Biology Reference
In-Depth Information
pseudo-F-test(s) in permutation tests, allows us to test the same types of GLM and
MANOVA models that are tested using the analytic methods.
In calculating these mean squares, we typically use the univariate degrees of freedom
rather than multiplying the univariate degrees of freedom by the dimensionality of the
data (e.g. 2K
7 for K landmarks in three
dimensions, etc.). It therefore seems reasonable to ask whether the multivariate degrees of
freedom should also be used in calculating means squares. There are two reasons why
they do not seem necessary. The first is that we chose to think of variance as a property of
the configuration, not of each coordinate separately. Consequently, dividing the sums of
squares by the univariate degrees of freedom gives a mean square that represents the con-
tribution that an individual makes to the variance rather than the contribution made by a
landmark coordinate. The second is that we are using F-ratios, and the numerator and
denominator both are altered by the same multiplicative factor so it does not matter
whether we use the multivariate or univariate degrees of freedom. Moreover, we are using
permutations to test the statistical significance of F, and the degrees of freedom do not
enter into those tests. However, the degrees of freedom are important if we use analytic
tests. When using analytic multivariate tests, the degrees of freedom should be the multi-
variate degrees of freedom.
4 for K landmarks in two dimensions, 3K
2
2
Permutation Tests Based on the Distance Matrix
When we want to test a particular null hypothesis, there is an implied statement about
the exchangeability of specimens if the null hypothesis is true (
Anderson, 2001a,b
). If we
want to test the significance of a particular factor (A), then the null hypothesis is that spe-
cimens belonging to different levels of this particular factor (A
i
) could be exchanged with
one another and, if the null hypothesis is true, this should not alter the sums of squares
predicted by the model in any significant way. So, to test the null hypothesis, we might
compute an F-ratio as described in the previous section, for the original labeling of speci-
mens. We might then randomize the labeling of specimens that indicate their level on A
i
,
leaving labels for other factors intact, and then recompute the F-ratio, doing this repeat-
edly to estimate a distribution of F-ratios. We would then compare the observed F-ratio to
that distribution to estimate a p-value for the observed F-ratio. One advantage of the use
of the distance matrix is that since the data in the distance matrix are distances between
specimens I and J, we can permute the rows and columns in the distance matrix without
having to permute the individuals themselves. As a result, we do not have to recompute
all distances between all specimens for each permutation. Instead, we can simply rear-
range the existing distances within the matrix (see
Anderson, 2001a,b
, for a complete
discussion of this approach).
An important feature of all permutation methods is that the units being permuted be
exchangeable, which is equivalent to saying that the variables are independent and identi-
cally distributed, although they need not be normally distributed (
Anderson and ter
Braak, 2003
). The assumption that error variances are equal among groups is also required,
just as it was in the classical analytic approaches to multivariate analogs of the F-ratio.
While the pseudo F-ratios do not explicitly address the variance
covariance matrix of the
