Richards's equation and nonlinear mixed models applied to avian growth: why use them?

Abstract

Postnatal growth is an important life‐history trait that varies widely across avian species, and several equations with a sigmoidal shape have been used to model it. Classical three‐parameter models have an inflection point fixed at a percentage of the upper asymptote which could be an unrealistic assumption generating biased fits. The Richards model emerged as an interesting alternative because it includes an extra parameter that determines the location of the inflection point which can move freely along the growth curve. Recently, nonlinear mixed models (NLMM) have been used in modeling avian growth because these models can deal with a lack of independence among data as typically occurs with multiple measurements on the same individual or on groups of related individuals. Here, we evaluated the usefulness of von Bertalanffy, Gompertz, logistic, U4 and Richards's equations modeling chick growth in the imperial shag Phalacrocorax atriceps. We modelled growth in commonly used morphological traits, including body mass, bill length, head length and tarsus length, and compared the performance of models by using NLMM. Estimated adult size, age at maximum growth and maximum growth rates markedly differed across models. Overall, the most consistent performance in estimated adult size was obtained by the Richards model that showed deviations from mean adult size within 5%. Based on AICc values, the Richards equation was the best model for all traits analyzed. For tarsus length, both Richards and U4 models provided indistinguishable fits because the relative inflection value estimated from the Richards model was very close to that assumed by the U4 model. Our results highlight the bias incurred by three‐parameter models when the assumed inflection placement deviates from that derived from data. Thus, the application of the Richards equation using the NLMM framework represents a flexible and powerful tool for the analysis of avian growth.

Bertalanffy-Pütter models for avian growth

“…Other approaches used spline-interpolation [ 59 , 60 ], time series methods [ 61 ], or stochastic (partial) differential equations [ 62 ]. More recently, mixed-effect models became popular [ 63 – 65 ], but to remain viable they were based on growth curves that could be parameterized by means of elementary functions such as for the Verhulst model [ 66 ] or the Richards model [ 67 ].…”

A unified-models analysis of the development of sexual size dimorphism in Damaraland mole-rats, Fukomys damarensis

“…Even so, they were the first to develop and propose a unifiedmodels framework that incorporated the wider sigmoid family Tjørve 2010, Tjørve andTjørve 2017a). In the intervening years this unified-models approach has been increasingly adopted in the avian literature (Tjørve and Tjørve 2017b, Svagelj et al 2019, Vrána et al 2019, no doubt stimulated by the relative ease with which growth data can be collected from birds, where individuals routinely reach adult mass in the 2-3 weeks that they are bound to the nest.…”

Sex‐specific environmental sensitivity on the postnatal growth of a sexually size‐dimorphic seabird

“…2016, Hildebrandt & Schaub 2018) and the development of Richards equation (Tjørve & Tjørve 2010, 2017a,b, Svagelj et al . 2019) have improved our analytic capabilities, allowing an unbiased evaluation of growth parameters in data with multiple measurements on the same individual or on groups of related individuals (Giudici et al . 2017, Svagelj & Quintana 2017, Tuero et al .…”

“…2018, Svagelj et al . 2019). Here, we used this approach to analyse diverse drivers of sex‐specific effects in the Imperial Shag Leucocarbo atriceps , a sexually dimorphic seabird with males being ~ 18 % heavier than females (Svagelj & Quintana 2007).…”