Potential biases in estimating the rate parameter of sigmoid growth functions

Abstract

Summary 1.We compare biases in estimates of the rate constant of the logistic and Gompertz functions applied to avian growth using three methods of parameter estimation. The methods differ in how the asymptote is determined, and consist of either floating asymptotes estimated from the full data (floating A) or data truncated at 70% of the adult mass (70% A), or fixing the asymptote at the adult mass (fixed A). 2. First, using data for two passerine species exhibiting different growth patterns, we truncated mass measurements at different ages to explore how each method of growth rate estimation responded to simulated differences in nestling period. We bootstrapped growth rates and error measures from these data to produce unbiased error terms, which we used to compare the different models. 3. We tested the effects of age truncation on both real and simulated data, and found that the fixed A method produced less bias and better fits than using estimates of floating asymptotes from the full growth curve (floating A) or truncated at 70% of the adult mass (70% A). Logistic and Gompertz models with a floating asymptote generally provided poorer fits than those with a fixed asymptote, although fits were improved by including a variable accounting for the ratio of asymptotic to adult mass. 4. To evaluate the performance of the methods across an array of species, we applied the three methods to growth data for 45 species of New World songbirds spanning diverse rates of growth, developmental periods and geographic origins. To determine whether choice of method influenced interpretation of regional and life-history differences in avian growth, we used model selection to estimate the influence of three independent variables (adult mass, nestling period and tropical vs. temperate region) with known effects on nestling growth rate calculated by each method. The coefficients of determination in these analyses suggest that the most appropriate method to estimate growth rates, regardless of the growth function (i.e. logistic, Gompertz), is the fixed A method, taking care to use accurate and appropriate estimates of adult mass.

Modelling avian growth with the Unified‐Richards: as exemplified by wader‐chick growth

“…A fledging mass that is higher than the population average (for adults), is typically found in species where chicks or young gain large fat stores before fledging, as in for example swifts, auks and petrels. The high fledgling values, in this example body mass of spotted thick‐knees, may make the fitted curve overshoot adult values (Austin et al ). The uneven spread of data points in older chicks, in particular the lower number of data points for body mass of spotted thick‐knees and that most of them are heavier, influences the growth curve resulting in the fitted curve overshooting the fixed one.…”

“…If the fitted A ‐value deviates considerably from an average for the adult population (or species), meaning that the fitted algorithm returns a biologically meaningless low or meaningless high upper asymptote (Koncagul and Cadirci , Tjørve et al , Tjørve and Tjørve , Austin et al ), one should consider constraining the A ‐parameter to a fixed value. A too low fitted A ‐value is typically caused by truncation with respect to the largest young or chick sizes, whereas prefledgings with higher than adult sizes (typically mass) may return fitted A ‐values that are too high, resulting in a curve that overshoots (Ricklefs , Tjørve et al , Austin et al ). A number of species have pre‐fledglings that are heavier than the adult birds, for example many seabirds and even passerines (Ricklefs , Remeš and Martin ).…”

“…Truncation often occurs in the post‐natal growth data of precocial or semi‐precocial species: the highest values (mass or length) in the oldest age classes being underrepresented and, thus, having a lower sampling probability. This happens because the young or chicks become progressively more difficult to catch closer to fledging or when they have left the nest (Koncagul and Cadirci , Austin et al ). However, fledging date or date for leaving the nest depends not only on age but also on the size of the young or chick, meaning that heavier or bigger individuals fledge or leave the nest earlier.…”

“…The growth‐rate coefficients, k L , k G and k B , of the traditional three‐parameter models are not directly comparable (Ricklefs , , Austin et al ). Because of how the models are parameterized, neither of their growth parameters measure relative growth rate (nor do they measure absolute growth rate).…”

“…We do not discuss pseudoreplication, or provide an in depth discussion of alternative fitting methods or algorithms (but see Tjørve et al , 2009b, Sofaer et al , for further discussion). Neither do we discuss data cut‐offs, that occur close to fledging, in order to avoid the effect of weight recessions of the growth curve (Remeš and Martin , Austin et al ).…”