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Key references Method: ,   CAP is a routine for performing canonical analysis of principal coordinates. The purpose of CAP is to find axes through the multivariate cloud of points that either: (i) are the best at discriminating among a priori groups (discr...

In some cases, we may know that there are differences among some pre-defined groups (for example, after performing a cluster analysis, or after obtaining a significant result in PERMANOVA), and our interest is not so much in testing for group differences as it...

Details of CAP and how it is related to other methods are provided by and . In brief, a classical canonical analysis is simply done on a subset of the PCO axes. Here, we provide a thumbnail sketch to outline the main features of the analysis. The important is...

We will begin with an example provided by Trevor Willis and Chris Denny (; ), examining temperate reef fish assemblages at the Poor Knights Islands, New Zealand. Divers have counted the abundances of fish belonging to each of p = 62 species in each of nine 25 ...

How did the CAP routine choose an appropriate number of PCO axes to use for the above discriminant analysis (m = 7)? The essential idea here is that we wish to include as much of the original variability in the data cloud as possible, but we do not wish to inc...

The procedure of pulling out one sample at a time and checking the ability of the model to correctly classify that sample into its appropriate group is also called cross-validation. An important part of the CAP output from a discriminant type of analysis is th...

CAP can be used to test for significant differences among the groups in multivariate space. The test statistics in CAP are different from the pseudo-F used in PERMANOVA. Instead, they are directly analogous to the traditional classical MANOVA test statistics, ...

It might seem confusing that both CAP and PERMANOVA can be used to test for differences among groups in multivariate space. At the very least, it begs the question: which test should one use routinely? The main difference between these two approaches is that C...

When using the CAP routine, it should come as no surprise that the hypothesis (usually) is evident in the plot. Indeed the role of the analysis is to search for the hypothesis in the data cloud. However, once faced with the constrained ordination, one might be...

A new utility of the windows-based version of the CAP routine in PERMANOVA+ is the ability to place new samples onto the canonical axes of an existing CAP model and (in the case of a discriminant analysis) to classify each of those new samples into one of the ...

So far, the focus has been on hypotheses concerning groups and the use of CAP for discriminant analysis. CAP can also be used to analyse how well multivariate data can predict the positions of samples along a continuous or quantitative gradient. As an example,...

In some cases, interest lies in finding axes through the cloud of points so as to maximise correlation with not just one X variable, but with linear combinations of multiple X variables simultaneously. In such cases, neither of these two sets of variables (i.e...

It was previously stated that CAP effectively “sphericises” the data clouds as part of the process of searching for inter-correlations between them (e.g., Fig. 5.11). The idea of “sphericising” a set of variables, rendering them orthonormal, deserves a little ...

So, how does CAP differ from dbRDA for relating two sets of variables? First, dbRDA is directional. Each set of variables has a role as either predictor variables (X) or response variables (Q), while for CAP (when there are multiple variables in X), the two se...

The relationship between dbRDA and CAP can also be seen if we consider their formulation using singular value decomposition (SVD). For simplicity, but without loss of generality, suppose that each of the variables in matrices Y and X are centred on their means...

A study by explored the relationships between two sets of variables: the abundances of hunting spiders (Lycosidae) obtained in pitfall traps and a suite of environmental variables for a series of sites across a dune meadow in the Netherlands. A subset of thes...

We wish to thank our many colleagues, whose ongoing research has supported this work by providing ideas and datasets. We trust that our citations in the text and associated with datasets provides ample evidence of the many researchers who have contributed towa...

Akaike (1973) Akaike H. 1973. ‘Information theory as an extension of the maximum likelihood principle’, pp. 267-281 in Petrov BN & Caski F (eds). Proceedings, 2nd International Symposium on Information Theory. Akademiai Kiado, Budapest. ...

Matrices and vectors A = matrix containing elements $a _ {ij} = - \frac{1}{2} d _ {ij} ^ 2 $ B = matrix of variables (N × s) that are linear combinations of normalised X variables having maximum correlation with CAP axes C = matrix of CAP axes (N × s), standa...

Below is an index to the data sets used in examples, listed in order of appearance in the text. With each dataset are given the name and location of the data file, the original reference, a description of its use as an example in the manual and the page number...