geomorph (version 2.0.1)

procD.lm: Procrustes ANOVA/regression for shape data

Description

Function performs Procrustes ANOVA with permutation procedures to assess statistical hypotheses describing patterns of shape variation and covariation for a set of Procrustes-aligned coordinates

Usage

procD.lm(f1, data = NULL, iter = 999)

Arguments

f1

A formula for the linear model (e.g., y~x1+x2)

data

An optional value specifying a data frame containing all data (not required)

iter

Number of iterations for significance testing

Value

• Function returns an ANOVA table of statistical results for all factors: df (for each factor), SS, MS, F ratio, Prand, and Rsquare.

Details

The function quantifies the relative amount of shape variation attributable to one or more factors in a linear model and assesses this variation via permutation. Data input is specified by a formula (e.g., y~X), where 'y' specifies the response variables (shape data), and 'X' contains one or more independent variables (discrete or continuous). The response matrix 'y' must be in the form of a two-dimensional data matrix of dimension (n x [p x k]), rather than a 3D array. It is assumed that the landmarks have previously been aligned using Generalized Procrustes Analysis (GPA) [e.g., with gpagen]. The function two.d.array can be used to obtain a two-dimensional data matrix from a 3D array of landmark coordinates. 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. 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 observed SS are evaluated through permutation, where the rows of the shape matrix are randomized relative to the design matrix. Procedurally, Procrustes ANOVA is identical to permutational-MANOVA as used in other fields (Anderson 2001). For several reasons, Procrustes ANOVA is particularly useful for shape data. First, covariance matrices from GPA-aligned Procrustes coordinates are singular, and thus standard approaches such as MANOVA cannot be accomplished unless generalized inverses are utilized. This problem is accentuated when using sliding semilandmarks. Additionally, GM datasets often have more variables than specimens (the 'small N large P' problem). In these cases, distance-based procedures can still be utilized to assess statistical hypotheses, whereas standard linear models cannot.

References

Anderson MJ. 2001. A new method for non-parametric multivariate analysis of variance. Austral Ecology 26: 32-46. Goodall, C. R. 1991. Procrustes methods in the statistical analysis of shape. Journal of the Royal Statistical Society B 53:285-339.

Examples

### MANOVA example for Goodall's F test (multivariate shape vs. factors)
data(plethodon)
Y.gpa<-gpagen(plethodon$land)    #GPA-alignment
y<-two.d.array(Y.gpa$coords)

procD.lm(y~plethodon$species*plethodon$site,iter=99)

### Regression example
data(ratland)
rat.gpa<-gpagen(ratland)         #GPA-alignment

procD.lm(two.d.array(rat.gpa$coords)~rat.gpa$Csize,iter=99)