Biology Reference
In-Depth Information
which will follow an F-distribution with degrees of freedom df
A
,df
error
. Working with uni-
variate data, we can simply look up this value in a table of F-values to find the associated
p-value. Notice that the F-ratio is the variance explained by the model divided by the
unexplained variance, just as when we used an F-ratio to determine the variance explained
by a regression model, or used Goodall's F-test to compare the mean shapes of two
groups. Because the expected mean term in the numerator is equal to the denominator
plus one additional term, the F-value will be larger than one if the additional term:
X
J
2
j
n
j
α
(8.30)
ð
J
1
Þ
2
j
1
5
is not zero. The null hypothesis is that this term actually is zero so, under the null hypoth-
esis, the F-ratio will be one so larger F values indicate lower probabilities that the null
hypothesis is true. If the null hypothesis is rejected, we may then interpret the MS
A
term
as the variance explained by the factor A, and compare it to the unexplained variance
estimate, MS
error
.
EXTENSION OF THE UNIVARIATE ANOVA TO MULTIVARIATE
SHAPE DATA
There is a simple approach to extending the single factor ANOVA to shape data (the
more complex approaches are discussed in the next chapter, never fear). Because we have
a well-understood measure of differences in shape, i.e. the Procrustes distance, sums of
squared Procrustes distances may be used to characterize variance in data sets as we dis-
cussed already in the context of regression and the comparison of two groups. To extend
the single factor univariate ANOVA to multiple groups, we simply replace all the summed
square differences in the equations above by summed square Procrustes distances around
the means of each level of A, and about the overall mean shape (
Klingenberg and
McIntyre, 1998; Rohlf, 2009
). Variance partitioning proceeds exactly as discussed in the
univariate case. The resulting ratio:
MS
A
MS
error
5
SS
A
=
df
A
F
5
(8.31)
=
SS
error
df
error
is referred to as a Generalized Goodall's F-test (
Rohlf, 2009
) or a pseudo-F-test (
McArdle
and Anderson, 2001
).
As in the case of linear regression and the pairwise comparison of means, we need not
rely on an analytic model of the distribution of the Generalized Goodall's F-statistic.
Instead, we can use a permutation approach to test the null hypothesis that the variance
explained by the factor is due to a random association between specimens and group
levels. To test this null hypothesis, we permute the group labels assigned to each speci-
men, randomly associating each specimen with a label. We then compute the F-ratio for
each permuted data set and the distribution of F values obtained over many permutations
can then be used to test the observed F-value at any desired
α
level. To reject the null
