Research Article Comparing covariance matrices: random skewers method compared to the common principal components model

Abstract

Comparisons of covariance patterns are becoming more common as interest in the evolution of relationships between traits and in the evolutionary phenotypic diversification of clades have grown. We present parallel analyses of covariance matrix similarity for cranial traits in 14 New World Monkey genera using the Random Skewers (RS), T-statistics, and Common Principal Components (CPC) approaches. We find that the CPC approach is very powerful in that with adequate sample sizes, it can be used to detect significant differences in matrix structure, even between matrices that are virtually identical in their evolutionary properties, as indicated by the RS results. We suggest that in many instances the assumption that population covariance matrices are identical be rejected out of hand. The more interesting and relevant question is, How similar are two covariance matrices with respect to their predicted evolutionary responses? This issue is addressed by the random skewers method described here.

Comparing G: multivariate analysis of genetic variation in multiple populations

“…Random skewers is a method used to compare differences in orientation among G (Cheverud, 1996;Cheverud and Marroig, 2007). In these approaches, random b vectors are placed into the multivariate breeders' equation with each G, and the vector correlations between the resulting Dz vectors are used as an indication of the differences among G. The test for the significance of the similarity or dissimilarity of the matrices is then evaluated by comparison of the distribution of observed vector correlations with a distribution of vector correlations conforming to a null model.…”

“…The null models are often generated by bootstrapping and represent cases where matrices have coincident spaces (vector correlations ofB1) (for example, Calsbeek and Goodnight (2009)), or cases where matrices have distinct spaces (vector correlations of B0) (for example. Cheverud and Marroig (2007)), depending on whether researchers are interested in convergence or divergence among G . Hansen and Houle (2008) described how, in addition to the differences in Dz, a random skewer approach can be used to examine differences among G in the magnitude of genetic variance (as well as metrics of variance such as evolvability and respondability) using matrix projection.…”

Diversidade craniana humana e suas implicações evolutivas

“…Já as comparações entre as matrizes V/CV foram realizadas através da técnica de "random skewers" (Cheverud, 1996b;Marroig & Cheverud, 2001;Cheverud & Marroig, 2007). Tal técnica consiste em estimar a similaridade entre matrizes através da comparação dos vetores-resposta resultantes da multiplicação entre vetores aleatórios e as matrizes que se deseja comparar.…”

“…As similaridades entre as matrizes de V/CV foram investigadas através da técnica de "random skewers" (Cheverud, 1996b;Marroig & Cheverud, 2001;Cheverud & Marroig, 2007;Revell, 2012), utilizando-se 1.000 vetores para cada comparação entre duas matrizes V/CV. A média das diferenças entre os "vetores-resposta" obtidos em cada uma das …”

“…No entanto, além da inferência direta da ocorrência dos diferentes padrões morfocranianos e da comparação entre estes diferentes padrões e suas implicações populacionais e demográficas, ou seja, suas reconstituições históricas (Berry & Berry, 1967;Sparks & Jantz, 2002;Neves & Hubbe, 2005;Bernardo, 2007), pouco têm sido feito, especialmente em relação aos humanos, em favor da investigação e de uma melhor compreensão dos mecanismos e eventos envolvidos no processo de diversificação do formato do crânio (Lynch, 1990;Marroig & Cheverud, 2001;Cheverud & Marroig, 2007;Roseman, 2004;Weaver et al, 2007;Hubbe et al, 2009;Perez & Monteiro, 2009, para alguns exemplos). É exatamente este tipo de inferência evolutiva a que se propõe a presente tese.…”