Ontogeny and Individual Variation: Analysis of Patterned Covariance Matrices with Common Principal Components

Abstract

Morphometric data from longitudinal growth studies with multiple measurements made in several growth stages on the same specimens confront researchers with difficult statistical problems because traits are correlated both within and across stages. Here, we introduce a statis? tical model especially designed to deal with this complexity. The common principal component (CPC) model for dependent random vectors is based on the assumption that the same pattern underlies both variation within stages and covariation across stages. Thus, a single transformation, when applied to all stages, renders the resulting CPCs uncorrelated not only within but also across stages. Because of these simplifying assumptions, the CPC model greatly reduces the num? ber of parameters to be estimated; it is thus an efficient tool for data reduction. This model is demonstrated using growth of the water strider Limnoporus canaliculatus. The CPCs can be inter? preted as patterns of size variation and contrasts between parts that are common to all stages, although there are minor deviations from the model. The size CPC accounts for most variation in all instars and is therefore an effective measure of overall growth. Moreover, the CPC model clarifies the link between static and ontogenetic variation by including both levels in a joint anal? ysis and can be used to study morphological integration and constraints on the evolution of ontogenies. (Allometry; common principal components; Gerridae; growth; longitudinal data; mul? tivariate morphometrics; size.)