# Chapter 1: Permutational ANOVA and MANOVA (PERMANOVA)

Key references:

Method: Anderson (2001a), McArdle & Anderson (2001)

Permutation techniques: Anderson (2001b), Anderson & ter Braak (2003)

#### 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...