Biology Reference
In-Depth Information
measured variables, the permutation methods do. That is because the individual measure-
ments are not exchangeable, instead, it is the individual organism with all its measure-
ments that is treated as a unit. It is the individual's landmark configuration that is
exchangeable, not individual landmarks. So the estimated significance of an observed
F-ratio is based on a permutation that preserves the variance
covariance pattern among
the individual elements in
Y
(i.e. landmark positions). Consequently, the observed variance
covariance matrix is taken as a given when assessing the significance of the F-ratio.
Types of Permutations
There is a range of possible approaches to the permutation of data to determine the sig-
nificance of a test statistic, such as the F-ratios discussed here. An exact statistical test is
one in which the Type I error rate (alpha) is exactly equal to the a priori chosen alpha value.
The approach to achieving an exact test for a one-way ANOVA (or MANOVA) seems to
be straightforward and well understood (
Good, 1994; Manly, 1997; Anderson and ter
Braak, 2003
). However, the approach is less straightforward when it is difficult to form
exact tests, such as when there are multiple factors (
Anderson and ter Braak, 2003
), and is
a subject of debate (
ter Braak, 1992; Edginton, 1995; Manly, 1997
). The simplest approach
is to permute the raw data, meaning that specimens are randomly reassigned to groupings
based on all factors at once. This approach is fine for a single factor, and can be used to
form an exact permutation test. However, once there is more than one factor, permutation
of residuals becomes an alternative approach.
One approach is to calculate the residuals based on the most complex model under con-
sideration (the full model), and then base the estimates of p-values for all simpler models
(meaning for models of individual factors) on the distributions of F-values obtained under
this permutation of the residuals from the full model. In a two factor model, with factors
A
,
B
and an interaction
A
B
,
so that the significance of the factor
A
would be based on the full residuals. In a permuta-
tion based on a reduced model, the factor
A
would be tested using residuals permuted with
a fixed
B
value. The permutation is done with all the terms other than the one(s) being
tested held fixed (
Anderson and ter Braak, 2003
), an approach also discussed by
Edginton
(1995)
.
Anderson and ter Braak (2003)
recommend using permutations under the reduced
model on the grounds that the power should be greater than or equal to that of the exact
test. Permutations of residuals always yield approximate tests, which should asymptoti-
cally approach the exact test.
B
, the residuals would be computed based on
A
B
A
3
1
1
3
MODELS WITH MULTIPLE FACTORS
To exemplify the analysis of models with multiple factors, we consider first the analysis
of alpine chipmunk jaw shape with two fixed factors, sex and region (the Yosemite region
and the southern Sierras). Rather than restricting the analysis to a balanced design, as we
did in the last chapter, we use all the animals collected between 1911 and 1919 from these
two regions. Because there are 117 animals, but 15 landmarks and 85 semilandmarks, we
