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

terminology:
Ho: There is no significant interaction effect
Computations for testing this hypothesis with
the use of an analysis of variance table are
presented below.
Symbolically, an observation must have three
indices specified to be completely identified:
position, time, and sample number. Thus there
are three subscripts: Xi jk is an observation at
position i, time j, and from sample k. A value of
1 for i is upstream; 2, downstream; 1 for j is
before; 2, after. A particular example is Xl 2 3,
the third sample upstream after the plant began
operation, or 22.1 pounds. A total (Table 10) is
specified by using the dot notation. For the
value of ~ j., then the individually sampled
values for position i, time j are totaled. It is a
total for a treatment combination. For example,
the value of XII. is 158.9, and the value of Xl..,
where samplings and times are both totaled to
give the total for upstream, is 290.7.
For a slight advantage in generality, let the
following additional symbols apply: t = number
of times of sampling (in this case t = 2); p =
number of positions samples (in this case p = 2);
s = number of samples per treatment combina-
tion; and n = the total number of observations.

The computations are:
Correction for mean (CT):

(~Xi jk)2 (529.2)2
n = -zn-
= 14002.63

Treatment Sum of Squares (SSTMT):
(~Xij.2)
s - CT
(158.9)2 (131.8)2 (144.1)2 (94.4)2
---s- + ----s- +--s- +---s- - 14002.63 = 456.69

(Note that the divisor (5) may be factored out
here, if desired, but where a different number of
samples is taken for each treatment combination
it should be left as above.)

Positions Sum of Squares (SSP):

~X' 2
1.. CT
-st -
(25~o7)2 + (23:05)2 _ 14002.63 = 136.24

Times Sum of Squares (SST):

~X.j.2_CT
sp

(3~~0)2 + (22:02)2 _ 14002.63 = 294.91

Interaction of Positions and Times Sum of
Squares (SSPT):

SSTMT - sSP - SST
456.69 - 136.24 - 294.91 = 25.54

Error Sums of Squares:

~ Xi jk2 - SSTMT - CT

15308.24 - 456.69 - 14002.63 = 848.92

Although not important to this example, the
main effects, positions and times, are tested for
significance. The F table is entered with df = 1
for effect tested, and df = 16 for error. The posi-
tions effect is not significant at any probability
usually employed. The times effect is significant
with probability greater than .95. The inter-
action effect is not significant, and we, there-
fore, conclude that no effect of the suspected
toxic effluent can be distinguished in this data.
Had the F value for interaction been large
enough, we would have rejected the null hy-
pothesis, and concluded that the effluent had a
significant effect (Table 11).

.

TABLE 11. ANALYSIS OF VARIANCE
TABLE FOR FIELD STUDY DATA
OF TABLE 9

Source df SS MS F
Treatments 3 456.69
Positions 1 136.24 136.24 2.56
Times 1 294.91 294.91 5.55*
Positions
X times 1 25.54 25.54 <1
Error 16 848.92 53.05

6.0 CONFIDENCE INTERVALS FOR MEANS
AND VARIANCES

When means are computed in field studies, the
desire often is to report them as intervals rather
than as fixed numbers. This is entirely reason-
able because computed means are virtually
always derived from samples and are subject to
the same uncertainty that is associated with the
sample.

.

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