Geoscience Reference
In-Depth Information
866
148
.
.
F
= =
5 851
.
Fortunately, critical values of the
F
ratio have been tabulated for frequently used significance lev-
els analogous to the χ
2
distribution. Thus, the result, 5.851, can be compared to the appropriate value
of
F
in the table. The tabular
F
for significance at the 0.05 level is 2.87 and that for the 0.01 level
is 4.43. As the calculated value of
F
exceeds 4.43, we conclude that the difference in height growth
between treatments is significant at the 0.01 level. (More precisely, we reject the hypothesis that
there is no difference in mean height growth between the treatments.) If
F
had been smaller than
4.43 but larger than 2.87, we would have said that the difference is significant at the 0.05 level. If
F
had been less than 2.87, we would have said that the difference between treatments is not significant
at the 0.05 level. Researchers should select their own levels of significance (preferably in advance
of the study), keeping in mind that significance at the α (alpha) level, for example, means this: If
there is actually no difference among treatments, then the probability of getting chance differences
as large as those observed is α or less.
7.16.1.5
t
Test vs. the Analysis of Variance
If only two treatments are compared, the analysis of variance of a completely randomized design
and the
t
test of unpaired plots lead to the same conclusion. The choice of test is strictly one of per-
sonal preference, as may be verified by applying the analysis of variance to the data used to illustrate
the
t
test of unpaired plots. The resulting
F
value will be equal to the square of the value of
t
that
was obtained (i.e.,
F
=
t
2
). Like the
t
test, the
F
test is valid only if the variable observed is normally
distributed and if all groups have the same variance.
7.16.2 m
ultiple
C
omparisons
In the example illustrating the completely randomized design, the difference among treatments was
found to be significant at the 0.01 probability level. This is interesting as far as it goes, but usually
we will want to take a closer look at the data, making comparisons among various combinations of
the treatments. Suppose, for example, that A and B involve some mechanical form of site prepara-
tion while C, D, and E are chemical treatments. We might want to test whether the average of A and
B together differ from the combined average of C, D, and E. Or, we might wish to test whether A
and B differ significantly from each other. When the number of replications (
n
) is the same for all
treatments, such comparisons are fairly easy to define and test.
The question of whether the average of treatments A and B differs significantly from the average
of treatments C, D, and E is equivalent to testing whether the linear contrast
(
)
−++
(
)
Q
=+
33
AB
222
CDE
differs significantly from zero (A = the mean for treatment A, etc.). Note that the coefficients of this
contrast sum to zero (3 + 3 - 2 - 2 - 2 = 0) and are selected so as to put the two means in the first
group on an equal basis with the three means in the second group.
7.16.2.1
F
Test with Single Degree of Freedom
A comparison specified in advance of the study (on logical grounds and before examination of the
data) can be tested by an
F
test with single degree of freedom. For the linear contrast
ˆ
QaXaXaX
=++
33
…
+
11
22
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