Nonlinear Mixed-Effect Models to Describe Growth Curves of Pepper Fruits in Eight Cultivars Including Group Effects

Abstract

Evaluating the behavior of fruit width and length characters throughout the pepper crop cycle is essential for researchers in decision-making aimed at developing appropriate management techniques and harvesting fruits at proper growth stages. The Nonlinear Mixed-Effect Models (NLME) method is recommended to jointly model the residuals and the entire database, including group effects, to describe growth curves. This work compared four nonlinear equations (Gompertz, Logistic, Richards, and von Bertalanffy) by including groups (pepper and bell pepper) to describe the pepper genotypes’ length and width growth. Of the eight genotypes used, three were bell pepper, and five were pepper. For each, fruit length and width were measured in 10 periods. According to the fit-quality measures studied, the best model for adjusting the length of the fruit was the Richards ($R_{adj.}^2=0.9960$), while for the width, it was the Logistic ($R_{adj.}^2=0.9957$). The estimated random effects showed that for asymptotic length and time to the inflection point presented a correlation of 0.75, indicating a positive association between these traits. For width, however, this result was different: −0.02. NLME adjustment allowed efficient prediction of values and efficient characterization of the studied genotypes, proving to be an efficient method for longitudinal data.

1. Introduction

Pepper fruits are a source of natural pigments and antioxidants, including vitamin C, flavonoids, and phenolic acids, as well as carotenoids [1]. Given the nutraceutical and anticancer properties of pepper compounds, they are important preventive factors against many diseases, e.g., cardiovascular disease, type II diabetes, and other aging-associated disorders [2,3,4]. Pepper cultivation is economically valuable, nutritionally beneficial, and supports job creation and rural livelihoods. It contributes to crop diversification, genetic diversity, and international trade, playing a significant role in global agriculture.

Understanding the pattern of length and growth of pepper (Capsicum annum L.) fruits is crucial for effective cultivation planning, yield estimation, quality assessment, pest management, and scientific research. It allows farmers to schedule planting, fertilization, and harvesting activities, estimate yields, assess fruit quality, detect pest or disease issues, and develop improved varieties. By tracking fruit growth, growers can optimize their practices, enhance productivity, and ensure successful pepper cultivation.

According to Rêgo et al. [5], fruit weight, length, and diameter are essential in many horticultural crops, including pepper plants. In this context, evaluating the behavior of these characteristics throughout the crop cycle is fundamental for the researcher when deciding to develop appropriate management techniques and harvest fruits at appropriate growth stages [6].

In many cases, it is desirable to quantify the growth of horticultural products with functions [7]. Usually, the development of living beings shows a distinct behavior, starting slowly, passing to an exponential phase, and tending to stabilize at the end. This fact makes the method of nonlinear models an excellent alternative to adjust such growth behaviors. For pepper data, we can cite some research examples that used NLM to fit growth curves, such as [6,7,8].

Usually, in its applications, the estimation method is done traditionally, whereby the growth characteristics are obtained through an adjustment per individual. In this case, information about all individuals must be evaluated at a second stage, regardless whether they belong to the same group. An alternative capable of solving this problem and still allowing the inclusion of individual and group effects in the same model is Nonlinear Mixed-Effect Models (NLME).

This method has some advantages over traditional (non-mixed) models. Among them, we can mention the allowing the individual modeling of accessions/individuals as random effects, together with the inclusion of the fixed effects of species, environment, management, or another source of variation. Lindstrom and Bates [9] claim that the notion that individuals’ responses all follow a similar functional form with varying parameters among individuals seems to be appropriate in many situations.

Several researchers have used NLME modeling in different contexts, including the development of animal growth curves, as can be seen in Alves et al. [10] for Guzerá cattle and in Araujo Neto et al. [11] for dairy buffaloes. In addition to applications in the animal field, this methodology has been recurrently applied in the silvicultural area, such as in the development of height-diameter models [12,13,14,15,16], crown-base-height models [17] and diameter model [18]. The findings of these studies converge to indicate the remarkable superiority of the accuracy of the NLME over other modeling approaches, such as Ordinary Least Squares regression. However, to our knowledge, NLME has not been used to fit pepper fruits’ growth data.

As the growth of the pepper plants depends on the cultivar and the growing conditions, when relevant information is available on such variables, their inclusion in a mixed nonlinear model as a fixed effect combined with the individual variation provided by random effects makes NLME an ideal method to model the growth of pepper fruits.

Given the above, this work aims to:

(i)

compare the non-linear models of Gompertz, Logístic, Richards, and von Bertalanffy using NLME to fit the length- and width-growth data of pepper and bell pepper fruits, both inserted as covariates (fixed effects), where, for each model, measures of goodness of fit will be calculated and the best fits for each characteristic will be identified;

(ii)

identify the growth patterns in each group according to the estimation of the fixed effects (pepper and bell pepper) and

(iii)

verify the correlations between the biological interpretable growth variables, evaluated individually by the random effects of the models.

2. Materials and Methods

2.1. Description of the Experiment

The experiment was conducted in a greenhouse in the Olericulture sector of the Department of Plant Science at the Federal University of Viçosa (UFV), Viçosa, Minas Gerais. Viçosa is in the Zona da Mata of Minas Gerais, at geographic coordinates 20°45′ south latitude and 42°51′ west longitude, with an average altitude of 650 m.

The experimental design was completely randomized with four replications, and the experimental unit consisted of one plant. Five pepper genotypes were evaluated: Vulcão, Cayenne, Peter, Picante (vase), and Jamaica Yellow, and three bell pepper genotypes: Quadrado, Ikeda Hard Shell, and Giant Rubi. The quantitative characters evaluated for each genotype were fruit length (CF, mm) and fruit width (LF, mm). Each of the eight experimental units was assessed at 10 different times: 0, 7, 14, 21, 28, 35, 42, 56, 63, and 70 days after flowering (DAF). For each time, the average of each plant was used as an observation to adjust the model, totaling N = 80 experimental observations. The means and standard deviations for length and width of the pepper and bell pepper genotypes can be seen in

Table 1. Mean and standard deviation of length and width of pepper and bell pepper fruits.

Pepper (n = 5)			Bell Pepper (n = 3)
t (DAF 1)	Length (mm)	Width (mm)	t (DAF)	Length (mm)	Width (mm)
0	6.4 $\pm$ 1.3	3.2 $\pm$ 1.6	0	8.31 $\pm$ 1.2	9.0 $\pm$ 0.9
7	15.2 $\pm$ 3.8	5.7 $\pm$ 3.2	7	25.2 $\pm$ 6.3	18.2 $\pm$ 5.9
14	27.5 $\pm$ 12.3	9.5 $\pm$ 4.8	14	48.9 $\pm$ 11.9	37.3 $\pm$ 12.2
21	34.7 $\pm$ 19.6	11.9 $\pm$ 5.7	21	56.3 $\pm$ 13.2	45.7 $\pm$ 13.2
28	37.5 $\pm$ 22.4	13.5 $\pm$ 6.8	28	62.0 $\pm$ 15.3	50.1 $\pm$ 12.3
35	38.6 $\pm$ 23.1	14.1 $\pm$ 7.3	35	64.0 $\pm$ 15.9	52.0 $\pm$ 12.6
42	39.2 $\pm$ 23.4	14.5 $\pm$ 7.9	42	67.4 $\pm$ 11.1	52.9 $\pm$ 12.6
56	39.5 $\pm$ 23.4	14.9 $\pm$ 8.5	56	68.6 $\pm$ 11.9	53.7 $\pm$ 12.9
63	39.6 $\pm$ 23.6	15.4 $\pm$ 9.3	63	68.6 $\pm$ 11.9	53.7 $\pm$ 12.9
70	39.6 $\pm$ 23.6	15.4 $\pm$ 9.3	70	68.6 $\pm$ 11.9	53.7 $\pm$ 12.9

¹ Days after flowering.

.

2.2. Nonlinear Mixed-Effect Models

The representation of NLME, according to Pinheiro and Bates [19], can be understood as a hierarchical model. In the present context, the j-th observation for the i-th access can be denoted by:

$$y_{ij}=f\left({\mathit{\varphi }}_i,x_{ij}\right)+{\epsilon }_{ij},$$

(1)

where $i$ represents the index for the number of individuals ($i=1,\dots , 30$), where each individual has a quantity $n_i$ of observations, which are indexed by $j$ ($j=1,\dots , 4$), $f$ is a nonlinear function of the vector of parameters ${\mathit{\varphi }}_i$ and the predictors $x_{ij}$; and ${\epsilon }_{ij}$ is the term that represents the error associated to the observation $y_{ij}$, being ${\epsilon }_{ij}~N\left(\mathbf{0},{\sigma }^2\mathit{I}\right)$. The function $f$ is nonlinear and at least one of the components of the parameter vector ${\varphi }_i$, which can be decomposed as follows:

$${\mathit{\varphi }}_i={\mathit{A}}_i\mathit{\beta }+{\mathit{B}}_i{\mathit{b}}_i,$$

(2)

where $\mathit{\beta }$ is a $p$-dimensional vector associated with the fixed effects, in this case representing the group effect (pepper or bell pepper) for each parameter of the model; ${\mathit{b}}_i$ is a $q$-dimensional vector associated with the random effect of i-th individual, ${\mathit{b}}_i~N\left(\mathbf{0},{\sigma }^2\mathit{D}\right)$, that in this case represents the individual deviations of each fruit’s length/width in relation to its type (pepper or bell pepper); ${\mathit{A}}_i$ and ${\mathit{B}}_i$ are, respectively, the incidence matrices of dimensions $rxp$ and $rxq$ associated with the fixed and random effects, allowing the inclusion of fixed and random effects, respectively, in a single fit; ${\sigma }^2\mathit{D}$ is the covariance matrix of ${\mathit{b}}_i$.

For this study, since there are two types of pepper (pepper and bell pepper), the effects of each group will be estimated through beta. Therefore, the incidence matrix will be ${\mathit{A}}_i={\left[\mathit{I} \mathbf{0}\right]}_{6\mathrm{x}3}$ for individuals of group 1 (pepper) and ${\mathit{A}}_i={\left[\mathbf{0} \mathit{I}\right]}_{6\mathrm{x}3}$ for individuals of group 2 (bell pepper), adapting what was described by Lindstrom and Bates [9]. The incidence matrix of the random effects aimed to identify individual differences between the biologically interpretable parameters of plants, that is, ${\mathit{B}}_i={\mathit{I}}_{3\mathrm{x}2}$, considering as random effects only the parameters which have biological interpretation (asymptotic length/width and the time until the inflection point).

If we consider, for instance, an individual that belongs to the pepper group in a model that contains three parameters, the parameter vector can be decomposed as follows:

$${\mathit{\varphi }}_i={\mathit{A}}_i\mathit{\beta }+{\mathit{B}}_i{\mathit{b}}_i=\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{\beta }_1\\ {\beta }_2\\ {\beta }_3\\ {\beta }_4\\ {\beta }_5\\ {\beta }_6\end{array}\right]+\left[\begin{array}{cc}1& 0\\ 0& 1\\ 0& 0\end{array}\right]\left[\begin{array}{c}{b}_{1i}\\ b_{2i}\end{array}\right],$$

(3)

and considering an element from the bell pepper group the decomposition is:

$${\mathit{\varphi }}_i={\mathit{A}}_i\mathit{\beta }+{\mathit{B}}_i{\mathit{b}}_i=\left[\begin{array}{cccccc}0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}{\beta }_1\\ {\beta }_2\\ {\beta }_3\\ {\beta }_4\\ {\beta }_5\\ {\beta }_6\end{array}\right]+\left[\begin{array}{cc}1& 0\\ 0& 1\\ 0& 0\end{array}\right]\left[\begin{array}{c}{b}_{1i}\\ b_{2i}\end{array}\right]$$

(4)

For both equations, ${\beta }_1$, ${\beta }_2$ and ${\beta }_3$ represent asymptotic length/width, time until the inflection point and the steepness parameter for the genotypes of group 1 (Pepper), and ${\beta }_4$, ${\beta }_5$ and ${\beta }_6$ represents the asymptotic length/width, time until the inflection point and the steepness parameter for the genotypes of group 2 (Bell Pepper). For the Richards model, the parameter vector $\mathit{\beta }$ contains eight parameters (${\beta }_7$ and ${\beta }_8$ represent the control of asymmetric growth for pepper and bell pepper, respectively). Thus, the vector of observations for the i-th individual ($i=1,\dots ,M$) can be represented in matrix form as follows: ${\mathit{y}}_i=\left[\begin{array}{c}\begin{array}{c}{y}_{i1}\\ y_{i2}\end{array}\\ \begin{array}{c}⋮\\ y_{i{n}_i}\end{array}\end{array}\right];{\mathit{\epsilon }}_i=\left[\begin{array}{c}\begin{array}{c}{\epsilon }_{i1}\\ {\epsilon }_{i2}\end{array}\\ \begin{array}{c}⋮\\ {\epsilon }_{i{n}_i}\end{array}\end{array}\right]$ e ${\mathit{\eta }}_i\left({\mathit{\varphi }}_i\right)=\left[\begin{array}{c}\begin{array}{c}f\left({\mathit{\varphi }}_i,x_{i1}\right)\\ f\left({\mathit{\varphi }}_i,x_{i2}\right)\end{array}\\ \begin{array}{c}⋮\\ f\left({\mathit{\varphi }}_i,x_{i{n}_i}\right)\end{array}\end{array}\right]$.

Considering the above information, the model for the i-th sample element can be represented in matrix form as follows:

$${\mathit{y}}_i={\mathit{\eta }}_i\left({\mathit{\varphi }}_i\right)+{\mathit{\epsilon }}_i,$$

(5)

in which ${\mathit{\epsilon }}_i~N\left(\mathbf{0},{\sigma }^2{\mathit{\Lambda }}_i\right)$. The matrix ${\mathit{\Lambda }}_i$ is the residual covariance matrix and can be described by an identity matrix $\mathit{I}$, an autoregressive process of order 1, compound symmetry, among other forms [19]. For this work, we considered ${\mathit{\Lambda }}_i=\mathit{I}.$ Considering M elements in the sample, the individual matrices should be grouped into a model in matrix form as follows:

$$\mathit{y}=\mathit{\eta }\left(\mathit{\varphi }\right)+\mathit{\epsilon },$$

(6)

With the elements being represented by: $\mathit{y}=\left[\begin{array}{c}\begin{array}{c}{\mathit{y}}_1\\ {\mathit{y}}_2\end{array}\\ \begin{array}{c}⋮\\ {\mathit{y}}_M\end{array}\end{array}\right]$; $\mathit{\epsilon }=\left[\begin{array}{c}\begin{array}{c}{\mathit{\epsilon }}_1\\ {\mathit{\epsilon }}_2\end{array}\\ \begin{array}{c}⋮\\ {\mathit{\epsilon }}_M\end{array}\end{array}\right]$ and $\mathit{\eta }\left(\mathit{\varphi }\right)=\left[\begin{array}{c}\begin{array}{c}{\mathit{\eta }}_1\left({\mathit{\varphi }}_1\right)\\ {\mathit{\eta }}_2\left({\mathit{\varphi }}_2\right)\end{array}\\ \begin{array}{c}⋮\\ {\mathit{\eta }}_M\left({\mathit{\varphi }}_M\right)\end{array}\end{array}\right]$, where: $\mathit{y}|\mathit{b}~N\left(\mathit{\eta }\left(\mathit{\Phi }\right),{\sigma }^2\mathit{\Lambda }\right), \mathit{\varphi }=\mathit{A}\mathit{\beta }+\mathit{B}{\mathit{b}}_i$, $\mathit{b}~N\left(\mathbf{0},{\sigma }^2\tilde{\mathit{D}}\right)$, $\mathit{\Lambda }=diag\left({\mathit{\Lambda }}_1,{\mathit{\Lambda }}_2,\dots ,{\mathit{\Lambda }}_M\right)$ and $\tilde{\mathit{D}}=diag\left(\mathit{D},\mathit{D},\dots ,\mathit{D}\right)$, $\mathit{b}=\left[\begin{array}{c}\begin{array}{c}{\mathit{b}}_1\\ {\mathit{b}}_2\end{array}\\ \begin{array}{c}⋮\\ {\mathit{b}}_M\end{array}\end{array}\right]$ and $\mathit{A}=\left[\begin{array}{c}\begin{array}{c}{\mathit{A}}_1\\ {\mathit{A}}_2\end{array}\\ \begin{array}{c}⋮\\ {\mathit{A}}_M\end{array}\end{array}\right]$.

Parameter estimation was done according to the method proposed by Lindstrom and Bates [9], which is divided into two steps: the first consists of minimizing the nonlinear sum of squares, known as the Penalized Nonlinear Least Squares (PNLS) Step, which makes use of the Gauss-Newton algorithm. The second resembles estimating the variance components of linear mixed models and is therefore called the Linear Mixed Effects (LME) Step [19].

2.3. Equations to Represent $f\left({\mathit{\varphi }}_i,x_{ij}\right)$

The equations for describing the length and width of pepper/bell pepper fruits are described below (

Table 2. Nonlinear equations for growth curves.

Model	Reference	Equation 1
Gompertz	[20]	$y_{ij}={\varphi }_{1i}exp\left\{-exp\left[-{\varphi }_{3i}\left(x_{ij}-{\varphi }_{2i}\right)\right]\right\}+{\epsilon }_{ij}$
Logistic	[21]	$y_{ij}=\frac{{\varphi }_{1i}}{1+exp\left[-{\varphi }_{3i}\left(x_{ij}-{\varphi }_{2i}\right)\right]}+{\epsilon }_{ij}$
Richards	[22]	$y_{ij}=\frac{{\varphi }_{1i}}{{\left\{1+{\varphi }_{4i}exp\left[-{\varphi }_{3i}\left(x_{ij}-{\varphi }_{2i}\right)\right]\right\}}^{\frac{1}{{\varphi }_{4i}}}}+{\epsilon }_{ij}$
von Bertalanffy	[23]	$y_{ij}={\varphi }_{1i}\left[1-{\varphi }_{2i}exp\left(-{\varphi }_{3i}{x}_{ij}\right)\right]+{\epsilon }_{ij}$

¹ Each parameter of the equations can be decomposed as showed on Equation (2).

):

The elements of the parameter vector for each model (${\mathit{\varphi }}_i$) can be decomposed as described in Equation (2) from the previous section. For these models, ${\varphi }_{1i}$ represents the asymptotic length/width (mm), which represents the length/width of the fruits if they grew indefinitely. Usually, this value approaches the maximum length/width; ${\varphi }_{2i}$ is the time when the curve reaches its inflection point, where growth stops being accentuated to occur less intensely. The inflection point may have different X-axis points according to different models; ${\varphi }_{3i}$ controls the steepness of the curve; ${\varphi }_{4i}$ is a constant associated with asymmetric growth, present only in the Richards model, where it is worth noting that when ${\varphi }_{4i}=1$, the Richards model is equivalent to the Logistic equation [24].

2.4. Fit Comparison

The quality of fit of the models was evaluated using likelihood-based measures (AIC, BIC), based on the fitted values (Mean Squared Error-MSE and Mean Absolute Error-MAE) and the adjusted coefficient of determination ($R_{adj.}^2$).

The Akaike criterion [25], known as AIC (Akaike Information Criterion), can be obtained by:

$$AIC=-2.ln\left(L\right)+2.p,$$

(7)

The Schwarz criterion [26], also named BIC (Bayesian Information Criterion), is described as follows:

$$BIC=-2.L+p.log\left(N\right),$$

(8)

For both criteria, L is the logarithm of the likelihood ratio obtained by estimation, $p$ represents the number of parameters in the model, and $N$ is the sample size. For both AIC and BIC, lower values indicate the best models.

The MAE and MSE measures measure the approximation between the observed values ($y_{ij}$) and values estimated by the respective models (${\widehat{y}}_{ij}$). The MSE is given by:

$$MSE={\displaystyle \sum }_{i=1}^M{\displaystyle \sum }_{j=1}^{n_i}\raisebox{1ex}{${\left(y_{ij}-{\widehat{y}}_{ij}\right)}^2$}\!\left/ \!\raisebox{-1ex}{$N$}\right.,$$

(9)

while the MAE is calculated as follows:

$$M\mathit{A}E={\displaystyle \sum }_{i=1}^M{\displaystyle \sum }_{j=1}^{n_i}\raisebox{1ex}{$\left|y_{ij}-{\widehat{y}}_{ij}\right|$}\!\left/ \!\raisebox{-1ex}{$N$}\right.,$$

(10)

In which $y_{ij}$ and ${\widehat{y}}_{ij}$ are, respectively, the observed and estimated values of length and width per plant ($Y$) for the i-th access ($i=1,\dots ,8$) at the j-th time ($j=0, 7, 14, 21, 28, 35, 42, 56, 63, 70$) and $N$ represents the sample size ($N=80$). The best models are those with the lowest AIC, BIC, MSE and MAE values.

The $R_{aj.}^2$ is used to check the proportion of the total variability explained by the regression model and is calculated as follows [27]:

$$R_{adj.}^2=1-\left[\frac{N-1}{N-p}\right]\left(1-R^2\right),$$

(11)

where $R^2$ is the coefficient of determination ($R^2=1-RSS/TSS$), TSS being the Total Sum of Squares and RSS the Residual Sum of Squares, the value of $N$ represents the same sample size as described above. Unlike the other measures of quality of fit already mentioned, the practical interpretation of $R_{aj.}^2$ varies in the range between $0$ and $1$, and the closer to $1$, the better the model.

2.5. Computational Tools

All analyses were performed in R software [28]. The data manipulation, the calculation of descriptive measures, and the generation of the graphs were performed with the aid of the packages dplyr [29] and ggplot2 [30], respectively. Each model was fitted considering all eight pepper plants previously described with the inclusion of the group effects, representing the fixed effects of type and random effects (individual per fruit) with the support of the NLME package [31].

3. Results

Figure 1 highlights the difference in length between the pepper and bell pepper groups that were considered for estimating the models’ fixed effects. The increase in fruit length occurs sharply until approximately the twentieth DAF, from where this difference between fruit types becomes clearer. The bell pepper species have an asymptotic length (at the end of the curve) higher than the pepper fruits (Figure 1), with average results close to 60 mm and 45 mm, respectively.

When the width of the fruits is descriptively analyzed (Figure 2), the difference between types of pepper again occurs both in the shape of the curve, which is more pronounced for pepper fruits, and for the asymptotic width, with mean values close to 50 mm for bell peppers and less than 20 mm for peppers. For both groups, the growth in width occurs markedly until approximately the tenth DAF, from when it happens in a less obvious form.

Both for the length and for the width of the fruits, there was convergence in the estimation of parameters for all the models used. Analyzing the comparison between models for the adjustment of growth in fruit length, we can see that the Gompertz, Logistic, and Richards models presented similar results according to the measures of goodness of fit (

Table 3. Fit quality measures for the four models considered in the fruit length fit.

Model 1	AIC	BIC	MSE	MAE	${\mathit{R}}_{\mathit{a}\mathit{d}\mathit{j}.}^2$
Gompertz	420.09	441.53	2.80	1.21	0.9949
Logistic	423.55	444.99	2.94	1.20	0.9946
Richards	412.61	432.82	2.32	1.06	0.9957
von Bertalanffy	678.76	700.20	226.24	12.03	0.5887

¹ The best models have lower values of AIC, BIC, MSE and MAE and higher values of $R_{adj.}^2$.

). The Richards equation presented the best results according to AIC (412.61), BIC (432.82), MSE (2.32), MAE (1.06) and $R_{adj.}^2$ (0.9957), showing a difference with lower values for the other models, except for $R_{adj.}^2$. The Gompertz and Logistic models also proved efficient for such measures. The von Bertalanffy equation showed divergent results, distancing from the others in terms of goodness of fit, showing higher values of both MSE (226.24) and MAE (12.03), respectively. The other results of the model quality assessment can be found in Table 3.

When observing the quality criteria of the models for adjusting the width of the fruits (

Table 4. Fit quality measures for the four models considered in the fruit-width adjustment.

Model 1	AIC	BIC	MSE	MAE	${\mathit{R}}_{\mathit{a}\mathit{d}\mathit{j}.}^2$
Gompertz	353.99	375.43	1.90	0.98	0.9949
Logistic	339.61	361.04	1.50	0.91	0.9959
Richards	342.21	368.41	1.47	0.92	0.9960
von Bertalanffy	569.99	591.43	58.09	6.38	0.8429

¹ The best models have lower values of AIC, BIC, MSE and MAE and higher values of $R_{adj.}^2$.

), the model that presented the best measures of quality of adjustment, in general, was the Logistic one, showing lower values of AIC, BIC, and MAE (339.61, 361.04 and 0.91, respectively). Analyzing the MSE and the $R_{adj.}^2$, the Richards model was superior (1.47 and 0.9960), despite presenting a slight difference from the Logistic model (1.50 and 0.9959). As in the length modeling, the two cited models and the Gompertz equation were superior in terms of goodness of fit compared to the von Bertalanffy one. The $R_{adj.}^2$ for all models, except for the von Bertalanffy equation, showed an almost perfect fit, always greater than 0.9949 (Table 4).

Considering the fixed-effects estimates (${\widehat{\beta }}_j$’s) together with their standard errors (between parentheses) for the length-adjustment models (

Table 5. Fixed-effect parameters estimated in fruit-length adjustments, with standard errors in parentheses.

Parameter 1	Gompertz	Logistic	Richards	von Bertalanffy
${\widehat{\beta }}_1$	39.68 (8.32)	39.26 (8.28)	39.27 (8.43)	40.07 (3.53)
${\widehat{\beta }}_4$	67.86 (10.82)	66.95 (10.70)	67.87 (10.88)	68.48 (4.59)
${\widehat{\beta }}_2$	4.82 (1.38)	7.89 (1.43)	7.72 (1.72)	0.50 (0.20)
${\widehat{\beta }}_5$	6.47 (1.69)	9.82 (1.76)	6.43 (2.06)	0.53 (0.16)
${\widehat{\beta }}_3$	0.14 (0.01)	0.20 (0.01)	0.20 (0.03)	0.10 (0.04)
${\widehat{\beta }}_6$	0.12 (0.01)	0.18 (0.01)	0.12 (0.01)	0.10 (0.03)
${\widehat{\beta }}_7$	-	-	0.96 (0.39)	-
${\widehat{\beta }}_8$	-	-	−0.01 (0.21)	-

¹ The asymptotic length, time to inflection point and slope parameter of the curve are being represented, respectively, by ${\widehat{\beta }}_1$, ${\widehat{\beta }}_2$ and ${\widehat{\beta }}_3$ for pepper and ${\widehat{\beta }}_4$, ${\widehat{\beta }}_5$ and ${\widehat{\beta }}_6$ for bell pepper. The Richards model is the only one that has the fourth parameter, which is represented by ${\widehat{\beta }}_7$ for pepper and ${\widehat{\beta }}_8$ for bell pepper.

), it is observed that the biologically interpretable parameters (${\widehat{\beta }}_1$ and ${\widehat{\beta }}_2$, respectively representing the asymptotic length and time to inflection point) are similar for all models. The asymptotic length was estimated for the group of peppers (${\widehat{\beta }}_1$) ranging from 39.26 mm (Logistic) to 40.07 mm (von Bertalanffy), while for bell peppers, this parameter ranged from 66.95 mm (Logistic) to 68.48 mm.

The Richards model was considered the best for adjusting this variable and the others and could estimate the difference between the studied groups for all parameters compared between pepper and bell pepper (Table 5). We can observe from the estimates of this model, for example, that bell pepper fruits have an asymptotic length (${\widehat{\beta }}_4$) of 67.87 mm, while the asymptotic length for pepper (${\widehat{\beta }}_1$) is close to 39.27 mm (Table 5). The difference is smaller when we consider the time to the point of inflection of fruit length in pepper (${\widehat{\beta }}_2$) and bell pepper (${\widehat{\beta }}_4$), where growth is maximized at 7.72 and 6.43 days, respectively. Estimates of the other parameters, with their respective standard errors, can be seen in Table 5.

Considering the parameter estimates for fruit width (

Table 6. Fixed-effect parameters estimated in fruit-width adjustments, with standard errors in parentheses.

Parameter 1	Gompertz	Logistic	Richards	von Bertalanffy
${\widehat{\beta }}_1$	15.57 (4.19)	15.28 (4.10)	15.51 (4.19)	15.45 (2.19)
${\widehat{\beta }}_4$	53.83 (5.40)	53.15 (5.29)	53.16 (5.40)	54.38 (2.37)
${\widehat{\beta }}_2$	6.39 (0.94)	11.16 (0.94)	7.26 (4.26)	0.44 (0.20)
${\widehat{\beta }}_5$	6.46 (0.43)	10.04 (0.50)	10.03 (1.03)	0.52 (0.10)
${\widehat{\beta }}_3$	0.08 (0.01)	0.12 (0.01)	0.09 (0.02)	0.08 (0.05)
${\widehat{\beta }}_6$	0.12 (0.01)	0.18 (0.01)	0.18 (0.02)	0.10 (0.02)
${\widehat{\beta }}_7$	-	-	0.15 (0.71)	-
${\widehat{\beta }}_8$	-	-	0.99 (0.34)	-

¹ The asymptotic width, time to inflection point and slope parameter of the curve are being represented, respectively, by ${\widehat{\beta }}_1$, ${\widehat{\beta }}_2$ and ${\widehat{\beta }}_3$ for pepper and ${\widehat{\beta }}_4$, ${\widehat{\beta }}_5$ and ${\widehat{\beta }}_6$ for bell pepper. The Richards model is the only one that has the fourth parameter, which is represented by ${\widehat{\beta }}_7$ for pepper and ${\widehat{\beta }}_8$ for bell pepper.

), the difference between the asymptotic-weight estimates of pepper and bell pepper is also evident, with width estimates varying around 15.50 and 53.50, respectively. The time to the inflection point is practically the same for both groups according to the Gompertz model (6.39 and 6.46, respectively) and shows similar estimates for the Logistic model (11.16 and 10.04, respectively), considered the best adjustment for this question. Therefore, according to this model, fruit-width growth is maximized at approximately 11 days for pepper fruits and ten days for bell peppers. The other results can be seen in Table 6.

Considering the criteria used as a quality of fit, the Richards equation proved to be more efficient for fruit-length adjustment because it was superior in all the measures used, while the Logistic model stood out for better adjusting the width growth, being superior in terms of AIC, BIC, and MAE.

Figure 3 shows the proximity between the observed values (points) and the adjusted values (solid line) according to the Richards model for fruit-length growth. We can see that the solid line for pepper (in blue) and bell pepper (in red) shows how close the estimated values are to the actual values. There was also a high variation within groups, mainly in terms of asymptotic length. It is more clearly observed that the Ikeda Hard Shell and Quadrado peppers have a higher asymptotic length than most other plants, with estimates greater than 70 mm (Figure 3). As for the same characteristic, the pepper fruits are smaller, with measurements close to 40 mm, except for the Cayenne pepper, the only pepper fruit that showed an asymptotic length greater than 60 mm.

Observing the curves adjusted by the Logistic model for fruit width (Figure 4), there is a noticeable proximity between the observed values and the model (continuous lines) for pepper and bell pepper, represented by the same colors. The growth occurs more markedly among the bell pepper fruits, reaching higher length and width in general compared to the pepper fruits. It can be seen graphically that most fruits’ tipping point is approximately ten days. The average asymptotic width for the bell pepper fruits is higher than for the pepper fruits. As well as the asymptotic length adjustment, a high variation was also observed within the groups considering the model for width fit.

The relationship between the zero-centered individual random-effects estimates corresponding to the asymptotic length (${\widehat{b}}_1$) and time to the inflection point (${\widehat{b}}_2$) is shown in Figure 5. The estimated Pearson correlation coefficient was 0.75, showing that regardless of the type of pepper, the relationship between these two growth parameters is positive and moderate. It can also be noted that individual differences between peppers of the same kind could be identified, such as the Cayenne and Peter peppers, which presented positive estimates indicating that they are above average for both ${\widehat{b}}_1$ and ${\widehat{b}}_2$. Considering bell peppers, however, the variation was smaller in relation to peppers. Quadrado and Ikeda Hard Shell were superior in terms of asymptotic length. At the same time, Giant Rubi reached the curve’s inflection point more quickly (Figure 5).

Considering the similar graph, now observing the analysis for width data (Figure 6), we can identify the most pronounced variability among bell pepper fruits. The correlation between random effects, unlike what happened with the length model, showed an almost zero correlation of −0.02. The Jamaica Yellow pepper was superior to the others in the same group regarding asymptotic width, while the Peter pepper took longer to reach the inflection point. Ikeda Hard Shell was shown to be inferior to other peppers. However, it takes longer to reach the tipping point. The behavior of the Quadrado bell pepper could be better, obtaining prominence for the asymptotic width and less time to the inflection point. The other estimates of random effects for the biologically interpretable parameters can be seen in Figure 6.

4. Discussion

When proposing the estimation method used in this work, Lindstrom and Bates [5] highlighted that procedures for selecting and criticizing models need to be developed and studied using real data. The flexibility of this methodology using data from real experiments proved to be true when it was possible to include in the model fixed effects of the group of peppers (pepper and bell pepper) and individual random effects for the biologically interpretable variables (asymptotic length and width and time to the point of curve inflection). The efficiency in terms of prediction could be observed according to the low MSEs and MAEs observed in Table 3 and Table 4 and Figure 3 and Figure 4, which illustrate estimated curves close to the actual observations for the two groups for length adjustment and fruit width. This methodology efficiency corroborated with some works that obtained good results, such as applications for eucalyptus spp. trees [32], pepper [7], Guzerá cattle [10], dairy buffaloes [11], dairy goats [33], heifers [34] and larch [16]. This shows the effectiveness of the methodology in different contexts.

The difference between the growth patterns for both the length and width of pepper and bell pepper fruits is evidenced by observing Figure 2 and Figure 4, respectively. For the three bell pepper fruits, both the length and the asymptotic width (final part of the growth phase) show different results, with the same behavior observed for the growth rate, where it is noted that bell pepper fruits increase in size more sharply. This notorious difference in behavior between the two types of fruit reinforces the need to use NLME with the insertion of the kind of fruit as a fixed effect. To confirm this need, it is interesting to observe the results of Table 5 and Table 6, which show that, both for length and width adjustment, the average fixed-effect coefficients (${\widehat{\beta }}_j$’s) presented different estimates between the groups, mainly in what concerns the asymptotic length and width, respectively, shown in Table 5 and Table 6. The difference between the width of pepper and bell pepper fruits is in accordance with what was observed by Rosado et al. [35], who, using genotypes like those of this work, estimated that the mean fruit width of bell pepper was statistically greater than in pepper by the Scott-Knott test considering a nominal significance level of 5%. However, when analyzing length, peppers and bell peppers obtained similar measurements.

Analyzing the comparison of models for adjusting fruit-length growth (Table 3), we can see that the best model in terms of all 5 measures of goodness of fit (AIC, BIC, MSE, MAE and, $R_{adj.}^2$) was that of Richards. This equation is the only one in this study that presents the coefficient ${\varphi }_{4i}$, which, according to Archontoulis and Miguez [24], has the function of dealing with asymmetric growth. This fact suggests that the addition of this parameter was able to improve the adjustment efficiency both in terms of likelihood when observing the AIC and BIC and in terms of prediction (MSE and MAE and $R_{adj.}^2$). The improvement of the model also suggests that fruit-length growth occurs asymmetrically, reinforcing what was observed in the graphical analysis (Figure 1), where the phenotypes reach the inflection point in approximately ten days. This can also be confirmed by looking at the estimates for the time to the tipping point for pepper (${\widehat{\beta }}_2$) and pepper (${\widehat{\beta }}_5$) in Table 5. Still about the length characteristic, the efficiency of the Logistic model was corroborated by Oliveira et al. [6], who observed that for three of the five pepper genotypes, the Logistic model was the best fit. In the comparison made by this work, however, the Richards model was not included.

Regarding the fruit width, the Logistic model presented more relevant results in terms of AIC, BIC, and MAE (Table 3), in addition to values of MSE and $R_{adj.}^2$ practically equal to those of the Richards model. This result shows that, despite having three parameters (one less than the Richards model), the Logistic model was the most efficient for describing the growth of this variable. This also shows that the fourth parameter of the Richards model, described earlier, did not make much difference in adjusting the fruit width since there were no gains compared to the Logistic model. Oliveira et al. [6], also working with pepper width, observed that the Logistic model presented better AIC values for three of the five genotypes used when evaluating the fruits’ width, corroborating this work’s results. Other similar works studying fruit growth also concluded that the Logistic model stood out from the other studies [8,36,37].

The proximity of zero estimated by the MSE was also found by other authors [6], who used Quantile Regression and Ordinary Least Squares (OLS) for parameter estimation. The authors used an adjustment per individual considering fixed effects and included in the sample genotypes like those of pepper (first group) found in this work. Considering the same area, the Gompertz, Logistic and von Bertalanffy models adjusted the data satisfactorily, as in the present work.

We can observe that, for the comparison among models for both length and width, the equations showed differences between high AICs between the best and worst models. It is known according to [38] that differences greater than 10 units in terms of AIC comparison between the best and worst models indicate that the one with the highest AIC is not suitable. Therefore, there is a reasonable difference in performance among the models studied on both occasions (Table 3 and Table 4). The good quality of fit with the type of pepper as a covariate also suggests that by adding other covariates in future work, such as thermal time, we may also find relevant results.

The estimates of parameter ${\beta }_1$ and ${\beta }_4$, both for length and width, were close in all models studied, whereas, for the estimate of parameter ${\beta }_2$ and ${\beta }_5$, more divergent results were obtained (Table 3 and Table 4). Considering ${\beta }_3$ and ${\beta }_6$, in all traits, there was a similarity in the parameter estimates between the Gompertz, Logistic, Richards, and von Bertalanffy models. Similar results can be observed in work by Wen et al. [39], who, using partridges, studied the body-weight curve using nine different nonlinear models, including four of the five used in our work. The authors obtained similar estimates of ${\beta }_1$ and ${\beta }_3$ among the Gompertz, Logistics, von Bertalanffy, and Richards models, whereas for ${\beta }_2,$ there were greater divergences of results when comparing the models.

The random effects of the parameters of the curves estimated by the Richards model for fruit length (Figure 5) and Logistic for fruit width (Figure 6) identify the individual estimates of these parameters for each genotype used. Looking at the relationship between asymptotic length (${\widehat{b}}_1$) and time to the inflection point (${\widehat{b}}_2$) in Figure 5, we note a positive and strong relationship of 0.75, indicating that fruits that reach a higher asymptotic weight tend to require more time to reach the tipping point. It is also noted that the Peter and Cayenne species tend to have a greater difference in relation to the estimated average effects for the first group (pepper), with these differences being higher in this group. Looking at Figure 6, we can see that the relationship between the asymptotic width (${\widehat{b}}_1$) and the time to the inflection point (${\widehat{b}}_2$) is close to zero correlation, with a value of −0.02. This result indicates that contrary to fruit length, there is no association between these variables when analyzing fruit width, indicating that the increase in asymptotic width is unrelated to time until the inflection point. As these are random effects derived from individual estimates of pepper fruits, regardless of the groups they are part of in this study, this result indicates that fruits that reach the inflection point more quickly will not necessarily have a high asymptotic weight. This represents the opposite of what happens if we consider the relationship between these variables for fruit length. It is also worth mentioning that in this case, there was a greater variation associated with group 2 (bell peppers).

5. Conclusions

The Nonlinear Mixed-Effect Models obtained convergence in the estimation of parameters for the four models used. They allowed the insertion of the fixed effect of the type of pepper (pepper and bell pepper) and individual estimates from a single fit per equation. Three of the four models used fitted the growth data satisfactorily. According to the criteria, the best model to describe fruit-length growth was the Richards model, while for fruit width, it was the Logistic model. Such models allowed the estimation and characterization of all pepper genotypes, with reports by the groups. The analysis of random effects allowed identifying the superior genotypes within each group and pointed out the relationship between biologically interpretable growth characteristics for both growth and fruit width. It is worth considering that the gain in this methodology was observed even in a small sample, which leads us to believe that its use to obtain information in a larger sample can provide excellent results, mainly in the context of analyzing longitudinal data for growth data.