The implementation of phylogenetic structural equation modeling for biological data from variance-covariance matrices, phylogenies, and comparative analyses

One statistical approach with a long history in the social sciences is a multivariate method called Structural Equation Modeling (SEM). The development of SEM followed the evolution of factor and path analyses, multiple regression analysis and MACOVA. One of the key innovations of factor analysis and SEM is that they group a set of multivariate statistical approaches that condense variability among a set of variables in fewer latent (unobserved) factors. Most biological systems are multivariate, which are not easily dissected into their component parts. However, most biologists use only univariate statistical methods, which have definitive limitations in accounting for more than a few variables simultaneously. Therefore, the implementation of methodologies like SEM into biological research is necessary. However, SEM cannot be applied directly to most biological datasets or generalized across species because of the hierarchical pattern of evolutionary history (i.e., phylogenetic non-independence or signal). This report includes the theoretical grounds for the development of Phylogenetic SEM in preparation of the development of utilitarian algorithms. I have divided this report in six parts: (1) a brief introduction to factor analysis and SEM from historical perspective and a brief description of its utility; (2) a summary of the implications of using biological data and the underlying hierarchical structure due to shared common ancestry or phylogeny; (3) a summary of the two most common comparative methods to incorporate the phylogeny in univariate analyses (i.e., phylogenetic independent contrasts and phylogenetic generalized least squares); (4) I describe how some intermediate output from both comparative methods can be used to estimate the variance–covariance matrix that has been corrected for phylogenetic signal; (5) I describe how to perform a exploratory factor analysis, specifically principal component analysis, with the corrected variance–covariance matrix; and (6) I describe the development of the phylogenetic confirmatory factor analysis and phylogenetic SEM. I hope that this report encourages other researchers to develop adequate multivariate analysis that incorporate the evolutionary principles in its analyses.