How to Define, Use, and Interpret Pagel's λ $$ \lambda $$ (Lambda) in Ecology and Evolution

Abstract

Aim

Pagel's $\lambda$ (lambda) is a useful tool in ecology and evolution for describing trait evolution, imputing missing species' data, and generalising ecological relationships beyond their study system. Here, we review the various applications and interpretations of $\lambda$, highlight common misconceptions, and show how confusion in defining and using $\lambda$ can mislead our interpretation of ecological and evolutionary processes.

Innovation

We highlight that: (1) as an index of phylogenetic signal applied to continuous traits, $\lambda$ typically (but not always) ranges between 0 and 1, and is a rate-independent measure of the degree to which closely-related species resemble one-another relative to a Brownian motion expectation. (2) $\lambda$ estimated on incompletely sampled clades assumes random species sampling, which is rarely the case in ecological data sets, and likely has large uncertainty. (3) High $\lambda$ is a necessary but not sufficient prerequisite for phylogenetic imputation. (4) $\lambda$ in Phylogenetic Generalised Least Squares (PGLS) models is estimated using model residuals and is not (in most cases) an index of phylogenetic signal of measured traits. (5) New hierarchical methods including intra-specific variation return metrics such as $h^2$ that are similar but not identical to $\lambda$; we show how these disparate approaches can be unified within a single framework.

Main Conclusions

Phylogenetic methods are increasingly integrated into ecological and evolutionary analyses. Pagel's $\lambda$, a phylogenetic scaling parameter that describes how shared evolutionary history structures species similarities and differences, is commonly used as both a metric of ‘phylogenetic signal’ and as a statistical correction for the evolutionary non-independence of species in phylogenetic comparative analysis. We show how to use $\lambda$ to resolve these conceptual and statistical discrepancies.