Comparison of Frequentist and Bayesian Generalized Linear Models for Analyzing the Effects of Fungicide Treatments on the Growth and Mortality of Piper Nigrum

Abstract

Foot rot disease poses a devastating threat to pepper agriculture. In Vietnam, conventionally, fungicides are the control methods used against the disease. However, the practical effectiveness of fungicide treatment has yet to be quantitatively assessed. To fill this gap a three-factorial experiment was conducted, the factors of which were fungicide application, soil type, and infection pathway, with plant mortality and plant growth as the target variables. Two of the most common fungicides were chosen, including Agrifos 400 (potassium phosphonate) and Aliette 800WG (fosetyl-Al). The two fungicides were used in multiple treatment plans, with soil drenching selected as the means of controlling foot rot disease on red basalt soil and red basalt soil added with organic matter in a greenhouse experiment. Three-month-old pepper (Piper nigrum) plants were treated with Agrifos (application interval of 10 and 20 days), Aliette (application interval of 30 and 60 days), and a combination of both fungicides at half doses. Pepper plants were infected with the fungus Pythium spp. from soil or by direct inoculation. To assess the effect of fungicides on foot rot mortality and the growth of pepper plants, multiple generalized linear models were set up using frequentist and Bayesian approaches. Generally, both procedures suggest the same conclusions for model selection in terms of the Akaike information criterion (AIC) and the deviance information criterion (DIC). Fungicide type was found to be the main factor that affected the survival of plants. Most of the treatments (except Aliette, with an interval of 60 days) effectively reduced the mortality. The usage of fungicides affected the growth of plants in interaction with soil types. Aliette suppressed the growth of plants, especially on red soil, while Agrifos had no effect on the growth of pepper plants. The combined application of Agrifos and Aliette with half doses proves to be a promising solution for balancing cost and effectiveness in protecting plants against foot rot pathogens without affecting their growth. In our case, under the inhomogeneity of variance and unbalanced samples, the Bayesian inference appeared to be the most useful because of its flexibility in terms of model structure.

1. Introduction

Black pepper (Piper nigrum L.) is a valuable perennial crop grown in the tropics and it is widely used as a spice around the world [1]. The production of black pepper is threatened by foot rot disease, also known as sudden wilt, which is one of the most common fungal-caused diseases [2,3]. Foot rot disease can have a devastating impact on crop yields, causing losses of 40–50% [4] which translate into severe economic losses, especially in Southeast Asia, including Vietnam—one of the world’s major pepper producers [1,5]. The symptoms of the disease are wilting and rapid defoliation, and these are certainly followed by the death of the infected plant [6]. Several surveys conducted in Vietnam discovered that foot rot occurrence was widespread and presented an alarming problem that can sweep away substantial areas of pepper plantations [7,8]. Therefore, promising new treatments were introduced, including biocontrol agents [2,9,10] and seed treatment with chemicals and bio-fungicides which can protect the seeds and enhance germination [11]. Agroforestry patterns were also found to offer a way of building resilience against foot rot disease [12], as was pre-planting soil treatment with sodium chloride to prevent the infectious spread of foot rot disease [13]. In Vietnam, however, the new revolution of bacterial antagonists is a long way from finding practical application. Currently, the popular disease controls implemented by farmers are fungicides and conventional cultivation practices, such as crop rotation and soil amendments [14]. For chemical fungicides, it has previously been observed that chemical fungicides and agricultural practices, though dated, are very effective methods in controlling fungal disease. Potassium phosphonate was found to be effective in protecting pepper plants from disease by soil drenching and inoculate treatments [14]. The experiment of Türkölmez and DerviŞ [15] showed that fosetyl-Al, metalaxy-M+ mancozeb, and phosphorous acid significantly reduced root rot incidence in apricot (Prunus armeniaca L.) and cherry (Prunus avium L.). However, long-term and overuse of fungicides was said to be one of the reasons that caused soil degradation and fungicide resistance [16,17,18].

Thus, on the one hand, it is necessary to have a high production of pepper, and on the other hand, it is necessary to protect the quality of agricultural areas. Hence, care should be taken to alternate the conventional chemical fungicide practices and re-evaluate their effectiveness in defending against the quick wilt pathogen after long-term usage.

Statistical methods have been applied widely to unravel the effect of examined factors. They play a vital role in the inference and interpretation of many processes [19]. Besides the classical statistical approach, recently, there has been a surge in Bayesian applications to evaluate processes [20,21,22,23]. A number of studies on Bayesian inference have identified its advantages compared to classical statistics [22,24,25]. Bayesian methods account for multiple sources of heterogeneity and uncertainty and are consistent with conditional probability logic incorporating available knowledge. The conventional statistical method, on the other hand, is at the risk of misinterpretation and neglects the uncertainty of estimations [26,27]. With advances in computational statistics, Bayesian hierarchical models are more flexible and can adapt to the high uncertainty of ecological processes, while accounting for the stochasticity that was ignored by deterministic models and experimental designs [28,29,30]. The Bayesian inference of hierarchical models is a potential alternative approach that has been applied to evaluate ecological models, such as fisheries [31] and forestry [32]. For the mortality disease caused by fungal disease, the frequentist and Bayesian approaches were employed and compared through the estimation of meta-analysis generic models in the work of Lázaro, et al. [33].

Although we have noted the aforementioned works, to our knowledge, there are no studies on the effect of treatments involving different fungicides and their combinations on the growth and mortality of pepper plants under different soil types and infectious pathways. To fill this gap, the objectives of our study are:

• An evaluation of the effectiveness of fungicide application plans involving the fungicides Aliette and Agrifos and combinations thereof on mortality due to quick wilt pathogens of pepper plants under different soil types and via different infectious pathways.

• An assessment of the impact of fungicide treatment on plant height increments.

• A comparison of the inference results of Bayesian and frequentist approaches.

• The issuing of treatment recommendations.

2. Materials and Methods

2.1. Experimental Design

Our experimental setup was based on treatment plans that were used in practice by farmers in Vietnam. The multifactorial experimental design consisted of:

• Fungicide treatments involving single fungicide and combinations thereof and timing of applications (F₁ to F₆);

• Soil type (S₁, S₂);

• Infectious pathway (I₁, I₂).

Agrifos 400 and Aliette 800WG are the two popular fungicides that are used by farmers in Vietnam. The former contains potassium phosphonate, the latter contains fosetyl-Al. Both of the fungicides provide phosphonate as a hydrolyzed product which is an effective antifungal agent [34]. Agrifos releases potassium in the soil along with phosphonate and it is believed to provide substantial nutrients to the soil. The price of Agrifos is higher than that of Aliette, which releases aluminum cations into the soil when being used. Multiple combinations of fungicides and application timings were investigated. Agrifos (1000 ppm) application time intervals were 10 days (F₁) and 20 days (F₄, double usage period from the user manual). The applications of Aliette (2500 ppm) with treatment intervals of 30 and 60 days were annotated by F₂ and F₅, respectively. Additionally, a group of treatments applying both fungicides with a half dose of each was notated as F₃. The baseline treatments did not use any of the fungicides and were denoted by F₆. We used two soil types, including the original red basalt soil from a pepper region and the soil added with organic matter (here notated as red soil S₁ and mixed soil S₂, respectively). The peppers were infected via two pathways including infected soil and direct inoculation. The two pathways were hereafter referred to as I₁ and I₂, respectively. Isolated Pythium spp. was obtained from infected field samples (pepper roots, leaves, branches). At the I₁ treatments, 50 mL of fungi-contaminated solution was sprayed into the soil while 10 mL was injected into the wounded stems of pepper plants using syringes at the I₂ treatments. The summary of notations can be found in the nomenclature.

The experiment was conducted in a greenhouse with a total number of 554 healthy 3-month-old pepper trees assigned to 24 treatments for two months (from 1 November to 29 December 2019). The trees were planted individually in plastic pots to isolate them from infection due to water flow and cross infection. The height increment of each tree was measured by subtracting the post- and pre-treatment height, and the number of dead trees in each treatment was also recorded.

2.2. The Generalized Linear Model

The effects of experimental factors (types of fungicide treatment, types of soil, and infectious pathways) on height increment and mortality were analyzed by the generalized linear model framework (GLM) [35]. The GLM defines the effect of predictors as the weighted sum of predictor variables and their effects will be estimated as a deflection from a reference class. The linear relationship of predictor variables is related to the response variables via link functions. The general formula of the linear model is presented by Equation (1):

(1)$\begin{array}{c}g\left({\mu }_i\right)={\beta }_0+{\beta }_{F_j}\cdot X_{F_{\mathrm{j}}\left[i\right]}+{\beta }_{S_{\mathrm{k}}}·X_{S_k\left[i\right]}+{\mathsf{\beta }}_{I_{\mathrm{l}}}·X_{I_{\mathrm{l}}\left[i\right]}+{\beta }_{F_{\mathrm{j}},S_k}·X_{F_{\mathrm{j}}\left[i\right]}·X_{S_k\left[i\right]}+{\beta }_{S_{\mathrm{k}},I_{\mathrm{l}}}·X_{S_{\mathrm{k}}\left[i\right]}\\ ·X_{I_{\mathrm{l}}\left[i\right]}+{\beta }_{F_j,I_{\mathrm{l}}}·X_{F_{\mathrm{j}}\left[i\right]}·X_{I_{\mathrm{l}}\left[i\right]}+\\ {\beta }_{F_{\mathrm{j}},S_k,I_l}·X_{F_{\mathrm{j}}\left[i\right]}·X_{S_k\left[i\right]}·X_{I_l\left[i\right]}\end{array}$

where:$i$ denotes tree number, ${\mu }_i$ denotes the expectation of the response variable (tree height increment or mortality), $F_{\mathrm{j}} \left(j=1,.., 6\right)$ denote fungicide treatments, $S_{\mathrm{k}} \left(k=1, 2\right)$ are soil types, and $I_{\mathrm{l}} \left(l=1, 2\right)$ are infectious pathways. ${\beta }_0$ is the intercept, ${\beta }_{F_j}, {\beta }_{S_k }and {\beta }_{I_l}$ are coefficients of the factors of fungicide applications $F_j$, soil types $S_{\mathrm{k}}$ and infectious pathways $I_{\mathrm{l}}$. ${\beta }_{F_{\mathrm{j}},S_k}$,${\beta }_{S_{\mathrm{k}},I_{\mathrm{l}}}$,${\beta }_{F_j,I_{\mathrm{l}}}{}_{ }$ are the two-way interaction coefficients between the two factors, and ${\beta }_{F_{\mathrm{j}},S_k,I_l}{}_{ }$ are the three-way interaction coefficients between the three factors. $X_{F_{\mathrm{j}}\left[i\right]}, X_{S_k\left[i\right]}, X_{I_{\mathrm{l}}\left[i\right]}$ are dummy variables (from the factors of fungicide types, soil types, and infection pathways). $g\left({\mu }_i\right)$ is the link function and it is substituted by the identity function when applied to the data on height increments. The application of the identity function is quite popular and is applied in many studies (for example, [36,37,38,39,40]. In the case of mortality data, $g\left({\mu }_i\right)$ was replaced by the logit function. The logit function was usually applied in the conventional logistic regression analysis and is suited for the case of mortality analysis [33,41].

We aim to detect the effects of influencing factors (fungicide types, soil types, and infection pathways) on the growth and the mortality of pepper plants (analogous to conventional ANOVA). Thus, we explored step-by-step all the factors in the model, from single main factors to the interaction of the factors (as expressed in Table 1). The three factors no fungicide application (F₆), mixed soil (S₂), and direct inoculation (I₂) were used as the references for comparison with the other treatment effects. There were in total 15 candidate models comprising nine models for height increment analysis and six models for mortality analysis. All models are labeled and summarized in Table 1.

2.3. Model Parameter Estimation in Frequentist Approach

The frequentist approach considers parameters of the model as unknown fixed values. The parameters of frequentist models, as shown in Table 1, were estimated using an iteratively reweighted least square fitting method provided by the function glm() in R software version 4.0.4. The Akaike information criterion (AIC) [42] was used to compare the performance of models. A smaller AIC value indicates a better model. The use of AIC for the comparison of model performance is popular; see, for example, the works of Cai, et al. [43],Krainski, et al. [44],Yano, et al. [45].

In the analysis of tree mortality, an odds ratio was used to interpret logistic model coefficients. This represents the ratio of the odds that mortality will occur given the presence of different levels of the predictor variables against the control group ($F_6$). The odds ratio of mortality comparing to the fungicide $F_6$ is expressed by Equation (2):

(2)$O{R}_{F_j}=\exp \left({\beta }_{F_j}\right)$

where ${\beta }_{F_j}$ with $j=1,\dots ,5$ are the coefficients of fungicide factor (F) in Equation (1).

2.4. Model Parameter Estimation in Bayesian Approach

The Bayesian approach, on the other hand, treats model parameters as random variables and assigns their probability distributions [21,46]. Within the Bayesian framework, the linear combination of predictors in Equation (1) does not directly equal the exact value of the predicted variable. Instead, it expresses the central tendency of the probability distribution of $\mu$. In the case of height increment analysis, ${\mu }_i$ (in Equation (1)) is an independent realization from the normal distribution with the variance $\tau$ that governs the spread of the distribution. The assumption of a normal distribution for a population (in this case plant height increment data) is well known and has been used in many researches (see, for example, the works of Hao, et al. [36],McCarthy [37],Pakbaz, et al. [38],Robinson, et al. [39],Welicky, et al. [40]. If the same $\tau$ is used across treatments, it will be equivalent to the assumption of balanced variance, which is assumed in models HB1, HB2, and HB3 (Table 1). Additionally, models HB1Het, HB2Het, and HB3Het in Table 1 take into account the inhomogeneity of variances by allocating different $\tau$ parameters to the treatments.

For the mortality analysis, the number of dead trees observed is independently drawn from a binomial distribution parameterized by the number of trials (n) and the probability of mortality, as expressed by Equation (1) with a logit linking function.

In this case, the interdependence of parameters in Equation (1) is the characteristic of the hierarchical model, with relationships between multilayer parameters and levels [47]. In a hierarchical model, the hyperparameters are assigned by the hyperprior distributions that reflect the expected distribution of parameters before the observation of data. The parameter distribution in one layer governs the distribution of the other layers [48]. Therefore, the parameters coexist in a joint parameter space. The Bayesian inference applies Bayes’ theorem to the joint parameter space and this process involves the calculation of multi-dimensional integrals for the denominator, which was the main barrier keeping complex models from having real-world applications [21].

Thus, the numerical approximation methods, such as the Markov chain Monte Carlo (MCMC) method [49,50], was used to generate samplers that investigate the joint parameter space. The JAGS software (version 4.3.0) which employs Gibbs sampling to approximate the distribution [51] was applied. All MCMC simulations were run with three chains comprised of 50,000 burn-in steps, followed by 200,000 iterations. The convergence and representative of chains were carefully investigated by graphical presentations with trace plots and autocorrelation plots with the effective sample size (ESS) values greater than 10,000 in order to ensure an effective exploration in the posterior distribution [47]. After MCMC simulations, the distribution and uncertainty of estimations for parameters were summarized by the median and 95% highest density intervals (HDIs) [52].

The uninformative normal distribution $N\left(0,{10}^{-6}\right)$ was used as the prior distribution for all coefficients in Equation (1). For the parameters ${\tau }_j,$ a vague gamma distribution $Gam({10}^{-3},{10}^{-3}$) was used as the prior.

The deviance information criterion (DIC) [53] was used to compare the relative performance of models. Analogous to the AIC of the frequentist counterpart, DIC evaluates goodness of fit while preventing overfitting, and smaller DIC values correspond to a better model. The models with DIC values of 2–4 units of the minimum DIC are considered as possible candidates in practice [48].

2.5. Residual Analysis

In order to evaluate the formulated candidate models, it is necessary to check the assumptions of the empirical model. The analysis of residuals was performed for both frequentist and Bayesian models. The results of this analysis are shown in Appendix B. All of the standardized residuals lined up diagonally with a neglectable deviation at the tail, which indicates that the residual of both models follows a Gaussian distribution, proving that they are reasonable to explain the data.

3. Results

3.1. Descriptive Statistics of Experimental Data

The height increment of each treatment is illustrated in Figure 1A. Two months after the onset of the experiment, differences of height increments are clearly distinguishable.

What stands out in Figure 1A is the monotonic relationship between the mean height increment and the variability. Obviously, the among group variances correlate with the mean values. Figure 1B shows the percentage of mortality in each group, together with the sample sizes. The highest mortality rate occurred under the F₆–S₂–I₁ treatment with a percentage of 17.6% (three deaths in a total of 17 trees). In general, Figure 1B suggests that mortality tended to occur under most of the treatments without fungicides (F₆) while treatments with fungicides was associated with relatively low mortality rates.

Although the original layout of the experiment was balanced, plant losses occurred due to foot rot disease and other causes before the onset of treatments. The highest number of samples was 28 trees in the F₄–S₁–I₁, F₄–S₁–I₂ and F₂–S₁–I₁ treatments, and the smallest number of samples was 12 trees in the F₆–S₁–I₁ and F₆–S₁–I₂ treatments. The unbalance of treatment samples is known to be error-prone in estimation in classical statistics.

3.2. Generalized Linear Models for Growth Data

3.2.1. Model Performance Evaluation

According to the frequentist results, the best model is HF3, which includes the three-way interaction terms with the smallest AIC value of 3337.53, followed by HF2 (3342.33) and HF1 (3377.35). Turning to the Bayesian result, between the equal variance models, HB3 has the smallest DIC (3339.53) compared to HB2 and HB1, with DIC values of 3343.69 and 3377.64, respectively. The values of AIC and DIC for frequentist and Bayesian results are quite similar with the same order of performance ranking between models.

However, the DIC values of models dramatically dropped by introducing heterogeneous variances to the model. HB3Het and HB2Het are both possible candidates for the best model because the DIC values of the two models are small and not sufficiently different. The DIC values of HB3Het and HB2Het are 3271.85 and 3272.12, respectively.

In general, all AIC and DIC values of the models (Table 2) imply that there are interactions between factors because the smallest AIC/DIC values always corresponded to the models which include interactions.

3.2.2. Statistical Modeling Inference Results

Even the best values of AIC and DIC corresponded to the models of three-way interaction. It is shown that, in the HF3 model of the frequentist estimation, the three-way interaction terms (F_j–S₁–I₁, j = 1 to 5) do not have any significant p-values. Similar to the frequentist result, HB3Het with Bayesian estimation suggests the same median of parameters for three-way interaction terms but the highest density intervals (HDI) of these terms do not substantially differ from zero. This implies that both approaches suggest the non-existence of three-way interactions, indicating that the effect of fungicides depends on soil types and infection pathways. The estimated parameters of the two-way and three-way interactions models from frequentist and Bayesian approaches are presented in Appendix A.

The less sophisticated models (HF2 and HB2Het) that contain the two-way interactions are preferable because most of them show significant interactions (p < 0.05). The interactions of F₁–S₁ and F₄–S₁ are found by both frequentist and Bayesian approaches (p < 0.05 for the HF2 and substantially different from zero for the HB2Het). These interactions decrease the height increments of pepper (−5.74 and −5.56 cm in HF2 and about −5.5 in HB2Het, respectively, Table 3). In the HF2 model, most of the interactions of soil and fungicides (F_j–S₁, j = 1 to 5) significantly decrease the height increments except the interaction F₃–S₁ (+ 4.35 cm, but with an insignificant p-value = 0.055), while most of the interactions of fungicides and infection type (F_j–I₁, j = 1 to 5) increase the height increments of pepper but with no significant degree (p > 0.05) except the interaction F₂–I₁. With the Bayesian method in the HB2Het model, the trends of increasing or decreasing plant height increments are similar to the HF2 model, as indicated by median values, but with lower confidences (except the interactions of F₁–S₁ and F₄–S₁).

The significant main effects of both the HF2 and HB2Het models indicate that fungicides inhibit the growth of pepper trees. All main effect coefficients of fungicides (F) are smaller than zero, with significant p-values in frequentist estimation, while some have high uncertainty with Bayesian estimation. The effect of soil type (S) does not significantly differ from zero (p = 0.142) but its interactions with other factors significantly impact the growth of trees.

The difference between the estimation of the HF2 model and the HB2Het model in Table 3 can be explained by the addition of the heterogeneity of variance terms in HB2Het that account for the uncertainty and the unbalance which occurred in the data. The estimated 95% HDI of variances by the treatments are illustrated in Figure 2. It is clear in Figure 2 that the standard deviations between treatments are not equal because multiple credible intervals do not overlap. The F₃–S₁–I₁ treatment and the control group (F₆) have greater variances than the other treatments. This is due to the small number of observations and high within-group variances in the observed samples, as illustrated in Figure 1. This result is particularly important because it shows the significance of including information on heteroscedasticity.

3.2.3. Marginal Mean Estimation and Pairwise Comparison

The differences between the effects of fungicides (F) for each soil type (S) and infection pathway (I) can be analyzed using pairwise comparison by holding other factors (S and I) unchanged. The frequentist (HF2) and Bayesian (HB2Het) results are shown in Figure 3 and Figure 4 for S₁ and S₂, respectively.

For the frequentist result, the left panels of Figure 3 and Figure 4 display the estimated marginal mean of each treatment by the center dot of each bar, while the whiskers indicate the 95% CI. Pairwise comparisons with Bonferroni adjustment were conducted for all pairs of treatments and are indicated by horizontal brackets between treatments. Only significant differences with a p-value < 0.05 are displayed in the plots.

For the Bayesian results, the right panels of Figure 3 and Figure 4 illustrate the HDI of the marginal posterior distribution of each treatment, where the center dots are the median of MCMC samples and the vertical caterpillar represents the 95% HDI. The relative density of posterior distributions is displayed behind each caterpillar. Under the Bayesian framework, making the pairwise comparison is straightforward. The posterior distribution of a comparison is sampled directly from the posterior distribution of each treatment and summarized by the horizontal brackets between treatments. The numbers are the median of difference in the posterior distribution. Only comparisons with substantial values (95% HDI did not overlap zero) are shown in Figure 3 and Figure 4.

The result revealed that for S₁ soil, the combination of fungicides (F₃) and control (F₆) groups have superior height increments compared to the rest, regardless of the infection pathway (I). This is indicated by the significance of the comparison between F₃ and F₆ groups to other treatments. For the S₂ soil, the inhibition of fungicides on tree growth is distinctive because the height increment values in the groups of fungicide treatment are significantly lower than those without fungicides. This inhibitory effect is proven by both approaches for both soil types. Concerning the factors of soil types (S) and infection pathways (I), Bayesian inference suggests a greater number of meaningful differences (31 pairs) than the frequentist method (24 pairs). These differences are probably due to the consideration of variance inhomogeneities in the HB2Het model.

3.3. Generalized Linear Models for Mortality Data

3.3.1. Model Performance Evaluation

According to AIC values (Table 4), the best model is MF1 with an AIC score of 121.82, followed by MF2 and MF3 with AIC values of 128.2 and 136.85, respectively. The Bayesian counterparts show a similar pattern. Model MB1 achieves the smallest DIC values of 60.66, which suggests that it is a better model than MB2 and MB3, with DIC values of 62.48 and 63.85, respectively. Both approaches indicate that the mortality probability depends only on the main effects of the experiment factors.

3.3.2. Statistical Modelling of Inference Results

The coefficients of the MF1 and MB1 models are shown in Table 5. In both approaches, the coefficients of the fungicide effects are negative. With the MF1 model, the coefficients for F₁, F₂, and F₄ have significant negative values of −2.21 (p = 0.046), −2.2 (p = 0.047), and −2.19 (p = 0.048), respectively. A similar pattern was also found in MB1 under the Bayesian framework. It is noted that under Bayesian inference, the effect of F₃ treatment is meaningful with a median of −2.35 (−5.33, −0.084), while it is not meaningful on the frequentist result (Table 5). These results imply that the mortality probability associated with fungicide application is effectively decreased. The effect of the other factors, the type of soil (S) and infection pathway (I), are not significant (Table 5).

The frequentist approach to mortality probability cannot be directly interpreted. Thus, an odds ratio (OR) was used. The MF1 frequentist model in Table 5 shows that the application of fungicides F₁, F₂, and F₄ decreased the odds ratio (OR) of mortality by a multiplier of OR_F1 = exp(−2.213) = 0.109; OR_F2 is 0.11 and OR_F4 is 0.11. This result indicates that the usage of F₁, F₂, and F₄ supports a reduction of odds of mortality compared to the control treatment (F₆) (indicated by the odds ratio < 1).

Figure 5 shows the mortality probability of fungicide treatments (F) for S₁ and I₁ by Bayesian inference. From the posterior distribution of the parameters, the marginal posterior distribution of mortality probability by fungicides can be directly sampled and compared.

In Figure 5A, the highest mortality probability occurs under no fungicide treatment (F₆), with a median of 0.066 and 95% HDI from 0.015 to 0.197. Compared to the no fungicide treatment, all other treatments (except F₅) effectively decrease the mortality probability to <0.005. The comparison of the effectiveness can be seen in Figure 5B with the non-overlapping zero pairwise comparisons posteriors with 95% HDI.

4. Discussion and Conclusions

In this study, we conducted an experiment on a total of 554 pepper trees to investigate the effects of fungicides on pepper growth and mortality caused by the foot rot pathogen. By investigating 15 candidate models, the results reveal the advantages and disadvantages of frequentist and Bayesian approaches within the GLM framework. The results also emphasize the effectiveness of fungicide usage against foot rot disease. However, plant growth rates are negatively affected by fungicide treatments depending on the combination of soil type and infection pathway.

Concerning the question of the usage of fungicides to prevent tree mortality, the estimated models indicate strong evidence that fungicide usage is an effective solution for managing the devastating destruction caused by foot rot pathogens, as was suggested by Erwin and Ribeiro [4]. We found that both fungicides effectively reduce the mortality rate. The application of F₁, F₂, and F₄ significantly decreased the mortality probability compared to the control treatment (Figure 5). The reduction of the mortality rate from fungal disease is achieved by the impact of phosphonate, which is one primary product from hydrolyzed Aliette and Agrifos [54]. Phosphonate is proven to be the key antifungal defense against foot rot disease [34]. The ingredients of Aliette include fosetyl-Al, which inhibits the growth of lesions and suppresses mycelial growth [55]. Similarly, fosetyl-Al was found to be effective on chile pepper plants [56] and citrus [57] in preventing plant mortality and increasing survival compared to the other plants which were not applied with the fungicide. Doubling the application period of Aliette to 60 days in our experiment (F₅) significantly decreased the effectiveness (Table 5). On the other hand, applying Aliette with a low concentration also reduced the effectiveness, as was reported by Thomas and Naik [58] in an in vitro environment. Agrifos, on the other hand, kept its effectiveness even with double application timing (20 days) in our F₄ treatment (Table 5); OR_F1/F4 by Equation (2) is 0.986. The application of Agrifos fungicide by soil drenching is proven to be effective in protecting pepper from foot rot. Our result is in agreement with the results of other authors [14,59]. Comparison of Agrifos (F₁) with Aliette (F₂) showed that both fungicides were equally effective, while Thomas and Naik [58] concluded that Aliette had lower fungi inhibition. This is presumably because in our treatments we used the concentration of 2500 ppm, which was higher compared to the 1000 ppm in the experiment of Thomas and Naik [58]. Another important finding is that the combined usage of both fungicides with half concentration (F₃) showed remarkable effectiveness in preventing mortality from foot rot pathogens, as shown in Table 5. This result encourages an alternative treatment plan that reduces the amount of fungicide residual in the soil, while protecting pepper plants from foot rot disease.

Although the usage of fungicide significantly increased the survival chance of pepper plants, the effect of fungicide on plant growth is negative (Table 3). It is encouraging to relate our results with the work of Förster, et al. [60], who found that treating tomatoes with phosphonate significantly reduced plant growth despite the fact that disease incidence was significantly reduced. In accordance with the present results, the experiment of [61] on chili pepper (Capsicum annuum L.) demonstrated that inoculated plants treated with a synthetic fungicide (Metalaxyl-M) had lower average heights. Ouimette and Coffey [54] explained that phosphonate and phosphate, despite having quite similar chemical formula, are in fact different, and the phosphonate produced by fungicides did not enhance growth response in plants, which is in line with the result of MacIntire, et al. [62]. Moreover, phosphonate is metabolically stable and is not converted into phosphate by the plants [54,63]. Possibly, phosphonate might be transformed to phosphate by microbial activity under certain conditions [64], and with sufficient phosphate concentration should stimulate the growth of pepper [65], but that process is slow and makes no practical contribution to phosphate enrichment [66]. This finding supports previous studies on other plants that phosphonate can be inhibitory to the growth of plants, such as the growth of seedlings of Arabidopsis [67] and tomato (Lycopersicon esculentum) [68]. Regarding Agrifos, besides phosphonate, K is another product that was found to stimulate plant growth [69]. However, the potassium from Agrifos did not result in a significant height difference compared to the Aliette group, as shown in our experiment. It is possible that the concentration of K was limited, which was suggested to be ineffective on the growth of chili pepper (Capsicum annuum Avicolare) and bell pepper (California Wonder) [70]. In addition to phosphonate, Aliette releases aluminum into the soil, and under low soil pH conditions (like the red soil S₁ in our experiment) Al can be soluble and may inhibit the growth and metabolic functions of plants [71,72]. Since there are inhibitors of plant growth from the breakdown products of both fungicides, phosphonate and Al, the overuse of these fungicides may affect plant growth and leave residues in the soil environment.

The results of this study suggest that the combined usage of both fungicides with a reducing dose (F₃) in soil S₁ (natural red soil) can effectively protect the pepper plant from foot rot disease (Figure 3), which is indicated by mortality analysis (Figure 5). The treatment F₃ is a promising potential fungicide application scheme since it helps reduce toxicity from fungicide residuals and enhances pepper growth. In addition, by applying the F₃ treatment, the farmers can reduce the cost of buying fungicides.

Our results indicate that the evaluations using both inferential methodologies yield similar practical advice. The values of AIC and DIC on frequentist and Bayesian models do not differ much. Given the similarity in the underlying model structures for frequentist and Bayesian approaches, the results are almost identical. This behavior is expected and in agreement with the conclusion of previous studies [24,33,73,74,75,76,77]. However, each approach has its pros and cons.

For the height increment data, AIC and DIC values suggest the existence of three-way (with no significant p-value) and two-way interactions, while only the main effects are suggested for the mortality data. A similar conclusion was drawn from the work of Zhang, et al. [77] when fitting the linear Gaussian model with tree biomass data. The narrower 95% CI intervals of frequentist estimation compared to the 95% HDI intervals of Bayesian estimation are attributed to the number of sources of uncertainty taken into account by the latter. Similar patterns were also obtained by other publications [24,78,79]. For the pairwise comparison analysis, the marginal estimation from the Bayesian model shows more substantial comparisons than frequentist results (Figure 3 and Figure 4), which is probably due to the violation of the assumption of homoscedasticity in the frequentist estimation. This highlights the key strength of the Bayesian hierarchical model with its flexibility to adapt to the data structure [74]. Bayesian approaches are more straightforward in addressing uncertainty. They do not differentiate between random and systematic effects, providing a better adaptation to the practical problems [80]. Posterior distributions can be characterized in terms of probability. However, Bayesian inference needs a prior specification for each parameter. It could be a weakness of this approach which may lead to different posterior distributions affecting the robustness of the inference. In addition, the computational demand of Bayesian models can be a problem when dealing with complicated and large-scale processes. It has been suggested to use a Bayesian approach along with the frequentist approach to find the best solution for a problem [81], since frequentist models are easy to construct and can be investigated with a variety of available statistical software packages, while Bayesian methods have an advantage in their inference results.

In conclusion, we have analyzed the effects of fungicide usage on pepper growth and mortality with multiple treatment strategies, soil types, and infection pathways. The results suggest an alternative spray program by combining the application of two fungicides, which reduces the mortality without the side effect of growth retardation. The extension of this work should be the conduction of numerous field trials to explore the wide range of cropping systems and plant diseases.

Author Contributions

Conceptualization, A.N.; data curation, B.V.Q.L. and T.T.N.; formal analysis, B.V.Q.L.; methodology, B.V.Q.L. and A.N.; supervision, A.N.; validation, O.R.; writing—original draft, B.V.Q.L.; writing—review and editing, A.N. and O.R. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by Vietnam National University Ho Chi Minh City (VNU-HCM) under grant number B2019-24-02.

Acknowledgments

This research is funded by Vietnam National University Ho Chi Minh City (VNU-HCM) under grant number B2019-24-02.

Conflicts of Interest

The authors declare that they have no known competing financial interest or personal relationships that could have appeared to influence the work reported in this paper.

Nomenclature

Symbol	Description
F	The factor of fungicide types
F 1	Usage of Agrifos 400 with application interval following the user manual (10 days)
F 2	Usage of Aliette 800WG with application interval following the user manual (30 days)
F 3	The combined usage of Agrifos 400 with Aliette 800WG with half concentration and each fungicide following the application interval according to the manufacturer
F 4	Usage of Agrifos 400 with double application interval (20 days)
F 5	Usage of Aliette 800WG with double application interval (60 days)
F 6	Control treatment without fungicide usage
I	The factor of fungal infection pathways
I 1	Fungi infected the soil
I 2	Direct inoculation of wounded stem
S	The factor of soil types
S 1	Red soil (Ferralsols)
S 2	Red soil added with organic matter
$${\beta }_0$$	The coefficient of the reference treatment with the factors F6, S2, I2
$$g$$	Link function in the generalized linear models
$$P$$	Distribution of data
$$\tau$$	Parameter of $$P$$ distribution
OR	Odds ratio
AIC	Akaike information criterion
DIC	Deviance information criterion
HDI	Highest density interval
CI	Confidence interval

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Figures and Tables

Figure 1. (A) Height increments of the treatments. (B) Mortality percentage of the treatments with total number of samples in a treatment.

Figure 2. Highest density intervals (95%) of the standard deviation of height increments for each treatment. White dots are the medians of posterior distributions.

Figure 3. (A) Frequentist pairwise comparison results for soil type S1. The white points and whiskers are marginal means and 95% CI; the horizontal brackets represent significant differences with a p-value < 0.05. (B) Bayesian pairwise comparison result for soil type S1. The white points and whiskers are medians of marginal posterior and 95% HDI; the horizontal brackets indicate the substantial differences with the corresponding medians (the numbers above each bracket).

Figure 4. (A) Frequentist pairwise comparison results for soil type S2. The white points and whiskers are marginal means and 95% CI; the horizontal brackets represent significant differences with a p-value < 0.05. (B) Bayesian pairwise comparison result of inference for soil type S2. The white points and whiskers are medians of marginal posterior and 95% HDI; the horizontal brackets indicate the substantial differences with the corresponding medians (the numbers above each bracket).

Figure 5. (A) Mortality probability from fungicide usage in S1 and I1. (B) Pairwise posterior comparison with the control treatment (F6). The horizontal caterpillars represent 95% HDI and the white dots are the medians of posterior distributions. The red line represents the reference of zero difference.

Table 1

Model summary and notations.

Dataset	Model Name	Description
Height increment	Frequentist models
	HF1	Equation (1) with only the main factors, all the interaction terms were set equal to 0.
	HF2	HF1 with the addition of the two-way interaction terms in Equation (1).
	HF3	HF2 with the addition of the three-way interaction terms in Equation (1).
	Bayesian models
	HB1	Equation (1) with only the main factors, all the interaction terms were set equal to 0. Observations follow a normal distribution and the variance between treatments are homogeneous.
	HB2	HB1 with the two-way interaction terms in Equation (1) and the variance between treatments are homogeneous.
	HB3	HB2 with the three-way interaction terms in Equation (1) and the variance between treatments are homogeneous.
	HB1Het	HB1 with the heterogeneity of variances.
	HB2Het	HB2 with the heterogeneity of variances.
	HB3Het	HB3 with the heterogeneity of variances.
Mortality	Frequentist models
	MF1	Equation (1) with only the main factors; all the interaction terms were set equal to 0, $$g\left( \right)$$ is the logit link function.
	MF2	MF1 with two-way interaction terms in Equation (1).
	MF3	MF2 with three-way interaction terms in Equation (1).
	Bayesian models
	MB1	Equation (1) with only the main factors; all the interaction terms were set equal to 0, $$g\left( \right)$$ is the logit link function, observations follow the binomial distribution.
	MB2	MB1 with two-way interaction terms in Equation (1).
	MB3	MB3 with three-way interaction terms in Equation (1).

Table 2

Model comparison for frequentist and Bayesian estimations.

Frequentist Models	AIC	Bayesian Models	DIC
HF1	3377.35	HB1	3377.64
HF2	3342.33	HB2	3343.69
HF3	3337.53	HB3	3339.53
		HB1Het	3289.22
		HB2Het	3272.12
		HB3Het	3271.85

Table 3

Parameter estimation results of the best models HF2 and HB2Het. CI (confidence interval), HDI (highest density interval).

Parameters	HF2 Frequentist Result				HB2Het Bayesian Result
	Estimate	95% CI Low	95% CI High	p-Value < 0.05	Median	95% HDI Low	95% HDI High	95% HDI Does Not Overlap Zero
$${\beta }_0$$	8.74	8.17	9.31	*	13.38	9.38	17.55	*
$${\beta }_{S_1}$$	0.87	−0.27	2.01		2.97	−2.17	8.02
$${\beta }_{F_1}$$	−7.04	−9.21	−4.87	*	−5.02	−9.85	−0.35	*
$${\beta }_{F_2}$$	−5.53	−7.69	−3.37	*	−5.96	−10.64	−1.14	*
$${\beta }_{F_3}$$	−3.79	−6.05	−1.54	*	−8.11	−12.43	−3.73	*
$${\beta }_{F_4}$$	−7.08	−9.24	−4.91	*	−4.57	−9.22	0.02
$${\beta }_{F_5}$$	−7.51	−9.72	−5.30	*	−4.81	−9.47	−0.25	*
$${\beta }_{I_1}$$	−0.41	−1.55	0.72		−2.81	−7.70	2.05
$${\beta }_{F_1,S_1}$$	−5.74	−10.07	−1.40	*	−5.50	−11.06	−0.25	*
$${\beta }_{F_2,S_1}$$	−6.00	−10.32	−1.69	*	−3.55	−9.61	2.15
$${\beta }_{F_3,S_1}$$	4.35	−0.16	8.86		4.59	−1.10	10.51
$${\beta }_{F_4,S_1}$$	−5.56	−9.89	−1.24	*	−5.52	−11.04	−0.19	*
$${\beta }_{F_5,S_1}$$	−5.46	−9.88	−1.05	*	−4.79	−10.13	0.72
$${\beta }_{S_1,I_1}$$	0.45	−1.77	2.67		1.77	−0.42	4.05
$${\beta }_{F_1,I_1}$$	1.19	−3.11	5.50		1.26	−4.00	6.50
$${\beta }_{F_2,I_1}$$	4.90	0.61	9.19	*	2.91	−2.92	8.81
$${\beta }_{F_3,I_1}$$	4.01	−0.47	8.48		4.14	−1.33	9.61
$${\beta }_{F_4,I_1}$$	0.78	−3.52	5.08		0.55	−4.73	5.65
$${\beta }_{F_5,I_1}$$	−0.74	−5.14	3.65		−1.03	−6.39	4.14

Note: * expresses that the p-Value < 0.05.

Table 4

Model comparison criteria result for frequentist and Bayesian estimation.

Models	Description
Frequentist results		AIC
MF1	Logistic model without interaction	121.82
MF2	MF1 added two-way interaction	128.2
MF3	MF2 added three-way interaction	136.85
Bayesian results		DIC
MB1	Logistic model without interaction	60.66
MB2	MB1 added two-way interaction	63.85
MB3	MB2 added three-way interaction	62.48

Table 5

Results of the parameter estimation of models MF1 and MB1.

Parameters	MF1 Frequentist Result				MB1 Bayesian Result
	Estimate	95% CI Low	95% CI High	p-Value < 0.05	Median	95% HDI Low	95% HDI High	95% HDI Did Not Overlap Zero
$${\beta }_0$$	−2.334	−3.628	−1.285	*	−2.458	−3.738	−1.313	*
$${\beta }_{F_1}$$	−2.213	−5.181	−0.351	*	−2.528	−5.472	−0.207	*
$${\beta }_{F_2}$$	−2.205	−5.173	−0.343	*	−2.526	−5.542	−0.281	*
$${\beta }_{F_3}$$	−2.037	−5.005	−0.178		−2.350	−5.333	−0.084	*
$${\beta }_{F_4}$$	−2.199	−5.168	−0.334	*	−2.509	−5.541	−0.257	*
$${\beta }_{F_5}$$	−1.034	−2.654	0.416		−1.094	−2.732	0.463
$${\beta }_I{}_1$$	0.031	−1.156	1.219		0.033	−1.159	1.277
$${\beta }_S{}_1$$	−0.209	−1.454	0.968		−0.229	−1.486	1.004

Note: * expresses that the p-Value < 0.05.

Appendix A. Parameter Estimation Results of the Two-Way and Three-Way Interaction Models

Table A1. Parameter estimation results for HF2 and HB2Het, CI (confidence interval), HDI (Highest density interval). Table A2. Parameter estimation results for HF3 and HB3Het, CI (confidence interval), HDI (Highest density interval).

Table A1

Parameter estimation results for HF2 and HB2Het, CI (confidence interval), HDI (Highest density interval).

Parameters	HF2				HB2Het
	Estimate	95% CI Low	95% CI High	p-Value < 0.05	Median	95% HDI Low	95% HDI High	95% HDI Does Not Overlap Zero
$${\beta }_0$$	8.744	8.1734	9.3146	*	13.37525	9.384302	17.54658	*
$${\beta }_{S_1}$$	0.8661	−0.2739	2.0062		2.967866	−2.17376	8.021664
$${\beta }_{F_1}$$	−7.0434	−9.2126	−4.8742	*	−5.02263	−9.85022	−0.34799	*
$${\beta }_{F_2}$$	−5.5318	−7.6927	−3.3708	*	−5.96315	−10.6444	−1.14421	*
$${\beta }_{F_3}$$	−3.7937	−6.0517	−1.5357	*	−8.10988	−12.4266	−3.73057	*
$${\beta }_{F_4}$$	−7.0767	−9.2405	−4.9128	*	−4.56627	−9.22079	0.024685
$${\beta }_{F_5}$$	−7.5116	−9.7233	−5.3	*	−4.80762	−9.46856	−0.24629	*
$${\beta }_{I_1}$$	−0.4118	−1.5477	0.7242		−2.80754	−7.70217	2.054169
$${\beta }_{F_1,S_1}$$	−5.7357	−10.0672	−1.4042	*	−5.49821	−11.0582	−0.24885	*
$${\beta }_{F_2,S_1}$$	−6.0021	−10.3163	−1.6878	*	−3.54615	−9.60581	2.149822
$${\beta }_{F_3,S_1}$$	4.3484	−0.1603	8.8571		4.589433	−1.10445	10.50893
$${\beta }_{F_4,S_1}$$	−5.5645	−9.8852	−1.2439	*	−5.51641	−11.0358	−0.18744	*
$${\beta }_{F_5,S_1}$$	−5.4623	−9.8782	−1.0465	*	−4.79468	−10.1317	0.721548
$${\beta }_{S_1,I_1}$$	0.4502	−1.7711	2.6714		1.77243	−0.41988	4.048004
$${\beta }_{F_1,I_1}$$	1.1947	−3.1145	5.504		1.261065	−4.00219	6.50492
$${\beta }_{F_2,I_1}$$	4.8995	0.6074	9.1915	*	2.908951	−2.92048	8.808656
$${\beta }_{F_3,I_1}$$	4.0064	−0.4679	8.4808		4.143038	−1.33135	9.609577
$${\beta }_{F_4,I_1}$$	0.7796	−3.5179	5.0771		0.553609	−4.73398	5.653307
$${\beta }_{F_5,I_1}$$	−0.742	−5.1362	3.6521		−1.03361	−6.38761	4.144957

* p-Value < 0.05.

Table A2

Parameter estimation results for HF3 and HB3Het, CI (confidence interval), HDI (Highest density interval).

Parameters	HF3 Frequentist Result				HB3Het Bayesian Result
	Estimate	95% CI Low	95% CI High	p-Value < 0.05	Median	95% HDI Low	95% HDI High	95% HDI Does Not Overlap Zero
$${\beta }_0$$	8.75	8.18	9.31	*	12.86	8.18	17.51	*
$${\beta }_{S_1}$$	0.85	−0.29	1.98		4.26	−3.28	11.51
$${\beta }_{F_1}$$	−7.02	−9.18	−4.87	*	−3.89	−9.95	1.94
$${\beta }_{F_2}$$	−5.53	−7.67	−3.38	*	−6.72	−12.07	−1.51	*
$${\beta }_{F_3}$$	−3.79	−6.03	−1.54	*	−7.67	−12.56	−2.68	*
$${\beta }_{F_4}$$	−7.06	−9.21	−4.91	*	−4.07	−9.53	1.57
$${\beta }_{F_5}$$	−7.50	−9.70	−5.31	*	−2.81	−8.48	2.25
$${\beta }_{I_1}$$	−0.44	−1.57	0.69		−1.83	−7.99	4.63
$${\beta }_{F_1,S_1}$$	−5.72	−10.03	−1.41	*	−7.55	−15.67	0.96
$${\beta }_{F_2,S_1}$$	−5.96	−10.25	−1.66	*	−2.69	−11.07	5.04
$${\beta }_{F_3,S_1}$$	4.39	−0.09	8.88		3.64	−4.80	11.89
$${\beta }_{F_4,S_1}$$	−5.55	−9.84	−1.25	*	−6.79	−14.72	1.48
$${\beta }_{F_5,S_1}$$	−5.42	−9.81	−1.02	*	−8.69	−16.87	−0.68	*
$${\beta }_{S_1,I_1}$$	0.48	−1.79	2.75		−0.69	−10.95	9.37
$${\beta }_{F_1,I_1}$$	1.23	−3.08	5.54		−0.59	−8.18	6.85
$${\beta }_{F_2,I_1}$$	5.07	0.78	9.36	*	8.40	0.22	16.54	*
$${\beta }_{F_3,I_1}$$	4.11	−0.38	8.60		3.39	−3.45	10.46
$${\beta }_{F_4,I_1}$$	0.82	−3.48	5.12		−0.44	−7.57	7.18
$${\beta }_{F_5,I_1}$$	−0.71	−5.10	3.68		−3.90	−10.82	3.07
$${\beta }_{F_1,S_1,I_1}$$	3.62	−5.00	12.24		3.64	−7.78	14.91
$${\beta }_{F_2,S_1,I_1}$$	−6.63	−15.21	1.96		−6.61	−18.16	6.05
$${\beta }_{F_3,S_1,I_1}$$	1.42	−7.55	10.39		1.40	−10.71	13.38
$${\beta }_{F_4,S_1,I_1}$$	2.42	−6.17	11.02		2.45	−8.54	13.72
$${\beta }_{F_5,S_1,I_1}$$	6.39	−2.40	15.17		6.44	−3.92	17.99

* p-Value < 0.05.

Appendix B

Models Residual - In the case of growth data, the identity link function was used and the population distribution was assumed to follow a Normal distribution. The residuals should be distributed as a Normal distribution [82].

For the Frequentist approach, the Q-Q plot of standardized residuals for the HF2 and HF3 models are illustrated in Figure A1 and Figure A2. Visually examination shows that all of the standardized residuals lined up diagonally with a neglectable deviation at the tail, which indicates that the residual of both models follows a Gaussian distribution. Therefore the proposed models reasonably explain the data.

Figure A1. Q-Q plot for HF2 residuals.

Figure A2. Q-Q plot for HF3 residuals.

For the Bayesian approach, the medians of the residuals were summarized in the Q-Q plot of the standardized median of residuals for the HB2Het and HB3Het models as illustrated by Figure A3 and Figure A4. As can be visually seen in Figure A3 and Figure A4, all standardized residuals follow the diagonal line of normal theoretical distribution with a slight deviation at the tail. These result also implies that the proposed models adequately explain the response variable.

Figure A3. Q-Q plot for HB2Het residuals.

Figure A4. Q-Q plot for HB3Het residual.

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