Given two matrices, A and B, each with n rows, and with p and q columns, respectively, a very common goal is the comparison of these matrices in a subspace of dimension s, 1 ≤ s ≤ min p, q. That is, assuming linear mappings of A and B into the subspace, the comparison involves the choice of two transformation matrices L and M of dimensions p by s and q by s, respectively, and the subsequent comparisons of the images AL and BM in some fashion.

Kaiser presented a method for finding a set of derived orthogonal variables which correlate maximally with a set of original variables. A simpler, more complete derivation of Kaiser's result is given and compared to related types of transformations. The transformation derived here suggests a direct method for finding the orthogonal factor solution which is maximally similar to a given oblique solution.

We analysed the morphological variations through geometric morphometric approaches of the dorsal and ventral views of growing male and female Callinectes danae Smith 1869, based on the hypothesis that swimming crabs present polymorphism during growth. Our research identified six instars for females, seven instars for males and one instar for unsexed individuals (young). The sixth instar of females and the seventh instar of males were composed of adults. We identified 20 landmarks from the dorsal view, and 16 landmarks from the ventral view. Based on canonical analysis and Procrustes distance, we observed the formation of clearly separated instars (both views), indicating a significant change during ontogeny, not only in the pubertal moult. The most prevalent changes occurred in the abdomen, with males thinning and females broadening the abdomen during growth. In the dorsal view, we observed a displacement of anterolateral spines to the posterior region throughout the growth period and that the anterolateral and frontal teeth of juveniles were more ornate than those of adults. The ontogenetic trajectories for males and females have similar origins and follow different directions over the instars, with maximum distance after the pubertal moult (P < 0.05).

The Hierarchical Personality Inventory for Children (HiPIC) is a developmentally appropriate parent-report measure of the Five Factor Model (FFM) that has been validated in several European languages but only recently in English. The English translation of the HiPIC was evaluated in an Australian context. Parent-rated HiPIC scores were obtained for 202 children (aged 5–14 years) via an online survey. Exploratory factor analysis indicated five factors that appeared reasonably congruent with the original Flemish HiPIC structure, though with some apparent differences particularly in regard to the Extraversion and Benevolence facets. A Procrustes targeted rotation was used to evaluate the congruence with the original Flemish structure. This indicated encouragingly high congruences for the overall model (.97), as well as high facet and factor congruence (.92–.99). These findings can be taken to reflect the robust nature of the HiPIC model, validating the instrument and more specifically confirming its applicability for use in practice and research investigating children's development and wellbeing in Australia.

The paper considers the bias of Bookstein's mean estimator for shape under the isotropic normal model. This work depends on certain distributional properties of shape variables. An alternative unbiased modified estimator is proposed and its performance is compared with various estimators, including Procrustes and the exact maximum likelihood estimator, in a simulation study.