Perhaps no other pair of variables in ecology has generated as much discussion as species richness and ecosystem productivity, as illustrated by the reactions by Pierce (2013) and others to Adler et al.'s (2011) report that empirical patterns are weak and inconsistent. Adler et al. (2011) argued we need to move beyond a focus on simplistic bivariate relationships and test mechanistic, multivariate causal hypotheses. We feel the continuing debate over productivity–richness relationships (PRRs) provides a focused context for illustrating the fundamental difficulties of using bivariate relationships to gain scientific understanding.
Pierce (2013) disputes Adler et al.'s (2011) conclusion that bivariate productivity–richness relationships (PRRs) are ‘weak and variable’. He argues, instead, that relationships in the Adler et al. data are actually strong and, further, that failure to adhere to the humped-back model (HBM; sensu Grime 1979) threatens scientists' ability to advise conservationists. Here, we show that Pierce's reanalyses are invalid, that statistically significant boundary relations in the Adler et al. data are difficult to detect when proper methods are used and that his advice neither advances scientific understanding nor provides the quantitative forecasts actually needed by decision makers.
We begin by examining Grimes' HBM through the lens of causal networks. We first translate the ideas contained in the HBM into a causal diagram, which shows explicitly how multiple processes are hypothesized to control biomass production and richness and their interrelationship. We then evaluate the causal diagram using structural equation modelling and example data from a published study of meadows in Finland. Formal analysis rejects the literal translation of the HBM and reveals additional processes at work. This exercise shows how the practice of abstracting systems as causal networks (i) clarifies possible hypotheses, (ii) permits explicit testing and (iii) provides more powerful and useful predictions.
Building on the Finnish meadow example, we contrast the utility of bivariate plots compared with structural equation models for investigating underlying processes. Simulations illustrate the fallibility of bivariate analysis as a means of supporting one theory over another, while models based on causal networks can quantify the sensitivity of diversity patterns to both management and natural constraints.
A key piece of Pierce's critique of Adler et al.'s conclusions relies on upper boundary regression which he claims to reveal strong relationships between production and richness in Adler et al.'s original data. We demonstrate that this technique shows strong associations in purely random data and is invalid for Adler et al.'s data because it depends on a uniform data distribution. We instead perform quantile regression on both the site-level summaries of the data and the plot-level data (using mixed-model quantile regression). Using a variety of nonlinear curve-fitting approaches, we were unable to detect a significant humped-shape boundary in the Adler et al. data. We reiterate that the bivariate productivity–richness relationships in Adler et al.'s data are weak and variable.
We urge ecologists to consider productivity–richness relationships through the lens of causal networks to advance our understanding beyond bivariate analysis. Further, we emphasize that models based on a causal network conceptualization can also provide more meaningful guidance for conservation management than can a bivariate perspective. Measuring only two variables does not permit the evaluation of complex ideas nor resolve debates about underlying mechanisms.