3.1 General description

Key references

Method: Torgerson (1958) , Gower (1966)

PCO is a routine for performing principal coordinates analysis ( Gower (1966) ) on the basis of a (symmetric) resemblance matrix. PCO is also sometimes referred to as classical scaling, with origins in the psychometric literature ( Torgerson (1958) ). PCO places the samples onto Euclidean axes (i.e., so they can be drawn) using only a matrix of inter-point dissimilarities (e.g., Legendre & Legendre (1998) ). Ordination using non-metric multi-dimensional scaling (MDS) focuses on preserving only the rank order of dissimilarities for an a priori chosen number of dimensions, whereas ordination using PCO (like PCA) is a projection onto axes, but (unlike PCA) in the space of the dissimilarity measure chosen. The user chooses the number of PCO axes to include in the output, but generally only the first two or three axes are drawn in ordination plots. A further utility provided in the PERMANOVA+ add-on that uses PCO axes is the option to calculate distances among centroids for levels of a chosen factor. This allows visualisation of centroids from an experimental design in the space of the resemblance measure chosen.

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)