Bivariate line‐fitting methods for allometry

Abstract

Fitting a line to a bivariate dataset can be a deceptively complex problem, and there has been much debate on this issue in the literature. In this review, we describe for the practitioner the essential features of line-fitting methods for estimating the relationship between two variables : what methods are commonly used, which method should be used when, and how to make inferences from these lines to answer common research questions.A particularly important point for line-fitting in allometry is that usually, two sources of error are present (which we call measurement and equation error), and these have quite different implications for choice of linefitting method. As a consequence, the approach in this review and the methods presented have subtle but important differences from previous reviews in the biology literature.Linear regression, major axis and standardised major axis are alternative methods that can be appropriate when there is no measurement error. When there is measurement error, this often needs to be estimated and used to adjust the variance terms in formulae for line-fitting. We also review line-fitting methods for phylogenetic analyses.Methods of inference are described for the line-fitting techniques discussed in this paper. The types of inference considered here are testing if the slope or elevation equals a given value, constructing confidence intervals for the slope or elevation, comparing several slopes or elevations, and testing for shift along the axis amongst several groups. In some cases several methods have been proposed in the literature. These are discussed and compared. In other cases there is little or no previous guidance available in the literature.Simulations were conducted to check whether the methods of inference proposed have the intended coverage probability or Type I error. We identified the methods of inference that perform well and recommend the techniques that should be adopted in future work.

Conservative whole‐organ scaling contrasts with highly labile suborgan scaling differences among compound eyes of closely related Formica ants

“…Additionally, MA/SMA removes assumptions concerning biological phenomenon being directly related (Stillwell et al., 2016). Major axis or standardized major axis regression lines should only be fitted when both X and Y variables are sampled randomly (Warton et al., 2006); however, we sampled a broad size range of ants to ensure appropriate coverage. The measurement error in our data is likely to be small compared with the (unavoidable) amount of equation error (i.e., data points not lying exactly on the regression line).…”

Weak coordination among petiole, leaf, vein, and gas‐exchange traits across Australian angiosperm species and its possible implications

“…The “shift” test in this R package was also used to test for significant dispersion of species from different sites along a common axis (rather than having different regression coefficients) (Warton et al. 2006). In these cases, mean site axis scores were calculated for each site and regressed against the climate data to determine whether climate differences were aligned with the observed across‐site species' differences.…”

Volumetric Analysis of the African Elephant Ventricular System

“…We then divided the data into five groups, these being elephants, primates, megachiropterans, insectivores, and microchiropterans and undertook three separate analyses of the transformed data. First, standardized major axis, or reduced major axis (see Warton et al, 2006 for discussion of the terminology), was used to describe and compare the bivariate relationship of the different groups. The software SMATR ver.…”