Estimating and comparing allometric growth of body mass with length using quantile regression

Background/Question/Methods

Information on allometric growth of animals is often used so that body condition (mass at length) can be compared on a common size scale, either for theoretical studies of morphometrics and physiology or for comparisons of changes associated with different environments.   The common allometric relationship used in most statistical models considers organism mass (M) as a power function of length (L), M = α(L)^β, with a multiplicative error structure that is made additive by taking logarithms of both M and L before estimation (log M = log α + β log L + ε).  Many comparisons of allometric growth of animals that depend on this fundamental relationship ignore the possibility that there is heterogeneity in allometric exponents (β) both within and among groups of animals being compared not addressed by conventional regression procedures (i.e., ε is not i.i.d. but may depend on L).  I demonstrate that a simple expansion of the log-log form of the allometric growth model estimated with quantile regression allows for easy incorporation of additional multiplicative heterogeneity both within and among groups of animals being compared.  This provides improved insights over procedures based on scaled mass indices and other condition indices (Fulton’s K, relative weight) that assume a common allometric exponent.   Heterogeneity in allometric exponents among groups implies that comparisons among groups will differ with length L and heterogeneity within groups implies differential changes by quantile.

Results/Conclusions

I demonstrate the quantile regression procedures with two applications to fish body condition.  Weight-length data (n = 6,857) collected on walleye (Sander vitreus) in Lake McConaughy, Nebraska were used to evaluate body condition before and after prey augmentation.  All quantiles of walleye weights increased in the years after prey augmentation, ranging from 1.01 – 1.12× weights in the years before prey augmentation, with greatest increases for the lower (<0.50) quantiles and greater lengths indicating greatest improvements in body condition associated with fish in poorer initial condition.  Blue sucker (Cycleptus elongatus) weight-length data (n = 8,462) were used to compare body condition at 19 large-river locations across their geographic range in the central USA.  Blue suckers at southern locations had larger allometric growth exponents and their weights at length were 1.08 – 1.30× the average across all locations.  Weights at length of blue suckers at northern locations were 0.84 – 1.00× the average across all locations.  Thus, there was as much as a 55% greater weight at length between blue suckers at different locations.