Evaluating scaling models in biology using hierarchical Bayesian approaches

Abstract

Theoretical models for allometric relationships between organismal form and function are typically tested by comparing a single predicted relationship with empirical data. Several prominent models, however, predict more than one allometric relationship, and comparisons among alternative models have not taken this into account. Here we evaluate several different scaling models of plant morphology within a hierarchical Bayesian framework that simultaneously fits multiple scaling relationships to three large allometric datasets. The scaling models include: inflexible universal models derived from biophysical assumptions (e.g. elastic similarity or fractal networks), a flexible variation of a fractal network model, and a highly flexible model constrained only by basic algebraic relationships. We demonstrate that variation in intraspecific allometric scaling exponents is inconsistent with the universal models, and that more flexible approaches that allow for biological variability at the species level outperform universal models, even when accounting for relative increases in model complexity.

Figure 1

Posterior distributions for the global exponents in the specialized allometry model (SPAM). The dashed vertical lines represent exponent values predicted by the universal models (Table 1). None of the universal models enjoys strong support across all allometries or all datasets. Bayesian credible intervals (BCI) and the exponent predictions from the universal models are reported in Table 2. Note that the elastic similarity model makes the same predictions as the model of West et al. (1999) for the scaling of mass and length. In addition, stress and elastic similarity models do not make predictions for the scaling of surface area.

Figure 2

Smoothed frequency histograms for the fraction of the Bayesian credible intervals (BCI) for each species-specific scaling exponent that include the exponent value indicated on the x-axis. The predicted exponent values from the universal models are plotted for reference (horizontal dashed lines). Note that the stress and elastic similarity models do not make predictions for the scaling of surface area.

Figure 3

Illustration of the improvement in predictive power with more flexible scaling models. The predicted mass values are the posterior means for replicated data. The black line in each figure is the 1 : 1 line. Note that the model of Price et al. (2007); PES) and the specialized allometry model (SPAM) have less scatter about the 1 : 1 line compared with the universal models, WBE model of West et al. (1999) and the geometric model (GEOM).