Biology Reference
In-Depth Information
CHAPTER
9
Ge
neral Linear Mod
els
In the last chapter, we examined two simple models, one a linear regression and the other
a comparison of two means. It is likely obvious that we will need more complex models to
answer questions left open by those two analyses. In particular, in our analyses of alpine
chipmunk (Tamias alpinus) jaw shape, we found that both size and sex have a statistically
significant impact. What we now want to know is whether sex has a significant impact on
shape, controlling for size, and whether alpine chipmunk jaws are sexually dimorphic in
their response to size. To answer these questions we need more complex models but, like
the simple models that we already presented, these more complex ones are all examples of
General Linear Models, which have important features in common. One common feature is
that we can test the statistical significance of the models using F-tests, and a second common
feature is that we can state how much of the variation is explained both by the model and by
error by variance partitioning. It turns out that a wide range of models can be analyzed using
the same general approach, including the models used in Analysis of Variance (ANOVA),
Analysis of Covariance (ANCOVA), Multivariate Analysis of Variance (MANOVA),
Multivariate Analysis of Covariance (MANCOVA), Multiple Regression (and so forth). This
list might not actually seem impressively long, but some of these procedures admit to
numerous designs (many of which have names and apparently specialized methods to fit).
But, despite their diversity, all can be viewed as cases of the General Linear Model. Because
some of the named models have multiple names (owing to the diversity of fields that have
used them), we will try to alert you to the multiple terms for a single model or concept.
General Linear Models comprise a broad class of predictive mathematical models
employed in statistics which, as is evident from their name, are linear. More specifically,
they are linear in their fitted parameters. What this means is that any change in the predicted
outcome is linearly related to a change in the fitted parameters. For example, if we change
a parameter by some value
δ
, resulting in a change of
Δ
in the predicted value, then a
change of 2
in the predicted value. The
most common of the general linear models is the univariate linear regression model:
δ
in the parameter will produce a change of 2
Δ
Y
mX
b
1
ε
(9.1)
5
1
