<b>Parameterization effects in nonlinear models to describe growth curves

Abstract

Various parameterizations of nonlinear models are common in the literature.In addition to complicating the understanding of these models, these parameterizations affect the nonlinearity measures and subsequently the inferences about the parameters. Bates and Watts (1980) quantified model nonlinearity using the geometric concept of curvature. Here we aimed to evaluate the three most common parameterizations of the Logistic and Gompertz nonlinear models with a focus on their nonlinearity and how this might affect inferences, and to establish relations between the parameters under the various expressions of the models. All parameterizations were adjusted to the growth data from pequi fruit. The intrinsic and parametric curvature described by Bates and Watts were calculated for each parameter. The choice of parameterization affects the nonlinearity measures, thus influencing the reliability and inferences about the estimated parameters. The most used methodologies presented the highest distance from linearity, showing the importance of analyzing these measures in any growth curve study. We propose that the parameterization in which the estimate of B is the abscissa of the inflection point should be used because of the lower deviations from linearity and direct biological interpretation for all parameters.Efeito da parametrização em modelos não lineares na descrição de curvas de crescimento RESUMO. Diferentes parametrizações de modelos não lineares são comuns na literatura, mas além de complicar seu entendimento, podem afetar as medidas de não linearidade e as inferências sobre os parâmetros. Bates and Watts (1980) quantificaram a não linearidade presente no modelo utilizando o conceito geométrico de curvatura. O objetivo deste trabalho foi avaliar as três parametrizações mais comuns dos modelos não lineares Logístico e Gompertz, quanto à sua não linearidade, as implicações nas inferências e estabelecer relações entre os parâmetros nas diferentes formas de expressar os modelos. Todas as parametrizações foram ajustadas a dados de crescimento do fruto de pequi do cerrado. Para cada parametrização foram calculadas as medidas de curvatura intrínseca e paramétrica descritas por Bates e Watts. A escolha da parametrização afeta as medidas de não linearidade, consequentemente, influencia na confiabilidade e nas inferências sobre os parâmetros estimados. As formas mais utilizadas na literatura apresentaram os maiores afastamentos da linearidade, evidenciando a importância de se analisar estas medidas em qualquer estudo sobre curva de crescimento. Devem ser utilizadas as parametrizações na qual a estimativa de B representa a abscissa do ponto de inflexão por apresentarem menores desvios de linearidade e interpretação biológica direta para todos os parâmetros.Palavras-chave: interpretação biológica, modelo Gompertz, modelo logístico, medidas de curvatura, não linearidade.

The use of the nonlinear models in the growth of pears of ‘Shinseiki’ cultivar

“…For the evaluators, residual standard deviation (DPR) and Akaike's information criterion (AIC), the logistic model revealed lower values, suggesting a better fit; also, according to the findings of FERNANDES et al, (2015) the lower AIC values suggested that this model best describes the data. Therefore, the adjustment quality evaluators highlight the superiority of the Logistic model with second order autoregressive errors to describe the pear fruit length increase.…”

“…The Gompertz and Logistic growth models were effective in describing the cacao fruit development , and the fruits of the cashew tree (MUIANGA et al, 2016), dopequi (FERNANDES et al, 2015), and coffee tree (FERNANDES et al, 2014), giving satisfactory results, for all instances. JESUS et al, (2008) in their study of the longan fruit growth, did a comparison of the Logistic and Quadratic Exponential models and observed that the Logistic model fitted better.…”

Nonlinear Models for Description of Cacao Fruit Growth With Assumption Violations

“…Nonlinear models have been used in plant science studies with satisfactory results, which allowed to estimate fruit growth parameters with biological interpretation, for example, on dwarf date palm (TERRA; SAVIAN; MUNIZ, 2010), dwarf green coconut (PRADO; SAVIAN; MUNIZ, 2013), the Cerrado pequi (FERNANDES et al, 2015) and cashew (MUIANGA et al, 2016). These models were also used in the description of the germination curve of seeds (SOUSA et al, 2014), plant growth (PEREIRA et al, 2014) and canopy diameter (WYZYKOWSKI et al, 2015) of coffee plants.…”

Description of the growth of pequi fruits by nonlinear models

“…Regarding the residual standard deviation (RSD) and corrected Akaike information criterion (AIC), the Gompertz model presented lower values and according to results obtained by Fernandes et al (2015), the lower AIC and RSD values suggest that the model best describes data. Therefore, adjustment quality evaluators point out superiority of the Gompertz model to describe the crosssectional growth of pequi fruits.…”

“…Considering the adjustment of nonlinear models, the estimation of parameters from the classical point of view is usually performed based on the minimization of the sum of squares of residuals, which leads to a system of normal equations with no analytical solution, so iterative processes should be used to obtain estimates (DRAPER, SMITH, 1998;SOUZA, 2007). Several iterative methods are used, especially Gauss-Newton (PEREIRA et al, 2005;MENDES et al, 2008;ZEVIANI et al, 2012;CARNEIRO et al, 2014;FERNANDES at al., 2015). In regression studies, it is usual to admit in the estimation process and inference under parameters that the errors are independent, which does not necessarily occur when working with time-ordered data that are potentially correlated (CASSIANO; SÁFADI, 2015).…”