Postharvest Disease Control Experiments: Challenges on Statistical Methodologies

Abstract

In postharvest disease (PHD) control experiments, treatments with the same observed values for all their replicates are frequently present, which leads to one or more treatments with zero observed variance—this type of data are referred to as data with zero-variance patterns. Such data patterns clearly violate the assumptions of classical parametric statistical models. These patterns are frequently overlooked in phytopathological studies, thus resulting in inadequate inference. In vivo experiments evaluating postharvest disease (PHD) control in fruits involve diverse response variables describing disease progress over time, including binary, discrete, and continuous data, which require different probability distributions for appropriate statistical modeling. These experiments typically follow a completely randomized design (CRD), with individual fruits serving as experimental units and being evaluated over a time interval (t, days) defined by the host–pathogen system. Disease progress is commonly quantified by incidence, severity (measured by discrete scales or mean lesion diameter, MLD), and the area under the disease progress curve (AUDPC). In PHD datasets, high/very low treatment efficacy often leads to zero/maximum values across all replications which produces zero-variance patterns, thus rendering the use of general linear models (GLM) unsuitable. Under such conditions, alternative nonparametric approaches or permutation tests are required, including Fisher’s exact test for incidence and tests for nonparametric contrasts for severity or AUDPC. Our objective is to contribute towards the adoption of more adequate statistical inferential methods for analyzing data from PHD experiments.

1. Introduction

The main challenges in adopting the correct statistical methods for analyzing the diversity of disease-related response variables arising from PHD experiments come from the occurrence of scenarios for which the applied treatments have either high or null efficacy in controlling the target disease, thus leading to equal observed values (minimum or maximum) for incidence, severity, and AUDPC in all treatment replicates. These special cases will be referred to as cases presenting a zero-variance pattern. Because of the equal observed values in at least one of the experimental treatments, their respective observed variances are zero, causing computational limitations in the process of model fitting, even for generalized linear models. The model fitting and associated parameter estimation requires the inversion of a matrix containing the estimates of the treatment variances in its diagonal and the covariance between pairs of treatments off the diagonal, the so-called model’s variance–covariance matrix (VarCov) [1]. Whenever there is at least an element of VarCov diagonal equal to zero, that matrix is not invertible, thus impairing model fitting and parameter estimation. Under the hypothesis of homogeneous variances (homoscedasticity) among treatments, the occurrence of zero observe values does not constitute a restriction for the VarCov inversion itself—the elements of the VarCov’s diagonal are equal for all treatments, an estimate of the pooled treatments’ variance (σ²) that will not be zero, unless all the estimated variances are zero for all treatments. However, whenever at least one treatment has zero observed variance and the other ones have a non-null variance estimate, it is highly unlikely that the homoscedasticity hypothesis is not rejected. Conversely, under the scenario of heterogeneous variances, the zero covariance pattern causes computational restrictions for the VarCov inversion, as mentioned before.

In the in vivo PHD experiments for fruit trees, the experimental units (fruits) are randomly allocated to the treatments, thus characterizing a completely randomized design (CRD). The fruits are inoculated with the causal agent of the target disease and evaluated during a t-day interval, which depends on fruit species and the disease. A variety of disease-related responses are commonly observed in these experiments: (a) disease incidence, a binary variable used to estimate the percentage of fruits infected by the studied plant disease; (b) disease severity, quantified via a digfvvv0-screte scale (e.g., 0 to 4) or via mean lesion diameter (MLD, mm) and, (c) AUDPC, the area under the disease progress curve representing the temporal evolution of the disease quantified by either incidence or severity [2,3].

In the absence of these zero-variance patterns, continuous variables like disease severity are measured by mean lesion diameter (MLD) and the summary measure AUDPC or severity, quantified by using an ordinal scale, are treated as continuous and are subjected to analysis of variance followed by multiple comparison of means [1] or regression model (LM) fitting in the case of quantitative treatments [4]. Whenever the hypothesis of homogeneous variances is not rejected via tests such as Levene’s [5] or Brown–Forsythe’s test [6], ordinary linear models (LMs) can be adequately used. Otherwise, general linear models (GLM) that can accommodate heterogeneous variances [1] are required. Conversely, as the previously commented methods are not applicable, rank-based nonparametric tests for contrasts are an adequate alternative option [7,8,9,10].

The disease incidence in each treatment, a binary variable (yes/no visible disease infection), is commonly modeled as a binomial variable—the estimation of incidence and associated uncertainties (standard errors, confidence intervals) as well as the comparison among treatments’ incidence are based on chi-square-type tests derived from weighted least squares or likelihood-based inferential methods [11,12]. Similarly, as in the cases of continuous variables, the presence of at least one treatment with zero or all fruits infected leads to zero-variance patterns that make the use of chi-square-based methods no longer suitable. In these cases, even the generalized linear models developed for dealing with zero-inflated variance data are not suitable. The presence of absolute expected frequencies inferior to five in any cell of the treatment vs. incidence status contingency table also renders the chi-square-based methods inadequate [13]. The alternative is the use of Fisher’s exact tests [14,15] for pairwise comparison of incidence between treatments with possible adoption of p-value adjustment to control the experiment-wise Type I error [16,17].

Here, our objective is to discuss common misuses of statistical methods in the analysis in PHD experimental data and to propose alternative statistical inferential methods adequate for the analysis of data, including zero-variance patterns. Four case studies are presented, including a variety of response variables and special data patterns commonly observed in experiments for controlling postharvest diseases in fruits.

2. Case Studies

In this section, we present four case studies including the types of response variables observed in postharvest disease experiments, as discussed before. The first two cases deal with the PHD incidence in real data. In the first one (Case 2.1.1) [18], the experimental data do not present zero-variance patterns and the analysis follows the traditional binomial-based approach; in the second one (2.1.2) [19], there is a treatment where none of the fruits were infected, resulting in observed zero-variance (data with zero-variance pattern), thus requiring the use of Fisher’s exact test. The two following cases (2.2.1 and 2.2.2) involves two severity-related variables: disease severity at the last evaluation time and the AUDPC, with absence [18] and presence [20] of zero-variance patterns, respectively.

The four case studies presented here come from three experiments [18,19,20] aiming to evaluate clean technologies as an alternative to the use of traditional chemical fungicides, which can lead to the risk of the presence of toxic residues in fruits. All the four experiments followed completely randomized design (CRDs) with one fruit as the experimental unit. The experiments were realized in the Postharvest and Microbiology Laboratories from Embrapa Meio Ambiente, Jaguariúna, SP, Brazil.

The SAS 9.4 and R codes developed to perform the case studies’ statistical analysis are presented in the Supplementary Materials. The experimental datasets are available upon request.

2.1. Binary Variables: Disease Incidence

Differently from continuous variables like disease lesion diameter, fruit mass, protein concentration, or enzymatic activity, the analysis of binary variables requires methods appropriate to their particular nature. A binary variable, as the name indicates, can present only two types of outputs for each experimental unit (e.g., yes/no, male/female, infected/not infected). Whenever independent binary variables are observed in each experimental unit, the natural choice of a probability distribution to be assumed in their statistical modeling is a binomial with two parameters: N_k (number of experimental units in the treatment k) and p (percent of success), denoted by Binomial (N_k,p_k) [11,12]. As the responses in each treatment are also supposed to be mutually independent, the joint distribution to be considered for the whole experiment is a product binomial probability distribution with the vector parameters N’ = (N₁, N₂, …, N_k)’ and p’ = (p₁, p₂, …, p_k)’. For balanced experiments, N₁ = N₂ = …= N_k = j, the fixed constant number of experimental units per treatment [11,12]. However, in some particular cases that commonly occur in PHD experiments, the use of the above-described methodology is no longer suitable, even for binary variables, thus requiring the use of adequate alternative methods that will be addressed later in this work.

In PHD experiments, the binary variable disease incidence is a preliminary way of assessing the efficacy of the disease control treatments along time [2]. Here, for simplicity, our analysis will be restricted to incidence at the last observation time. Each fruit is classified as infected or not infected depending on the presence/absence of visible disease symptoms. These primary data are used to estimate the percentage of infected fruits (incidence) in each treatment. Statistical inferential methods based on binomial distributions are thus required to estimate adequately the incidence in each treatment, their associated uncertainties, and to perform comparisons among treatments [11,12]. However, in some particular cases, tests derived from binomial distributions, such as Wald chi-square tests, are no more adequate even for binomial variables. In this session, we discuss such special situations and present alternative methods suitable for adoption.

Consider an experiment with k treatments (disease control methods) and j replications (fruit)—the disease incidence data can be arranged in a k × 2 table, referred to as a contingency table, presenting the observed absolute frequency of infected/no infected fruits in each treatment. Whenever none of these frequencies is inferior to five, ꭓ²-based tests are adequate to the overall hypotheses of equal incidence in the k treatments. Contrasts between pairs of treatments regarding incidence can be performed after fitting a generalized linear model by maximum likelihood or the weighted least squares method, including methods to deal with zero-inflated variance data. Wald ꭓ² tests are employed to evaluate the significance of each contrast [11]. Conversely, if at least one of the absolute frequencies in the referred contingence table is inferior to five, the ꭓ²-type tests are no more adequate [18]: the alternatives are the Exact Fisher’s test for the global hypothesis of equality among treatments [14,15] and pairwise Exact Fisher’s tests for the contrasts of interest. This second scenario (frequency < 5) includes the cases of zero observed frequency of infected (or no infected) fruits in at least one treatment.

• Case Study 2.1.1: Evaluation of combined application of modulated UV-C radiation and an emulsion of bioactive constituents of oregano oil for the control of anthracnose in papaya.

This experiment aimed to evaluate the effectiveness of the combined application of modulated UV-C radiation and a chitosan-stabilized emulsion of oregano bioactive constituents (

Table 1. Hypothetical incidence data derived from real experimental data of anthracnose incidence in papaya caused by the fungus C. gloeosporioides observed for the following treatments: check, modulated UV-C (0.44 kJ m⁻²), chitosan-based emulsion containing carvacrol, p-cymene, and thymol (5.0 μL mL⁻¹), and the combined treatment UV-C + emulsion [18]. The new data were generated keeping the ratio infected/not infected observed in the real experiment (Silva et al., in preparation), but considering the number of replications equal to 27 fruits.

Treatment	Number of Fruits		Incidence 1 (%)	SE 2 (%)
	Infected	Not Infected
T0: Check	16	2	0.8889 b	0.0741
T1: UV-C radiation	10	8	0.4444 a	0.1111
T2: Emulsion	6	12	0.6667 a	0.1111
T3: Emulsion + UV-C (Combined)	6	12	0.6667	0.0741

¹ Incidence values followed by (a) do not differ from the combined treatment (T₃); the check incidence value, followed by (b), differs from the mean incidence of the other treatments (T₁, T₂ and T₃) according to the Wald chi-square test for contrasts at 0.05 significance level; ² Standard error of the estimated percentage of infected fruit (incidence).

) on the incidence of the fungal postharvest disease anthracnose in papayas, caused by C. gloeosporioides [18]. The binary variable disease incidence is coded as 0 (no visible infection) or 1 (visible infection). The number of fruit per replicate was nine.

The data we analyzed in this case study were derived from the observed data from the experiment previously described in this case study, but considering the number of replications as being 27 fruits instead of nine. This number of replications was used in order to show the application of chi-square-based methods, which is not adequate for cases with at least one expected absolute frequency less than five.

The contrasts indicate that the disease incidence of the combined use of emulsion and UV-C radiation did not improve the effectiveness of anthracnose control in papayas when compared with the use of these isolated components (

Table 2. Estimates of contrasts between incidences or mean incidence of groups of treatments tested for control of the postharvest disease anthracnose in papaya. The treatments were: check, modulated UV-C radiation (UV-C, 0.44 kJ m⁻²), chitosan-based emulsion containing carvacrol, p-cymene, and thymol (emulsion, 5.0 μL mL⁻¹), and the combined treatment UV-C + emulsion (combined). The data used in this case study were generated keeping the ratio infected/not infected observed in the real experiment (Silva et al. in preparation), but considering the number of replications equal to 18 fruits.

Contrast	Estimate	SE 1	Wald Chi-Square	p-Value 2
Emulsion vs. Combined	0.2222	0.1335	2.77	0.0961
UV-C × Combined	−0.2222	0.1614	1.89	0.1687
Check × Others	0.6667	0.0630	112.16	<0.0001

¹ Standard error of the contrast estimate; ² Nominal significance value of the Wald chi-square test used to compare anthracnose incidence between treatments or groups of treatments.

; Wald test, p > 0.0961). However, the disease incidence in fruits submitted to these three treatments, on average, was inferior to the check incidence in about 75%, thus indicating the high effectiveness of the use combined or isolate of the emulsion and the UV-C radiation.

The incidence estimates, respective standard errors and Wald Chi-square tests for contrasts were obtained using the CATMOD Procedure of the SAS/STAT^® [21]. The SAS codes are available in Figure S1.

• Case Study 2.1.2: Evaluation of UV-C radiation modulation frequency on the incidence of sour rot in orange caused by Geotrichum citri-aurantii.

In the UV-C radiation experiment, the total exposure time was kept constant (30 s), and the modulation frequencies tested were 15, 30, or 45 Hz. The accumulated doses also remained constant (0.99 kJ m⁻²). In the continuous irradiation treatment (0 Hz), the accumulated dose was 1.99 kJ m⁻², corresponding to approximately twice that the one generated under modulation. The experiment was carried out using a completely randomized design (CRD) with five treatments and eight replicates (

Table 3. Raw experimental data: the binary variable incidence (infected/not infected) of sour rot in orange caused by Geotrichum citri-aurantii in each experimental unit (fruit) evaluated at thirteen days after disease inoculation (INCID13). The treatments evaluated were control (fruits not irradiated) and UV-C radiation frequencies 0, 15, 30, or 45 Hz [19].

Treatment (UV-C Frequency)	Type of UV-C Radiation	Sour Rot Incidence 1
		Replicate (Fruit) Number
		F1	F2	F3	F4	F5	F6	F7	F8
Check	-	1	1	1	1	0	0	0	1
0 Hz	Continuous	0	0	0	0	0	0	0	0
15 Hz	Modulated	0	1	0	1	0	0	0	1
30 Hz	Modulated	0	1	0	0	0	0	0	1
45 Hz	Modulated	0	0	0	0	0	0	1	0

¹ Data highlighted in red come from treatments for which the observed values of INCID13 (0—not infected; 1—infected) were zero for all replications, leading to zero observed variance for the treatment continuous UV-C radiation (0 Hz).

). The treatments consisted of a check (no radiation) and four UV-C radiation treatments: 0 Hz (continuous radiation) and the modulated radiation frequencies 15, 30, or 45 Hz [19].

None of the oranges (n = 8) submitted to continuous UV-C radiation (frequency 0 Hz) presented sour rot infection (

Table 4. Incidence of sour rot in orange caused by Geotrichum citri-aurantii subjected to different UV-C radiation frequencies, evaluated at thirteen days after treatment inoculation [19].

Treatment (UV-C Frequency)	No. Infected Fruits	No. Not Infected Fruits	Incidence (1) (%)	SE (2)
Check	5	3	62.50	17.10
0 Hz	0	8	0.00 (*)	0.00
15 Hz	3	5	37.50	17.10
30 Hz	2	6	25.00	15.30
45 Hz	1	7	12.50	11.70

⁽¹⁾ Incidence estimates (n = 8) followed by (*) differ from the control incidence according to the lower-tailed Fisher’s exact test for comparing each UV-C treatment with the check at a significance level of 0.05; ⁽²⁾ Standard error of the estimated percentage of infected fruit (incidence).

), leading to a zero sour rot incidence estimate with also zero standard error. The presence of such a data pattern makes the use of chi-square-based methods unsuitable, as discussed before. As our objective is to contrast each treatment incidence with the control incidence, we applied the lower-tailed pairwise Fisher’s exact tests because the control incidence is expected to be greater than the ones in any of the UV-C radiation treatments. The radiation treatments are expected to reduce the disease incidence by impairing the fungal activity—this prior knowledge allows us to use the unilateral version of Fisher’s test, which considerably improves its statistical power. By choosing this test version, the probability of not rejecting the hypothesis of null UV-C influence on the sour rot incidence for scenarios of true treatment efficacy (Type II error) is increased.

The pairwise Fisher’s exact tests were performed by using the NPAR1WAY Procedure (Proc NPAR1WAY) of the statistical software SAS/STAT^® [21]. In this SAS Procedure, unilateral tests (lower- or upper-tailed) are available only for 2 × 2 contingency tables, as is the case for the table UV-C treatment/control versus presence/absence of visible disease symptoms. The SAS codes are available in the Figure S2.

The only treatment, which lead to significant disease incidence reduction, was the application of continuous UV-C radiation on the oranges (Table 2). The estimated incidence was zero, indicating the high efficacy of this type of modulation. The lack of significant evidence of disease reduction for the modulated UV-C treatments (lower-sided Fisher’s exact test, p > 0.05), even for numerical differences of around 37 percentage points in incidence, points to the adoption of a small sample size (n = 8), which negatively affects the statistical power of the adopted test (Table 2).

2.2. Continuous Variables: Mean Lesion Diameter (MLD) and AUDPC

A useful metric to quantify disease severity is the mean lesion diameter (MLD) [7]. To measure this variable, fruits are wounded at the equatorial region, using a stainless-steel needle, and then inoculated with spores of the disease causal agent under investigation. The lesion evolution in terms of mean diameter (measured in two perpendicular directions) is then monitored during a time interval that depends on the host–pathogen system. In the absence of treatments presenting zero-variance patterns, these continuous variables are analyzed via general linear models (GLM). It is necessary to check the assumption of equal variances among treatments (homoscedasticity) to decide on the type of variance–covariance matrix to be considered in the analysis. The F-test of the GLM’s Anova indicates the overall significance of the model. The analysis to be run following the Anova whenever the F-test is significant depends on the type of treatment: (a) unstructured qualitative treatments require pairwise comparison between means that can be performed via tests such as Tukey or Duncan; (b) for qualitative structured treatments, t-tests for contrasts are used to compare pre-defined comparisons between means or group of means [9], and (c) for quantitative treatments, linear or nonlinear models can be fit to represent the relationship between the response variable and the treatment levels [11].

For cases including at least one treatment with continuous variables a presenting zero-variance pattern, GLMs are no longer adequate due to computational constraints related to the estimation of the variance/covariance matrix. Even generalized linear models that can accommodate overdispersion cannot be used because of the problem related to the VarCov matrix inversion. The alternative is the use of nonparametric contrasts for comparisons between pairs or groups of treatments [9,10].

• Case Study 2.2.1: Evaluation of combined application of modulated UV-C radiation and a chitosan-stabilized emulsion of bioactive constituents on the anthracnose severity in papaya.

The experiment followed a CRD with nine treatments and five replications (papayas). The treatments were check (fruits inoculated, not treated; modulated UV-C radiation at 30 Hz for 20 s (dose 0.44 kJ m⁻²), chitosan-based emulsion containing carvacrol, p-cymene, and thymol (5.0 μL mL⁻¹), and the combined treatment UV-C + emulsion (

Table 5. Estimates of the treatment means and respective standard errors (SE) for the response variables anthracnose severity at the 6th day after inoculation in papaya fruits, expressed by the mean lesion diameter at the 6th day after inoculation (MLD6), and the respective area under the disease progress curve (AUDPC). Treatments: check, modulated UV-C (0.44 kJ m⁻²), chitosan-based emulsion containing carvacrol, p-cymene, and thymol (5.0 μL mL⁻¹), and the combined treatment UV-C + emulsion (combined). Source: [18].

Treatment	Response Variable
	MLD6 (mm)		AUDPC (mm·day)
	Mean	SE 1	Mean	SE 1
Check	12.77	1.77	25.94	3.47
Emulsion	4.32	1.77	6.56	3.47
UV-C radiation	7.23	1.77	9.84	3.47
Emulsion + UV-C (Combined)	4.30	1.77	3.30	3.47

¹ The standard errors (SE) of the estimated means are equal for all treatments because the hypothesis of homogeneity of variances was not rejected for none of the response variables.

). The objective was to evaluate the treatment’s effectiveness in the control of anthracnose in papayas, a fungal postharvest disease caused by C. gloesporioides [18]. The severity-related variables analyzed were the disease severity at the 6th day after inoculation, quantified by the mean lesion diameter (MLD6), and the summary measure area under the disease severity progress curve (AUDPC). The raw experimental data are presented in the Table S1.

As both variables are continuous and do not present zero-variance patterns, the first steps were to run an ordinary Anova and check the hypothesis of homogeneous variances via the Levene test. In CRDs, F-tests are not too sensitive to lack of normality. The hypothesis of homoscedasticity was not rejected for MLD6 (Levene test, p = 0.4771) and AUDPC (Levene test, p = 0.3308).

The Anova’s F-tests for the overall hypothesis of no treatment influence on both variables were rejected (SEV6, p = 0.0055; AUDPC, p = 0.0003), thus leading to the need for more tests for discrimination among treatments. The treatment’s structure itself indicates the a priori contrasts to be tested (

Table 6. Estimates of contrasts defined a priori among individual treatment means or groups of means, and respective standard errors (SE) for the response variables anthracnose severity at the 6th day after inoculation (MLD6) in papaya fruits, expressed by the mean lesion diameter at the 6th day after inoculation (MLD6), and the area under the disease progress curve (AUDPC). Treatments: check, modulated UV-C radiation (UV-C, 0.44 kJ m⁻²), chitosan-based emulsion containing carvacrol, p-cymene, and thymol (emulsion, 5.0 μL mL⁻¹), and the combined treatment UV-C + emulsion (combined). Source: [18].

Response Variable	Contrast	Estimate	SE	t-Statistic	p-Value 1
MLD6 (mm)	UV-C vs. Combined	0.01	2.50	0.01	0.4977
	Emulsion vs. Combined	2.93	2.50	1.17	0.1254
	Check vs. Others	7.48	2.04	3.66	0.0005
AUDPC (mm day)	UV-C vs. Combined	3.26	4.91	0.66	0.2558
	Emulsion vs. Combined	6.54	4.91	1.33	0.0964
	Check vs. Others	19.37	4.01	4.83	<0.0001

¹ Nominal significance value of the upper-tailed t-test for contrasts. p-values highlighted in red indicates a contrast significant at the 0.001 level.

). Firstly, we want to know if the anthracnose control in the combined treatment (emulsion + UV-C) is more effective than the control in each of the isolated ones, the emulsion and the UV-C treatments alone. As the severity is expected to be lower in the combined treatment when contrasted with the isolated ones, the differences between the means of the combined and each isolated treatment (UV-C or emulsion) is expected to be positive. This a priori knowledge allows the use of unilateral tests; in this case, this is the upper-tailed version, which is more powerful than the corresponding bilateral test. As the hypothesis of superiority of the combined treatment was not rejected for both variables (Table 6; SEV6, p > 0.1254; AUDPC, p > 0.0964), the next step was to contrast the overall mean of the three non-check treatments with the check mean, aiming to verify if, on average, the treatments combined or isolated are effective in the disease control. Similarly, the difference between the check mean and the other treatment means is also expected to be positive (as in the previous contrasts), leading to the use of upper-tailed t-tests for both variables. The SAS codes developed for running Anova using the Proc GLM and t-tests for contrasts are shown in the Figure S3. To obtain the contrast estimates, respective standard errors, and the results for the respective t-tests, the Proc GLM statement ‘estimate’ was used. By using this statement, the p-values computed are for bilateral t-tests (p_bi). They were later converted to the corresponding ones for unilateral upper-tailed tests (p_uni) via the expression p_uni = p_bi/2.

The results confirmed the superiority of the non-check treatments in controlling the anthracnose in papayas; regardless, UV-C and emulsion were used in a combined or isolated way (Table 6; SEV6, p = 0.0005; AUDPC, p = 0.0002). The mean severity in the non-check treatments was 7.48 mm inferior to the check severity (Table 6; SEV6, p = 0.0005) and the AUDPC mean, and inferior to the check AUDPC by about 7 mm·day⁻¹, which corresponds to reductions of around 60 and 75%, respectively.

• Case Study 2.2.2. Influence of UV-C radiation doses and modulation frequencies on the anthracnose severity in “Suprema” guavas

The experiment followed a CRD with seven treatments and five replications (guavas). The treatments were combinations of two UV-C doses (0.66 kJ m⁻² and 0.99 kJ m⁻²), corresponding to 30 and 45 s exposition times, with three frequency levels (15, 30, and 45 Hz) plus a control treatment (without UV-C application), thus corresponding to a 2 × 3 factorial arrangement with an additional treatment (

Table 7. Experimental data: anthracnose severity at the 6th day after inoculation in “Suprema” guavas expressed by the mean lesion diameter at the 6th day after inoculation (MLD6) and respective area under the disease progress curve (AUDPC). Source: [19].

Severity-Related Metric		Treatment (UV-C)		UV-C Dose (kJ m−2)	Replicate (Fruit)
	Label	Frequency (Hz)	Time (s)		1	2	3	4	5
MLD6 1 (mm)	T0	-	-	0.00	0.20	0.15	0.00	0.10	2.80
	T1	15	30	0.66	0.00	0.00	0.00	1.10	0.00
	T2	15	45	0.99	1.40	0.00	0.00	0.00	0.00
	T3	30	30	0.66	0.00	0.00	0.00	0.00	2.00
	T4	30	45	0.99	0.00	0.00	0.00	0.00	0.00
	T5	45	30	0.66	0.00	0.00	0.00	0.00	0.00
	T6	45	45	0.99	0.00	0.00	1.00	0.00	0.00
AUDPC 1 (mm day)	T0	-	-	0.00	0.50	0.38	0.00	0.25	5.33
	T1	15	30	0.66	0.00	0.00	0.00	1.35	0.00
	T2	15	45	0.99	1.40	0.00	0.00	0.00	0.00
	T3	30	30	0.66	0.00	0.00	0.00	0.00	2.10
	T4	30	45	0.99	0.00	0.00	0.00	0.00	0.00
	T5	45	30	0.66	0.00	0.00	0.00	0.00	0.00
	T6	45	45	0.99	0.00	0.00	1.00	0.00	0.00

¹ The observed values from treatments for which the observed values of MLD6 and AUDPC were zero for all replications are highlighted in red. The observed variance for both response variables in those treatments is zero.

). The objective was to evaluate the treatment’s effectiveness in the control of the postharvest fungal disease anthracnose in guavas [19].

In this case study, we focus on two anthracnose severity-related variables: severity at six days after inoculation (MLD6) expressed as the mean lesion diameter (mm) and the AUDPC corresponding to the area under the curve representing the evolution of anthracnose severity along six days after disease inoculation.

The treatments corresponding to the frequency/exposition time combinations of 30 Hz/45 s and 45 Hz/30 s were highly effective in the control of the target disease, leading to no infected guavas and consequently causing severity and AUDPC equal zero for all treatment’s replications (Table 3). These zero-data for all replicates present in the two referred treatments leaded to zero-variance patterns for both variables (Table 3). In these zero-variance cases, the continuous variables could be both subjected to the KW test for the global hypothesis of no treatment effect. However, as one-sided KW versions are not available for that global hypothesis, we choose to go straight to defining nonparametric contrasts and apply (unilateral) nonparametric tests corresponding to each scientific hypothesis. Using unilateral tests, we take the advantage of their improved power provided by the previous information on the expected decrease in disease severity metrics caused by UV-C radiation in relation to the check.

The nonparametric version of the upper-sided Dunnett’s test [9] was then adopted for comparing the severity metrics between each UV-C treatment and the check. The tests were performed using the R package 4.5.2 nparcomp [10], specifying the contrast matrix by type = ‘userdefined’ (

Table 8. Matrices of contrast coefficients corresponding to the options (a) type = ‘userdefined’ and (b) type = ‘Dunnett’ of the R package nparcomp [9,10] are used to perform nonparametric multiple comparisons of metrics related to the severity of the fungal postharvest disease in Guavas between each UV-C treatment and the check (not irradiated fruits). The matrices were used for both analyses: disease severity at the 6th day after inoculation expressed by the mean lesion diameter (MLD6) and the summary measure area under the disease progress curve (AUDPC).

	Contrast Matrix Options in the R Package Nparcomp 1
(a) Type = ‘userdefined’								(b) Type = ‘Dunnett’
T0	T1	T2	T3	T4	T5	T6		T0	T1	T2	T3	T4	T5	T6
1	−1	0	0	0	0	0		−1	1	0	0	0	0	0
1	0	−1	0	0	0	0		−1	0	1	0	0	0	0
1	0	0	−1	0	0	0		−1	0	0	1	0	0	0
1	0	0	0	−1	0	0		−1	0	0	0	1	0	0
1	0	0	0	0	−1	0		−1	0	0	0	0	1	0
1	0	0	0	0	0	−1		−1	0	0	0	0	0	1

¹ Each matrix line corresponds to the comparison between the check mean (μ₀) and each mean of the six UV-C radiation treatments (μ₁, μ₂, μ₃, μ₄, μ₅, μ₆).

) and the option for the unilateral test by alternative = ‘less’. Alternatively, we could use type = ‘dunnett’ (Table 8) and alternative = ‘greater’—the results are equivalent and the choice of lower- or upper unilateral tests depends on the way the matrix of contrasts is defined. The significance level adopted was 0.05. The R codes developed for this analysis are presented in the Figure S4.

A contrast is any linear combination of the treatment means. A set of contrasts is usually represented by a matrix for which each line corresponds to the coefficients of the respective contrast (Table 8). For example, the comparison between the T₀ (check) mean and the T₁ mean (μ₀-μ₁) is expressed by the linear combination 1μ₀-1μ₁ + 0μ₂ + 0μ₃ + 0μ₄ + 0μ₅ + 0μ₆ whose coefficients will constitute the first line of the matrix presented in Table 8. As the severity for T₀ (check) is expected to be greater than the one for T₁ (check), (μ₀-μ₁) is expected to be positive and, consequently, the correct unilateral t-test will be upper-tailed. The construction of the remaining matrix lines (2 to 6) presented in Table 8 (type = “userdefined”) follows the same logic explained before for the first line.

Conversely, if the symmetrical representation (μ₁-μ₀) is adopted (as in Table 8, type = “Dunnett”), the difference is expected to be negative, and the corresponding unilateral test will be lower-tailed. This second option corresponds to the option type = ‘Dunnett’ in the nparcomp package—in this case, it is not necessary to provide the matrix to run the contrasts, as it is automatically generated when the option type = ‘Dunnett’ is informed.

According to the nonparametric contrasts, only two treatments had the means of both severity-related variables, MLD6 and AUDPC, which significantly different from the respective control means, corresponding to the frequency/exposition time combinations of 30 Hz/45 s and 45 Hz/30 s (

Table 9. Estimates of nonparametric contrasts defined for comparing the check with each one of the UV-C irradiated treatments, regarding two severity metrics: anthracnose severity at the 6th day after inoculation in Suprema guavas, expressed by the mean lesion diameter at the 6th day after inoculation (MLD6), and the area under the disease progress curve (AUDPC). The treatments were combinations of two UV-C doses (0.66 kJ m⁻² and 0.99 kJ m⁻²), corresponding to 30 and 45 s exposition times, with three frequency levels (15, 30, and 45 Hz) plus a control treatment (without UV-C application). Source: [19].

Contrast	MLD6		AUDPC
	Estimate (mm)	p-Value	Estimate (mm Day)	p-Value
Check vs. F15_T30	0.43	0.2650	1.02	0.2900
Check vs. F15_T45	0.37	0.2651	1.01	0.2900
Check vs. F30_T30	0.25	0.2652	0.87	0.2900
Check vs. F30_T45	0.65	0.0247	1.29	0.0272
Check vs. F45_T30	0.65	0.0247	1.29	0.0272
Check vs. F45_T45	0.63	0.0638	1.09	0.2901

p-values highlighted in bold indicates a contrast significant at the 0.01 level.

; MLD6, p = 0.0247; AUDPC, p = 0.0272). None of the fruits in both treatments presented visible infection symptoms. Consequently, the estimated incidence, mean severity and mean AUDPC were zero for these treatments (

Table 10. Estimates of the mean anthracnose severity at the 6th day after inoculation in “Suprema” guavas were expressed by the mean lesion diameter at the 6th day after inoculation (MLD6) and the respective mean area under the disease progress curve (AUDPC). Source: [19].

Treatment (UV-C)			Modulated UV-C Dose (kJ m−2)	Estimated Mean 1,2
Label	Frequency (Hz)	Time (s)		MLD6 (mm)	AUDPC (mm·day)
Check (T0)	-	-	0.00	0.65	1.29
F15_T30 (T1)	15	30	0.66	0.22	0.27
F15_T45 (T2)	15	45	0.99	0.28	0.28
F30_T30 (T3)	30	30	0.66	0.40	0.42
F30_T45 (T4)	30	45	0.99	0.00 *	0.00 *
F45_T30 (T5)	45	30	0.66	0.00 *	0.00 *
F45_T45 (T6)	45	45	0.99	0.02	0.20

¹ Number of replications per treatment (n = 5 guavas); ² Means highlighted in bold correspond to the treatments for which the observed values of MLD6 and AUDPC were zero for all replications leading to zero observed variance patterns for both response variables; * indicates UV-C treatments that differ from check (not irradiated guavas) according to the nonparametric version of the lower-tailed Dunnett test at a 0.05 significance level.

), indicating a high effectiveness in controlling the postharvest anthracnose.

3. Discussion

The type of approaches explored here for analysis of PHD control experimental data are restricted to univariate analysis and represent the simplest cases in terms of statistical methodology. Our objective is to highlight the frequent misuse of well-known methods based on analysis of variance (Anova) associated with ordinary linear models (OLMs) that require normality, independence, and homoscedasticity assumptions. Methods such as OLMs followed by multiple comparisons of treatment means are of widespread use in biological and agronomic studies, even when the referred assumptions are clearly violated.

The use of inadequate inferential statistical tools can lead to erroneous scientific conclusions with practical consequences for the effective management of postharvest diseases. In some common experimental outputs, including both zero infection (or all replicates infected) and responses at intermediate levels (incidence close to 50% and AUDPC with high internal variance), the hypothesis of equal variances among treatments is clearly violated. The misuse of OLM leads to over- and underestimation of treatment variances that can lead to incorrected results of multiple comparisons of means. In the presence of zero-variance patterns, even general linear models that accommodate heterogeneous variances or generalized linear models for zero-inflated data present computational constraints for the inversion of variance–covariance matrix that do not allow for obtaining the parameter estimates and running the inferential procedures.

In our case studies, contrasts between treatment means or groups of means were used. By using these comparisons, the maximum probability of Type I error (significance level) is controlled only for each comparison, not for the whole experiment. For controlling the experimental wise error, it is necessary to adopt a correction for the p-values of individual comparisons such as Bonferroni, Sidak or Holm [16]. Some authors do not recommend controlling the experimental wise error because in this process the statistical power is drastically reduced making the tests highly conservative [17].

In this paper, we explored the analysis of severity as measured by the MLD. However, the severity can also be quantified via discrete scores with each fruit being classified in a class or category (e.g., a five-class scale varying from 0 to 4). These scores can also be treated as continuous variables, especially if the number of replications is not too small. As this variable is not naturally continuous, a rigorous analysis of residues is necessary to check model assumptions. In the case of inadequacy of the proposed GLM, an approach based on multinomial distribution is recommended.

Statistical methods similar to the ones recommended for analyzing incidence, such as a binary variable, can also be used for multinomial variables. A mean score response function can also be constructed for each treatment and also compared via ꭓ2-type tests derived from weighted least square methods [11,12]. Note that this approach is different from the one that uses GLMs by considering the normal approximation for the mean scores.

In the case of experiments with (all) treatments of a quantitative nature (e.g., doses of fungicide or UV-C radiation), linear or nonlinear regression modeling is recommended to describe the relationship between the response variables and the levels of the independent variables corresponding to the treatments [4]. These cases were not addressed here.

Another important methodological complexity in PHD studies arises from the longitudinal nature of the disease metrics that are commonly quantified over time in the same experimental unit (fruit). In some epidemiological studies it is important to characterize the disease progress adjustment of nonlinear models. In those longitudinal studies, the patterns of time correlations need to be modeled, and the model corresponding to the best pattern can be selected based on criteria such as the Akaike’s Information Criterion (AIC, [22]). GLMs accounting for the longitudinal nature of repeated measures are adequate whenever multivariate normality is not violated; otherwise, for non-normal multivariate distributions, generalized linear models approaches with repeated measures are the option. This type of repeated measures approach for statistical modeling is much more complex and will be addressed in future work.

4. Concluding Remarks

The high variety of metrics used for the evaluation of postharvest diseases affecting fruit and their diverse probabilistic representations constitutes a challenge for phytopathology researchers dealing with advanced statistical methodologies adequate for each particular case.

The alternative methods proposed here are readily available in statistical software, making their implementation easy given the necessary knowledge background and availability of guides for non-statistician users. The codes developed to analyze the data from case studies are available in the Supplementary Materials.

The methods approached here constitute a primer on a general guidance to the correct use of statistical tools in the analysis of PHD experiments. Future studies will include multivariate approaches, including models developed to adequately account for the longitudinal nature of PHD data in which response variables such as incidence and severity are measured over time to capture the pattern of the disease progress.