# 6.12 Two-way ordered ANOSIM designs

Under the non-parametric framework adopted in this manual (and in the PRIMER package) three forms of 2-way ANOSIM tests were presented on page 6.5: 2-factor nested, B within A (denoted by B(A)); 2-factor crossed (denoted A$\times$B); and a special case of A$\times$B in which there are no replicates, either because only one sample was taken for each combination of A and B, or replicates were taken but considered to be ‘pseudo-replicates’ (*sensu*
Hurlbert (1984)
) and averaged.^{¶}

The principle of these tests, and their permutation procedures, remain largely unchanged when A or B (or both factors) are ordered. Previously, the test for B under the nested B(A) model (page 6.6) averaged the 1-way $R$ statistic for each level of A, denoted $\overline{R}$, and the same form of averaged statistic was used for testing B under the crossed A$\times$B model with replicates (page 6.7); without replicates the crossed test used the special (and less powerful) construction of page 6.8, with test statistic the pairwise averaged matrix correlation, $\rho_{av}$. (There was no test for B in the nested model, in the absence of replicates for B). If B is now ordered, $R$ is replaced by $R^{Oc}$ where there are replicates (becoming $\overline{R}^{Oc}$ when averaged across the levels of A), or by $R^{Os}$ where there are not (becoming $\overline{R}^{Os}$); there is no longer any necessity to invoke the special form of test based on $\rho_{av}$ when the factor is ordered. The same substitutions then happen for the test of A, if it too is ordered: $\overline{R}$ and $\rho_{av}$ are replaced by $\overline{R}^{Oc}$ and $\overline{R}^{Os}$. If A is not ordered, any ordering in B does not change the way the tests for A are carried out, e.g. for A$\times$B, the A test is still constructed by calculating the appropriate 1-way statistic for A, separately for each level of B, and then averaging those statistics.

Such a plethora of possibilities are best summarised in a table, and the later Table 6.3 lists all the possible combinations of 2-way design, factor ordering (or not) and presence (or absence) of replicates, giving the test statistic and its method of construction, listing whether or not pairwise tests make sense^{†}, and then giving some examples of marine studies in which the factors would have the right structure for such a test.

We have already seen unordered examples of 1-way tests (1a, Table 6.3) in Figs. 6.3 & 6.5, 2-way crossed (2a) in Fig. 6.7 and, without replication (2b), in Figs. 6.10 & 6.12; Fig. 6.6 is 2-way nested (2g). Examples of 2-way crossed without replicates, with one (2d) or both (2f) factors ordered, now follow.

^{¶} *An example of the latter might be ‘replicate’ cores from a multi-corer deployed only once at each of a number of sites (A) for the same set of months (B); these multiple cores are neither spatially representative of the extent of a site (a return trip would result in multi-cores from a slightly different area within the site) nor, it might be argued, temporally representative of that month.*

^{†} *If they do make sense, the PRIMER7 ANOSIM routine will give them. Performing such a 2-(or 3-) way analysis is much simpler than reading these tables! It is simply a matter of selecting the form of design (all likely combinations of 1-, 2- or 3-factor, crossed or nested) and then specifying which factors are to be considered ordered – the factor levels must be numeric in that case but only their rank order is used. Analyses that use specific numerical levels (unequally-spaced) can be catered for in many cases within the expanded RELATE routine, utilising a r statistic, see Chapter 15.*