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Key references Method: , Permutation techniques: ,   PERMANOVA is a routine for testing the simultaneous response of one or more variables to one or more factors in an analysis of variance (ANOVA) experimental design on the basis of any resemblance measu...

Getting in touch with us PERMANOVA+ for PRIMER was produced as a collaborative effort between Professor Marti Anderson (New Zealand Institute for Advanced Studies, Massey University, Albany, Auckland, New Zealand and current Director of PRIMER-e) and Ray Gorle...

Rationale PERMANOVA+ is an add-on package which extends the resemblance-based methods of PRIMER to allow the analysis of multivariate (or univariate) data in the context of more complex sampling structures, experimental designs and models. The primary reasons ...

The new Windows interface All of the original DOS routines have been fully re-written, translated from their original FORTRAN into the new Microsoft .NET environment, as used by PRIMER v6. This gives the software a fully modern Windows user interface. The ...

Typographic conventions The typographic conventions for this manual follow those used by for PRIMER v6, as follows: Text in bold indicates the menu items that need to be selected, > denotes cascading sub-menu items, tab choices, dialog boxes or sub-boxes, $\b...

We shall begin by considering the balanced one-way (single factor) ANOVA experimental design. A factor is defined as a categorical variable that identifies several groups, treatments or levels which we wish to compare. Imagine that we have one factor with a gr...

This partitioning is fine and perfectly valid for Euclidean distances. But what happens if we wish to base the analysis on some other dissimilarity (or similarity) measure? This is important because Euclidean distance is generally regarded as inappropriate for...

We can now consider the structure of a distance/dissimilarity matrix and how sums of squares for a one-way multivariate ANOVA partitioning would be calculated (Fig. 1.4). Fig. 1.4. Calculation of sums of squares directly from a distance/dissimilarity matrix. ...

Once the partitioning has been done we are ready to calculate a test statistic associated with the general multivariate null hypothesis of no differences among the groups. For this, following R. A. Fisher’s lead, a pseudo-F ratio is defined as: $$ F = \frac{ ...

An appropriate distribution for the pseudo-F statistic under a true null hypothesis is obtained by using a permutation (or randomization) procedure (e.g., , ). The idea of a permutation test is this: if there is no effect of the factor, then it is equally like...

Recall that for traditional one-way ANOVA, the assumptions are that the errors are independent, that they are normally distributed with a mean of zero and a common variance, and that they are added to the treatment effects. In the case of a one-way analysis, t...

Our first real example comes from a study by , who studied changes in community structure of soft-sediment benthic macrofauna in relation to oil-drilling activity at the Ekofisk oil platform in the North Sea. These data consist of p = 174 species sampled by...

We shall formally test the hypothesis of no differences in community structure among the four groups (where, in this case, “differences in community structure” is defined by the Bray-Curtis measure on fourth-root transformed data) using PERMANOVA. For any anal...

To run PERMANOVA on the Ekofisk data, click on the resemblance matrix and select PERMANOVA+ > PERMANOVA. In the PERMANOVA dialog box (Fig. 1.9), leave the defaults for all options, except (Num. permutations: 9999) & (Permutation method: $\bullet$Unrestricted p...

Pair-wise comparisons among all pairs of levels of a given factor of interest are obtained by doing an additional separate run of the PERMANOVA routine. This is appropriate because which particular comparisons should be done, in most cases, is not known a prio...

In some situations, there are not enough possible permutations to get a reasonable test. Consider the case of two groups, with two observations per group. There are a total of 4 observations, so the total number of possible re-orderings (permutations) of the 4...

The analysis of similarities (ANOSIM), described by is also available within PRIMER and can be used to analyse multivariate resemblances according to one-way and some limited two-way experimental designs19. Not surprisingly, ANOSIM and PERMANOVA will tend to ...

The primary advantage of PERMANOVA is its ability to analyse complex experimental designs. The partitioning inherent in the routine allows interaction terms in crossed designs to be estimated and tested explicitly. As an example, consider a manipulative experi...

What do we mean by an “interaction” between two factors in multivariate space? Recall that for a univariate analysis, a significant interaction means that the effects of one factor (if any) are not the same across levels of the other factor. An interaction can...

Central to an understanding of what an interaction means for linear models25 is the idea of additivity. Consider the example of a two-way crossed design for a univariate response variable, where the cell means and marginal means are as shown in Fig. 1.19a. Not...