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As for the one-way case, the distribution of each of the pseudo-F ratios in a multi-way design is generally unknown. Thus, a permutation test (or some other approach using re-sampling methods) is desirable. When there is more than one factor, situations common...

Recall that we assume for the analysis of a one-way design by PERMANOVA that the multivariate observations are independent and identically distributed (i.i.d.) under a true null hypothesis. When more complex (multi-way) designs are analysed, a few more assumpt...

In some cases, what is of interest in a particular experimental design is not necessarily the comparisons among all pairs of levels of some factor found to be significant, but rather to compare one or more groups (or levels) together versus one or more other g...

All of the factors considered so far have been fixed, but factors can be either fixed or random. In univariate ANOVA the choice of whether a particular factor is fixed or random has important consequences for the assumptions underlying the model, the expected ...

For any given ANOVA design, PERMANOVA identifies a component of variation for each term in the model, denoted as ‘S(*)’ for the fixed terms and ‘V(*)’ for the random terms. This notation is used because, in the analogous univariate case, components of variati...

An important consequence of the choice made for each factor as to whether it be fixed or random is identified by examining the expected mean squares (EMS) for each term in the resulting model. This is vitally important because the EMS’s are used to identify an...

The determination of the EMS’s gives a direct indication of how the pseudo-F ratio should be constructed in order to isolate the term of interest to test a particular hypothesis (e.g., Table 1.1). Consider the test of the term ‘Block’ (‘Bl’) in the above mixed...

The denominator mean square of the pseudo-F ratio for any particular term in the analysis is important not just because it isolates the component of interest in the numerator for the test: it also identifies the exchangeable units needed to obtain a correct te...

It is worthwhile pausing to consider how the above tests correspond to meaningful hypotheses for the mixed model. What is being examined by $F _ {Tr}$ is the extent to which the sum of squared fixed effects can be detected as being non-zero over and above the ...

Given the fact that so many important aspects of the results (pseudo-F ratios, P-values, power, the inference space, etc.) depend so heavily on the experimental design (information given in the design file), one might wish to examine the qualities of various d...

We have seen how a crossed design is identifiable by virtue of every level of one factor being present in every level of the other factor, and vice versa (e.g., Fig. 1.14). We can contrast this situation with a nested design. A factor is nested within another ...

The EMS’s also yield another important insight: they provide a direct method to get unbiased estimates of each of the components of variation in the model. PERMANOVA estimates these components using mean squares, in a directly analogous fashion to the unbiased...

For a given design file, PERMANOVA, by default, will do a partitioning according to all terms that are directly implied by the experimental design. For multi-factor designs, PERMANOVA will assume that all factors are crossed with one another, unless nesting is...

A topic related to the issue of pooling is the issue of designs that lack replication. Familiar examples are some of the classical experimental designs, primarily from the agricultural literature, such as randomised blocks, split plots or latin squares (e.g. M...

Another special case of a design lacking appropriate replication is known as a split-plot design. These designs usually arise in an agricultural context, where the experimenter has applied the treatment levels for a factor (say, factor A) randomly to whole plo...

While randomised blocks, latin squares and split-plot designs lack spatial replication, a special case of a design lacking temporal replication (and which occurs quite a lot in ecological sampling) is the repeated measures design (e.g., , ). In essence, these ...

Virtually all of the examples thus far have involved the analysis of what are known as $balanced$ experimental designs. For these situations, there is equal replication within each level of a factor (or within each cell). Even data that lack replication have a...

When the design is unbalanced, there will be a number of different ways to do the partitioning, which will depend to some extent on our hypotheses and how we wish to treat the potential overlap among the terms. The different ways of doing the partitioning are ...

A topic that is related (perhaps surprisingly) to the topic of unbalanced designs is the analysis of covariance, or ANCOVA. There are some situations where the experimenter, faced with the analysis of a set of data in response to an ANOVA-type of experimental ...

Several aspects of the above analysis demonstrate its affinity with unbalanced designs. Note that: (i) the multipliers for components of variation in each EMS are not whole numbers; and (ii) the multipliers for a given component of variation are not the same i...