Analogue of univariate means plots

For this final section, we consider only cases in which the samples form a 1-way layout, i.e. there is a single (a priori defined) factor whose levels divide the data into groups, the samples in a group being considered replicates of that factor level. This factor could, of course, be from a combined factor for a higher-way crossed layout, e.g. all combinations of sites and times with (real) replicates for each combination. For a univariate response, such as a diversity index, the first step would be to test if there were significant differences among the groups and, if so, this is followed by a means plot, showing the respective means for each group with some measure of how reliably that mean has been determined, usually a confidence interval surrounding the mean. This is the type of output generated by all statistical packages (and by the Plots>Means Plot routine, seen in Section 15 for the taxonomic distinctness diversity index). It is uncommon, for example, to see a plot of the actual replicate values juxtaposed for each group in the univariate case. A glance at Fig. 18.1 of CiMC will soon tell you why! By comparison with a means plot, displaying the replicates will often make it difficult to see that pattern among the means clearly. Yet, in the multivariate context, there seems to be a (false) perception that the correct ordinations to plot have always to be ones that contain a point for each sample, in spite of the high stress solutions that often result – with that stress often due to random sampling variability within each group rather than to an inherently high-d structure in the relationship among groups. nMDS is surprisingly successful in capturing underlying group structure from replicate data, partly because it is able to non-linearly stretch and squeeze the dis-similarities into the low-d MDS distance scale, compressing the scale for smaller dissimil¬arities, which are often those from within groups. However, the logic from univariate practice (and it is sound logic) is that for a priori defined structures, we should run a testing procedure (ANOSIM, RELATE, PERMANOVA, …) to justify interpreting the data and follow this with an ordination means plot, to display and interpret the among-group patterns. On several occasions in this manual and in CiMC, we have followed through this logic to good effect. We have also seen that there are a number of ways of producing a means plot, i.e. different ways of defining a measure of central tendency (to use the accepted jargon of statistics). This is particularly so in the multi¬variate context, where we have already referred to relationships among means calculated from: averaging the raw data, the transformed data matrix, the similarities, the rank similarities, and (in PERMANOVA+) generating centroids in the high-d PCO space. Chapter 18 of CiMC discusses the pros and cons of some of the different methods, but the practical reality (fortunately) is that the relationship among means is often very similar whichever method is used. The possible exception to this is averaging raw data in cases where a strong trans¬form¬ation is appropriate – the means can then get hijacked by species with large outlying counts in a single replicate, something which the usual process of trans-formation before computation of similarities is able to resolve. Ordination plots of means, based on averaging a data matrix with Tools>Average (rather than averaging biological-type similarities – also by Tools>Average, but on the resem¬blance matrix) are therefore preferably better carried out on the transformed form of that data sheet. (This is especially true if that averaging is over the levels of other factors.)