUNDS OF FISH CAUGHT
PER 10 HOURS OVERNIGHT SET OF A
125-FOOT, 1Yz-INCH-MESH GILL NET
Times
Before
(T 1)
After
(T2)
Upstream (PI)
28.3
33.7
38.2
41.1
17.6
15.9
29.5
22.1
37.6
26.7
Positions
Downstream (P 2 )
29.0
28.9
20.3
36.5
29.4
19.2
22.8
24.4
16.7
11.3
TABLE 10. TREATMENT TOTALS FOR
THE DATA OF TABLE 9
Total
Before
After
Positions
totals
Upstream
Positions
Downstream
158.9 144.1
131.8 94.4
290.7 238.5
Times totals
303.0
226.2
Grand total
529.2