Biology Reference
In-Depth Information
which is equivalent to measuring the variance of each coordinate, summed over all the
coordinates:
P
n
T
1
ð
X
ij
X
i
Þ
ð
X
ij
X
i
Þ
2
2
j
5
Var
(12.3)
5
N
i
2
1
If we have multiple groups, such as multiple treatments, and wish to test the hypothesis
that one genotype (or population) is canalized over multiple treatments, we would calcu-
late a pooled within-group variance by:
P
P
a
n
T
1
ð
X
ij
2
X
i
Þ
ð
X
ij
2
X
i
Þ
j
1
j
5
5
Var
pooled
5
(12.4)
P
a
N
i
1
2
i
5
1
In the case of
Var
pooled
, the summed squared distances of
j
individuals from a treatment
mean are summed over all treatments. Confidence intervals can be placed on these mea-
sures of variance just as they are on measures of disparity, a topic discussed in depth in
Chapter 10.
One approach for testing the null hypothesis that two samples do not differ in their var-
iance, using permutations of residuals, was discussed above in context of tests for plastic-
ity. Another approach uses a
t
-test, computing the variance for each group at each
iteration of a bootstrap or permutation procedure, then subtracting one group's variance
from that of the other, and iterating the calculation of the variances and the difference
between them at each iteration to generate the distribution of the difference between the
two variances. If the confidence interval for that difference includes zero, we could not
reject the null hypothesis that the two populations do not differ in their variances.
Example: Ontogenetic Decrease in Variance of Skull Shape
We exemplify an analysis of canalization by testing the hypothesis that variance
diminishes over ontogeny in the absence of any selective deaths in the population. To that
end, we compare the variance of skull shape across four ages, 10-, 15-, 20-, and 25-days
postnatal, of the randombred Hsd/ICR strain of the house mouse (
Mus musculus domesti-
cus
). The superimposed landmarks for each sample are shown in
Figure 12.6
; the estimates
for the variance in shape at each age, and standard errors of the estimate, are given in
Table 12.1
. To compare the levels of variance between successive ages, we use the
t
-test
described above to evaluate the difference between variances relative to the pooled stan-
dard errors of those variances. Over the initial 5-day interval, variance is halved; the dif-
ference in the variances for the two samples is 0.000279. The 95% upper bound on the
confidence interval for the difference between the two variances is 0.00015639, and the
magnitude of the observed difference was exceeded by none of the 200 permutations.
After that point, variance is stable
no statistically significant differences are found
between successive age classes later. The initial reduction of variance, in the absence of
