Invariance and Meaningfulness in Phenotype spaces

Abstract

Mathematical spaces are widely used in the sciences for representing quantitative and qualitative relations between objects or individuals. Phenotype spaces—spaces whose elements represent phenotypes—are frequently applied in morphometrics, evolutionary quantitative genetics, and systematics. In many applications, several quantitative measurements are taken as the orthogonal axes of a Euclidean vector space. We show that incommensurable units, geometric dependencies between measurements, and arbitrary spacing of measurements do not warrant a Euclidean geometry for phenotype spaces. Instead, we propose that most phenotype spaces have an affine structure. This has profound consequences for the meaningfulness of biological statements derived from a phenotype space, as they should be invariant relative to the transformations determining the structure of the phenotype space. Meaningful geometric relations in an affine space are incidence, linearity, parallel lines, distances along parallel lines, intermediacy, and ratios of volumes. Biological hypotheses should be phrased and tested in terms of these fundamental geometries, whereas the interpretation of angles and of phenotypic distances in different directions should be avoided. We present meaningful notions of phenotypic variance and other statistics for an affine phenotype space. Furthermore, we connect our findings to standard examples of morphospaces such as Raup’s space of coiled shells and Kendall’s shape space.

Appendix: Affine Invariant Statistics

Incidence relations, averages, and also distances relative to a statistical distribution are affine invariant. Consequently, many statistical tests can be applied to data with an affine structure. For example, the classical test to compare multivariate means is based on the Hotelling’s T ² statistic

$$ T^2=n(\bar{{\mathbf{x}}}-\varvec{\mu}_0)^T {\mathbf{S}}^{-1}(\bar{{\mathbf{x}}}-\varvec{\mu}_0), $$

where n is the number of cases, $\bar{{\bf x}}$ a p-dimensional vector representing the estimated mean, $\varvec{\mu}_{0}$ the hypothesized mean, and S is a p × p sample covariance matrix. If x is a random variable with a multivariate normal distribution and S has a Wishart distribution with m = n − 1 degrees of freedom and is independent of x, then T ² has a Hotelling’s T ² distribution with the parameters p and m. The Hotelling’s T ² statistic is equal to the squared Mahalanobis distance (11) multiplied by n . We have shown in (12) that Mahalanobis distance is invariant to affine transformation and hence T ² is affine invariant too. It follows that multivariate means can be tested meaningfully even if their underlying geometric structure is affine instead of Euclidean.

An extension of the Hotelling’s T ² distribution is the Wilks’ lambda distribution. For two covariance matrices $\mathbf{S}_1$ and $\mathbf{S}_2$ with Wishart distributions of m and n degrees of freedom, Wilks lambda is

$$ \Uplambda=\frac{\det({\mathbf{S}}_1)}{\det({\mathbf{S}}_1+{\mathbf{S}}_2)}= \frac{1}{\det({\mathbf{I}}+{\mathbf{S}}_1^{-1}{\mathbf{S}}_2)}=\prod\left(\frac{1}{1+ \lambda_i}\right)\sim\Uplambda(p,m,n), $$

where I is the identity matrix and λ _i is the ith eigenvalue of $\mathbf{S}_1^{-1}\mathbf{S}_2.$ In several likelihood ratio tests (e.g., in the context of MANOVA) $\mathbf{S}_1$ is the error variance and $\mathbf{S}_2$ the variance explained by some model. As shown in (11), the eigenvalues of $\mathbf{S}_1^{-1}\mathbf{S}_2$ are equal to the eigenvalues of $(\mathbf{A}^T \mathbf{S}_1\mathbf{A})^{-1}\mathbf{A}^T\mathbf{S}_2 \mathbf{A},$ so that $\Uplambda$ is affine invariant.

For the likelihood ratio test of homogeneity of covariance matrices, i.e., of H ₀: $\varvec{\Upsigma}_1=\varvec{\Upsigma}_2=\cdots=\varvec{\Upsigma}_k,$ the maximum likelihood estimate of $\varvec{\Upsigma}_i$ is $\mathbf{S}=n^{-1} \sum n_i \mathbf{S}_i$ under H ₀ and $\mathbf{S}_i$ under the alternative, where n _i is the sample size of the ith group and n = ∑ n _i . The likelihood ratio

$$ - 2\,\log \,\lambda _{a} = \sum\limits_{{i = 1}}^{k} {n_{i} } \log \det (S_{i}^{{ - 1}} S), $$

has an asymptotic χ² distribution with $\frac{1}{2} p(p+1)(k-1)$ degrees of freedom (Mardia et al. 1979, p. 140). As the eigenvalues of $\mathbf{S}_i^{-1}\mathbf{S}$ are affine invariant (11), this test is meaningful also for data with an affine structure. Similarly, the metric for covariance matrices

$$ d_{cov}({{\mathbf{S}}_1,{\mathbf{S}}_2})=\sqrt{\sum_{i=1}^{p}(\rm{log} \;\lambda_i)^2}, $$

where λ _i is the ith eigenvalues of $\mathbf{S}_2^{-1} \mathbf{S}_1,$ is invariant to affine transformation (Mitteroecker and Bookstein 2009).

For the multivariate multiple regression of $\mathbf{Y}$ on $\mathbf{Z},$ the least squares estimates of the regression coefficients are given by $\varvec{\beta}=(\mathbf{Z}^T \mathbf{Z})^{-1}\mathbf{Z}^T \mathbf{Y}$ and the predicted values are $\hat{\mathbf{Y}}=\mathbf{Z} \varvec{\beta}.$ Affine transformation of the predictors $\mathbf{Z}$ has no affect on the prediction because

$$ \begin{aligned} \hat{{\mathbf{Y}}}&={\mathbf{Z A}} ({\mathbf{A}}^T {\mathbf{Z}}^T {\mathbf{Z A}})^{-1}{\mathbf{A}}^T {\mathbf{Z}}^T {\mathbf{Y}}\\ &={\mathbf{Z A A}}^{-1} ({\mathbf{Z}}^T {\mathbf{Z}})^{-1} ({\mathbf{A}}^T)^{-1} {\mathbf{A}}^T {\mathbf{Z}}^T {\mathbf{Y}}\\ &={\mathbf{Z}} ({\mathbf{Z}}^T {\mathbf{Z}})^{-1}{\mathbf{Z}}^T {\mathbf{Y}}. \end{aligned} $$

As Fisher’s linear discriminant function is computationally equivalent to a multiple regression of a group variable on the phenotypic variables, the success of linear discrimination is unaffected by affine transformations (see Mitteroecker and Bookstein 2011, for an explicit proof). When the dependent variables $\mathbf{Y}$ are transformed into $\mathbf{YA},$ the predicted values result from the same transformation $\hat{\mathbf{Y}}\mathbf{A}=\mathbf{Z} (\mathbf{Z}^T \mathbf{Z})^{-1}\mathbf{Z}^T \mathbf{Y A}.$

We have argued that only ratios of generalized variance (determinant of the covariance matrix) are affine invariant, whereas generalized variance itself and also total variance (trace of the covariance matrix) and ratios of total variance are affected by affine transformations. We demonstrate this here by an example. Consider two covariance matrices

$$ {\mathbf{S}}_1=\left(\begin{array}{cc}2.3 & 1.2 \\ 1.2 & 1.8\end{array}\right), {\mathbf{S}}_2=\left(\begin{array}{cc}2.1 & 0.6 \\ 0.6 & 2.4\end{array}\right) $$

and the transformation matrix

$$ {\mathbf{A}}=\left(\begin{array}{cc}0.8 & -0.3 \\ 0.2 & 1.2\end{array}\right). $$

The total variance of $\mathbf{S}_1$ is $\mathrm{Tr}(\mathbf{S}_1)=4.10$ and that of the transformed data $\mathrm{Tr}(\mathbf{A}^T \mathbf{S}_1 \mathbf{A})=3.86,$ and similarly for the generalized variances $\mathrm{Det}(\mathbf{S}_1)=2.70$ and $\mathrm{Det}(\mathbf{A}^T \mathbf{S}_1 \mathbf{A})=2.81.$ The ratios of total variance are $\mathrm{Tr}(\mathbf{S}_1)/\mathrm{Tr}(\mathbf{S}_2)=0.91$ and $\mathrm{Tr}(\mathbf{A}^T \mathbf{S}_1 \mathbf{A})/\mathrm{Tr}(\mathbf{A}^T \mathbf{S}_2 \mathbf{A})=0.80,$ whereas the ratio of the generalized variance is affine invariant: $\mathrm{Det}(\mathbf{S}_1) / \mathrm{Det}(\mathbf{S}_2)=\mathrm{Det}(\mathbf{A}^T \mathbf{S}_1 \mathbf{A})/\mathrm{Det}(\mathbf{A}^T \mathbf{S}_2 \mathbf{A})=0.58.$ Ratios of total variance are affine invariant only if $\mathbf{S}_2=k\mathbf{S}_1.$