2STAGE for displaying ‘interactions’

A very different way of using 2nd stage matrices is best accessed through the alternative entry option in the dialog box for 2STAGE, namely to specify a single similarity matrix with factors defining a 2-way crossed layout of samples (e.g. of sites and times), and allow 2STAGE to select the sub-matrices on which to calculate the second-stage correlations. To motivate this, return to the Phuket coral data at the start of this section, in which the spatial pattern of assemblage change over an onshore-offshore transect was compared for two years, 1983 and 1987. The rank correlation (Spearman) between the two Bray-Curtis similarity matrices underlying these profiles was only  = 0.08, indicating a poorly matching sequence, the conclusion being that the sedimentation from dredging for a deep-water port in 1986 and 87 had disrupted the spatial pattern of the assemblages. In fact, that study has data from 13 years over 1983 to 2000 (the merged file for which was created in Section 11). This period included a further potentially disruptive event in 1998, a prolonged high pressure anomaly creating a period of low sea levels, increasing the frequency of desiccation. If the transect patterns for all pairwise sets of years are now matched, a correlation matrix of $\rho$ values is produced, which is the second stage matrix. These ‘similarities’ between years can be input to an MDS or clustering to give a visual summary of the inter-annual changes, not of the community as such (i.e. not of the average assemblage, or the assemblage at one fixed point on the transect – that would be a first-stage MDS) but of the internal pattern of assemblage change along the transect. Years which are anomalous in terms of their spatial pattern should stand out as outliers on this 2nd stage MDS or 2nd stage cluster analysis. If the inter-annual differences do not disrupt the internal spatial structuring but simply, for example, increase the abundance of all species down the transect in some years, relative to others, then the 2nd stage plot will show nothing whatsoever – that type of signal will be seen in a (1st stage) plot of yearly changes in the community, when averaged over the whole transect. In a sense, what the 2nd stage plot does is to remove ‘main effects’ of years (to use familiar univariate terminology) and concentrate on ‘interactions’, the changes in the internal spatial gradient for some years compared with others. This example is now implemented but is also discussed, along with other examples, in Clarke KR, Somerfield PJ, Airoldi L, Warwick RM 2006, J Exp Mar Biol Ecol 338: 179-192, and at the end of Chapter 16 of CiMC.