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...