Prior to geomorph 3.0.0, the function, plotAllometry, was used to perform linear regression
of shape variables and size, and produce plots to visualize shape allometries. This function was deprecated
when procD.allometry was launched with geomorph 3.0.0, which performed homogeneity of slopes tests to determine
if a common allometry or unique group allometries were more appropriate as a model. The S3 generic, plot.procD.allometry
provided the same plotting as plotAllometry before it. In geomorph 3.1.0, procD.allometry has been deprecated in favor of using
procD.lm and pairwise for analyses, which can include additional variables,
thus eliminating plot.procD.allometry. This function coalesces a few plotting options found in other functions,
as a wrapper, for the purpose of retaining the plot.procD.allometry options in one place.
There are fundamentally two different kinds of allometry plots: those based on linear models and those that do not have a linear
model basis (more detail below). The common allometric component (CAC) and size-shape PCA (Mitteroecker et al. 2004) are plotting
strategies that do not have results
that vary with linear model parameters. By contrast, prediction lines (PredLine, Adams and Nistri 2010) and regression scores
(RegScore, Drake and Klingenberg 2008) are based on fitted values and regression coefficients, respectively, to visualize allometric patterns.
The plotAllometry function will extract necessary components from a procD.lm fit to calculate these various statistics
(although the variables used in the procD.lm fit are inconsequential for CAC and size-shape PCA; only the shape variables are used).
There are multiple ways to visualize allometry. One way is to simply append a size variable to shape variables and perform a principal component analysis
(PCA). In the event that size and shape strongly covary, the first PC scores might reflect this (Mitteroecker et al. 2004). Alternatively, the major
axis of covariation between size and shape can be found by a singular value decomposition of their cross-products, a process known as two-block partial
least squares (PLS; Rohlf and Corti 2000). This major axis of variation is often referred to as the common allometric component
(CAC; Mitteroecker et al. 2004). Neither of these methods is associated with a model of allometric shape change, especially as such change might vary
for different groups. As such, these methods have limited appeal for comparing group allometries (although color-coding groups in plots might reveal
different trends in the plot scatter).
By contrast, describing a linear model (with procD.lm) that has an explicit definition of how shape allometries vary by group can be
more informative. The following are the three most general models:
simple allometry: shape ~ size
common allometry, different means: shape ~ size + groups
unique allometries: shape ~ size * groups
However, other covariates can be added to these models. One could define these models with procD.lm
and use anova.lm.rrpp to explicitly test which model
is most appropriate. The function, pairwise can also be used to test pairwise differences among least-squares means or slopes.
To visualize different allometric patterns, wither prediction lines (PredLine; Adams and Nistri 2010) or regression scores
(RegScore; Drake and Klingenberg 2008) can be used. The former plots first PCs of fitted values against size; the latter calculates a regression score
as a projection of data on normalized vector that expresses the covariation between shape and the regression coefficients for size, conditioned
on other model effects. For a simple allometry model, CAC and RegScore are the same (Adams et al. 2013) but RegScore, like PredLine but unlike CAC,
generalizes to complex models.
Either PredLine or RegScore can help elucidate divergence in allometry vectors among groups.
If the variable for size is used in the procD.lm fit, the plot options will resemble past allometry plots found in
geomorph. However, with this updated function philosophy, the model fit does not have to necessarily contain size. This might be useful if
one wishes to visualize whether shape, size, and some other variable covary in some way (by first performing a procD.lm fit
between shape and another covariate, then performing plotAllometry with that fit and size). For example, one can entertain the question,
"Are species differences in shape merely a manifestation of shape allometry, when species differ in size?" By fitting a model, shape ~ species,
then using plotAllometry for the model fit (with either PredLine or RegScore), the plot will help reveal if allometry and species effects are confounded.
The following are brief descriptions of the different plotting methods, with references.
If "method = PredLine" (the default) the function calculates fitted values from a procD.lm fit, and
plots the first principal component of the "predicted" values versus size as a stylized graphic of the
allometric trend (Adams and Nistri 2010). This method is based on linear models and
can allow for other model variable to be incorporated.
If "method = RegScore" the function calculates standardized shape scores
from the regression of shape on size, and plots these versus size (Drake and Klingenberg 2008).
For a single allometry, these shape scores are mathematically identical to the CAC (Adams et al. 2013).
This method is based on linear models and can allow for other model variable to be incorporated.
If "method = size.shape" the function perform principal components analysis on a data space containing both shape
and size (sensu Mitteroecker et al. 2004). This method is not based on linear models and results will not be changed by
changing the allometry model.
If "method = CAC" the function calculates the
common allometric component of the shape data, which is an estimate of the average allometric trend
for group-mean centered data (Mitteroecker et al. 2004). The function also calculates the residual shape component (RSC) for
the data. This method is not based on linear models and results will not be changed by
changing the allometry model.
The function returns values that can be used with picknplot.shape or
shape.predictor and plotRefToTarget to visualize shape changes in the plot.