## Introduction

The past several decades have seen a resurgence of interest in the field of biological scaling. The publication of several compendia of allometric relationships for animals (Peters 1983; Calder 1984) and plants (Niklas 1994) have highlighted what appear to be recurrent scaling patterns within and across taxa. Examples of allometric relationships that address organismal form and function include: relationships between morphological traits, such as tree diameter and tree height (McMahon & Kronauer 1976; Niklas & Spatz 2004), or relationships between organism size and physiology, such as body mass and metabolic rate (Kleiber 1932; Heusner 1982; White & Seymour 2003; Savage *et al.* 2004).

The existence of such recurrent scaling patterns has motivated attempts to model the scaling of biological phenomena based on physical first principles. In the case of plants, several scaling models have garnered significant attention due to their proposed generality and because they yield multiple, testable predictions (Table 1). These include the biomechanical models for the scaling of ‘life’s dimensions’ first introduced by McMahon (1973) and McMahon & Kronauer (1976) and more recent efforts invoking fractal branching networks (West *et al.* 1997, 1999; Price & Enquist 2007; Price *et al.* 2007). Understanding how well these models characterize allometric scaling behaviour provides important insights into the processes underlying observed allometries and the level of model complexity necessary for addressing particular biological scaling questions.

Model (category) | r | l | M |
---|---|---|---|

Dashes denote the symmetric or isometric elements. NA indicates that the model does not make specific predictions for the corresponding scaling exponent. WBE, model of West *et al.*; PES, model of Price*et al.*; SPAM, specialized allometry model.
| |||

Elastic similarity (universal) | |||

r | – | – | – |

l | 2/3 | – | – |

M | 8/3 | 4 | – |

A | NA | NA | NA |

Stress similarity (universal) | |||

r | – | – | – |

l | 1/2 | – | – |

M | 5/2 | 5 | – |

A | NA | NA | NA |

Geometric similarity (universal) | |||

r | – | – | – |

l | 1 | – | – |

M | 3 | 3 | – |

A | 2 | 2 | 2/3 |

WBE (universal) | |||

r | – | – | – |

l | 2/3 | – | – |

M | 8/3 | 4 | – |

A | 2 | 3 | 3/4 |

PES (constrained) | |||

r | – | – | – |

l | b/a | – | – |

M | (2a + b)/a | (2a + b)/b | – |

A | 1/a | 1/b | 1/(2a + b) |

SPAM (specialized) | |||

r | – | – | – |

l | η | – | – |

M | ϕ | ϕ/η | – |

A | λ | λ/η | λ/ϕ |

Empirical tests of these scaling models typically rely on traditional approaches that fit simple linear regressions to bivariate plots of log-transformed data for a single predicted relationship (i.e. for one particular property vs. another). The confidence intervals for key parameters (e.g. slopes) are examined to determine whether or not they contain a particular scaling model’s predicted value. This approach ignores the fact that many allometric models make predictions for a suite of interconnected relationships among multiple properties and does not allow for exploration of varying degrees of model complexity. Another issue is that classical methods for estimating the coefficients describing how a particular property of an organism scales with another property either ignore uncertainty in one of the variables (e.g. the ‘*x*-variable’) or employ relatively restrictive assumptions about variance terms when accounting for uncertainty in both variables (Warton *et al.* 2006). To address these issues, we describe a hierarchical Bayesian (HB) approach that simultaneously evaluates multiple predicted scaling relationships and explicitly accounts for uncertainty in all measured traits. This approach is applied to compare intraspecific differences in allometric relationships of plant morphology based on whole-plant and leaf datasets.

The allometric models we considered can be divided into three major categories: universal, constrained, and specialized (Table 1). Universal models are derived from physical first principles and are expected to be universally applicable both within and across species. These models yield specific numerical predictions for a suite of allometric exponents, and the numerical values are assumed to be the same across all individuals and species. In constrained models, the scaling exponents may take on a wide array of numerical values, but these values are ‘constrained’ by physical design principles. That is, assumptions about biological limitations result in the scaling exponents for one allometry to be expressed as a function of the exponents describing other allometries. In contrast, specialized models are highly flexible ones that do not arise from underlying physical or biological assumptions. In these models, the allometric exponents are only constrained by simple logical (i.e. algebraic) relationships such that each species may take-on unique (or ‘specialized’) exponent values. Our objective is to compare the predictive power of different scaling models, representing different levels of complexity, while accounting for the fact that universal models inherently involve fewer free parameters than constrained models, which involve fewer free parameters than specialized models. We utilize three large allometric datasets of plant and leaf traits containing in total 2362 individuals from 110 species to evaluate the ability of the universal, constrained, and specialized models to fit observed data and to determine if the universal models satisfactorily capture observed allometric patterns.

We first define the scaling models to be compared and highlight the predictions made by each model. Next, we describe an HB approach for evaluating the predictive power of scaling models of varying complexity. We compare the performance of the different scaling models in two primary ways: (i) we compare the posterior distributions of the population-level scaling exponents to predictions from universal models, and (ii) we rigorously evaluate the ability of each scaling model to predict the observed data via model goodness-of-fit comparisons and estimates of posterior predictive loss.