The response matrix 'y' can be in the form of a two-dimensional data
matrix of dimension (n x [p x k]) or a 3D array (p x k x n). It is assumed that the landmarks have previously
been aligned using Generalized Procrustes Analysis (GPA) [e.g., with gpagen]. The names specified for the
independent (x) variables in the formula represent one or more
vectors containing continuous data or factors. It is assumed that the order of the specimens in the
shape matrix matches the order of values in the independent variables. Linear model fits (using the lm function)
can also be input in place of a formula. Arguments for lm can also be passed on via this function.
The function performs statistical assessment of the terms in the model using Procrustes distances among
specimens, rather than explained covariance matrices among variables. With this approach, the sum-of-squared
Procrustes distances are used as a measure of SS (see Goodall 1991). The SS between models is evaluated through
permutation. In morphometrics this approach is known as a Procrustes ANOVA (Goodall 1991), which is equivalent
to distance-based anova designs (Anderson 2001). Unlike procD.lm, this function is strictly for comparison
of two nested models. (Use of procD.lm will be more suitable in most cases.)
A residual randomization permutation procedure (RRPP) is utilized
for reduced model residuals to evaluate the SS between models (Collyer et al. 2015). Effect-sizes (Z-scores) are
computed as standard deviates of the SS or pairwise statistic sampling
distributions generated, which might be more intuitive for P-values than F-values (see Collyer et al. 2015). If a phylogeny is
used, the ANOVA Z-score is calculated from the sampling distributions of the F value, as the total SS will vary among permutations.
For ANOVA Z-scores, a log-transformation is performed first, to assure a norammly distributed sampling distribution.
Pairwise tests are only performed if formulae are provided to compute such results.
The generic functions, print, summary, and plot all work with advanced.procD.lm.
The generic function, plot, produces diagnostic plots for Procrustes residuals of the linear fit.
Notes for geomorph 3.0.4 and subsequent versions
Compared to previous versions of geomorph, users might notice differences in effect sizes. Previous versions used z-scores calculated with
expected values of statistics from null hypotheses (sensu Collyer et al. 2015); however Adams and Collyer (2016) showed that expected values
for some statistics can vary with sample size and variable number, and recommended finding the expected value, empirically, as the mean from the set
of random outcomes. Geomorph 3.0.4 and subsequent versions now center z-scores on their empirically estimated expected values and where appropriate,
log-transform values to assure statistics are normally distributed. This can result in negative effect sizes, when statistics are smaller than
expected compared to the avergae random outcome. For ANOVA-based functions, the option to choose among different statistics to measure effect size
is now a function argument.
An optional argument for including a phylogenetic tree of class phylo is included in this function. ANOVA performed on separate PGLS models is analagous
to a likelihood ratio test between models (Adams and Collyer 2017). Pairwise tests can also be performed after PGLS estimation of coefficients but users
should be aware that no formal research on the statistical properties (type I error rates and statistical power) of pairwise statistics with PGLS has yet
been performed. Using PGLS and analysis of pairwise statistics, therefore, assumes some risk.