Bayesian inference applied to the response of Amaranthus species to herbicides under different likelihood functions

Inferência Bayesiana aplicada à resposta de espécies de Amaranthus a herbicidas sob diferentes distribuições de verossimilhança

ABSTRACT:

This study evaluated the fit of the nonlinear logistic model to herbicide dose-response data in weed species of the genus Amaranthus, using different likelihood function under the Bayesian approach. Plants of Amaranthus hybridus and Amaranthus retroflexus were exposed to different doses of the trifloxysulfuron-sodium herbicide: 16D, 4D, D, 1/4D, 1/16D, 1/64D, and a control without application, where D is the recommended dose of the product, applied in two experimental conditions (3.75 g ha^-1 and 7.5 g ha^-1). Model fitting was performed in the R software using the nonlinear logistic model under Bayesian inference, considering four distributions for the likelihood function: normal, Weibull, gamma, and exponential. The goodness of fit was assessed using the deviance information criterion and the ordered predictive density criterion. There was no evidence of non-convergence of the chains. Parameter estimates were similar among the distributions; the normal likelihood showed the best fit. However, the overlap of the HPD intervals suggested that other likelihood functions could also describe the dose-response data.

Key words:
Weibull and Gamma distributions; exponential and normal distributions; MCMC (Markov Chain Monte Carlo); regression

RESUMO:

Este estudo teve como objetivo avaliar o ajuste do modelo não linear logístico a dados de dose-resposta de herbicidas em plantas daninhas do gênero Amaranthus, utilizando diferentes distribuições para a verossimilhança na abordagem bayesiana. A fim de atingir o objetivo, foram analisadas duas espécies: Amaranthus hybridus e Amaranthus retroflexus. As plantas foram submetidas a diferentes doses do herbicida trifloxysulfuron-sodium, sendo elas: 16D, 4D, D, 1/4D, 1/16D, 1/64D e ausência de aplicação, sendo D a dose recomendada do produto, aplicada em duas conduções experimentais (3,75 g ha^-1 e 7,5 g ha^-1). Além disso, os ajustes foram realizados no software R, com o modelo não linear logístico sob inferência bayesiana, considerando quatro distribuições para a função de verossimilhança: a) normal, b) Weibull, c) gama e d) exponencial. A qualidade dos ajustes foi avaliada pelos critérios de informação da deviance e o critério da densidade preditiva ordenada. Os resultados mostram que não houve evidências de não convergência das cadeias. As estimativas dos parâmetros foram semelhantes entre as distribuições, com destaque para a verossimilhança normal, que apresentou melhor ajuste. No entanto, a sobreposição dos intervalos HPD sugere que outras distribuições da função de verossimilhança também podem ser consideradas na descrição dos dados de dose-resposta.

Palavras-chave:
distribuição Weibull e gama; distribuição exponencial e normal; MCMC; regressão

INTRODUCTION

The genus Amaranthus (Amaranthaceae), commonly known as pigweed or amaranth, comprises weed species that infest agricultural areas and damage crops such as soybeans, corn, sugarcane, vegetables, and orchards and some are hosts for pests and insects, providing a green bridge and food source during the off-season (BRAZ & TAKANO, 2022; RESENDE et al., 2022).

These weeds are mainly controlled with herbicides, the most widely used method due to its efficiency and lower cost compared to other strategies. However, indiscriminately using these products causes some weed biotypes to develop resistance, including species of the genus Amaranthus (MATHIONI et al., 2022; BRAZ & TAKANO, 2022; YANNICCARI et al., 2023; LAMEGO et al., 2024).

The dose-response concept describes the relationship between the amount of a substance applied, such as an herbicide, and the observed biological effects. This relationship is also studied in areas such as biology, medicine, chemistry, toxicology, and pharmacology. In the case of herbicides, these effects may include plant mortality, for example (MA et al., 2020; YANG et al., 2021).

This relationship typically resembles an S-shaped sigmoid curve and can be described by nonlinear regression models (KESHTKAR et al., 2021; AZARIAS et al., 2023). These models have the advantage of having a practical interpretation of some of their parameters and being more parsimonious because they have fewer parameters (MISCHAN & PINHO, 2014; KESHTKAR et al., 2021).

Nonlinear models fit well and make it possible to understand how plants react to different concentrations of herbicides. Additionally, this analysis helps to estimate appropriate dosages, avoiding the misuse of chemicals. This reduces production costs, minimizes environmental impacts, and decreases the emergence of herbicide-resistant plants (BRITO et al., 2018; FRANCISCHINI et al., 2019; KESHTKAR et al., 2021; AZARIAS et al., 2024; AZARIAS et al., 2026).

The logistic nonlinear regression model, proposed by STREIBIG (1988), is frequently used to model dose-response curves, including those involving herbicides. CARVALHO et al. (2015) applied this model to evaluate the susceptibility of Amaranthus palmeri to the herbicide glyphosate; KÜPPER et al. (2017) used it to confirm and characterize the multiple resistance of Amaranthus viridis to both glyphosate and ALS inhibitors; and ALBRECHT et al. (2020) demonstrated the multiple resistance of Conyza sumatrensis to the herbicides paraquat, glyphosate, and chlorimuron using this model.

An approach that has proven efficient for estimating the parameters of nonlinear models is Bayesian inference, which is based on Bayes theorem and considers the model parameters as random variables (TURKMAN et al., 2019; LAMBERT, 2018). In Bayesian inference, prior knowledge, represented by the prior density P(θ) , is multiplied by the likelihood function of the data L(Y│θ) to obtain the posterior distribution P(θ│Y). This posterior distribution represents the updated probability, or belief, based on the data (REIS et al., 2008; SAVIAN et al., 2009; TURKMAN et al., 2019; VAN DE SCHOOT et al., 2021; KAPLAN, 2023).

In statistical analyses, it is common to assume that the data follows a normal distribution, justifying the use of a normal likelihood function in Bayesian inference (FACCIN & ROSSI, 2024; MOURA et al., 2025). However, this assumption does not always reflect the actual behavior of the data because, in many cases, the data do not exhibit the characteristic symmetry of a normal distribution. Sometimes, the observed variables present asymmetries, heavy tails, extreme values (outliers), or support only positive numbers (PINO, 2014).

For example, this is the case in dose-response studies with herbicides, where the observed variables are always positive (NETTO et al., 2016; FRANCISCHINI et al., 2019; ALBRECHT et al., 2020; AZARIAS et al., 2024), a possible solution in the context of Bayesian inference is to change the distribution used in the likelihood function, replacing the normal distribution with a distribution that better aligns with the characteristics of the observed data. The gamma, Weibull, and exponential distributions are appropriate for modeling continuous and positive variables (RIGBY et al., 2019; ALMEIDA et al., 2021).

In this sense, the objective of this study was to evaluate the fit of the nonlinear logistic model to herbicide dose-response data on weeds by assuming different likelihood function distributions using the Bayesian approach, to identify the most suitable distribution for describing the data.

MATERIALS AND METHODS

The data used were extracted from CARVALHO et al. (2006). The experiment was conducted in a greenhouse at the Department of Plant Production of the Luiz de Queiroz College of Agriculture - ESALQ/USP, located in Piracicaba, state of São Paulo, between September and December 2005.

The study analyzed five weed species of the genus Amaranthus. However, only two were considered here: A. hybridus (green amaranth) and A. retroflexus (red root amaranth). Doses of the trifloxysulfuron-sodium herbicide equivalent to 16D, 4D, D, 1/4D, 1/16D, 1/64D, were applied, where D are the recommended doses of the product. Two applications were carried out, with doses of 3.75 in the first and 7.5 g ha^-1 in the second. A control without herbicide application was also used, which was evaluated 20 days after application (DAA), where 0 indicates no control and 100% indicates plant death.

For the description of the dose-response curve data of the trifloxysulfuron-sodium herbicide, we used the non-linear logistic model proposed by STREIBIG (1988):

$y_i=\frac{a}{(1+{\left(\frac{x_i}{b}\right)}^c)}+{\mathrm{\epsilon }}_i$

where, i = 1, 2,...,n; y _i is the i-th observation of the dependent variable, corresponding to the percentage of control; x _i is the i-th observation of the independent variable, representing the herbicide doses; a, b and c, are the model parameters: a is the horizontal asymptote, a mathematical estimate of the amplitude, i.e., the maximum control minus the minimum control; b is the dose that provides 50% herbicide control; c is related to the slope of the curve; ɛ_i are the random errors attributed to the model. This definition of the model is used for all likelihoods presented.

Parameters were estimated using Bayesian inference, based on Bayes Theorem, which combines the prior distribution for the model parameters, P(a, b, c, σ), with the likelihood function, L(y│a, b, c, σ). This combination results in the posterior distribution, P(a, b, c, σ│y), and is expressed by the relation P(a, b, c, σ) ∝ L(y│a, b, c, σ) P(a, b, c, σ│y), indicating that the posterior distribution is proportional to the product of the likelihood function and the prior distribution of each parameter (VAN DE SCHOOT et al., 2021; KAPLAN, 2023; INCHAUSTI, 2023).

The prior distributions were defined using maximum entropy priors, following the approach adopted by AZARIAS et al. (2026). This principle states that, among all distributions that satisfy the known constraints, the one that maximizes entropy should be selected, thus representing the least informative distribution conditional on the available information. The hyperparameters were defined based on mean values extracted from dose-response studies, including the works of FRANCISCHINI et al. (2013), FRANCISCHINI et al. (2014), FRANCISCHINI et al. (2019), and NETTO et al. (2016).

Since the possible values of parameter a, which represents the horizontal asymptote, and parameter b, which corresponds to the dose responsible for 50% of the herbicide response, are positive, a truncated Normal distribution was adopted as the prior distribution, ensuring that these parameters remain within biologically interpretable ranges, according to equations (1) and (2). For parameter c, associated with the slope of the curve and which, according to the literature, can assume negative or positive values, a normal distribution was adopted as a prior, according to equation (3). Thus:

$P\left(a\right|{\mathrm{\mu }}_a,{\mathrm{\sigma }}_a^2)\propto \exp \{-\frac{1}{2{\mathrm{\sigma }}_a^2}{(a-{\mathrm{\mu }}_a)}^2\},a\in (0,\infty )$(1)

$P\left(b\right|{\mathrm{\mu }}_b,{\mathrm{\sigma }}_b^2)\propto \exp \{-\frac{1}{2{\mathrm{\sigma }}_b^2}{(b-{\mathrm{\mu }}_b)}^2\},b\in (0,\infty )$(2)

$P\left(c\right|{\mathrm{\mu }}_c,{\mathrm{\sigma }}_c^2)\propto \exp \{-\frac{1}{2{\mathrm{\sigma }}_c^2}{(c-{\mathrm{\mu }}_c)}^2\},c\in (-\infty ,\infty )$(3)

being (μ_a , σ _a ² ), (μ_b , σ _b ² ) and (μ_c , σ _c ² ) hyperparameters associated with parameters a, b and c, respectively.

For the dispersion parameter, an exponential distribution was considered as the prior distribution, with hyperparameter $𝛿=\frac{1}{0.001}=1000$, according to equation (4). Figure 1 shows the behavior of the prior densities of parameters a, b, c and σ. The prior distributions were assumed independent, thus P(a, b, c, σ) = P(a)P(b)P(c)P(σ).

$P\left(𝜎\right|\mathrm{\delta })\propto \exp \{-\frac{𝜎}{\mathrm{\delta }}\},𝜎\in (0,\infty )$(4)

Figure 1
Density of prior distributions assigned to parameters a, b, c and σ.

In the likelihood function, the normal, gamma, exponential, and Weibull distributions were used. The last three are indicated for positive data, and the normal distribution was included for its flexibility in modeling continuous variables.

Consider that the likelihood function L(y│θ) can be expressed as the product of the individual conditional distributions, equation (5), assuming that the observations y ₁,…,y _n are conditionally independent given the parameters θ. In this case, θ represents the set of parameters for the model, and f _Y corresponds to the probability density function (pdf). Thus, the likelihood can be approximated as:

$L\left(𝒀\right|𝜽)\approx \prod _{i=1}^n{f}_Y\left(y_i\right|𝜽)$(5)

Normal likelihood (μ,σ)

The parameterization adopted for the probability density function (pdf) of the normal (or Gaussian) distribution, denoted by N(μ,σ), is

$f_Y(y\mid 𝜇,𝜎)=\frac{1}{\sqrt{2𝜋}𝜎}exp[-\frac{{(y-𝜇)}^2}{2{𝜎}^2}]$

where - ∞ < μ < ∞ and σ > 0 . The mean of E(Y) = μ and the variance by Var(Y) = σ². The distribution N(μ, σ²) is symmetrical around y = μ.

Assuming that the observations y _i are conditionally independent given the vector of model parameters θ = (a, b, c, σ), and that $y_i\sim N({𝜇}_i,{𝜎}^2),\text{with}{𝜇}_i=\frac{a}{1+{\left(\frac{x_i}{b}\right)}^c}$, $L\left(𝒀\right|𝜽)=\prod _{i=1}^n\frac{1}{\sqrt{2𝜋}𝜎}exp[-\frac{{(y_i-{𝜇}_i)}^2}{2{𝜎}^2}]$.

Gamma likelihood (μ,σ)

RIGBY et al. (2019) proposed a reparameterization of the gamma distribution, presented by JOHNSON et al. (1994):

$p_y\left(y\right)=\frac{x^{𝛼-1}{e}^{-x/𝛽}}{{𝛽}^{𝛼}𝛤\left(𝛼\right)},y>0;𝛼>0,𝛽>0$

Defining α = 1/σ² and β = μσ², we have that μ = α β and σ² = 1/ α. The probability density function (pdf) of the gamma distribution, gamma (μ, σ), defined by RIGBY et al. (2019) is:

$f_Y\left(y\right|𝜇,𝜎)=\frac{y^{1/{𝜎}^2-1}{e}^{-y/\left({𝜎}^2𝜇\right)}}{{\left({𝜎}^2𝜇\right)}^{1/{𝜎}^2}𝛤\left(1/{𝜎}^2\right)}$

for y > 0, where μ > 0 is the scale parameter and σ > 0 is the coefficient of variation. Given E(Y) = μ and variance Var (Y) = σ²μ.

Assuming that the observations y _i are conditionally independent given the vector of parameters θ, and that $y_i\sim \text{gamma}({𝜇}_i,𝜎),{𝜇}_i=\frac{a}{1+(x_i/b{)}^c}$according to the reparameterization presented by RIGBY et al. (2019), the likelihood function with gamma distribution, can be expressed as the product of the conditional distributions of the observations:

$L\left(𝒀\right|𝜽)=\prod _{i=1}^n\frac{y_i^{1/{𝜎}^2-1}{e}^{-y_i/\left({𝜎}^2{𝜇}_i\right)}}{{\left({𝜎}^2{𝜇}_i\right)}^{1/{𝜎}^2}𝛤\left(1/{𝜎}^2\right)}$

Weibull likelihood (μ, σ)

RIGBY et al. (2019) present three parameterizations for the Weibull distribution, WEI (μ, σ). In this study, the parameterization where μ represents the mean of the distribution was adopted. This parameterization, represented by the authors as WEI3 (μ, σ), is defined as:

$f_Y\left(y\right|𝜇,𝜎)=\frac{𝜎y^{𝜎-1}}{{𝜇/𝛤(1/𝜎+1)}^{𝜎}}\exp \{-{\left(\frac{y}{𝜇/𝛤(1/𝜎+1)}\right)}^{𝜎}\}$

for y > 0, where μ > 0 is the scale parameter and σ > 0.

Assuming that the observations y _i are conditionally independent given the vector of parameters θ, and that $y_i\sim \text{Weibull}({𝜇}_i,𝜎),{𝜇}_i=\frac{a}{1+(x_i/b{)}^c}$ according to the reparameterization presented by RIGBY et al. (2019), the likelihood function with Weibull distribution, can be expressed as the product of the conditional distributions of the observations:

$L\left(𝒀\right|𝜽)=\prod _{i=1}^n\frac{𝜎y_i^{𝜎-1}}{{({𝜇}_i/𝛤(\frac{1}{𝜎}+1))}^{𝜎}}\mathrm{ex}\mathrm{p}\{-{\left(\frac{y_i}{{𝜇}_i/𝛤(\frac{1}{𝜎}+1)}\right)}^{𝜎}\}$

Exponential likelihood (μ )

The exponential distribution is a special case of the gamma and Weibull distributions when σ² = 1 and σ = 1, respectively. Therefore, the probability density function of the exponential distribution, denoted by EXP (μ) is given by:

$f_Y\left(y\right|𝜇)=\frac{1}{𝜇}exp(-\frac{y}{𝜇})$

for y > 0, where μ > 0 is the scale parameter. Thus, E(Y) = μ and Var (Y) = σ²μ.

Assuming that the observations y _i are conditionally independent given the vector of parameters θ, and $y_i\sim \text{Exp}\left({𝜇}_i\right),{𝜇}_i=\frac{a}{1+(x_i/b{)}^c}$ that according to the reparameterization presented by RIGBY et al. (2019), the likelihood function with exponential distribution, can be expressed as the product of the conditional distributions of the observations:

$L\left(𝒀\right|𝜽)=\prod _{i=1}^n\frac{1}{{𝜇}_i}exp(-\frac{y_i}{{𝜇}_i})$

Based on Bayes’ Theorem, the joint posterior distribution is proportional to the likelihood function multiplied by the joint distribution of the priors, considered independent, for each parameter of the model. The joint posterior distribution for the different likelihoods is given by:

$P(a,b,c,𝜎\mid Y)\propto \prod _{i=1}^n{f}_Y\left(y_i\right|𝜽)\times N({𝜇}_a,{𝜎}_a)\mathrm{T}(0,+\infty )\times$

$N({𝜇}_b,{𝜎}_b)\mathrm{T}(0,+\infty )\times N({𝜇}_c,{𝜎}_c)\times exp(-\frac{\mathrm{\tau }}{\mathrm{\delta }})$

To obtain the marginal posterior distribution of the parameter θ _i (for i = 1,…, n), it is necessary to integrate the joint posterior distribution P(θ/Y) over all other parameters in the vector. However, due to the complexity of the analytical solution of these integrals, simulation methods are used, such as Markov Chain Monte Carlo (MCMC) (REIS et al., 2008; SAVIAN et al., 2009; GELMAN et al., 2014; LYNCH, 2021; KAPLAN, 2023). In this context, the Metropolis-Hastings (MH) algorithm, proposed by METROPOLIS et al. (1953) and generalized by HASTINGS (1970), can be used to generate approximate samples of the joint posterior distribution, thus obtaining estimates of the marginal distribution (TURKMAN et al., 2019).

Based on the marginal distributions of each model parameter, the mean, standard deviation, and highest posterior density (HPD) interval of 95% credibility were calculated as summary measures. To select the model that best fits the data, the deviance information criterion (DIC) and the log pseudo-marginal likelihood function (LPML; IBRAHIM et al., 2001), obtained from conditional predictive ordinate (CPO), were used.

All computational aspects of this study were performed using the free statistical software R (R DEVELOPMENT CORE TEAM, 2024). The packages used were Convergence Diagnostics and Output Analysis (CODA) (PLUMMER et al., 2006), Bayesian Output Analysis Program (BOA) (SMITH, 2007), and extraDistr (WOLODZKO, 2023).

RESULTS AND DISCUSSION

The inference of the model parameters, considering the four forms assumed for the likelihood function (normal, Weibull, gamma and exponential distributions), was performed based on the marginal posterior distributions of the parameters. The derivation of these distributions is based on Bayes’ theorem, according to which the joint posterior is proportional to the product of the likelihood function and the prior distributions.

From the joint posterior, the full conditional distributions of the parameters were obtained. In cases where the likelihood assumes a normal form, some of these conditionals presented known analytical expressions, as described in AZARIAS et al. (2026).

For the models whose likelihood was specified by the Weibull, exponential, and gamma forms, the full conditional distributions do not have known analytical expressions. Therefore, the Metropolis-Hastings algorithm was used to generate samples from these conditionals and, consequently, approximate their marginal posterior distributions.

A total of 905,000 iterations were generated. After discarding the first 5,000 iterations (burn-in), samples were collected at 30-iteration intervals (thinning) to obtain approximately independent samples. A total of 30,000 samples were used to calculate the subsequent quantities of interest. Then, the RAFTERY & LEWIS (1992) and GEWEKE (1992) tests, available in the BOA package, were applied to check the convergence of the chains.

According to the results in table 1 for convergence analysis, the dependence factor (DF) was close to 1 for all parameters, considering the two weed species, and the P-values of the Geweke test were greater than 0.05. These results indicated that there is no evidence of non-convergence of the chains.

Table 1
Dependence factor (DF) of the Raftery and Lewis criterion and P-value of the Geweke criterion for the logistic model fitted to two weed species of the genus Amaranthus under different likelihood function.

Likelihood	Parameter	Raftery and Lewis (DF)	Geweke (P-value)
--------------------------------------------------------------------------A. hybridus-----------------------------------------------------------------------------
Normal	a	1.0339	0.9623
	b	1.0109	0.9415
	c	1.0467	0.9315
	σ	1.4054	0.7799
gamma	a	1.0376	0.1710
	b	1.0205	0.3177
	c	1.0040	0.8117
	σ	1.0459	0.6691
Weibull	a	1.0093	0.0716
	b	1.0040	0.7566
	c	1.0082	0.3325
	σ	1.0178	0.3722
exponential	a	1.0093	0.9580
	b	1.0149	0.4812
	c	1.0093	0.7026
--------------------------------------------------------------------------A. retroflexus---------------------------------------------------------------------------
Normal	a	1.0088	0.6720
	b	1.0088	0.9475
	c	1.0109	0.4477
	σ	1.2279	0.8924
gamma	a	1.0961	0.1492
	b	1.0178	0.8097
	c	1.0317	0.8436
	σ	1.0899	0.9769
Weibull	a	1.0488	0.3325
	b	1.0178	0.4421
	c	1.0010	0.9274
	σ	1.0603	0.3803
exponential	a	1.0696	0.9438
	b	1.0290	0.8119
	c	1.0010	0.2034

The criteria proposed by RAFTERY & LEWIS (1992) and GEWEKE (1992) are widely used in evaluating the convergence of MCMC chains. This step is fundamental to ensure that the generated samples adequately represent the posterior distribution of the model parameters. Several studies have employed these criteria as part of the convergence diagnosis in Bayesian analyses, such as in AZARIAS et al. (2026), AZARIAS et al. (2025), EVANGELISTA et al. (2024), FERREIRA et al. (2022), SILVA et al. (2022), SILVA et al. (2023).

Table 2 presents the estimates of the mean, standard deviation, and HPD interval with 95% confidence for the parameters of the logistic model. These summaries are used in Bayesian analyses to describe the estimates of the posterior distribution of the parameters (REIS et al., 2011; GELMAN et al., 2014; SILVA et al., 2022; SILVA et al., 2020; EVANGELISTA et al., 2024). However, other summaries may be adopted, depending on the context of the study and the objectives of the analysis (REIS et al., 2009; MCELREATH, 2018).

Table 2
Mean, standard deviation, and highest posterior density interval (HPD) of the model parameters (LL: lower limit and UL: upper limit) for the logistic model fitted to two weed species of the genus Amaranthus under different likelihood function.

Likelihood	Parameter	Mean	Standard deviation	-------------------HP D 95% ------------------
				LL	UL
--------------------------------------------------------------------------A. hybridus----------------------------------------------------------------------------
Normal	a	100.0781	0. 4256	99.2311	100.9003
	b	0.2968	0.0066	0.2836	0.3096
	c	-1.0858	0.0242	-1.1333	-1.0383
	σ	0.9998	0.0100	0.9798	1.0189
gamma	a	98.9866	0.9423	97.1270	100.8319
	b	0.3027	0.0386	0.2275	0.3787
	c	-1.0811	0.0959	-1.2714	-0.8950
	σ	0.9993	0.0098	0.9799	1.0184
Weibull	a	98.9780	0.9451	97.1060	100.8039
	b	0.3025	0.0384	0.2278	0.3786
	c	-1.0812	0.0961	-1.2718	-0.8935
	σ	1.0008	0.0099	0.9812	1.0202
exponential	a	99.0004	0.9478	97.1500	100.8691
	b	0.3030	0.0383	0.2269	0.3773
	c	-1.0809	0.0969	-1.2683	-0.8893
-------------------------------------------------------------------------A. retroflexus--------------------------------------------------------------------------
Normal	a	98.7805	0.4263	97.9580	99.6244
	b	0.1987	0.0046	0.1897	0.2079
	c	-1.0507	0.0263	-1.1028	-0.9988
	σ	0.9992	0.0100	0.9799	1.0193
gamma	a	98.9861	1.0028	97.0092	100.9363
	b	0.2045	0.0289	0.1471	0.2606
	c	-1.1734	0.0982	-1.3632	-0.9787
	σ	0.9991	0.0099	0.9798	1.0187
Weibull	a	98.9926	0.9954	97.0911	100.9902
	b	0.2045	0.0289	0.1485	0.2621
	c	-1.1733	0.0978	-1.3670	-0.9835
	σ	1.0006	0.0099	0.9809	1.0199
exponential	a	98.9970	1.0034	97.0688	100.9811
	b	0.2042	0.0288	0.1464	0.2595
	c	-1.1723	0.0983	-1.3713	-0.9865

In the Bayesian approach, all inferences are made from the posterior distribution, which provides the densities that allow summarizing the information through measures such as mean, median, mode, and variance, for example (REIS et al., 2009; MARTINS FILHO et al., 2008; TURKMAN et al., 2019; SILVA et al., 2020).

All parameters were significantly different from zero (Table 2), as their HPD intervals did not include the value zero. The likelihood with normal distribution presented an interval with a smaller amplitude compared to the others, which may suggest less uncertainty in the estimates of its parameters. However, there was overlap in most HPD intervals, indicating that there was no statistically significant difference between the model parameters according to the different distributions studied.

The influence of the choice of likelihood function has also been highlighted in previous studies in dose-response experiments. As in ENGLEHARDT & SWARTOUT (2006), who demonstrated how the predictive Bayesian model with the Pareto type II dose-response distribution can be used to model and update dose-response relationships for human pathogens, such as Cryptosporidium parvum, as new data are acquired.

Similarly, HAMZA et al. (2021) proposed a Bayesian dose-response meta-analysis model, assuming normal and binomial likelihood function and considering exposures grouped into clusters. According to the authors, the model with binomial likelihood function showed less bias compared to the model with normal likelihood function and the one-stage frequentist model, especially in studies with small sample sizes. These results reinforce that the chosen distribution can affect precision and robustness, even when point estimates remain similar.

Gamma, exponential, and Weibull distributions have been widely used to model data with positive skewness, especially in contexts such as survival and reliability analyses (BRUNELLO & NAKANO, 2015; TOMAZELLA et al., 2020; MUSE et al., 2021; NOOR et al., 2021; ALMEIDA et al., 2021). The data analyzed in this type of study included events such as the death of a patient, remission of a disease, reaction to a medication, failure of an electronic device, burnt-out of a light bulb, among others.

Bayesian inference has also been widely used in fitting these distributions. BRUNELLO & NAKANO (2015) fitted the discrete Weibull distribution to a dataset on the time to death of men diagnosed with AIDS (Acquired Immunodeficiency Syndrome). NOOR et al. (2021) applied an exponential mixture model to cancer data, involving incidences and deaths among residents in Pakistan. ABUJARAD & KHAN (2018) used the Bayesian approach to fit the exponential model to lifetime data, employing Stan in lung cancer data.

For both species, considering the different likelihood function, the mean parameter estimates were close. In the case of the normal distribution, the estimated value for parameter a was approximately 100.08%, while in the other distributions, the estimated values were close to 99%. Furthermore, the likelihood function with normal distribution showed the lowest standard deviation (0.43 versus 0.94), which indicates greater precision in its estimates. Similar results were reported by MANGUEIRA et al. (2016), FACCIN & ROSSI (2024), and MOURA et al. (2025), who also found that the different likelihood choices did not significantly affect the estimates.

The parameter a of the logistic model is related to the percentage of maximum control minus the minimum control of the plants and indicates the asymptotic value referring to control, that is, the expected level of control as the dose of the applied product increases. In the literature, estimated values of 99% and, in some cases, exceeding 100% are observed for this parameter (FRANCISCHINI et al., 2013; NETTO et al., 2016; FRANCISCHINI et al. 2019).

The efficacy of the trifloxysulfuron-sodium herbicide on Amaranthus species was also evaluated by FRANCISCHINI et al. (2013). According to the authors, control of the species was greater than 80%. The biotypes of A. hybridus and A. lividus reached 80% control with doses of 0.17 and 0.10 g ha^-1, respectively. In addition, A. hybridus was more sensitive to the herbicide than A. viridis. Efficient control was obtained for all species, except A. viridis, which showed less sensitivity when subjected to reduced doses of the herbicide.

In the context of dose-response, one of the main interests is in identifying the dose required to achieve 50% control, which is observed by the b parameter, designated as C₅₀. The results of this parameter allow the evaluation of the susceptibility of weeds to the applied herbicide, since lower C₅₀ doses indicate greater sensitivity to the product (FRANCISCHINI et al., 2013; RAIMONDI et al., 2015; NETTO et al., 2022).

As presented in table 2, the mean estimates of the b (C₅₀) parameter for A. hybridus, considering all distributions, were close to 0.30 g ha^-1, while for A. retroflexus, it was 0.20 g ha^-1. These results indicated greater susceptibility of A. retroflexus to the herbicide, as it requires a lower dose to achieve 50% control. Furthermore, the likelihood function assuming a Normal distribution provided greater precision in the estimation of this parameter, as evidenced by the smaller standard deviations for both species.

A. retroflexus biotypes from three cotton-producing states in Brazil were studied by FRANCISCHINI et al. (2014). Two biotypes showed high C₅₀ values, being 22.63 and 42.13 times higher than the doses used in the susceptible biotype. Furthermore, to achieve 50% control, biotypes GO 3, GO 4, and GO 6 required doses of 0.0104, 0.0282, and 0.0407 kg ha^-1, respectively, lower than those obtained in the present study (0.30 g ha^-1 and 0.20 g ha^-1). According to the authors, these results suggested that this resistance may have been caused by the continued use of herbicides with the same mechanism of action.

For parameter c, which is related to the slope of the curve, the mean estimates were similar; that is, the rate of change in control as a function of increasing dose was comparable for both species. However, the standard deviation was lower for the likelihood function with normal distribution (0.0242 and 0.0263) for A. hybridus and A. retroflexus, respectively.

Regarding the precision parameter σ, it was observed that it presented similar values across the analyzed distributions, being approximately 0.99 in all cases. Considering that the exponential distribution constitutes a particular case of the Weibull distribution when the shape parameter is equal to 1, and also of the gamma distribution when the shape parameter assumes a unit value, the proximity of the σ estimates to this value suggests that the three distributions exhibited very similar behavior in fitting the data. This justifies why the model fits were very close to each other, since the likelihood functions based on the Weibull distribution and gamma distribution approach the particular case of the exponential distribution.

The results obtained suggested that, for both species, both the likelihood function with normal distribution and the other distributions used in the likelihood function are adequate to describe the data. Although the likelihood function with normal distribution presents smaller standard deviations and HPD intervals with smaller amplitude, the overlap of these intervals suggests that other approaches can also be considered in the description of dose-response data.

The results of the deviance information criterion (DIC) and the log pseudo-marginal likelihood function (LPML) are available in table 3. For both species, A. hybridus and A. retroflexus, the normal distribution presented the lowest DIC value and the highest LPML value among the distributions evaluated. These results indicate that the model with normal likelihood showed better predictive performance and goodness of fit compared to the other distributions considered.

Table 3
Estimates of DIC and LPML considering the fit of the logistic model for different likelihood function, considering dose-response data of the weed species A. hybridus and A. retroflexus.

Likelihood	DIC	LPML
---------------------------------------------------------------------------A. hybridus----------------------------------------------------------------------------
normal	37.5422	-17.5839
gamma	106.2534	-62.3520
Weibull	106.2661	-62.3519
exponential	106.2738	-62.3519
--------------------------------------------------------------------------A. retroflexus--------------------------------------------------------------------------
normal	30.5346	-13.3485
gamma	106.9087	-63.3397
Weibull	106.9238	-63.3389
exponential	106.9417	-63.3391

In the frequentist approach, it is common to verify whether the residuals approximately follow the normal distribution as part of model diagnostics. In several studies, residual analysis has indicated behavior consistent with normality, as observed in the works of AZARIAS et al. (2023), MELLO et al. (2022), and SILVA et al. (2021). However, it is important to note that in studies with small sample sizes, the usual diagnostic procedures have limited power to detect moderate deviations from normality. Consequently, different distributions assumed for the likelihood function may produce similar fits, which may explain the good performance of models that assume a normal distribution for the likelihood when compared with the other distributions considered.

Figures 2 and figure 3 show the prior (in blue) and posterior (in red) densities of the logistic model fitted with normal likelihood function for the dose-response data of the species A. hybridus and A. retroflexus, respectively. One of the main advantages of Bayesian inference is precisely the obtaining of these probability distributions, which reflect knowledge before and after data analysis (PEREIRA et al., 2022; SILVA et al., 2022; AZARIAS et al., 2025; AZARIAS et al., 2026). The posterior densities are more concentrated around the mean, compared to the prior densities, suggesting that the combination of the prior distribution and the likelihood function of the data resulted in less uncertainty in the parameter estimation.

Figure 2
Prior density (in blue) and posterior density (in red) of the logistic model considering the likelihood function with normal distribution for the dose-response data of the weed A. hybridus.

Figure 3
Prior density (in blue) and posterior density (in red) of the logistic model considering the likelihood function with normal distribution for the dose-response data of the weed A. retroflexus.

CONCLUSION

According to the selection criteria used, the logistic model with a normally distributed likelihood function best describes the dose-response data for the species A. hybridus and A. retroflexus. Nevertheless, for both species evaluated, both the likelihood function with normal distribution and the other distributions used are adequate to describe the data. Although the normally distributed likelihood presented smaller standard deviations and HPD intervals with smaller amplitude, the overlap of the HPD intervals suggests that other approaches could be considered for describing the dose-response data.

The specie A. hybridus required approximately 0.30 g ha^-¹ of the herbicide to achieve 50% control, whereas A. retroflexus required about 0.20 g ha^-¹, indicating greater susceptibility of the latter species to the product. The estimates of parameter a were close to 99% for both species, suggesting a high maximum level of control. Parameter c, associated with the slope of the dose-response curve, showed similar values between species, indicating comparable rates of change in control as the herbicide dose increased.

ACKNOWLEDGMENTS

The authors thank Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) (141799/2023-4) and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) for their financial support (001).

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• CR-2025-0609.R1
• DATA AVAILABILITY STATEMENT
  The data sets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
• DECLARATION OF USE OFARTIFICIAL INTELLIGENCE
  The authors did not employ any artificial intelligence in this study.

Edited by

• ASSOCIATE EDITOR:
  Alessandro Dal’Col Lúcio (0000-0003-0761-4200)

• SCIENTIFIC EDITOR:
  Alberto Cargnelutti Filho (0000-0002-8608-9960)

Data availability

The data sets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Publication Dates

• Publication in this collection
  14 Sept 2026
• Date of issue
  2026

History

• Received
  25 Nov 2025
• Accepted
  11 May 2026
• Reviewed
  25 June 2026