-Cacao (Theobroma cacao L.) is an important fruit in the Brazilian economy, which is mainly cultivated in the southern State of Bahia. The optimal stage for harvesting is a major factor for fruit quality and the knowledge on its growth curves can help, especially in identifying the ideal maturation stage for harvesting. Nonlinear regression models have been widely used for description of growth curves. However, several studies in this subject do not consider the residual analysis, the existence of a possible dependence between longitudinal observations, or the sample variance heterogeneity, compromising the modeling quality. The objective of this work was to compare the fit of nonlinear regression models, considering residual analysis and assumption violations, in the description of the cacao (clone Sial-105) fruit growth. The data evaluated were extracted from Brito and Silva (1983), who conducted the experiment in the Cacao Research Center, Ilheus, State of Bahia. The variables fruit length, diameter and volume as a function of fruit age were studied. The use of weighting and incorporation of residual dependencies was efficient, since the modeling became more consistent, improving the model fit. Considering the first-order autoregressive structure, when needed, leads to significant reduction in the residual standard deviation, making the estimates more reliable. The Logistic model was the most efficient for the description of the cacao fruit growth.Keywords: Biological interpretation. Biometric measurement. Logistic model. Theobroma cacao L.. MODELOS NÃO LINEARES NA DESCRIÇÃO DO CRESCIMENTO DE FRUTOS DE CACAU COM VIOLAÇÕES DOS PRESSUPOSTOSRESUMO -O cacau é um importante produto para a economia brasileira, sendo cultivado principalmente no sul da Bahia. O ponto ótimo de colheita é um dos principais fatores de perda na qualidade do fruto e o conhecimento de suas curvas de crescimento pode auxiliar principalmente na identificação deste ponto. Os modelos de regressão não linear tem sido amplamente utilizados na descrição de curvas de crescimento. No entanto, em várias pesquisas nesse sentido não é realizada a análise de resíduos, não consideram a existência de uma possível dependência entre as observações longitudinais e nem a heterogeneidade de variâncias amostrais, comprometendo a qualidade da modelagem. Assim, este trabalho objetivou comparar o ajuste de modelos de regressão não lineares, considerando os possíveis desvios de pressuposição sobre os resíduos, na descrição do crescimento dos frutos de cacaueiro do clone Sial-105. Os dados analisados foram extraídos de Brito e Silva (1983) e correspondem a um experimento realizado no Centro de Pesquisa de Cacau, em Ilhéus-BA. As variáveis estudadas foram o comprimento, diâmetro e volume do fruto tomados em função da sua idade. O uso da ponderação e incorporação da dependência residual foi eficiente pois tornou a modelagem mais coerente melhorando a qualidade do ajuste. Considerar a estrutura auto-regressiva de primeira ordem, quando necessária, leva a redu...

RESUMO -o objetivo do trabalho foi avaliar o ajuste dos modelos gompertz e logístico, com estrutura de erros independentes e autorregressivos, no desenvolvimento de frutos de caju, com base em medidas de comprimento e largura do fruto, tomados ao longo do tempo. a estimação dos parâmetros foi feita por meio de rotinas no software R, utilizando-se do método dos mínimos quadrados e o processo iterativo de gaussNewton. Os modelos foram comparados pelos critérios: coeficiente de determinação ajustado (R 2 aj ), desvio padrão residual (DPR), critério de informação Akaike (AIC) e critério bayesiano de Schwarz (BIC). Para os dois modelos, os dados apresentaram autocorrelação residual positiva, tanto para a variável comprimento como para a largura do fruto, descrita por processo autorregressivo de primeira ordem. em geral, por todos critérios de avaliação da qualidade de ajuste, os dados se ajustaram ao modelo logístico com uma estrutura autorregressiva da primeira ordem, havendo no entanto superestimação do tamanho do fruto nas últimas idades, tanto no crescimento em comprimento (cm) como em largura (cm). termos para indexação: Modelo de crescimento. Medidas biométricas. Regressão não linear. Cajú. DESCRIPTION OF THE GROWTH CURVE OF CASHEW FRUITS IN NONLINEAR MODELSABSTRACT -The objective of this study was to evaluate the adjustment of gompertz and logistic models, with independent and autoregressive error structure in cashew fruits growth, based on the length and width measurements of the fruit, taken through the time. The estimation of the parameters was done in R software routines using the method of minimal squares and the interactive process of gauss-newton. the models were compared by the following criteria: the adjusted coefficient of determination, residual standard deviation, Akaike information criterion and Schwarz Bayesian criterion. For both models, the data showed positive residual autocorrelation for both variables: length and width of the fruit, described in autoregressive process of order one. in general, both models adjusted to the data, however there was an overestimation of the fruit size at the late stages. the logistic model was the most adequate to describe the cashew fruit growth in length and width.

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.

This study aimed to verify if the growth pattern of coffee berries, considering fresh mass accumulation over time, is double sigmoid and to select the most suitable nonlinear model to describe such behavior. Data used consisted of fourteen longitudinal observations of average fresh mass of coffee berries obtained in an experiment with the cultivar Obatã IAC 1669-20. The fits provided by the Logistic and Gompertz models were compared in their single and double versions. Parameters were estimated using the least squares method using the Gauss-Newton algorithm implemented in the nls function of the R software. It can be concluded that the growth pattern of the coffee fruit, in fresh mass accumulation, is double sigmoid. The double Gompertz and double Logistic models were adequate to describe such a growth curve, with a superiority of the double Logistic model.

The goal of this study was to elucidate the growth and development of the Asian pear fruit, on the grounds of length, diameter and fresh weight determined over time, using the non-linear Gompertz and Logistic models. The specifications of the models were assessed utilizing the R statistical software, via the least squares method and iterative Gauss-Newton process (DRAPER & SMITH, 2014). The residual standard deviation, adjusted coefficient of determination and the Akaike information criterion were used to compare the models. The residual correlations, observed in the data for length and diameter, were modeled using the second-order regression process to render the residuals independent. The logistic model was highly suitable in demonstrating the data, revealing the Asian pear fruit growth to be sigmoid in shape, showing remarkable development for three variables. It showed an average of up to 125 days for length and diameter and 140 days for fresh fruit weight, with values of 72mm length, 80mm diameter and 224g heavy fat.

Blackberry is a small fruit with several properties beneficial to human health and its cultivation is an alternative for small producers due to its fast and high financial return. Studying the growth of fruits over time is extremely important to understand their development, helping in the most appropriate crop management, avoiding post-harvest losses, which is one of the aggravating factors of blackberry cultivation, being a short shelf life fruit. Thus, growth curves are highlighted in this type of study and modeling through statistical models helps understanding how such growth occurs. Data from this study were obtained from an experiment conducted at the Federal University of Lavras in 2015. The aim of this study was to adjust nonlinear, double Logistic and double Gompertz models to describe the diameter growth of four blackberry cultivars (‘Brazos’, ‘Choctaw’, ‘Guarani’ and ‘Tupy’). Estimations of parameters were obtained using the least squares method and the Gauss-Newton algorithm, with the “nls” and “glns” functions of the R statistical software. The comparison of adjustments was made by the Akaike information criterion (AICc), residual standard deviation (RSD) and adjusted determination coefficient (R2 aj). The models satisfactorily described data, choosing the Logistic double model for ‘Brazos’ and ‘Guarani’ cultivars and the double Gompertz model for ‘Tupy’ and ‘Choctaw’ cultivars.

The objective of this work was to evaluate how the parameterization and the application of different allometric values affect the obtention of the most adequate fit of von Bertalanffy’s model, in the description of the growth curve of meat-producing mammals (bovine, pigs, rabbits, and sheep). Among the nonlinear models, von Bertalanffy’s has been very often applied in several areas, with different parameterizations. This model has been commonly used with an allometric value of m = 2/3; however, for mammals, it is believed that this value can be m = 3/4. The analyzed data referring to the mass of meat-producing mammals according to their age were obtained from research institutions and from the literature. The results showed that von Bertalanffy’s model, with the allometric value of m = 3/4 and the used parameterization, provided better adjustments to quality evaluators. Besides, the model softened the overestimation of parameter a, giving a direct interpretation of parameter b, with the lowest values for curvature measurements, mainly for the parametric ones, and provided more reliable adjustments. Von Bertalanffy’s model can be used in the description of the growth curves of meat-producing mammals.

The growth curves of animals, in general, have an “S” shape, also known as sigmoidal curves. This type of   curve is well fitted by nonlinear regression models, including von Bertalanffy’s model, which has been widely  applied in several areas, being presented in literature through different parameterizations, which in practice, can complicate its understanding, affect nonlinearity measures and inferences about parameters. To quantify  the nonlinearity present in a Bates and Watts model, a geometric concept of curvature has been used. The aim of this work was to analytically develop three parameterizations of the von Bertalanffy’s nonlinear model  referring to its nonlinearity, implications for inferences and to establish relationships between parameters in the different ways of expressing the models. These parameterizations were adjusted to the growth data of sheep. For each parameterization, the intrinsic and parametric curvature measurements described by Bates and Watts were calculated. The parameterization choice affects nonlinearity measures, consequently, influences the reliability and inferences about estimated parameters. The forms most used in literature showed the greatest deviations from linearity, showing the importance of analyzing these measures in any growth curve study. Parameterization should be used in which the b estimate represents the abscissa of the inflection point, as it presents minor linearity deviations and direct biological interpretation for all parameters.