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#### Introduction and overview

#### Chapter 1: Permutational ANOVA and MANOVA (PERMANOVA)

Key references: Method: Anderson (2001a), McArdle & Anderson (2001) Permutation techniques: Anderson (2001b), Anderson & ter Braak (2003)

#### Chapter 2: Tests of homogeneity of dispersions (PERMDISP)

Key reference Method: Anderson (2006)

#### Chapter 3: Principal coordinates analysis (PCO)

Key references Method: Torgerson (1958), Gower (1966)

#### Chapter 4: Distance-based linear models (DISTLM) and distance-based redundancy analysis (dbRDA)

Key references Method: Legendre & Anderson (1999), McArdle & Anderson (2001) Permutation methods: Freedman & Lane (1983), Anderson & Legendre (1999), Anderson & Robinson (2001), Anderson (2001b)

#### Chapter 5: Canonical analysis of principal coordinates (CAP)

Key references Method: Anderson & Robinson (2003), Anderson & Willis (2003)

#### Appendices

#### 2. Installing PRIMER software

#### 4. Getting data in to PRIMER

#### 5. Example analysis pathway (CLUSTER, MDS, ANOSIM)

#### 3. Download manuals and examples

(from within the PRIMER software)

#### 6. Some Wizards

#### 7. Run a PERMANOVA

#### 2. Let's consider using R

#### 1. Let's consider using PRIMER

#### 4. Take-home messages

#### 3. PERMANOVA vs adonis2 in R

#### 3.4 Recommendations

Hierarchical clustering with group-average linking, based on sample similarities or dissimilarities such as Bray-Curtis, has proved a useful technique in a number of ecological studies of the past half-century. It is appropriate for delineating groups of si...

#### 3.5 Similarity profiles (SIMPROF)

Given the form of the dendrogram in Fig. 3.3, with high similarities in apparently tightly defined groups and low similarities among groups, there can be little doubt that some genuine clustering of the samples exists for this data set. However, a statistical ...

#### 3.6 Binary divisive clustering

All discussion so far has been in terms of hierarchical agglomerative clustering, in which samples start in separate groups and are successively merged until, at some level of similarity, all are considered to belong to a single group. Hierarchical divisive cl...

#### 3.7 k-R clustering (non-hierarchical)

Another major class of clustering techniques is non-hierarchical, referred to above as flat clustering. The desired number of clusters (k) must be specified in advance, and an iterative search attempts to divide the samples in an optimal way into k groups, in ...

#### 4.1 Ordinations

An ordination is a map of the samples, usually in two or three dimensions, in which the placement of samples, rather than representing their location in space (or time), reflects the similarity of their biological communities. To be more precise, distances bet...

#### 4.2 Principal components analysis

The starting point for PCA is the original data matrix rather than a derived similarity matrix (though there is an implicit dissimilarity matrix underlying PCA, that of Euclidean distance). The data array is thought of as defining the positions of samples in ...

#### 4.3 Example: Garroch Head macrofauna

Fig. 4.1 shows the result of applying PCA to square-root transformed macrofaunal biomass data from the 65 species¶ found in subtidal sediments at 12 sites (1-12) along an E-W transect in the Firth of Clyde, Scotland ({G}, map at Fig. 1.5). A central site of th...

#### 4.4 PCA for environmental data

The above example makes it clear that PCA is an unsatisfactory ordination method for biological data. However, PCA is a much more useful in the multivariate analysis of environmental rather than species data¶. Here variables are perhaps a mix of physical para...

#### 4.5 Example: Dosing experiment, Solbergstrand mesocosm

An example of this final point for a real data set can be seen in Fig. 4.2. This is of nematode data for the dosing experiment {D} in the Solbergstrand mesocosms, at the GEEP Oslo Workshop (). Box core samples were collected from Oslofjord and held for three ...

#### 5.1 Other ordination methods

Principal Co-ordinates Analysis The two main weaknesses of PCA, identified at the end of Chapter 4, are its inflexibility of dissimilarity measure and its poor distance-preservation. The first problem is addressed in an important paper by , describing an exte...

#### 5.2 Non-metric multidimensional scaling (MDS)

The method of non-metric MDS was introduced by and , for application to problems in psychology; a useful introductory text is , though the applications given are not ecological. Generally, we use the term MDS to refer to Kruskal’s non-metric procedure (though...

#### 5.3 Diagnostics: Adequacy of MDS representation

Is the stress value small? By definition, stress reduces with increasing dimensionality of the ordination; it is always easier to satisfy the full set of rank order relationships among samples if there is more space to display them. The scree plot of best st...

#### 5.4 EXAMPLE: Dosing experiment, Solbergstrand

The nematode abundance data from the dosing experiment {D} at the GEEP Oslo Workshop was previously analysed by PCA, see Fig. 4.2 and accompanying text. The analysis was likely to be unsatisfactory, since the % of variance explained by the first two principal...

#### 5.5 Example: Celtic Sea zooplankton

In situations where the samples are strongly grouped, as in Figs. 5.4 and 5.5, both clustering and ordination analyses will demonstrate this, usually in equally adequate fashion. The strength of ordination is in displaying a gradation of community composition ...

#### 5.6 Example: Amoco-Cadiz oil spill, Morlaix

Benthic macrofaunal abundances of 251 species were sampled by at 21 times between April 1977 and February 1982 (approximately quarterly), at station ‘Pierre Noire’ in the Bay of Morlaix. Ten grab samples (1m²) of sediment were collected on each occasion and po...

#### 5.7 MDS strengths and weaknesses

MDS strengths MDS is simple in concept. The numerical algorithm is undeniably complex, but it is always clear what non-metric MDS is setting out to achieve: the construction of a sample map whose inter-point distances have the same rank order as the corresp...

#### 5.8 Further nMDS/mMDS developments

Higher dimensional solutions MDS solutions can be sought in higher dimensions and we noted previously that the stress will naturally decrease as the dimension increases. Fig 5.9a shows the scree plot of this decreasing stress (y axis) against the increasing nu...

#### 5.9 Example: Okura estuary macrofauna

describe macrofauna samples from the Okura estuary {O}, on the northern fringes of urban Auckland, NZ, taken inter-tidally at 2 times in each of 3 seasons under 3 sedimentary regimes (High, Medium and Low sedimentation levels), each regime represented by 5 si...

#### 5.10 Example: Messolongi lagoon diatoms

sampled 17 lagoons in E Central Greece for diatom communities (193 species), and also recorded a suite of 12 water-column variables: temperature, salinity, DO$_2$, pH, PO$_4$, total P, NH$_3$, NO$_2$, NO$_3$, inorganic N and SiO$_2$. After global square root ...

#### 5.11 Recommendations

Non-metric MDS can be recommended as the best general ordination technique available (e.g. ). Important early studies comparing ordination methods for community data gave nMDS a high rating (e.g. ) and improvements in computing power since those early studies...