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A new feature of the PERMANOVA+ add-on package is the ability to add vector overlays onto graphical outputs. This is offered as purely an exploratory tool to visualise potential linear or monotonic relationships between a given set of variables and ordination ...

Principal components analysis (PCA) is described in detail in chapter 4 of . As stated earlier, PCO produces an equivalent ordination to a PCA when a Euclidean distance matrix is used as the basis of the analysis. Consider the environmental data from the Firth...

In chapter 1, the difficulty in calculating centroids for non-Euclidean resemblance measures was discussed (see the section entitled Huygens’ theorem). In chapter 2, the section entitled Generalisation to dissimilarities indicated how PCO axes could be used in...

We recommend that, for routine ordination to visualise multivariate data on the basis of a chosen resemblance measure, non-metric MDS is the method of choice (; ). MDS is robust to outliers, and it explicitly uses a criterion of preserving rank orders of dissi...

Key references Method:, Permutation methods: , , ,   DISTLM is a routine for analysing and modeling the relationship between a multivariate data cloud, as described by a resemblance matrix, and one or more predictor variables. For example, in ecology, t...

Just as PERMANOVA does a partitioning of variation in a data cloud described by a resemblance matrix according to an ANOVA model, DISTLM does just such a partitioning, but according to a regression (or multiple regression) model. For an ANOVA design, the predi...

Consider an (N × p) matrix of response variables Y, where N = the number of samples and p = the number of variables. Consider also an (N × q) matrix, X, which contains q explanatory (predictor) variables of interest (e.g. environmental variables). The purpose ...

In our first example of DISTLM, we will examine the relationship between the Shannon diversity (H′) of macrofauna and log copper concentration from benthic sampling at 12 sites along the Garroch Head dumpground in the Firth of Clyde, using simple linear regres...

More generally, when X contains more than one variable, we may also be interested in conditional or partial tests. For example, if X contains two variables X$_1$ and X$_2$, or (more generally) two sets of variables X$_1$ and X$_2$, one may ask: how much of th...

To demonstrate conditional tests in DISTLM, we will consider the number of species inhabiting holdfasts of the kelp Ecklonia radiata in the dataset from New Zealand, located in the file hold.pri in the ‘HoldNZ’ folder of the ‘Examples add-on’ directory. These ...

Thus far, we have only done examples for a univariate response variable in Euclidean space, using DISTLM to fit linear models, but with tests being done by permutation. However, the fact that any resemblance measure can be used as the basis of the analysis in ...

In many situations, a scientist may have measured a large number of predictor variables that could be potentially important, and interest lies in determining which ones are best at explaining variation in the response data cloud and also whether particular com...

Before proceeding, a few cautionary notes are appropriate with respect to building models. First, the procedures of forward selection, backward elimination and step-wise selection are in no way guaranteed to find the best overall model. Second, even if the sea...

We shall now use the DISTLM tool to identify potential parsimonious models for benthic macrofauna near the Ekofisk oil platform in response to several measured environmental variables. The response data (in file ekma.pri in the ‘Ekofisk’ folder of the ‘Example...

We may wish to visualise a given model in the multivariate space of our chosen resemblance matrix. The ordination method of PCO was introduced in chapter 3 as a way of obtaining orthogonal (independent) axes in Euclidean space that would represent our data clo...

Something which certainly should come as no surprise is to see the X variables playing an important role in driving the variation along dbRDA axes. Of course, the X variables must feature strongly here, because it is from these that the fitted variation is der...

Let us examine the constrained dbRDA ordination for the parsimonious model obtained earlier using DISTLM on the Ekofisk data. The parsimonious model we had settled on included three predictor variables: (Distance)^(0.25), sqr(Ba) and sqr(Sr). Go to the ekevt w...

In some situations, it is useful to be able to partition variability in the data cloud according to sets of predictor variables, rather than treating each variable individually. For example, discussed the partitioning of variation in multivariate species data...

Sometimes the predictor variables of interest are not quantitative, continuous variables, but rather consist of categories or groups, called categorical or nominal variables. There are also situations where we have mixtures of variable types that we want to in...

On the face of it, the DISTLM routine might be thought of as playing a similar role to PRIMER’s BEST routine in the analysis of multivariate species data. More particularly, the BEST (BIOENV or BVSTEP) procedure in PRIMER is designed to find a combination of e...