Editing Morphometrics (section)

==Analyzing data==
Multivariate statistical methods can be used to test statistical hypotheses about factors that affect shape and to visualize their effects.  To visualize the patterns of variation in the data, the data need to be reduced to a comprehensible (low-dimensional) form. [[Principal component analysis]] (PCA) is a commonly employed tool to summarize the variation. Simply put, the technique projects as much of the overall variation as possible into a few dimensions.  See the figure at the right for an example.  Each axis on a PCA plot is an [[eigenvector]] of the covariance matrix of shape variables.  The first axis accounts for maximum variation in the sample, with further axes representing further ways in which the samples vary.  The pattern of clustering of samples in this morphospace represents similarities and differences in shapes, which can reflect [[phylogenetic relationship]]s. As well as exploring patterns of variation, Multivariate statistical methods can be used to test statistical hypotheses about factors that affect shape and to visualize their effects, although PCA is not needed for this purpose unless the method requires inverting the variance-covariance matrix.

Landmark data allow the difference between population means, or the deviation an individual from its population mean, to be visualized in at least two ways.  One depicts vectors at landmarks that show the magnitude and direction in which that landmark is displaced relative to the others.  The second depicts the difference via the [[thin plate splines]], an interpolation function that models change ''between'' landmarks from the data of changes in coordinates ''of'' landmarks.  This function produces what look like deformed grids; where regions that relatively elongated, the grid will look stretched and where those regions are relatively shortened, the grid will look compressed.