## Introduction

Redundancy analysis (RDA, Rao 1964) and canonical correspondence analysis (CCA, ter Braak 1986, 1987) are two forms of asymmetric canonical analysis widely used by ecologists and palaeoecologists. ‘Asymmetric’ means that the two data matrices used in the analysis do not play the same role: there is a matrix of response variables, denoted **Y**, which often contains community composition data, and a matrix of explanatory variables (e.g. environmental), denoted **X**, which is used to explain the variation in **Y**, as in regression analysis. Contrast this with canonical correlation and co-inertia analyses where the two matrices play the same role in the analysis and can be interchanged; see, however, Tso (1981) for an asymmetric interpretation of canonical correlation analysis. RDA and CCA produce ordinations of **Y** constrained by **X**.

This paper deals with methods to test the significance of the canonical axes that emerge from this type of analysis. The canonical axes are those that are formed by linear combinations of the predictor variables; they are sometimes referred to as ‘constrained axes’. The section ‘Background: the algebra of redundancy analysis’ will show how they are computed. Individual canonical axes may be tested when the overall relationship (*R*^{2}) between **Y** and **X** has been shown to be significant. We will concentrate on RDA; our conjecture is that the conclusions derived from our simulations should apply to CCA as well.

As we deal with complex, multivariate data influenced by many factors, it is to be expected that several independent structures coexist in the response data. If these structures are linearly independent, they should appear on different canonical axes. Each one should be identifiable by a test of significance. Canonical axes that explain no more variation than random should also be detected; they do not need to be further considered in the interpretation of the results.

It is not always necessary to test the significance of the canonical axes when there are only a few. However, researchers often face situations where there is a large number of canonical axes; they may want to know how many canonical axes should be examined, plotted, and interpreted. Spatial modelling is a good example of these situations: when analysing the spatial variation of species-rich communities (hundreds of species, hundreds of sites) by spatial eigenfunctions, we may end up with hundreds of canonical axes. Different types of spatial eigenfunctions have been described in recent years to model the spatial structure of multivariate response data: Griffith’s (2000) spatial eigenfunctions from a connection matrix of neighbouring regions or sites; Moran’s eigenvector maps (MEM, Borcard & Legendre 2002; Borcard *et al.* 2004; Dray, Legendre, & Peres-Neto 2006); asymmetric eigenvector maps (AEM, Blanchet, Legendre, & Borcard 2008). It is then of interest to determine which of the canonical axes derived from these eigenfunctions represent variation that is more structured than random. This is the objective of the tests of significance of the canonical axes. Variation along the (hopefully few) significant axes can be mapped, used to draw biplots or interpreted through subsequent analyses. The nonsignificant axes can be dropped because they do not represent variation more structured than random. The eigenvalue associated with each axis expresses the variance accounted for by the canonical axis; the fraction of the response variation that each axis represents is a useful supplementary criterion: axes that account for less than, say, 1% or 5% of the variation may not need to be further analysed even if they are statistically significant.

Three methods have been proposed to test the significance of canonical axes in RDA. We will compare them using simulated data to determine which ones, if any, have correct levels of type I error (defined in the section ‘Simulation method’). We will also compare these methods and two permutation procedures in terms of power. This paper provides the first formal description of these methods.