Multivariate Allometry and Removal of Size with Principal Components Analysis

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Abstract
A method is presented that constrains principal components analysis (PCA) to extract a first component that, by definition, summarizes isometric size alone. The remaining information is partitioned according to variation in shape. Size-constrained and conventional procedures are compared with analyses of data on sexual dimorphism in the painted turtle (Chrysemys picta marginata) and a crayfish (Cambarus bartoni). Contrary to results from standard analyses using covariance or correlation matrices, the size-constrained technique shows that turtles are dimorphic in isometric size but not in shape. Conventional methods do not completely isolate variation in isometric size from variation in shape. Analysis of the crayfish data confirms that PCA with the correlation matrix separates size from shape more effectively than analysis with the covariance matrix. Secondary shape components (i.e., the third and subsequent components) differ markedly, suggesting that incomplete partitions of isometric size and shape by the traditional methods dramatically affect the results.