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

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.

产后生长是鸟类中变异广泛的重要生活史性状，已有多种S形方程用于其建模。经典三参数模型的拐点固定在上渐近线的某一百分比处，这一假设可能不切实际，从而导致有偏拟合。理查兹模型（Richards model）作为一种极具吸引力的替代方案应运而生，因其包含一个额外参数，可决定拐点位置，使其能沿生长曲线自由移动。近年来，非线性混合模型（nonlinear mixed models, NLMM）被应用于鸟类生长建模，这类模型可处理数据间的非独立性——这在对同一个体或相关个体群进行多次测量时十分常见。本研究评估了冯·贝塔朗菲（von Bertalanffy）、Gompertz、logistic、U4及理查兹模型在皇鸬鹚（Phalacrocorax atriceps）雏鸟生长建模中的适用性。我们对常用形态性状（包括体重、喙长、头长及跗跖长）的生长过程进行建模，并利用NLMM比较各模型的表现。不同模型对成体大小、最大生长年龄及最大生长速率的估计存在显著差异。总体而言，理查兹模型在成体大小估计上表现最为一致，其与平均成体大小的偏差在5%以内。基于校正赤池信息准则（Akaike Information Criterion corrected, AICc）值，理查兹方程是所有分析性状的最优模型。对于跗跖长，理查兹模型与U4模型的拟合效果无显著差异，因为理查兹模型估计的相对拐点值与U4模型假设的值非常接近。研究结果强调，当假设的拐点位置与数据推导的位置存在偏差时，三参数模型会产生有偏估计。因此，在NLMM框架下应用理查兹方程是分析鸟类生长的一种灵活且强有力的工具。