Biology Reference
In-Depth Information
where
Σ
P
is the variance
covariance matrix of the predicted values of Y at a value of X in
the data set, det is the determinant of the matrix, and
is the variance
covariance matrix
for the original set of variables (e.g. partial warp scores or principal component scores or
any other complete set of scores for our data). Wilk's lambda can also be computed as a
function of the eigenvalues of the inverse of the variance
covariance matrix of the
residual
Σ
Σ
R
)
2
1
(
times the variance
covariance matrix of
the predicted values (
Σ
P
).
2
value.
Several other conventional multivariate test criteria can also be used, including Roy's max-
imum root test, Pillai's trace, Hotelling
Approximations are available to convert the
Λ
value into an F-statistic or a
χ
Lawley trace) (
Rencher, 1995
), all of which give
the same results when there is only one independent variable and the sample size is large.
When the sample size is small (relative to the number of landmarks), the authors' experi-
ence is that Wilk's
can substantially overestimate the variance explained by the regres-
sion model, perhaps due to difficulties associated with estimating variance
covariance
matrices at small sample sizes. The other analytic tests could be expected to share this
behavior. It is not clear what sample size must be used to obtain consistent results but the
general rule of thumb is that there should be five times as many observations as estimated
parameters. The number of parameters rises rapidly for landmark data, especially for
three-dimensional landmark data, and even more so for semilandmark data, making sam-
ple size a substantial concern. For that reason, results of these analytic tests should be
viewed with caution unless sample sizes are large relative to the number of parameters.
Λ
DIST
ANCE-BASED METHODS OF HYPOTHESIS TEST
ING
There is an alternative approach to testing statistical hypotheses, which uses variances
expressed in units of Procrustes distance rather than variance
covariance matrices of the
partial warp scores or other variables (
Goodall, 1991
). By this approach, we use the
Procrustes distance between each individual's observed shape and its expected value
given that individual's value on the independent variable. The summed squared
Procrustes distances give a measure of the variance in shape that is not explained by X.
Thus, it is a measure of the residual, i.e. the variance not explained by the regression,
because the distances being squared and summed are the deviations from the regression,
hence they are not explained by the model. This distance-based approach has the advan-
tage of expressing deviations in terms of the familiar (and meaningful) units of Procrustes
distance and it also has the large advantage of expressing variance as a simple squared
Procrustes distance
a scalar rather than a matrix. Using distance metrics in statistics has
proven quite useful in other contexts as well (e.g.
Anderson, 2001a,b
).
The generalized form of this test is an F-ratio of the variance explained by the regres-
sion model relative to that not explained by the regression model, in which the variance is
expressed as a summed square Procrustes distance.
Goodall's (1991)
original derivation (to
be discussed later), was an F-test for the difference in the means of the two groups relative
to the variance within each one. The test does make restrictive assumptions about the vari-
ance at each landmark, specifically, that it is normally, independently and identically dis-
tributed at each landmark. Rather than use a test that depends on this restrictive model,
we can instead use permutation tests based on the concept of exchangeability of the
