Bivariate and multivariate growth allometry: statistical and biological considerations

Bivariate and multivariate analyses of 55 cranial dimensions were completed on ontogenetic series of Pygmy chimpanzees ( Pan paniscus ), Common chimpanzees ( Pan troglodytes ), and gorillas ( Gorilla gorilla ). Subanalyses described here were specifically designed to compare and crosscheck quantitative assessments of relative growth as determined using the bivariate and multivariate (principal components analysis—PCA) approaches. Results indicate a strong concordance between the bivariate and multivariate patterns, empirically supporting the claim that PCA provides an effective multivariate approach to analysing growth allometry. Comparison of bivariate and multivariate results also suggests that in multi-group PCA, the first component summarizes shape variation resulting from the sharing and extension of common patterns of growth allometry, while the second and subsequent components summarize shape variation resulting from divergent growth trajectories, reflected in bivariate comparisons as either vertical shifts and/or slope differences. Examination of non-normalized first component loadings, plus a comparison with estimates of logarithmic growth in the cranial dimensions, reveals that the non-normalized loadings are proportional to coefficients of specific growth. This finding further links the bivariate and multivariate approaches, grounding both in Huxley's theoretical notions of multiplicative and relative growth.