Maximization of Complementary Richness (MCR)
Similar problems arise for another method that has some similarities to ED. Arponen et al. (2008) introduced the 'maximization of complementary richness' (MCR) method, described by the authors as the first “successful community-level strategy”. Arponen et al. developed their approach based on an assumption of unimodal responses for species centred at different positions in environmental space. It is logical, therefore, to assess whether their method succeeds in counting-up species or features under this unimodal model.
Arponen et al. did not report the similarities of MCR to the ED methods. Without proper comparisons and contrasts with ED, it remains unclear whether MCR offers advantages over the similar ED calculations. Their MCR method shares with ED several useful properties, including a similar unimodal model, an ordination space, variants of p-median, plus ED's GDM and richness-weighting options (for discussion of ED options, see Faith and Walker 1994; Faith and Ferrier 2002; Faith et al. 2003, 2004).
Arponen et al. claimed that MCR has unique properties, but some of these in fact also are shared by ED. For example, Arponen et al. (2008, p. 1438) claimed MCR is “different from the previous use of ordinations”, because, in using richness weighting and GDM, it “accounts for gradients in species richness and non-constant turnover rates of community composition”. However, the existing ED framework already uses these options (see Faith et al. 2004). Further, MCR, like ED, uses points described as “demand” points, served by one or more selected sites. In fact, both methods seek to minimise the degree to which species at demand points are not covered by selected sites. Although Arponen et al. describe MCR as maximising a summation of 'Ci' values (and each Ci value is to reflect the degree to which demand point i is covered by selected sites), each Ci equals one minus a product term. Thus, MCR is minimising the sum of product terms, and so minimising the degree to which demand points are not covered by selected sites. This property again matches ED methods.
Similarities aside, there are critical differences between the two methods. Simple examples will highlight the fact that MCR does have some novel properties relative to ED – but these properties de-grade the counting-up property that surely is critical to any truly “successful community-level strategy”.
Novel properties of MCR's basic selection criterion are well-revealed in the simple case where species richness is assumed equal at all sites. MCR then uses the product of a demand point's dissimilarities to all selected sites, and seeks to minimise the sum, over demand points, of these products. Single-gradient scenarios (Fig. 6a) highlight weaknesses of this calculation. Suppose there are two candidate sites for selection, A and B. Selection depends on which site most reduces the MCR product score. Note that when a demand point becomes a selected site, it makes no contribution to the sum of products (as its distance to itself is 0, making its product contribution equal to 0). Selecting site A removes its large product (=.05 × 0.60 × 0. 65 × 0.70 = 0.014) from the product sum (Fig. 6a). Also, it reduces the product score for non-selected sites (site B), with a reduction equal to (1–0.4) times the previous product value for B of (0.45 × 0.20 × 0.25 × 0.30 = 0.007), yielding a reduction of 0.004. Thus, selecting site A reduces the score by about 0.018 (0.014 + 0.004). In contrast, selecting site B implies removal of a product term equal to 0.007 (see above), and a reduction in the A product contribution of (1–0.4) times 0.0137 = 0.008. Thus, selecting site B reduces the MCR score by only 0.015, and MCR selects site A. We also can ask whether site A or B is best to lose (smallest features loss), assuming all sites initially are protected. Loss of B would add a new term to the MCR product sum equal to 0.45 × 0.40 × 0.20 × 0.25 × 0.30 = 0.003. Loss of A would add a larger term (0.05 × 0.40 × 0.60 × 0.65 × 0.70 = 0.005). MCR prefers to retain site A and lose site B. MCR prefers site A over site B, whether adding or removing sites – yet this does not accord with MCR's own model of random distributions of features in the environmental space.
Fig. 6 (a) A hypothetical gradient (for example, from GDM) with selected sites (solid circles), and two candidate sites for selection, A and B (hollow circles). Numbers along gradient are distances between sites. ED-complementarity of site B (areas with vertical stripes) is 0.045, while that for site A (areas with horizontal stripes) is only 0.015, reflecting its close proximity to an already-selected site. ED prefers site B, reflecting the greater count in number of branch/lineages gained. In contrast, MCR, to minimise its product score, selects site A. For MCR, selecting site B reduces the MCR product score by only 0.015, while selecting site A reduces the score by a higher value of about 0.018. For MCR, the greatest reduction in the product score implies the greatest branch/lineages gain, and so MCR prefers site A. For further information, see text. (b) Given two candidate sites (hollow circles) and already-selected sites (solid circles), MCR assigns a higher preference weight to site A, reflecting the large distance from A to the selected site at the other end of the gradient. ED identifies site B as the site that would fill the largest gap and provide the greatest gain in branch/lineages representation. (c) There are two candidate sites for selection, A and B (hollow circles). ED-complementarity values of A and B are shown by respective striped areas. Site B, selected by ED, provides more new branch/lineages. However, MCR cannot distinguish between the two sites
ED correctly prefers site B, in accord with the unimodal model and counting-up property. ED-complementarity for the gain of site B (vertical striped area; Fig. 6a) is 0.045, while that for site A (horizontal stripes) is only 0.015, reflecting the site's close proximity to an already-selected site. The difference is 0.03, and is the same value when determining the best site to lose, illustrating how ED provides a consistent counting-up of features in comparing the two sites under different scenarios. Thus, site B, filling a large gap, is expected to contribute more features (Fig. 6a).
This example highlights general MCR weaknesses: a site can be wrongly preferred because MCR is misled by the site's many large dissimilarities to other sites. Arponen et al. attempted to overcome one weakness of their core selection criterion – possible near-duplication of previously selected sites – by applying a downweighting of those candidate sites close to already-selected sites. The weighting, equal to the product of the site's dissimilarity to all selected sites, does not solve this problem. For example, a site very close to an already-selected site, nevertheless may receive higher weight because it is so far away from other selected sites (Fig. 6b).
MCR's failure to identify gaps is exacerbated by its use of actual sites as demand points (so mimicking 'discrete ED'; Faith and Walker 1996a). MCR consequently cannot take into account portions of the environmental space that do not have recorded sites. An example shows how ED, but not MCR, will give an edge site deserved priority (Fig. 6c), countering Arponen et al.'s claim that a particular advantage of MCR is that it gives priority to sites on the edge of environmental space.
ED succeeds, and MCR fails, in counting-up features under the basic unimodal model. While ED successfully has incorporated, in a consistent way, useful options relating to richness, extent of space, GDM, and other options, the MCR calculations degrade the counting-up of features. This contrast between MCR and ED has important implications for applications. Suppose we interpret the example (Fig. 6a) as a planning decision, in which the best site, A or B, will be removed from protection for non-conservation uses. MCR prefers to give away the site (B) implying a greater features loss. Thus, MCR would be a poor basis for the systematic conservation planning required to reduce rates of biodiversity loss; use of MCR in such conservation planning could inadvertently increase the rate of biodiversity loss. I conclude that MCR, like the Ferrier et al. method, will not provide an effective way to analyse PD-dissimilarities for assessments of PD representation and calculation of gains and losses.