PERMANOVA+ is an add-on package for PRIMER 6. It was produced as a collaborative effort between Marti Anderson (Department of Statistics, University of Auckland, New Zealand) and Ray Gorley & Bob Clarke (PRIMER-E Ltd, Plymouth, UK). It 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. By adopting a more parametric approach it allows

• partitioning variability according to one or more explanatory variables or factors
• measuring or testing interactions among factors (which can only be defined by reference to modelled main effects)
• developing explicitly quantitative models with explanatory, discriminatory or predictive uses

It retains robustness by being totally resemblance and permutation based.

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PERMANOVA - permutational ANOVA/MANOVA

Analyses univariate or multivariate data in response to factors, groups or treatments in an experimental design. PERMANOVA can be used as a better ANOVA/MANOVA. Whereas ANOVA/MANOVA assumes normal distributions and, implicitly, Euclidean distance, PERMANOVA works with any distance measure that is appropriate to the data, and uses permutations to make it distribution free. It carries this generalisation through to include most of the options you would expect from modern ANOVA/MANOVA implementation. For example, new theoretical work allows the handling of complex unbalanced designs, also including covariables.

PERMDISP - homogeneity of dispersions

Tests the homogeneity of multivariate dispersions within groups, on the basis of any resemblance measure. One application is to help in interpreting the results from a PERMANOVA analysis, which makes the implicit assumption (as for ANOVA and ANOSIM) that dispersions are roughly constant across groups.

PCO - principal coordinates analysis

Unconstrained ordination of multivariate data, projection-based (like principal components), but using any chosen resemblance measure.

DistLM - distance-based linear models

Analyses and models the relationship between a multivariate data cloud, and one or more predictor variables, with various options for model selection. As with the other methods here, it is based on a resemblance matrix and uses permutations, rather than the restrictive Euclidean distance and normality assumptions which underlie the standard approach to linear modelling. For example, in ecology, the resemblance matrix commonly describes dissimilarities (or similarities) among a set of samples on the basis of multivariate species abundance data, and interest may lie in modelling the relationship between this data cloud and one or more environmental variables that were measured for the same set of samples.

dbRDA - distance based redundancy analysis

Ordination and visualisation of fitted models (such as from DISTLM)

CAP - canonical analysis of principal coordinates

Constrained ordination, discriminating among a priori groups, or predicting values along a gradient. Distance-based canonical correlation.

Seamless interface with PRIMER 6

• Explorer tree navigation and workspace
• Data handling, multiple input/output formats (e.g., to or from Excel/Access, etc.)
• No limits on sizes of data matrices, numbers of factors, etc.
• 2d or 3d ordination graphics, with label/symbol control, vector overlays, reflections, rotations, spin, etc.
• Choose from more than 50 resemblance measures and a host of user-specified transformations/standardisations for individual variables or sets of variables.

PERMANOVA

• Dissimilarity/distance-based analysis of univariate or multivariate data in response to ANOVA experimental/sampling designs
• P-values by permutation or Monte Carlo asymptotic distributions
• Complex experimental designs (fixed or random factors, nested terms, hierarchies, interaction terms, mixed models)
• Correct construction of pseudo-F test statistic based on multivariate analogues to expected mean squares (EMS), including linear combinations of mean squares if required
• Choice of parameterisation of fixed effects in mixed models
• Choice of permutation method
• Estimation of sizes of components of variation
• Designs lacking replication (e.g., randomised blocks, split-plots, repeated measures, latin squares, etc.)
• Pair-wise comparisons among treatments or groups, also within levels or combinations of levels of other factors in a full model
• Specification and tests of contrasts and their interaction with other terms
• Pooling or excluding terms, and specify the order of terms in the model
• Unbalanced designs (including choice of the Type of SS: I, II or III)
• Asymmetrical designs (e.g., one impact site vs multiple control sites)
• Designs with covariates, including interactions with factors

PERMDISP

• Test of homogeneity of multivariate dispersions
• Distances to centroids or spatial medians in the space of the resemblance measure
• Test by permutation or classical tables
• Comparisons of beta-diversity using presence/absence data

PCO

• Principal coordinates analysis
• 2d and 3d ordinations with choice of vector overlays
• Correct treatment of negative eigenvalues
• Windows graphical interface and easy cutting and pasting

DistLM/dbRDA

• Distance-based linear models, distance-based redundancy analysis
• Models relating species (response) to environmental (predictor) variables
• Partitioning of variation, multivariate multiple regression
• Selection criteria: multivariate analogues to R2, adjusted R2, AIC, AICc, BIC
• Selection procedures: forward, backward, step-wise or "best"
• Fit variables individually or in sets
• Specify the order of fit
• Marginal and conditional or sequential tests by permutation
• Ordination of fitted values
• Superimpose vector overlays for base variables or some other set of variables, as either Pearson or Spearman raw correlations or as multiple partial correlations.

CAP

• Canonical analysis of principal coordinates
• Find axes through multivariate data clouds to predict a priori groups or quantitative gradients
• Explore inter-correlations between two sets of variables
• Discriminant analysis, canonical correlation analysis
• Leave-one-out misclassification errors (or residual SS) for model assessment
• Placement and allocation of new samples into existing models
• Ordination of canonical models, with choice of vector overlays
• Classification and prediction