Biology Reference
In-Depth Information
being right-handed or male. Whereas the right-handed sample is biased toward males,
the left-handed sample is biased toward females. As a result, it is difficult to conclude
that any difference found between right- and left-handed samples is due purely to hand-
edness. Part of the difference could instead be due to sex. In effect, the two factors are
confounded by the experimental design, making it difficult to separate the effects of the
factors and also to estimate the terms.
Unfortunately, unbalanced designs are far more common in biological studies than
balanced ones. In some cases, the design might have been balanced at the outset of the
experiment, but it becomes unbalanced when the organisms die or escape or when speci-
mens are damaged in the process of capture or preparation. In field studies, animals may
wander away from the study plots or be uncatchable on any given day. In observation of
museum collections, the number of specimens that can be used to estimate the levels of a
factor depends on the number of undamaged specimens at those levels contained in the
collection(s).
It is possible to ensure having a balanced design by restricting the number of indivi-
duals in each cell to the number contained in the smallest cell. For example, to obtain a
balanced design for our analysis of sexual dimorphism in the last chapter, we removed
the excess females to equalize the numbers of males and females. Had we added another
factor to the design, such as whether the animals came from the Yosemite area or the
southern Sierras, we would have ended up with only 24 animals per cell because there are
only 24 males from the southern Sierras. Balancing a design by randomly leaving out spe-
cimens is sometimes recommended as an alternative to using an unbalanced design
(
Underwood, 1997
). The reason for recommending a balanced design, even if that means
leaving out data, is not that sums of squares cannot be calculated. Rather, it is that the
results can be difficult to interpret. Additionally, the fact that the factors are confounded
raises problems for the analysis and there is some controversy about the best procedure to
use for calculating sums of squares for unbalanced designs (an overview of some proce-
dures is given below, in Unbalanced Designs and Sums of Squares).
DESIGN MATRICES
Design matrices play a crucial role in the statistical analysis because, as is obvious from
their name, they encode the design of the study. The design matrix contains the informa-
tion about the number of factors, the levels of each factor, the number of distinct combina-
tions of factors in each interaction term, etc. This matrix is often called the “
X
” matrix
because it is represented by
X
in the following expression, which is a strikingly compact
equation that applies to a wide range of models, both univariate and multivariate. It may
appear to be complex because it is written in terms of matrices, but the pay-off is the abil-
ity to formulate virtually any linear model in the following form:
Y
XB
(9.5)
5
1 ε
In this expression,
Y
is the centered data matrix for the dependent variable (typically
shape in our case). Because the data are centered, the mean of each column is zero and
