## Introduction

The interplay between ecological and evolutionary processes is increasingly recognized to shape the distribution of species in space and time. In addition, larger and more detailed phylogenies containing signatures of past evolutionary processes that led to contemporary biodiversity are becoming more rapidly available. As a result, many studies now use these phylogenies to account for the relatedness of species and the resulting dependencies of observations, for example, in the fields of comparative analyses, community ecology and macro-ecology (as reviewed by Blomberg, Garland & Ives 2003; Lavergne *et al.* 2010). One central concept in these studies is the statistical non-independence among species trait values because of their phylogenetic relatedness (Felsenstein 1985; Revell, Harmon & Collar 2008). This non-independence can be measured by the ‘phylogenetic signal’, hereafter defined as the ‘tendency for related species to resemble each other more than they resemble species drawn at random from the tree’ (Blomberg & Garland 2002, p. 905). Phylogenetic signal has been used to investigate questions in a wide range of research areas: How strongly are certain traits correlated with each other (Felsenstein 1985)? Which processes drive community assembly (Webb *et al.* 2002)? Are niches conserved along phylogenies (Losos 2008) and is vulnerability to climate change clustered in the phylogeny (Thuiller *et al.* 2011)?

Along with the different types of applications, a variety of indices has been proposed to measure and test for phylogenetic signal in a quantitative trait (cf. Table 1 and Table A1 in the Appendix for a selection of common indices and tests). Among these are indices that were originally developed within the context of spatial autocorrelation and have later been adapted to phylogenetic applications (e.g. Moran’s *I*, Moran 1950; Gittleman & Kot 1990; Pavoine *et al.* 2008; Revell, Harmon & Collar 2008). These phylogenetic autocorrelation indices have in common that their calculated values are not originally designed to offer a quantitative interpretation (Li, Calder & Cressie 2007 show this in a spatial context). Other indices explicitly relate to a Brownian motion (BM) model of trait evolution (Martins 1996; Pagel 1999; Blomberg, Garland & Ives 2003) and are designed to allow for the comparison of observed values among different phylogenies (Blomberg, Garland & Ives 2003). In the BM model, trait evolution follows a random walk along the branches of the phylogenetic tree, with the variance in the distribution of trait values being directly proportional to branch length. To test the null hypothesis of no phylogenetic signal, the observed value of the focal index can be compared with values expected under random trait distribution. Random trait distributions can either be numerically simulated by random permutations of the trait values among the tips of the phylogenetic tree or can be derived analytically, for example, by assuming a chi-square distribution for likelihood ratio tests (cf. Table 1 and Materials and methods section).

Approach | Directly model based? | Branch length considered? | Commonly applied test | |
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^{1}The test statistic for Blomberg’s test is not Blomberg’s*K*but the variance of standardized phylogenetic independent contrasts (see Materials and methods for more details).^{2}When calculated based on phylogenetic independent contrasts, it assumes Brownian motion; when it is based on generalized least squares, it depends on how the variance–covariance matrix of species dissimilarities is built from branch lengths (commonly this is performed under the assumption of Brownian motion).^{3}Only true if the definition of the weighting matrix is based on phylogenetic distances.
| ||||

Abouheif ‘s C_{mean} | Autocorrelation | No | No | Permutation |

Moran’s I | Autocorrelation | No | Yes^{3} | Permutation |

Pagel’s λ | Evolutionary | Yes | Yes | Maximum likelihood |

Blomberg’s K | Evolutionary | Yes | Yes | – |

Blomberg’s test^{1} | Evolutionary | Yes^{2} | Yes | Permutation |

Even though all indices have been developed to quantify and test for phylogenetic signal, they are calculated following different approaches. Consequently, all of these indices measure different aspects of phylogenetic signal and have been shown to respond differently to inaccurate phylogenetic information, low sample sizes and the absence of branch length information (Blomberg, Garland & Ives 2003; Cavender-Bares, Keen & Miles 2006). However, in the literature, they are used for the same ecological questions, and guidelines for selecting the most appropriate method are missing. To make the best use of these indices, it is essential to assess how estimates of strength and tests of phylogenetic signal are influenced by different properties of the data (Revell, Harmon & Collar 2008). Ultimately, in each specific situation, an educated decision on which index to use is necessary. Here, we compare four indices which have been commonly used in evolutionary ecology studies: Moran’s *I* (Gittleman & Kot 1990; applied e.g. in Nabout *et al.* 2010), Abouheif’s *C*_{mean} (Abouheif 1999; applied e.g. in Thuiller *et al.* 2011), Blomberg’s *K* (Blomberg, Garland & Ives 2003; applied e.g. in Krasnov, Poulin & Mouillot 2011) and Pagel’s *λ* (Pagel 1999; applied e.g. in Thuiller *et al.* 2011). Moran’s *I* (Gittleman & Kot 1990) and Abouheif’s *C*_{mean} (Abouheif 1999) are autocorrelation indices and are not based on an evolutionary model. The resulting values do not offer any quantitative interpretation when comparing values between different phylogenetic trees because the expected value of the statistic under the assumed model is unknown a priori. However, stronger deviations from zero indicate stronger relationships between trait values and the phylogeny. Blomberg’s *K* (Blomberg, Garland & Ives 2003) and Pagel’s *λ* (Pagel 1999) assume a BM model of trait evolution. For both indices, a value close to zero indicates phylogenetic independence and a value of one indicates that species’ traits are distributed as expected under BM. In most cases, the upper limit of Pagel’s *λ* is close to one (see Materials and methods for details), while Blomberg’s *K* can take higher values indicating stronger trait similarity between related species than expected under BM. All four indices have been shown to perform well for the specific aspects and range of phylogenetic signal they were developed for. However, for the typical applications in evolutionary ecology, some indices are more appropriate than others. The aim of our comparison is to provide guidelines for an adequate choice.

To develop these guidelines, we create synthetic data using numerical simulations to control for different strength of expected phylogenetic signal and compare both the response of the selected indices and the power of the associated statistical tests. Furthermore, we investigate the sensitivity of the four indices to the size of phylogenies (small vs. very large species number), the resolution of tree structure (phylogenies with and without polytomies) and the availability of branch length estimates (phylogenies with branch length information vs. phylogenies with uniform branch lengths). Finally, we account for more complex models of trait evolution such as Ornstein–Uhlenbeck processes and models that slow-down or speed-up the rate of trait evolution over evolutionary time.