Biology Reference
In-Depth Information
In addition to being a very simple looking expression even when the model is complex,
the approach based on matrix methods is relatively straightforward to implement in
software.
-TESTS AND MEAN SQUARES
F
We can write out expressions for the mean squares for the model or submodels (discussed
in more detail below), based on the assumption that the model is true. These terms are just
the sum of squares divided by the degrees of freedom in the model. It will turn out that the
mean square terms of the various subsets of the model are typically related to one another in
simple additive ways, meaning that models that differ only by additions of terms will be
identical in their predicted mean square (EMS) values for terms common to all the models
(at least for balanced designs). So, if we have two models that differ only by one factor, the
model that contains that factor would be termed the “full model” and the model lacking it
would be known as the “reduced model”; the ratio of the MS values for these two models, is
the F-ratio for the two models, just as it is when the MS of one factor is compared to error
MS. Both F-ratios can be assessed for their statistical significance using either analytic or
resampling methods, just as in the case of a linear regression model. When the design is
unbalanced, the situation is more complex because, depending on the approach taken, either
the expressions depend on the sample size in each cell or the sums of squares for the terms
will not add up to the total sums of squares. We will therefore defer consideration of unbal-
anced designs until we have more fully discussed the simpler case of balanced designs.
In the case of univariate data, the sum of squares and the mean squares are simple scalar
values, so forming F-ratios is simple and obvious. That is not the case for multivariate data.
There is a number of approaches to working with the sum of squares and cross products
(SSCP) matrices that arise from the matrix methods discussed above. Alternatively, it is pos-
sible to use permutation-based approaches based on summed squared Procrustes distances,
using the “outer sum” method developed by Anderson and colleagues (
Anderson, 2001a,b;
Anderson and Robinson, 2001
), discussed in more detail below. Rather than tackling the sev-
eral related topics all at once (i.e. how to calculate mean squares, the relevant F-ratios for
both balanced and unbalanced designs and the extension of these procedures to multivariate
data), we instead start by looking at a few types of models, applying them to univariate
data. After laying this foundation, we discuss unbalanced designs. Thereafter, we review a
range of experimental designs and give the expected mean squares and F-tests. We then
extend the models to multivariate data and then explain the permutation tests.
Univariate Data with One Factor
In this section, we examine the development of a GLM for a univariate variable Y that
is hypothesized to depend on a single fixed factor A. We show one approach to calculating
variances based on sums of squares, using the estimated means of the specimens in each
level of the factor A. This approach is conceptually easy to understand, but it is not in the
matrix notation presented earlier, and it is not the approach used in most computer-based
