2.1 General description

Key reference

Method: Anderson (2006)

PERMDISP is a routine for testing the homogeneity of multivariate dispersions on the basis of any resemblance measure. The test is a dissimilarity-based multivariate extension of Levene’s test ( Levene (1960) ), following the ideas of van Valen (1978) , O’Brien (1992) and Manly (1994) , who used Euclidean distances. In essence, the test uses the ANOVA F statistic to compare (among different groups) the distances from observations to their group centroid. The user has the choice of whether to perform the test on the basis of distances to centroids or distances to spatial medians ( Gower (1974) ). The user also has the choice either to use traditional tables or to use permutation of appropriate residuals (either least-squares residuals if centroids are used or least-absolute-deviation residuals if spatial medians are used) in order to obtain P-values.

0.1 Title page

0.2 Contact details and installation of the PERMANOVA+ software

0.3 Introduction to the methods of PERMANOVA+

0.4 Changes from DOS to PERMANOVA+ for PRIMER

0.5 Using this manual

1.1 General description

1.2 Partitioning

1.3 Huygens’ theorem

1.4 Sums of squares from a distance matrix

1.5 The pseudo-F statistic

1.6 Test by permutation

1.7 Assumptions

1.8 One-way example (Ekofisk oil-field macrofauna)

1.9 Creating a design file

1.10 Running PERMANOVA

1.11 Pair-wise comparisons

1.12 Monte Carlo P-values (Victorian avifauna)

1.13 PERMANOVA versus ANOSIM

1.14 Two-way crossed design (Subtidal epibiota)

1.15 Interpreting interactions

1.16 Additivity

1.17 Methods of permutations

1.18 Additional assumptions

1.19 Contrasts

1.20 Fixed vs random factors (Tasmanian meiofauna)

1.21 Components of variation

1.22 Expected mean squares (EMS)

1.23 Constructing $F$ from EMS

1.24 Exchangeable units

1.25 Inference space and power

1.26 Testing the design

1.27 Nested design (Holdfast invertebrates)

1.28 Estimating components of variation

1.29 Pooling or excluding terms

1.30 Designs that lack replication (Plankton net study)

1.31 Split-plot designs (Woodstock plants)

1.32 Repeated measures (Victorian avifauna, revisited)

1.33 Unbalanced designs

1.34 Types of sums of squares (Birds from Borneo)

1.35 Designs with covariates (Holdfast invertebrates, revisited)

1.36 Linear combinations of mean squares (NZ fish assemblages)

1.37 Asymmetrical designs (Mediterranean molluscs)

1.38 Environmental impacts

2.1 General description

2.2 Rationale

2.3 Multivariate Levene’s test (Bumpus’ sparrows)

2.4 Generalisation to dissimilarities

2.5 $P$-values by permutation

2.6 Test based on medians

2.7 Ecological example (Tikus Island corals)

2.8 Choice of measure

2.9 Dispersion as beta diversity (Norwegian macrofauna)

2.10 Small sample sizes

2.11 Dispersion in nested designs (Okura macrofauna)

2.12 Dispersion in crossed designs (Cryptic fish)

2.13 Concluding remarks

3.1 General description

3.2 Rationale

3.3 Mechanics of PCO

3.4 Example: Victorian avifauna

3.5 Negative eigenvalues

3.6 Vector overlays

3.7 PCO versus PCA (Clyde environmental data)

3.8 Distances among centroids (Okura macrofauna)

3.9 PCO versus MDS

4.1 General description

4.2 Rationale

4.3 Partitioning

4.4 Simple linear regression (Clyde macrofauna)

4.5 Conditional tests

4.6 (Holdfast invertebrates)

4.7 Assumptions & diagnostics

4.8 Building models

4.9 Cautionary notes

4.10 (Ekofisk macrofauna)

4.11 Visualising models: dbRDA

4.12 Vector overlays in dbRDA

4.13 dbRDA plot for Ekofisk

4.14 Analysing variables in sets (Thau lagoon bacteria)

4.15 Categorical predictor variables (Oribatid mites)