Affine invariant analysis of multivariate shape variation, with an example from human craniofacial growth

While the variation of observations around their mean is often considered noise in disciplines such as physics and engineering, it is an important signal in biology. Phenotypic variation is affected by genetic and environmental heterogeneity as well as by developmental instability, and is the substrate on which natural selection acts. In recent years, numerous studies on morphological integration, canalization, and evolution investigated multivariate variation of organismal shape. However, the exploratory analysis and comparison of multivariate variance patterns still remain challenging. Standard multivariate tools such as principal component analysis typically do not permit biological insight into the phenomena producing or canalizing phenotypic variation. Here, I briefly review an effective exploratory strategy for the comparison of variancecovariance patterns based on relative eigenvalues and eigenvectors. I demonstrate its usefulness in a geometric morphometric application to human craniofacial growth. What characterises a scientifically meaningful statement? One necessary criterion has been put forward by Louis Narens and others (yet outside the natural sciences): the invariance to the relevant class of transformations (e.g., Narens 2002; for a broader discussion see also Bookstein 2014). For example, in many biometric contexts, including traditional morphometrics, the units of the variables are arbitrary and thus a scientific conclusion should not depend on the scale of the variables; it should be invariant to linear transformations of the variables. This is particularly important in multivariate biometrics when pooling variables with different units. Even for variables with the same unit, the weight that variables should have in a statistic – reflecting their scientific relevance – as well as the numerosity of redundant or highly correlated variables – e.g. the spacing of landmarks – often is ambiguous, which likewise can be translated into the demand for invariance to linear transformations (Huttegger & Mitteroecker 2011). Formally, we wish that a function f of the n⇥p data matrix X is invariant to the transformation by a diagonal matrix D: f(X) = f(XD). For certain datasets, the variables may be geometrically dependent (such as interlandmark distances sharing a landmark, measurements scaled by a common size measure, or the shape coordinates of spatially adjacent landmarks) and the correlations may not all be directly interpretable. If such relationships are ambiguous or not interpretable, the statistics should be invariant to all linear transformations, including joint scaling of variables (shearing of the data space). Hence, we require the invariance to the transformation by an arbitrary nonsingular p⇥ p matrix A: f(X) = f(XA). If also the origin of the coordinate system is arbitrary (as for landmark coordinates), statistics should be invariant to all affine transformations (linear transformation and translation). In Mitteroecker & Huttegger (2009) and Huttegger & Mitteroecker (2011) we suggested that in many phenotype spaces only the affine invariant geometry should be interpreted biologically. The variance Var(x) = n 1 P (xi x̄) of a variable x is affected by linear scaling: Var(ax) = aVar(x). However, ratios of variance are affine invariant: Var(ax)/Var(ay) = Var(x)/Var(y).