Residual randomization in permutation procedures (RRPP) is an appropriate means of generating empirical sampling distributions for ANOVA statistics and linear model coefficients, using ordinary or generalized least‐squares estimation. This is an especially useful approach for high‐dimensional (multivariate) data. Here, we present an r package that provides a comprehensive suite of tools for applying RRPP to linear models. Important available features include choices for OLS or GLS coefficient estimation, data or dissimilarity matrix analysis capability, choice among types I, II, or III sums of squares and cross‐products, various effect size estimation methods, and an ability to perform mixed‐model ANOVA. The lm.rrpp function is similar to the lm function in many regards, but provides coefficient and ANOVA statistics estimates over many random permutations. The S3 generic functions commonly used with lm also work with lm.rrpp. Additionally, a pairwise function provides statistical tests for comparisons of least‐squares means or slopes, among designated groups. Users have many options for varying random permutations. Compared to similar available packages and functions, RRPP is extremely fast and yields comprehensive results for downstream analyses and graphics, following model fits with lm.rrpp. The RRPP package facilitates analysis of both univariate and multivariate response data, even when the number of variables exceeds the number of observations.

The analysis of phenotypic change is important for several evolutionary biology disciplines, including phenotypic plasticity, evolutionary developmental biology, morphological evolution, physiological evolution, evolutionary ecology and behavioral evolution. It is common for researchers in these disciplines to work with multivariate phenotypic data. When phenotypic variables exceed the number of research subjects-data called 'high-dimensional data'-researchers are confronted with analytical challenges. Parametric tests that require high observation to variable ratios present a paradox for researchers, as eliminating variables potentially reduces effect sizes for comparative analyses, yet test statistics require more observations than variables. This problem is exacerbated with data that describe 'multidimensional' phenotypes, whereby a description of phenotype requires high-dimensional data. For example, landmark-based geometric morphometric data use the Cartesian coordinates of (potentially) many anatomical landmarks to describe organismal shape. Collectively such shape variables describe organism shape, although the analysis of each variable, independently, offers little benefit for addressing biological questions. Here we present a nonparametric method of evaluating effect size that is not constrained by the number of phenotypic variables, and motivate its use with example analyses of phenotypic change using geometric morphometric data. Our examples contrast different characterizations of body shape for a desert fish species, associated with measuring and comparing sexual dimorphism between two populations. We demonstrate that using more phenotypic variables can increase effect sizes, and allow for stronger inferences.

Many evolutionary studies require an understanding of phenotypic change. However, while analyses of phenotypic variation across pairs of evolutionary levels (populations or time steps) are well established, methods for testing hypotheses that compare evolutionary sequences across multiple levels are less developed. Here we describe a general analytical procedure for quantifying and comparing patterns of phenotypic evolution. The phenotypic evolution of a lineage is defined as a trajectory across a set of evolutionary levels in a multivariate phenotype space. Attributes of these trajectories (their size, direction, and shape), are quantified, and statistically compared across pairs of taxa, and a summary statistic is used to determine the extent to which patterns of phenotypic evolution are concordant across multiple taxa. This approach provides a direct quantitative description of how patterns of phenotypic evolution differ, as well as a statistical assessment of the degree of repeatability in the evolutionary responses to selection among taxa. We describe how this approach can quantify phenotypic trajectories from many ecological and evolutionary processes, whose data encode multivariate characterizations of the phenotype, including: phenotypic plasticity, ecological selection, ontogeny and growth, local adaptation, and biomechanics. We illustrate the approach by examining the phenotypic evolution of several fossil lineages of Globorotalia.

Analyses of two-state phenotypic change are common in ecological research. Some examples include phenotypic changes due to phenotypic plasticity between two environments, changes due to predator/nonpredator character shifts, character displacement via competitive interactions, and patterns of sexual dimorphism. However, methods for analyzing phenotypic change for multivariate data have not been rigorously developed. Here we outline a method for testing vectors of phenotypic change in terms of two important attributes: the magnitude of change (vector length) and the direction of change described by trait covariation (angular difference between vectors). We describe a permutation procedure for testing these attributes, which allows non-targeted sources of variation to be held constant. We provide examples that illustrate the importance of considering vector attributes of phenotypic change in biological studies, and we demonstrate how greater inference can be made than by evaluating variance components with MANOVA alone. Finally, we consider how our method may be extended to more complex data. Abstract. Analyses of two-state phenotypic change are common in ecological research. Some examples include phenotypic changes due to phenotypic plasticity between two environments, changes due to predator/non-predator character shifts, character displacement via competitive interactions, and patterns of sexual dimorphism. However, methods for analyzing phenotypic change for multivariate data have not been rigorously developed. Here we outline a method for testing vectors of phenotypic change in terms of two important attributes: the magnitude of change (vector length) and the direction of change described by trait covariation (angular difference between vectors). We describe a permutation procedure for testing these attributes, which allows non-targeted sources of variation to be held constant. We provide examples that illustrate the importance of considering vector attributes of phenotypic change in biological studies, and we demonstrate how greater inference can be made than by evaluating variance components with MANOVA alone. Finally, we consider how our method may be extended to more complex data.

Recent years have seen increased interest in phylogenetic comparative analyses of multivariate data sets, but to date the varied proposed approaches have not been extensively examined. Here we review the mathematical properties required of any multivariate method, and specifically evaluate existing multivariate phylogenetic comparative methods in this context. Phylogenetic comparative methods based on the full multivariate likelihood are robust to levels of covariation among trait dimensions and are insensitive to the orientation of the data set, but display increasing model misspecification as the number of trait dimensions increases. This is because the expected evolutionary covariance matrix (V) used in the likelihood calculations becomes more ill-conditioned as trait dimensionality increases, and as evolutionary models become more complex. Thus, these approaches are only appropriate for data sets with few traits and many species. Methods that summarize patterns across trait dimensions treated separately (e.g., SURFACE) incorrectly assume independence among trait dimensions, resulting in nearly a 100% model misspecification rate. Methods using pairwise composite likelihood are highly sensitive to levels of trait covariation, the orientation of the data set, and the number of trait dimensions. The consequences of these debilitating deficiencies are that a user can arrive at differing statistical conclusions, and therefore biological inferences, simply from a dataspace rotation, like principal component analysis. By contrast, algebraic generalizations of the standard phylogenetic comparative toolkit that use the trace of covariance matrices are insensitive to levels of trait covariation, the number of trait dimensions, and the orientation of the data set. Further, when appropriate permutation tests are used, these approaches display acceptable Type I error and statistical power. We conclude that methods summarizing information across trait dimensions, as well as pairwise composite likelihood methods should be avoided, whereas algebraic generalizations of the phylogenetic comparative toolkit provide a useful means of assessing macroevolutionary patterns in multivariate data. Finally, we discuss areas in which multivariate phylogenetic comparative methods are still in need of future development; namely highly multivariate Ornstein-Uhlenbeck models and approaches for multivariate evolutionary model comparisons.