Biology Reference
In-Depth Information
means, which measures the dispersion of treatment means from the grand mean for that
group:
P
a
T
1
ð
X
i
X
Þ
ð
X
i
X
Þ
2
2
i
5
P
i
5
1
ð
D
A
(12.1)
5
N
i
1
Þ
2
where there is a total of
a
treatments,
X
i
is mean for the
i
th treatment, X is the grand mean
over all treatments, and
N
is the sample size for the group (e.g. for an ecomorph of
L. saxatilis
).
The statistical tests of
Littorina
disparity were done by randomly assigning snails to
treatments within their own group, then computing the treatment means from those ran-
domized data, then computing the disparity from those randomized data, iterating the
procedure 9999 times. Adding the observed value to those 9999 gives 10 000 values. The
number of times that a value as large as or larger than the observed one could be obtained
by chance can then be calculated by counting the number of values equal to or exceeding
the observed one.
To determine which particular treatments have a large effect relative to other treat-
ments, and also to determine which treatments have a large effect in one group compared
to that same treatment's effect in another group, distances between treatments were
compared within and between groups. These are the Procrustes distances between means.
For four treatments, there are six possible vectors extending between means: (1) control
mean to predator effluent-treatment mean; (2) control to conspecific effluent-treatment
mean; (3) control to simulated waves-treatment mean; (4) predator effluent-treatment
mean to conspecific effluent-treatment mean; (5) predator effluent-treatment mean to sim-
ulated waves-treatment mean; and (6) conspecific effluent-treatment mean to simulated
waves-treatment mean. As well as analyzing the lengths of these vectors, they also ana-
lyzed their directions by measuring the angles/correlations between them (the comparison
of directions by the analysis of angles or correlations between them is discussed in detail
in Chapter 11).
These comparisons of vector lengths and directions yielded 45 tests of vector lengths
within ecotypes, 18 tests of vector lengths between ecotypes, 45 tests of vector correlations
within ecotypes, and 18 tests of vector correlations between ecotypes. In light of the large
number of tests, a sequential Bonferroni test (
Rice, 1989
) was used. A Bonferroni test
divides the critical value,
α
, by the number of tests. So if there are 10 tests, and the desired
α
table-wide value of
is 0.05, 0.05 is divided by 10, yielding 0.005. A sequential Bonferroni
test is less conservative. Using this test,
is initially divided by the total number of tests
and the smallest p-value is judged against that, then
α
is divided by the remaining num-
ber of tests and the next smallest p-value is judged against that and so forth until reaching
one that is not significant.
Considering the large number of tests, especially the 45 tests of vector correlations
within ecotypes, it is remarkable that any differences were statistically significant.
With rare exceptions, the differences were very large (and statistically significant) or very
small (and therefore not statistically significant). An interesting result is that, in general,
both species exhibited similar magnitudes and directions of responses, contrary to the
α
