PERMANOVA+ for PRIMER: Guide to Software and Statistical Methods
M J Anderson, R N Gorley & K R Clarke (2008)
Introduction and overview
0.1 Title page
0.2 Contact details and installation of the PERMANOVA+ software
Getting in touch with us PERMANOVA+ for PRIMER was produced as a collaborative effort between Pro...
0.3 Introduction to the methods of PERMANOVA+
Rationale PERMANOVA+ is an add-on package which extends the resemblance-based methods of PRIMER t...
0.4 Changes from DOS to PERMANOVA+ for PRIMER
The new Windows interface All of the original DOS routines have been fully re-written, translat...
0.5 Using this manual
Typographic conventions The typographic conventions for this manual follow those used by for PRI...
Chapter 1: Permutational ANOVA and MANOVA (PERMANOVA)
Key references: Method: Anderson (2001a), McArdle & Anderson (2001) Permutation techniques:...
1.1 General description
Key references Method: , Permutation techniques: , PERMANOVA is a routine for testing the...
1.2 Partitioning
We shall begin by considering the balanced one-way (single factor) ANOVA experimental design. A f...
1.3 Huygens’ theorem
This partitioning is fine and perfectly valid for Euclidean distances. But what happens if we wis...
1.4 Sums of squares from a distance matrix
We can now consider the structure of a distance/dissimilarity matrix and how sums of squares for ...
1.5 The pseudo-F statistic
Once the partitioning has been done we are ready to calculate a test statistic associated with th...
1.6 Test by permutation
An appropriate distribution for the pseudo-F statistic under a true null hypothesis is obtained b...
1.7 Assumptions
Recall that for traditional one-way ANOVA, the assumptions are that the errors are independent, t...
1.8 One-way example (Ekofisk oil-field macrofauna)
Our first real example comes from a study by , who studied changes in community structure of soft...
1.9 Creating a design file
We shall formally test the hypothesis of no differences in community structure among the four gro...
1.10 Running PERMANOVA
To run PERMANOVA on the Ekofisk data, click on the resemblance matrix and select PERMANOVA+ > PER...
1.11 Pair-wise comparisons
Pair-wise comparisons among all pairs of levels of a given factor of interest are obtained by doi...
1.12 Monte Carlo P-values (Victorian avifauna)
In some situations, there are not enough possible permutations to get a reasonable test. Consider...
1.13 PERMANOVA versus ANOSIM
The analysis of similarities (ANOSIM), described by is also available within PRIMER and can be u...
1.14 Two-way crossed design (Subtidal epibiota)
The primary advantage of PERMANOVA is its ability to analyse complex experimental designs. The pa...
1.15 Interpreting interactions
What do we mean by an “interaction” between two factors in multivariate space? Recall that for a ...
1.16 Additivity
Central to an understanding of what an interaction means for linear models25 is the idea of addit...
1.17 Methods of permutations
As for the one-way case, the distribution of each of the pseudo-F ratios in a multi-way design is...
1.18 Additional assumptions
Recall that we assume for the analysis of a one-way design by PERMANOVA that the multivariate obs...
1.19 Contrasts
In some cases, what is of interest in a particular experimental design is not necessarily the com...
1.20 Fixed vs random factors (Tasmanian meiofauna)
All of the factors considered so far have been fixed, but factors can be either fixed or random. ...
1.21 Components of variation
For any given ANOVA design, PERMANOVA identifies a component of variation for each term in the mo...
1.22 Expected mean squares (EMS)
An important consequence of the choice made for each factor as to whether it be fixed or random i...
1.23 Constructing $F$ from EMS
The determination of the EMS’s gives a direct indication of how the pseudo-F ratio should be cons...
1.24 Exchangeable units
The denominator mean square of the pseudo-F ratio for any particular term in the analysis is impo...
1.25 Inference space and power
It is worthwhile pausing to consider how the above tests correspond to meaningful hypotheses for ...
1.26 Testing the design
Given the fact that so many important aspects of the results (pseudo-F ratios, P-values, power, t...
1.27 Nested design (Holdfast invertebrates)
We have seen how a crossed design is identifiable by virtue of every level of one factor being pr...
1.28 Estimating components of variation
The EMS’s also yield another important insight: they provide a direct method to get unbiased esti...
1.29 Pooling or excluding terms
For a given design file, PERMANOVA, by default, will do a partitioning according to all terms tha...
1.30 Designs that lack replication (Plankton net study)
A topic related to the issue of pooling is the issue of designs that lack replication. Familiar e...
1.31 Split-plot designs (Woodstock plants)
Another special case of a design lacking appropriate replication is known as a split-plot design....
1.32 Repeated measures (Victorian avifauna, revisited)
While randomised blocks, latin squares and split-plot designs lack spatial replication, a special...
1.33 Unbalanced designs
Virtually all of the examples thus far have involved the analysis of what are known as $balanced$...
1.34 Types of sums of squares (Birds from Borneo)
When the design is unbalanced, there will be a number of different ways to do the partitioning, w...
1.35 Designs with covariates (Holdfast invertebrates, revisited)
A topic that is related (perhaps surprisingly) to the topic of unbalanced designs is the analysis...
1.36 Linear combinations of mean squares (NZ fish assemblages)
Several aspects of the above analysis demonstrate its affinity with unbalanced designs. Note that...
1.37 Asymmetrical designs (Mediterranean molluscs)
Although a previous section has been devoted to the analysis of unbalanced designs, there are som...
1.38 Environmental impacts
Some further comments are appropriate here regarding experimental designs to detect environmental...
Chapter 2: Tests of homogeneity of dispersions (PERMDISP)
Key reference Method: Anderson (2006)
2.1 General description
Key reference Method: PERMDISP is a routine for testing the homogeneity of multivariate dis...
2.2 Rationale
There are various reasons why one might wish to perform an explicit test of the null hypothesis o...
2.3 Multivariate Levene’s test (Bumpus’ sparrows)
proposed doing an analysis of variance (ANOVA) on the absolute values of deviations of observati...
2.4 Generalisation to dissimilarities
Of course, in many applications that we will encounter (especially in the case of community data)...
2.5 $P$-values by permutation
The other hurdle that must be cleared is to recognise that, in line with the philosophy of all of...
2.6 Test based on medians
Levene’s test (for univariate data) can be made more robust (i.e. less affected by outliers) by u...
2.7 Ecological example (Tikus Island corals)
An ecological example of the test for homogeneity is provided by considering a study by on coral...
2.8 Choice of measure
An extremely important point is that the test of dispersion is going to be critically affected by...
2.9 Dispersion as beta diversity (Norwegian macrofauna)
When used on species composition (presence/absence) data in conjunction with certain resemblance ...
2.10 Small sample sizes
There is one necessary restriction on the use of PERMDISP, which is that the number of replicate ...
2.11 Dispersion in nested designs (Okura macrofauna)
In many situations, the experimental design is not as simple as a one-way analysis among groups. ...
2.12 Dispersion in crossed designs (Cryptic fish)
When two factors are crossed with one another, there may be several possible hypotheses concernin...
2.13 Concluding remarks
PERMDISP is designed to test the null hypothesis of no differences in dispersions among a priori ...
Chapter 3: Principal coordinates analysis (PCO)
Key references Method: Torgerson (1958), Gower (1966)
3.1 General description
Key references Method: , PCO is a routine for performing principal coordinates analysis () ...
3.2 Rationale
It is difficult to visualise patterns in the responses of whole sets of variables simultaneously....
3.3 Mechanics of PCO
To construct axes that maximise fitted variation (or minimise residual variation) in the cloud of...
3.4 Example: Victorian avifauna
As an example, consider the data on Victorian avifauna at the level of individual surveys, in the...
3.5 Negative eigenvalues
The sharp-sighted will have noticed a conundrum in the output given for the Victorian avifauna sh...
3.6 Vector overlays
A new feature of the PERMANOVA+ add-on package is the ability to add vector overlays onto graphic...
3.7 PCO versus PCA (Clyde environmental data)
Principal components analysis (PCA) is described in detail in chapter 4 of . As stated earlier, P...
3.8 Distances among centroids (Okura macrofauna)
In chapter 1, the difficulty in calculating centroids for non-Euclidean resemblance measures was ...
3.9 PCO versus MDS
We recommend that, for routine ordination to visualise multivariate data on the basis of a chosen...
Chapter 4: Distance-based linear models (DISTLM) and distance-based redundancy analysis (dbRDA)
Key references Method: Legendre & Anderson (1999), McArdle & Anderson (2001) Permutatio...
4.1 General description
Key references Method:, Permutation methods: , , , DISTLM is a routine for analysing and ...
4.2 Rationale
Just as PERMANOVA does a partitioning of variation in a data cloud described by a resemblance mat...
4.3 Partitioning
Consider an (N × p) matrix of response variables Y, where N = the number of samples and p = the n...
4.4 Simple linear regression (Clyde macrofauna)
In our first example of DISTLM, we will examine the relationship between the Shannon diversity (H...
4.5 Conditional tests
More generally, when X contains more than one variable, we may also be interested in conditional ...
4.6 (Holdfast invertebrates)
To demonstrate conditional tests in DISTLM, we will consider the number of species inhabiting hol...
4.7 Assumptions & diagnostics
Thus far, we have only done examples for a univariate response variable in Euclidean space, using...
4.8 Building models
In many situations, a scientist may have measured a large number of predictor variables that coul...
4.9 Cautionary notes
Before proceeding, a few cautionary notes are appropriate with respect to building models. First,...
4.10 (Ekofisk macrofauna)
We shall now use the DISTLM tool to identify potential parsimonious models for benthic macrofauna...
4.11 Visualising models: dbRDA
We may wish to visualise a given model in the multivariate space of our chosen resemblance matrix...
4.12 Vector overlays in dbRDA
Something which certainly should come as no surprise is to see the X variables playing an importa...
4.13 dbRDA plot for Ekofisk
Let us examine the constrained dbRDA ordination for the parsimonious model obtained earlier using...
4.14 Analysing variables in sets (Thau lagoon bacteria)
In some situations, it is useful to be able to partition variability in the data cloud according ...
4.15 Categorical predictor variables (Oribatid mites)
Sometimes the predictor variables of interest are not quantitative, continuous variables, but rat...
4.16 DISTLM versus BEST/ BIOENV
On the face of it, the DISTLM routine might be thought of as playing a similar role to PRIMER’s B...
Chapter 5: Canonical analysis of principal coordinates (CAP)
Key references Method: Anderson & Robinson (2003), Anderson & Willis (2003)
5.1 General description
Key references Method: , CAP is a routine for performing canonical analysis of principal co...
5.2 Rationale (Flea-beetles)
In some cases, we may know that there are differences among some pre-defined groups (for example,...
5.3 Mechanics of CAP
Details of CAP and how it is related to other methods are provided by and . In brief, a classica...
5.4 Discriminant analysis (Poor Knights Islands fish)
We will begin with an example provided by Trevor Willis and Chris Denny (; ), examining temperate...
5.5 Diagnostics
How did the CAP routine choose an appropriate number of PCO axes to use for the above discriminan...
5.6 Cross-validation
The procedure of pulling out one sample at a time and checking the ability of the model to correc...
5.7 Test by permutation (Anderson’s irises)
CAP can be used to test for significant differences among the groups in multivariate space. The t...
5.8 CAP versus PERMANOVA
It might seem confusing that both CAP and PERMANOVA can be used to test for differences among gro...
5.9 Caveats on using CAP (Tikus Island corals)
When using the CAP routine, it should come as no surprise that the hypothesis (usually) is eviden...
5.10 Adding new samples
A new utility of the windows-based version of the CAP routine in PERMANOVA+ is the ability to pla...
5.11 Canonical correlation: single gradient (Fal estuary biota)
So far, the focus has been on hypotheses concerning groups and the use of CAP for discriminant an...
5.12 Canonical correlation: multiple X’s
In some cases, interest lies in finding axes through the cloud of points so as to maximise correl...
5.13 Sphericising variables
It was previously stated that CAP effectively “sphericises” the data clouds as part of the proces...
5.14 CAP versus dbRDA
So, how does CAP differ from dbRDA for relating two sets of variables? First, dbRDA is directiona...
5.15 Comparison of methods using SVD
The relationship between dbRDA and CAP can also be seen if we consider their formulation using si...
5.16 (Hunting spiders)
A study by explored the relationships between two sets of variables: the abundances of hunting s...
Appendices
A1 Acknowledgements
We wish to thank our many colleagues, whose ongoing research has supported this work by providing...
A2 References
Akaike (1973) Akaike H. 1973. ‘Information theory as an extension of the maximum l...
A3 Index to mathematical notation and symbols
Matrices and vectors A = matrix containing elements $a _ {ij} = - \frac{1}{2} d _ {ij} ^ 2 $ B =...
A4 Index to data sets used in examples
Below is an index to the data sets used in examples, listed in order of appearance in the text. W...