Biology Reference
In-Depth Information
GENERALIZING AND EXTENDING THE SIMPLE UNIVARIATE
ANOVA
The simple univariate case with only one factor is only a starting point, but it neverthe-
less illustrates all the analytic procedures used by GLM methods. In all cases, we would
follow the same basic steps, albeit adapting them to multivariate data:
1.
Find the sum of squares due to each term in the model and use them to determine the
variance explained by each term (when possible). When working with multivariate
shape data, there are two approaches to finding these sums of squares: (a) the sum of
squares and cross products matrix (SSCP) or (b) the summed squared Procrustes
distances between specimens. Either can be used as the multivariate equivalent to the
simple sum of squares. This can be complex when the design is unbalanced, as
discussed below.
2.
Form F-ratios of different models to test the significance of the model. In the univariate
example given above, the numerator model states that factor A influences Y. The
denominator model states that Y is just a random value with variance
σ
e
2
. When we
discussed that case above, we referred to the denominator as the MS
error
, but this is
really just a special case. In general, it is preferable to think of the denominator as
another possible model because the denominator will not always be MS
error
. The
important idea is that the expected mean square in the numerator differs from the
expected mean square denominator only by a single term, the term that represents the
factor of interest. We want to test the significance of that factor and to construct the
appropriate test we need to derive the expected mean squares for each part of the
model. As we consider more complex models with more factors and covariates, we will
need an F-ratio for each term. Fortunately, there are many published tables that list the
expected mean squares for various models (although usually only for balanced
designs). The excellent text by
Lorenzen and Anderson (1993)
, for example, describes
how to form sums of squares and F-ratios for a wide range of univariate, balanced
designs, which may be adapted with reasonable care to other analyses. For multivariate
data, more complex analytic approaches are needed due to the use of sums of squares
and cross products (SSCP) matrices.
3.
Test the significance of the F-ratios. The tests can use sums of squares and cross products
matrices or distance based measures of sums of squares. Analytic tests are available for
the first and permutation methods are available for both, such as when Procrustes
distances are used to estimate sums of squares.
MODELS
Univariate Two Factor Balanced Design
We will now consider a case in which we have two factors, with a univariate dependent
variable Y, which depends on two factors A and B (which have p and q levels respectively).
As an example of a model of this sort, consider the alpine chipmunks that we analyzed in
