Multiple-trait, random regression, and compound symmetry models for analyzing multi-environment trials in maize breeding

Igor Ferreira Coelho; Marco Antônio Peixoto; Jeniffer Santana Pinto Coelho Evangelista; Rodrigo Silva Alves; Suellen Sales; Marcos Deon Vilela de Resende; Jefferson Fernando Naves Pinto; Edésio Fialho dos Reis; Leonardo Lopes Bhering

doi:10.1371/journal.pone.0242705

. 2020 Nov 20;15(11):e0242705. doi: 10.1371/journal.pone.0242705

Multiple-trait, random regression, and compound symmetry models for analyzing multi-environment trials in maize breeding

Igor Ferreira Coelho ¹, Marco Antônio Peixoto ¹, Jeniffer Santana Pinto Coelho Evangelista ¹, Rodrigo Silva Alves ², Suellen Sales ¹, Marcos Deon Vilela de Resende ³, Jefferson Fernando Naves Pinto ⁴, Edésio Fialho dos Reis ⁴, Leonardo Lopes Bhering ^1,^*

Editor: Diogo Borges Provete⁵

PMCID: PMC7678961 PMID: 33216796

Abstract

An efficient and informative statistical method to analyze genotype-by-environment interaction (GxE) is needed in maize breeding programs. Thus, the objective of this study was to compare the effectiveness of multiple-trait models (MTM), random regression models (RRM), and compound symmetry models (CSM) in the analysis of multi-environment trials (MET) in maize breeding. For this, a data set with 84 maize hybrids evaluated across four environments for the trait grain yield (GY) was used. Variance components were estimated by restricted maximum likelihood (REML), and genetic values were predicted by best linear unbiased prediction (BLUP). The best fit MTM, RRM, and CSM were identified by the Akaike information criterion (AIC), and the significance of the genetic effects were tested using the likelihood ratio test (LRT). Genetic gains were predicted considering four selection intensities (5, 10, 15, and 20 hybrids). The selected MTM, RRM, and CSM models fit heterogeneous residuals. Moreover, for RRM the genetic effects were modeled by Legendre polynomials of order two. Genetic variability between maize hybrids were assessed for GY. In general, estimates of broad-sense heritability, selective accuracy, and predicted selection gains were slightly higher when obtained using MTM and RRM. Thus, considering the criterion of parsimony and the possibility of predicting genetic values of hybrids for untested environments, RRM is a preferential approach for analyzing MET in maize breeding.

Introduction

Maize (Zea mays L.) is the most cultivated crop worldwide, with a global yield of 1.1 billion tons in the 2018/2019 crop year [1]. An important advantage of this crop is that it can be cultivated across a range of environments and seasons. However, such aspects lead to differential responses of genotypes to varied environmental conditions, which is known as genotype-by-environment interaction (GxE) [2] or phenotypic plasticity [3].

Due to the significant influence of environmental effects on the expression of quantitative traits, multi-environment trials (MET) are an important tool to assess such effects. Variations in phenotypic performance depend on the magnitude of GxE, which occurs when there is dissimilarity in a genotype’s performance in different environments [4]. An environment can be defined by biotic or abiotic factors to which plants are exposed, and can include other characteristics, such as level of technology used or plant population density [5].

To assess GxE, the compound symmetry model (CSM) is the most simple and parsimonious, while the multiple-trait model (MTM) is the most complex and complete [6]. With CSM, a small number of parameters can be estimated, however the assumptions of this model are limited, such as the genetic correlation between environments being equal to 1 [7]. On the other hand, MTM tends to present problems in relation to convergence due to the large number of parameters estimated [8]. Thus, the MTM became prohibitive when the number of environments is high [6].

Random regression models (RRM) estimate the same genetic parameters as MTM but with less parameterization [9] and capture the continuous change of a trait over time or environmental gradient [8,10]. Furthermore, this approach is a powerful tool to predict reaction norms [11,12] and, consequently, predict genotypic values of genotypes for untested environments [7]. The reaction norms consider the effects of GxE and can identify the underlying causes, as they are plotted over an environmental gradient [13].

Selective accuracy is the most suitable approach to compare statistical methods in genetic improvement [14], because it is directly related to the reliability of genetic selection [15]. Moreover, selection gains and heritability are also used to compare statistical methods for analyzing GxE in maize, as maximizing selection gains is the main objective of maize breeding programs [16].

Considering the importance of GxE in maize breeding [16], effective and informative statistical methods are necessary to assess MET, with the goal of improving selective accuracy and, consequently, genetic gains with selection [8]. Recently, MTM and RRM models have been successfully used in plant breeding to analyzing the GxE [11,17]. Thus, the objective of this study was to compare MTM, RRM, and CSM in terms of analyzing MET for maize breeding.

Material and methods

Experimental data

The MET was carried out between January and July 2018, in Goiás State, Brazil (S1 Table). The climate of the region is classified as humid temperate (Cwa), with dry winters and hot summers [18]. The average annual temperature is around 21.5°C and average rainfall is between 1400 and 2000 mm year^-1. The agricultural practices used in the trials are based on those commonly employed for maize cultivation in the region [19].

A data set including 84 maize hybrids evaluated for the trait grain yield (GY) in four environments (E1, E2, E3, and E4) was used (S1 Table). The trials were established using a complete block design with three replications and 44 plants per plot. Plots consisted of four 4 m rows, with a spacing of 0.40 m between plants and 0.45 m between rows. To eliminate the competition effect, only the two central rows were evaluated. The GY was calculated considering the weight of the grain produced per plot, converted to kilograms per hectare (kg ha^-1).

Statistical analyses

Three classes of statistical models, with homogeneous and heterogeneous residual variance structures, were considered: (i) CSM; (ii) MTM; and (iii) RRM fitted through Legendre polynomials. The estimation of variance components and prediction of the genotypic values for GY were conducted via the restricted maximum likelihood/best linear unbiased prediction (REML/BLUP) procedure [20].

(i) Compound symmetry models

CSM was determined by the following equation:

y = X r + Z g + W g e + e,

where: y is the vector of phenotypes; r is the vector of block-environment combinations (assumed to be fixed), which encompasses the effects of environment and block within environment, added to the overall mean; g is the vector of genotypic effects (assumed as random); ge is the vector of GxE effects (random); and e is the vector of residuals (random). Uppercase letters refer to the incidence matrices for the respective effects.

In this model $g ~ N (0, I σ_{g}^{2}), g e \sim N (0, {I σ}_{g e}^{2}),$ and e~N(0,R), where: $σ_{g}^{2}$ is the genotypic variance between hybrids; $σ_{g e}^{2}$ is the GxE variance; I is an identity matrix with appropriate order to the respective effect; and R refers to a diagonal matrix of residual variances (homogeneous or heterogeneous).

(ii) Multiple-trait models

MTM was determined by the following equation:

y = X r + Z g + e .

In this model g~N(0,Σ_g⊗I) and e~N(0,R), where: Σ_g is the genotypic covariance matrix; I is an identity matrix with the appropriate order; ⊗ is the Kronecker product; and R refers to a diagonal matrix of residual variances (homogeneous or heterogeneous).

(iii) Random regression models

In order to use Legendre polynomials, the phenotypic mean of each environment (μ_i) must be scaled to a range of -1 to +1. The environmental gradient values (E_i) were obtained with the following expression [21]:

E_{i} = - 1 + 2 [(μ_{i} - μ_{m i n}) / (μ_{m a x} - μ_{m i n})] .

The RRM were fitted through Legendre polynomials, considering all possible degrees of fit, as follows:

Y_{i j k} = μ + S_{j} + R (S_{j k}) + \sum_{d = 0}^{D} α_{i d} Φ_{i j d} + e_{i j k},

where: Y_ijk is the i^th genotype (i = 1, 2, …, 84) in the j^th environment (j = 1, 2, 3, 4) in the k^th block (k = 1, 2, 3); μ is the overall mean; S_j is the fixed effect of environment j; R(S_jk) is the fixed effect of block k within environment j; d is the Legendre polynomial degree, ranging from 0 to D (D = number of environments—1); α_id is the random regression coefficient for the Legendre polynomial for the genotype effects; Φ_ijd is the d^th Legendre polynomial for the j^th environment for the i^th genotype; and e_ijk is the residual random effect associated with Y_ijk.

In the matrix notation, the above model was described using the following equation:

y = X r + Z g + e .

In this model g~N(0,K_g⊗I) and e~N(0,R), where: K_g is the covariance matrix for the coefficients of genotypic effects; ⊗ is the Kronecker product; I is an identity matrix with the appropriate order; and R refers to a diagonal matrix of residual variances (homogeneous or heterogeneous).

Model selection

In order to compare the residual variance structures (homogeneous and heterogeneous) of CSM, MTM, and RRM the Akaike information criterion (AIC) [22] was used. The difference among the AIC values (Δ_AIC) [23] were calculated to indicate which model provided the best fit. Significance of the random effects of CSM, MTM, and RRM were tested using the likelihood ratio test (LRT) [24].

Variance components and genetic parameters

For CSM, phenotypic variance ( ${\hat{σ}}_{p}^{2}$ ), broad-sense heritability (h²), coefficient of determination of GxE ( $c_{g e}^{2}$ ), and genotypic correlation across environments (r_gloc), were estimated using the following expressions:

{\hat{σ}}_{p}^{2} = {\hat{σ}}_{g}^{2} + {\hat{σ}}_{g e}^{2} + {\hat{σ}}_{e}^{2},

h^{2} = \frac{{\hat{σ}}_{g}^{2}}{{\hat{σ}}_{p}^{2}},

c_{g e}^{2} = \frac{{\hat{σ}}_{g e}^{2}}{{\hat{σ}}_{p}^{2}}, and

r_{g l o c} = \frac{{\hat{σ}}_{g}^{2}}{{\hat{σ}}_{g}^{2} + {\hat{σ}}_{g e}^{2}} .

For RRM, the estimates of genotypic variance ( ${\hat{σ}}_{g}^{2}$ ) and predicted genotypic values ( ${\tilde{g}}_{i j}$ ) at the original scale, were estimated/predicted as follows [21,25]:

{\hat{σ}}_{g}^{2} = Φ_{i j d} {\hat{K}}_{g} Φ_{i j d}^{'}, and

{\tilde{g}}_{i j} = \sum_{d = 0}^{D} {\hat{α}}_{i d} {Φ'}_{i j d} .

For MTM and RRM, phenotypic variance ( ${\hat{σ}}_{p}^{2}$ ) and broad-sense individual heritability (h²) were estimated as [6]:

{\hat{σ}}_{p}^{2} = {\hat{σ}}_{g}^{2} + {\hat{σ}}_{e}^{2}, and

h^{2} = \frac{{\hat{σ}}_{g}^{2}}{{\hat{σ}}_{p}^{2}} .

The selective accuracies ( $r_{\hat{g} g}$ ) were calculated for CSM, MTM, and RRM, with the following expressions, respectively [6]:

r_{\hat{g} g} = \sqrt{1 - \frac{[1 - (1 - \frac{P E V}{{\hat{σ}}_{g}^{2}})] [1 - (1 - \frac{P E V}{{\hat{σ}}_{g e}^{2}})]}{1 - (1 - \frac{P E V}{{\hat{σ}}_{g}^{2}}) (1 - \frac{P E V}{{\hat{σ}}_{g e}^{2}})}};

r_{\hat{g} g} = \sqrt{1 - \frac{P E V}{{\hat{σ}}_{g}^{2}}}; and

r_{\hat{g} g} = \sqrt{1 - \frac{Φ_{i j d} P E V Φ_{i j d}^{'}}{{\hat{σ}}_{g}^{2}}};

where PEV is the prediction error variance, obtained by the diagonal elements of the inverse of the coefficient matrix (information matrix) of the mixed model equations.

The experimental coefficient of variation (CV_e) was calculated with the following expression:

{C V}_{e} = \frac{{\hat{σ}}_{e}}{μ_{i}},

where μ_i is the phenotypic mean of environment i.

Genetic selection

The genotypes were ranked for each environment by each model. Then, selection gains (SG) were predicted considering the selection of 5, 10, 15, and 20 genotypes, based on the following expression [26]:

S G (%) = (\frac{[(Σ_{i = 1}^{n} {G V}_{i}) / n] - μ_{p}}{μ_{p}}) x 100,

where: GV_i is the predicted genotypic value of genotype i; n is the number of genotypes selected; and μ_p is the overall mean.

The coincidence index was calculated to determine the similarity of the ranking of genotypes by the compared models in each environment. This index was calculated as the number of similarly ranked genotypes in the two compared models in each environment, divided by the total number of compared genotypes. The values were given in percentage for 5, 10, 15 or 20 selected genotypes, for the three models in the four environments.

Software

Statistical analyses were performed using ASReml 4.1 Software [20] and ASReml-R package [27] of the R Software [28].

Results

Model selection

Based on the differences among the AIC values (Δ_AIC), the best fit CSM was CS.Rhe, the best fit MTM was MT.Rhe, and the best fit RRM was RR.2.Rhe, since a difference below 2 suggests a competitive model with the best fit (Δ_AIC = 0) [29] (Table 1). Thus, the selected models (CS.Rhe, MT.Rhe, and RR.2.Rhe) were used to estimate the variance components and to predict the genotypic values. Considering all models (CSM, MTM, and RRM), the best fit was RR.2.Rhe. In addition, according to the LRT, significant genotypic effects were detected by all models and GxE effects were detected, explicitly, by CSM (Table 1).

Table 1. Model, Akaike information criterion (AIC), difference among AIC values (Δ_AIC), and likelihood ratio test (LRT) for genotypic effects and GxE effects (in parentheses), for grain yield (GY) evaluated for 84 maize hybrids in four environments.

Model^a	k^b	AIC	Δ_AIC	LRT
CS.Rho	3	14904.06	8.49	72.16^ (14.11^)
CS.Rhe	6	14895.57	0	74.06^ (11.92^)
MT.Rho	11	14888.15	6.44	211.74^**
MT.Rhe	14	14881.71	0	203.61^**
RR.1.Rho	2	14764.32	27.56	82.95^**
RR.2.Rho	4	14751.88	15.12	91.17^**
RR.3.Rho	7	14754.96	18.2	92.63^**
RR.4.Rho	11	14736.76	0	105.73^**
RR.1.Rhe	5	14753.98	17.22	81.08^**
RR.2.Rhe	7	14738.36	1.6	90.89^**
RR.3.Rhe	10	14741.68	4.92	92.23^**
RR.4.Rhe	14	14748.26	11.5	92.94^**

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^a: CS.R_ refers to compound symmetry models, MT.R_ refers to multiple-trait models, and RR.O.R_ refers to random regression models, where R_ is assumed to be homogeneous (Rho) or heterogenous (Rhe) residual variance structures, and O represents the Legendre polynomial order fit for the genetic effects.

^b: Number of estimated parameters.

**: Significant at 0.01 probability of error type I by the chi-square test; the null hypothesis was that random effects did not differ from zero.

Variance components and genetic parameters

Based on CSM, single estimates of genotypic and GxE variance were obtained (Table 2). On the other hand, one estimate of genotypic variance in each environment was obtained using MTM and RRM, but with no estimate of GxE variance (Table 2). For MTM and RRM, the highest estimates of genotypic variance were detected in environments E1 and E2, respectively. Environment E4 presented the lowest estimates of genotypic variance, and E3 presented the highest estimates of residual variance (Table 2).

Table 2. Variance components and their standard errors (in parentheses) and genetic parameters considering the compound symmetry (CSM), multiple-trait (MTM), and random regression (RRM) models for grain yield (GY) evaluated for 84 maize hybrids in four environments.

Component/Parameter	CSM	MTM	RRM
${\hat{σ}}_{g 1}^{2}$	324884.60 (± 67100.27)	785802.30 (± 170462.29)	467421.77 (± 122300.34)
${\hat{σ}}_{g 2}^{2}$		537714.80 (± 125347.84)	533106.91 (± 148594.44)
${\hat{σ}}_{g 3}^{2}$		313339.20 (± 115232.96)	391987.53 (± 92283.01)
${\hat{σ}}_{g 4}^{2}$		250269.40 (± 80175.67)	226533.74 (± 29299.47)
${\hat{σ}}_{e 1}^{2}$	971531.00 (± 105422.84)	852670.90 (± 94406.49)	1054350.00 (± 104702.09)
${\hat{σ}}_{e 2}^{2}$	743312.90 (± 77480.13)	752053.70 (± 82076.15)	753868.00 (± 76071.44)
${\hat{σ}}_{e 3}^{2}$	1092919.30 (± 111712.22)	1125631.50 (± 124218.84)	1208480.00 (± 115423.11)
${\hat{σ}}_{e 4}^{2}$	678742.60 (± 71483.92)	690943.60 (± 75721.01)	702445.00 (± 77022.48)
${\hat{σ}}_{g e}$	124696.50 (± 40296.2)	-	-
$h_{1}^{2}$	0.23	0.48	0.31
$h_{2}^{2}$	0.27	0.42	0.41
$h_{3}^{2}$	0.21	0.22	0.24
$h_{4}^{2}$	0.29	0.27	0.24
$c_{g e 1}^{2}$	0.11	-	-
$c_{g e 2}^{2}$	0.14	-	-
$c_{g e 3}^{2}$	0.10	-	-
$c_{g e 4}^{2}$	0.16	-	-
$\bar{{r_{\hat{g} g}}_{1}}$	0.88	0.89	0.91
$\bar{{r_{\hat{g} g}}_{2}}$		0.91	0.90
$\bar{{r_{\hat{g} g}}_{3}}$		0.82	0.91
$\bar{{r_{\hat{g} g}}_{4}}$		0.83	0.81
CV_e1	13.04	12.21	13.58
CV_e2	11.01	11.07	11.09
CV_e3	14.53	14.74	15.28
CV_e4	14.74	14.87	14.99
r_gloc	0.72	-	-
μ₁	7560.18
μ₂	7832.41
μ₃	7196.73
μ₄	5590.70

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${\hat{σ}}_{g i}^{2}$ : Genotypic variance in environment i; ${\hat{σ}}_{e i}^{2}$ : Residual variance in environment i; ${\hat{σ}}_{g e}^{2}$ : GxE interaction variance; $h_{i}^{2}$ : Broad-sense heritability in environment i; $c_{g e i}^{2}$ : Coefficient of determination of GxE interaction effect in environment i; $\bar{{r_{\hat{g} g}}_{i}}$ : Mean selective accuracy in environment i; CV_ei: Experimental coefficient of variation in environment i; r_gloc: Genotypic correlation across environments; μ_i: Phenotypic mean in environment i.

Broad-sense heritability estimates did not follow any pattern across the evaluated environments. Considering CSM, the estimates of broad-sense heritability varied from 0.21 (E3) to 0.29 (E4), with similar results obtained for MTM and RRM (Table 2). Regarding the mean selective accuracies, a single estimate was obtained using CSM (0.88), while MTM and RRM provided one estimate for each environment (Table 2 and S2 Table). E4 was the only environment in which CSM presented a higher mean selective accuracy compared to MTM and RRM (Table 2 and S2 Table). In the other environments, the mean selective accuracies estimated by MTM or RRM were slightly higher than CSM (Table 2 and S2 Table). Based on CSM, the genotypic correlation across environments was 0.72.

Reaction norms

The reaction norms (Fig 1), fitted through Legendre polynomials of degree two, confirmed the significance of the GxE interaction effects detected by CSM, because the reaction norms intersected, diverged, or converged (Fig 1B). Therefore, the ranking of genotypes changed across the environmental gradient (S3 Table) [30].

Genetic selection

Based on the three selected models, the greatest selection gains were obtained in environments E1 and E2 (Table 3), while MTM provided the highest selection gains for all selection intensities (5, 10, 15, and 20 hybrids). In general, the predicted selection gains using RRM were slightly lower than those predicted by MTM and higher than those predicted by CSM. As expected, selection gains decreased as the number of selected genotypes increased (Table 3).

Table 3. Predicted selection gains in percentage by the compound symmetry (CSM), multiple-trait (MTM) and random regression (RRM) models for grain yield (GY) evaluated for 84 maize hybrids in four environments.

Selection intensity	E1	E2	E3	E4
	CSM
5	21.79	20.07	13.11	13.49
10	15.65	14.48	8.95	10.76
15	13.17	11.02	7.82	8.92
20	11.43	9.33	6.45	6.98
	MTM
5	25.54	20.29	13.91	15.69
10	19.29	15.35	10.78	12.26
15	16.05	12.38	8.92	10.50
20	13.75	10.56	7.74	9.12
	RRM
5	25.06	20.29	13.00	14.74
10	18.86	15.35	9.78	11.45
15	14.86	12.29	8.43	10.27
20	12.95	10.48	7.16	8.90

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The coincidence indices between the selected genotypes for CSM, MTM, and RRM and each pair of environments are shown in the Fig 2.

Our results show a decrease in coincidence index values with an increase in selection intensity, and we can identify which environment had lowest coincidence values compared to other environments and across models. This information can be used to evaluate the most divergent environment isolated and study it deeper, trying to understand its characteristics and being able to draw different breeding strategies to it.

Discussion

Model selection

Although both MTM and RRM enable the assessment of genetic and residual (co)variance structures [6], RRM is considered a reduced and simplified MTM, in that it provides estimates of the same genetic parameters of interest (heritability, genetic correlation), but with less parameterization and computational effort [31]. Effective modeling of genetic and residual effects enables breeders to mitigate the adverse effects of GxE and maximize selective accuracy [32]. Resende et al. [6] highlight the importance of testing different residual variance structures, since these structures can directly affect the estimation of genetic parameters and the prediction of genetic values.

The selected models (CS.Rhe, MT.Rhe, and RR.2.Rhe) fit heterogeneous residuals (i.e., one residual variance for each environment). Heterogeneous residuals in MET analyses were also reported by Melo et al. [32] in analyzing progeny of the common bean, and Alves et al. [12] in analyzing eucalyptus clones. In this study, CSM, MTM, and RRM provided estimates for 6, 14, and 7 parameters, respectively. Thus, the parsimony (i.e., fewer parameters to be estimated) of CSM offers a significant advantage over MTM and RRM. Despite the efficacy of the Average Information REML [33], there are some issues regarding convergence for MTM (an unstructured model), particularly when the number of environments is high [6]. In the present study, RRM with the highest order (order four) was equivalent to MTM. Therefore, RRM (RR.2.Rhe) outperformed MTM (MT.Rhe) in terms of parsimony [7]. In addition, RRM enabled us to capture changes in GY continuously over the environmental gradient.

Variance components and genetic parameters

The estimates of variance components and genetic parameters using MTM and RRM are more realistic because these models consider the heterogeneity of genetic variances among environments [6]. Based on the scale proposed by Resende and Alves [15], GY showed heritabilities of moderate magnitude (0.15<h²<0.50) in all environments, independently of the model used.

Selective accuracy is one of the most relevant parameters in evaluating the effectiveness of inferences of predicted genetic values [12]. This parameter is defined as the correlation between the predicted and true genotypic values and offers evidence of experimental quality, since it considers residual and genetic variances [14]. Based on Resende and Duarte [14], the mean selective accuracy magnitudes ranged from high ( $0.70 < \bar{r_{\hat{g} g}} < 0.90$ ) to very high ( $\bar{r_{\hat{g} g}} \geq 0.90$ ), with RRM providing very high selective accuracies in three (E1, E2, and E3) of the four evaluated environments.

The genotypic correlation across environments (0.72) estimated by CSM, underscores the need for more robust models for genetic selection [34], as CSM assumes a constant genetic variance and a correlation across environments equal to one. However, these assumptions are not prerequisites in MTM and RRM [9]; as such, the latter two models are preferred [34].

Reaction norms

Based on CSM, GxE variance was estimated directly, and its significance was verified by LRT. For MTM and RRM, GxE variance cannot be estimated directly, since these effects are confounded with the genetic effects ( ${\hat{σ}}_{g_{i}}^{2} = {\hat{σ}}_{g_{i}}^{2} + {\hat{σ}}_{g e_{i}}^{2}$ ). In the random regression context, the significance of GxE effects can be determined through reaction norms; the interaction is significant when the reaction norms intersect, diverge, or converge [30]. Herein, the significance of the GxE effects is very clear because the ranking of genotypes was different across the environments, independently of the model used.

Phenotypic plasticity is essential for genotype performance in changing environments [10]. The reaction norms in the present study show that the evaluated hybrids present various forms of phenotypic plasticity. In this context, phenotypic plasticity can be considered favorable or unfavorable changes for genotype adaptedness [30].

Genetic selection

The difference among the ranking of genotypes observed in all models is due to the GxE effects, which have an impact on genotype performance in different environments [13]. As mentioned by van Eeuwijk et al. [30], when genes are expressed differently in different environments, GxE occurs.

Regarding genetic selection, in general MTM and RRM indicate slightly higher genetic gains than the results obtained with CSM. This result can be attributed to the best statistical properties of MTM and RRM [6]. Environments E1 and E2 stood out in terms of genetic gains, when considering the same selection intensity, independently of the model used. These differences among selection gains can be explained by the estimates of broad-sense heritability, which were higher in environments E1 and E2, except for CSM in environment E4, which presented the highest estimate of broad-sense heritability. Furthermore, E1 and E2 presented the lowest coefficients of experimental variation.

The coincidence index demonstrates some characteristics of the environmental correlations. Firstly, we can see a decrease in the coincidence index with an increase in the number of selected genotypes. Secondly, the less coincident an environment i is with the others, the greater the variability of an environment i across the three models, and, the higher is the CV_ei of the environment i, the poorer experimental quality this environment i presents.

Furthermore, CSM, MTM, and RRM can be fitted through Bayesian inference or through Hierarchical Generalized BLUP (HG-BLUP) [35], both in the phenotypic and in the genomic context. Thus, relevant studies can be carried out using these approaches.

Conclusion

The RRM presented the best fit and, consequently, provided more accurate estimates of genetic parameters and predicted genetic values. Furthermore, this model enabled to generate realistic reaction norms. Thus, the results suggest that among the three classes of statistical models, RRM is a preferential approach for analyzing MET in maize breeding.

Supporting information

S1 Table. Location of the four environments (E1, E2, E3, and E4) with their respective geographic coordinates and altitudes.

(DOCX)

Click here for additional data file.^{(13.3KB, docx)}

S2 Table. Selective accuracy for each genotype and mean selective accuracy (below the solid line) in each environment (E1, E2, E3, and E4) based on the compound symmetry (CSM), multiple-trait (MTM), and random regression (RRM) models.

(DOCX)

Click here for additional data file.^{(32.9KB, docx)}

S3 Table. Genotype ranking in each environment (E1, E2, E3, and E4), based on the compound symmetry (CSM), multiple-trait (MTM), and random regression (RRM) models.

(DOCX)

Click here for additional data file.^{(34.1KB, docx)}

Data Availability

All relevant data are within the paper and its Supporting Information files.

Funding Statement

The authors received no specific funding for this work.

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PLoS One. doi: 10.1371/journal.pone.0242705.r001

Decision Letter 0

Diogo Borges Provete

14 Jul 2020

PONE-D-20-15130

Multiple-trait, random regression, and compound symmetry models for analysis of multi-environment trials in maize breeding

PLOS ONE

Dear Dr. Bhering,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

I'm sorry it took so long for you to receive this decision, but it was really difficult to find reviewers. At the end, I was able to secure only one review, but it was quite comprehensive. I second the reviewer that this paper might have a large impact, but authors must do a much better job in accomodating the questions into the structure of the text by making it more concise and straightforward. Your main goal is relevant by itself, so avoid trying to fit in other parallel topics into the manuscript. I have provided a fully commented PDF attached.

Below I listed my major concerns:

1) The abstract needs to be fully revised. It doesn't state clearly the context of the study and provide a good summary of the results and conclusions.

2) most part of the Results is about comparing the parameters estimated in each of the three models. Since it seems important for the paper, why don't you also provide an estimate of uncertainty around each parameter? Consider fitting models under a Bayesian framework.

3) the discussion must be almost entirely re-written and restructured. It barely cites other papers in the fiels, even when there're a couple of reviews on the topic. Most of paragraphs have only one or two citations, this is not good practice in scientific writing. You have to promote a dialogue between your results and previous studies.

Most importantly, English language grammar and writing style need to be carefully revised throughout the whole manuscript. There're sentences that are really hard to understand, the reviewer also complained about it. Some parts of the text are also too much verbose, especially in the Results, try to reduce them to make it straightforward.

I hope authors are able to incorporate these suggestions to the manuscript and submit a revised version soon.

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We look forward to receiving your revised manuscript.

Kind regards,

Diogo Borges Provete, PhD

Academic Editor

PLOS ONE

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Reviewer #1: Partly

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Reviewer #1: Coelho et al present a detailed multi-environmental trial analysis of 84 maize hybrids. Their goal, as stated in the abstract, is to better understand the extent to which GEI interactions might be relevant to maize breeding programs. In order to do so, the authors use a model-fitting approach, comparing the model fit of three main model classes of widespread use (CSM, MTM and RRM).

Overall, the goal of this paper is clearly laid out, and the set of analyses is properly aimed at answering this key aspect of genetic variation patterns. Likewise, the amount of effort required to produce such dataset is commendable.

Having said that, this version of the manuscript also has some significant shortcomings, especially with regards to organization/clarity and in placing the work in a greater context. A significant rewrite is likely required.

In terms of organization and clarity, a substantive issue is the fact that the manuscript still tries to tackle too many issues at the same time, making it hard to follow. As made clear in the introduction, the goal of the manuscript is straightforward: compare these three model classes and infer the practical consequences of choosing one over the other (in terms of GEI). While the goal is clear, I feel that the manuscript falls short of such a goal, especially with regards to the interpretation of the results. While the manuscript reports detailed estimates of genetic variance, it does not address the ways in which all of these estimates have very different interpretations. For example, the estimate of GEI variance in the CSM model has no clear relationship to any of the parameters in the remaining models. However, it does have a potentially useful and clear interpretation. It can be not only interpreted as the portion of the total phenotypic variance that is due to GEI interactions, but can also be compared across traits or species, becoming therefore particularly useful.

The question that emerges is then, what is the practical significance of the parameters in MTM and RRM models? To what extent can they provide a intuitive understanding of the GEI interaction?

Instead, right now the manuscript goes back and forth between different topics. As such, the manuscript would greatly benefit from some streamlining. The measures of selective accuracy and selection gain are, for example, rather tangential to the main goal. Selective accuracy, in particular, is purely a theoretical expectation and follows directly from the heritability estimates. Essentially, the higher the heritability of a trait, the lower the prediction error variance. Perhaps eliminating some of these tangential aspects would help clarify the main thread. As is, the manuscript is an ensemble of variance estimates and the discussion does not help clarify their interpretation.

I would also add that the manuscript would greatly benefit from clearer figures. Most notably, Figure 1 seems to suggest that most environments have similar rankings of maize hybrids. While some hybrids have lower or higher yield across environments, their ranking seems to remain essentially the same. Instead, the remaining of the manuscript argues for the opposite of that. Given that changes in ranking have different implications than pure increases in the yield spread, clarifying this aspect of the manuscript might be particularly important. If all the hybrids that perform well in a single environment, perform (comparatively) as well in others, then GEI interactions (while still capable of generating variance) are of less importance for practical purposes, since all one would need to do is to select the ones with the highest yield at a single environment.

Finally, I would emphasize that ‘less is more’ in this case. The more clearly defined the manuscript becomes, the higher the impact it will have in its field.

Minor comment :

- The colors in most figures are hard to read and follow.

**********

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PLoS One. 2020 Nov 20;15(11):e0242705. doi: 10.1371/journal.pone.0242705.r002

Author response to Decision Letter 0

13 Aug 2020

Dear Dr. Diogo Borges Provete, we are sending the revised version of the manuscript PONE-D-20-15130 - Multiple-trait, random regression, and compound symmetry models for analysis of multi-environment trials in maize breeding.

We would like to thank you and the reviewer for the excellent contributions to improvement of this manuscript. The changes, based on the questions and comments, were highlighted in blue (the same color of the answers in this letter).

Below we respond individually each comment and we are available for any questions.

Reviewer #1:

“…. Second the reviewer that this paper might have a large impact, but authors must do a much better job in accomodating the questions into the structure of the text by making it more concise and straightforward. Your main goal is relevant by itself, so avoid trying to fit in other parallel topics into the manuscript. I have provided a fully commented PDF attached. Below I listed my major concerns:”

1. Line 148: “Degrees of fit?”

Response: In the random regression analyses, the degrees of fit are given by the number of environments (or repeated measures)

2. “no need to state the formula of AIC, which is widely used. Question: why didn't you use AICc instead?” …. Table 1 (line 236) “provide deltaAICc instead of raw AIC values, which are meaningless”

Response: The AIC and BIC are the standard criteria for models choice (Cavanaugh & Neath, 2019; Neath & Cavanaugh, 2012). Models with the same fixed terms are readily compared by these criteria (Verbyla, 2019). The AIC and BIC have been widely used in model selection in plant breeding data analyses (Alves et al., 2020; Melo et al., 2020).

We appreciate the suggestion, and we already cut off the AIC formula from the manuscript. We did not use the AICc because the n was large enough, 985. According to Brewer et al. (2016), the AICc is a correction in case of a small n, providing a stronger penalty than AIC for smaller sample. Although, Burnham and Anderson (2004) infers that AICc → AIC as n → ∞.

3. “The abstract needs to be fully revised. It doesn't state clearly the context of the study and provide a good summary of the results and conclusions?”

We have re-written the abstract.

4. “Most part of the Results is about comparing the parameters estimated in each of the three models. Since it seems important for the paper, why don't you also provide an estimate of uncertainty around each parameter? Consider fitting models under a Bayesian framework.”

As commented by the other reviewer, this article is already carrying a lot of information, in fact, would be enriching to provide an uncertainty measure about the parameter, but we believed we have enough information to discuss about the parameters. The use of Bayesian inference, with no prior information, conducts to the same results (Gianola & Fernando, 1986). In addition, the implementation of these models under Bayesian framework requires more computational efforts (processing capacity and time). The idea of fit models under Bayesian framework is very good and we will consider in a future work.

5. “The discussion must be almost entirely re-written and restructured. It barely cites other papers in the fiels, even when there're a couple of reviews on the topic. Most of paragraphs have only one or two citations, this is not good practice in scientific writing. You have to promote a dialogue between your results and previous studies.”

We re-wrote and restructured the discussion as required.

6. “Most importantly, English language grammar and writing style need to be carefully revised throughout the whole manuscript. There're sentences that are really hard to understand, the reviewer also complained about it. Some parts of the text are also too much verbose, especially in the Results, try to reduce them to make it straightforward.”

We made these changes. See the Results.

Thank you very much for the review and for the excellent contributions to improvement of this manuscript.

Reviewer #2:

Coelho et al present a detailed multi-environmental trial analysis of 84 maize hybrids. Their goal, as stated in the abstract, is to better understand the extent to which GEI interactions might be relevant to maize breeding programs. In order to do so, the authors use a model-fitting approach, comparing the model fit of three main model classes of widespread use (CSM, MTM and RRM). Overall, the goal of this paper is clearly laid out, and the set of analyses is properly aimed at answering this key aspect of genetic variation patterns. Likewise, the amount of effort required to produce such dataset is commendable. Having said that, this version of the manuscript also has some significant shortcomings, especially with regards to organization/clarity and in placing the work in a greater context. A significant rewrite is likely required. In terms of organization and clarity, a substantive issue is the fact that the manuscript still tries to tackle too many issues at the same time, making it hard to follow. As made clear in the introduction, the goal of the manuscript is straightforward: compare these three model classes and infer the practical consequences of choosing one over the other (in terms of GEI). While the goal is clear, I feel that the manuscript falls short of such a goal, especially with regards to the interpretation of the results.

1. “While the manuscript reports detailed estimates of genetic variance, it does not address the ways in which all of these estimates have very different interpretations. For example, the estimate of GEI variance in the CSM model has no clear relationship to any of the parameters in the remaining models. However, it does have a potentially useful and clear interpretation. It can be not only interpreted as the portion of the total phenotypic variance that is due to GEI interactions, but can also be compared across traits or species, becoming therefore particularly useful. The question that emerges is then, what is the practical significance of the parameters in MTM and RRM models? To what extent can they provide a intuitive understanding of the GEI interaction?”

Response: We made the changes (see Discussion, specifically lines 326-333).

“In the random regression context, the GEI effects are detected through reaction norms (Van Eeuwijk et al., 2016). The GEI occurs when the reaction norms intersect, diverge, or converge (Van Eeuwijk et al., 2016). In the multiple-trait models, the GEI effects can not be estimated, since these effects are confounded with the genetic effects (σ ^_(g_i)^2=σ ^_(g_i)^2+σ ^_(ge_i)^2). However, if the genotype ranking differs among environments, the complex GEI occurs.”

2. “Instead, right now the manuscript goes back and forth between different topics. As such, the manuscript would greatly benefit from some streamlining. The measures of selective accuracy and selection gain are, for example, rather tangential to the main goal. Selective accuracy, in particular, is purely a theoretical expectation and follows directly

from the heritability estimates. Essentially, the higher the heritability of a trait, the lower the prediction error variance. Perhaps eliminating some of these tangential aspects would help clarify the main thread. As is, the manuscript is an ensemble of variance estimates and the discussion does not help clarify their interpretation.”

Response: The selective accuracy is the most important parameter for the genetic recommendation in plant breeding (Resende & Duarte, 2007). This parameter corresponds to the correlation between predict and true genetic values (Resende & Duarte, 2007) and it is a reliability measure.

3. “I would also add that the manuscript would greatly benefit from clearer figures. Most notably, Figure 1 seems to suggest that most environments have similar rankings of maize hybrids. While some hybrids have lower or higher yield across environments, their ranking seems to remain essentially the same. Instead, the remaining of the manuscript argues for the opposite of that. Given that changes in ranking have different implications than pure increases in the yield spread, clarifying this aspect of the manuscript might be particularly important. If all the hybrids that perform well in a single environment, perform (comparatively) as well in others, then GEI interactions (while still capable of

generating variance) are of less importance for practical purposes, since all one would need to do is to select the ones with the highest yield at a single environment.”

Response: In fact, in this study, the best genotypes in one environment are also the best in the others. However, considering all genotypes in all environments the GEI is very clear, since the reaction norms cross. We cleared the figure. Similar results were found by Alves et al. (2020) evaluating Eucalyptus clones, for the Pylodin penetration trait. Besides that, these authors verified that for traits with higher coefficients of determination of GEI variances, the genotypes raking varies more along the environmental gradient.

4. “Finally, I would emphasize that ‘less is more’ in this case. The more clearly defined the manuscript becomes, the higher the impact it will have in its field.”

“Minor comment: - The colors in most figures are hard to read and follow.”

Response: We re-made the Results and Discuss almost entirely. (See Results and Discussion)

Thank you very much for the review and for the excellent contributions to improvement of this manuscript.

Alves, R. S., Resende, M. D. V., Azevedo, C. F., Silva, F. F., Rocha, J. R. do A. S. de C., Nunes, A. C. P., Carneiro, A. P. S., & Santos, G. A. (2020). Optimization of Eucalyptus breeding through random regression models allowing for reaction norms in response to environmental gradients. Tree Genetics & Genomes, 16(2), 38. https://doi.org/10.1007/s11295-020-01431-5

Brewer, M. J., Butler, A., & Cooksley, S. L. (2016). The relative performance of AIC, AIC C and BIC in the presence of unobserved heterogeneity. Methods in Ecology and Evolution, 7(6), 679–692. https://doi.org/10.1111/2041-210X.12541

Burnham, K. P., & Anderson, D. R. (2004). Multimodel inference: understanding AIC and BIC in model selection. Sociological Methods and Research, 33, 261–304.

Cavanaugh, J. E., & Neath, A. A. (2019). The Akaike information criterion : Background , derivation , properties , application , interpretation , and refinements. Wiley Computational Statistics, 11(January), 1–11. https://doi.org/10.1002/wics.1460

Gianola, D., & Fernando, R. L. (1986). Bayesian methods in animal breeding theory. Journal of Animal Science, 63(1), 217–244.

Melo, V. L. de, Marçal, T. de S., Rocha, J. R. A. S. de C., dos Anjos, R. S. R., Carneiro, P. C. S., & Carneiro, J. E. de S. (2020). Modeling (co)variance structures for genetic and non-genetic effects in the selection of common bean progenies. Euphytica, 216(5), 77. https://doi.org/10.1007/s10681-020-02607-9

Neath, A. A., & Cavanaugh, J. E. (2012). The Bayesian information criterion: background, derivation, and applications. Wiley Interdisciplinary Reviews: Computational Statistics, 4(2), 199–203. https://doi.org/10.1002/wics.199

Resende, M. D. V. De, & Duarte, J. B. (2007). Precisão e controle de qualidade em experimentos de avaliação de cultivares. Pesquisa Agropecuária Tropical, 37(3), 182–194. https://doi.org/10.5216/pat.v37i3.1867

Van Eeuwijk, F. A., Bustos-Korts, D. V., & Malosetti, M. (2016). What should students in plant breeding know about the statistical aspects of genotype × Environment interactions? Crop Science, 56(5), 2119–2140. https://doi.org/10.2135/cropsci2015.06.0375

Verbyla, A. P. (2019). A note on model selection using information criteria for general linear models estimated using REML. Australian and New Zealand Journal of Statistics, 61(1), 39–50. https://doi.org/10.1111/anzs.12254

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Submitted filename: Response letter.docx

Click here for additional data file.^{(34.1KB, docx)}

PLoS One. doi: 10.1371/journal.pone.0242705.r003

Decision Letter 1

Diogo Borges Provete

25 Sep 2020

PONE-D-20-15130R1

Multiple-trait, random regression, and compound symmetry models for analysis of multi-environment trials in maize breeding

PLOS ONE

Dear Dr. Bhering,

Again, apologies for the delay in sending the decision. I noticed that the abstract and discussion were almost entirely re-written, as requested, and they look much clearer now. Unfortunately, I will have to agree with the reviewer. Authors did not implemented most of the changed they could in this revised version. I still believe this paper could have a big impact but the writing has to improve. Your goal is to compare methods to infer GER, then provide practical guidance to users so they can not only choose the most adequate method, but discuss their limitations and biological interpretation of model parameters. I see you replied to my concern about parameter uncertainty in the rebuttal letter, but I haven't seen any sentence about it in the actual manuscript. This is areal concern and most readers will think about it when they read your paper. So even if you're not willing to implement any Bayesian technique, at least talk about the limitations of your protocol to compare methods.

#---Specific questions:

1) You still need to provide the difference between the best model and the other models in your model selection table. This is the deltaAIC. If you include an additional colum with dAIC you don't need to indicate the best model using #, which is kind of wierd. Notice that the criteria to select models in the information theoretical framework is not the raw AIC value per se (since the AIC is dimensionless, because it's derived from the log likelihood), but the difference in AIC between competing models (see Burnham & Anderson p. 70-2). Usually a diference in AIC higher than 2 demonstrates a unequivocal support for the model with the lowest AIC.

2) The discussion still contains sentences about AIC and LRT, whcih to me should be removed. Also, some sentences of the discussion repeats parts of the methods, which doesn't make sense. Discussion is still very lengthy to the amount of results you have and must be shortened. See my comments in the pdf attached. Avoid citing tables in the discussion.

Please submit your revised manuscript by Nov 09 2020 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

Please include the following items when submitting your revised manuscript:

A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'.
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We look forward to receiving your revised manuscript.

Kind regards,

Diogo Borges Provete, PhD

Academic Editor

PLOS ONE

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Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.

Reviewer #1: (No Response)

**********

2. Is the manuscript technically sound, and do the data support the conclusions?

Reviewer #1: Partly

**********

3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: No

**********

4. Have the authors made all data underlying the findings in their manuscript fully available?

Reviewer #1: Yes

**********

5. Is the manuscript presented in an intelligible fashion and written in standard English?

Reviewer #1: No

**********

6. Review Comments to the Author

Reviewer #1: I have now reviewed the revised version of Coelho et al, in which the authors present a detailed multi-environmental trial analysis of 84 maize hybrids. As mentioned in my previous review, the goal of the paper is clear. The manuscript clearly aims to better understand the extent to which GEI interactions might be relevant to maize breeding programs. The approach is also sound. The authors use a simple model-fitting approach and directly compare the model fit of three main model classes (CSM, MTM and RRM).

Having said that, I have now read this paper multiple times and have postponed writing my review to gather my thoughts. As a reviewer, I strive to make suggestions that will allow authors to improve their manuscript. My sincere hope is that the revised version will be clearer and make a stronger argument in favor of the manuscript. In my view, the revised version of this manuscript does not succeed in addressing most of the comments raised by the editor and me in its previous iteration.

I still find the manuscript to have significant shortcomings in regard to organization/clarity and in placing the work in a greater context.

As I mentioned before, the goal of the manuscript is straightforward: compare these three model classes and infer the practical consequences of choosing one over the other (in terms of GEI). While the manuscripts presents in detail the difference between these models in terms of their estimated components, it spends little to no time exploring the practical consequences of choosing one over the other. Yes, the parameters obtained by the different models are different, but what does that mean in practical terms ? To what degree will it change the standard agricultural practice if the genotype rankings are so similar and if the metrics of selective accuracy are nearly identical between at least two of the models? Also, as mentioned by the AE, what are the uncertainties around these parameters? Uncertainty in parameters is essential for a proper interpretation of the results. Should we observe a similar type of discrepancy between the three models if we were working with another population in different environments? In other words, the manuscript needs to go beyond reporting the technical details and expand on the biological implications. As admitted in the main text of the manuscript, some of these models do not even have parameters that are capable of estimating GEI components directly (such as the MTM), despite that being the main goal of the manuscript.

Another issue that was not properly addressed was the use of figures. Other than the addition of few red lines in one of the figures (Figure 1), there was no noticeable attempt to make the figures clearer. Why not highlight the crossing interactions (if that is the main argument being made) in Figure 1? Also, the figures came without caption this time. The figure with large matrices (which has no number) is very hard to read.

In short, the revised version of Coelho et al. remains a potentially interesting contribution to the field, but falls short of making a stronger case at this point in time.

**********

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PLoS One. 2020 Nov 20;15(11):e0242705. doi: 10.1371/journal.pone.0242705.r004

Author response to Decision Letter 1

6 Nov 2020

Response letter

Authors did not implement most of the changed they could in this revised version. I still believe this paper could have a big impact, but the writing has to improve. Your goal is to compare methods to infer GE, then provide practical guidance to users so they can not only choose the most adequate method but discuss their limitations and biological interpretation of model parameters. I see you replied to my concern about parameter uncertainty in the rebuttal letter, but I haven't seen any sentence about it in the actual manuscript. This is a real concern and most readers will think about it when they read your paper. So even if you're not willing to implement any Bayesian technique, at least talk about the limitations of your protocol to compare methods.

We agree, and we made several changes/corrections in the manuscript. The aim of this study was to compare several statistical models in the Fisherian context (REML/BLUP procedure) for MET analysis in maize breeding. Currently, REML/BLUP is the standard procedure for estimation of variance components and optimal selection in plant breeding (Resende 2016). We included one paragraph highlighting the possibilities to fit the same models through Bayesian inference or through Hierarchical Generalized BLUP (HG-BLUP). Please see the text highlight in blue in the manuscript.

Thank you very much for the review and for the excellent contributions to improvement of this manuscript.

Reviewer #1:

1. “You still need to provide the difference between the best model and the other models in your model selection table. This is the deltaAIC. If you include an additional column with dAIC you don't need to indicate the best model using #, which is kind of weird. Notice that the criteria to select models in the information theoretical framework is not the raw AIC value per se (since the AIC is dimensionless, because it's derived from the log likelihood), but the difference in AIC between competing models (see Burnham & Anderson p. 70-2). Usually a difference in AIC higher than 2 demonstrates a unequivocal support for the model with the lowest AIC.”

We agree, and we described the AIC and the ΔAIC in the Material and Methods section. Also, we included a column of ΔAIC in Table 1 and we highlight the importance of the ΔAIC in the selection of models in the Discussion section. Please see the text highlight in blue in the manuscript.

2. “The discussion still contains sentences about AIC and LRT, which to me should be removed. Also, some sentences of the discussion repeats parts of the methods, which doesn't make sense. Discussion is still very lengthy to the amount of results you have and must be shortened. See my comments in the pdf attached. Avoid citing tables in the discussion.

We agree, and we removed the sentences about AIC and LRT in the discussion section. We removed the citations of Tables in the discussion section. Also, we re-written almost everything in the discussion section, trying to be clearer and objective. Please see the text highlight in blue in the manuscript.

3. (line 302-304) “and then what? What's the implication of this result? You're simply restating the result here in the discussion, when you should be indeed **explaining** them... Also, there's no citation in this paragraph...”

We re-written the discussion section. At this point we include a topic of variance components and genetic parameters. We discuss deeper about these parameters and their importance in maize breeding. Please see the text highlight in blue in the manuscript.

Thank you very much for the review and for the excellent contributions to improvement of this manuscript.

Reviewer #2:

“I have now reviewed the revised version of Coelho et al, in which the authors present a detailed multi-environmental trial analysis of 84 maize hybrids. As mentioned in my previous review, the goal of the paper is clear. The manuscript clearly aims to better understand the extent to which GEI interactions might be relevant to maize breeding programs. The approach is also sound. The authors use a simple model-fitting approach and directly compare the model fit of three main model classes (CSM, MTM and RRM).

1. “I still find the manuscript to have significant shortcomings in regard to organization/clarity and in placing the work in a greater context.”

We agree, and we made several changes/corrections in the manuscript. Please see the text highlight in blue in the manuscript.

2. “As I mentioned before, the goal of the manuscript is straightforward: compare these three model classes and infer the practical consequences of choosing one over the other (in terms of GEI). While the manuscript presents in detail the difference between these models in terms of their estimated components, it spends little to no time exploring the practical consequences of choosing one over the other.

2.1) “Yes, the parameters obtained by the different models are different, but what does that mean in practical terms?”

In practical terms, the best fitted model can provide more accurate estimates of genetic parameters and predicted genetic values, and consequently can maximize the efficacy of the breeding program. This is the first study that apply the RRM for analysis of MET in maize breeding and demonstrated the great advantage of this models in MET analysis.

2.2) “To what degree will it change the standard agricultural practice if the genotype rankings are so similar and if the metrics of selective accuracy are nearly identical between at least two of the models?”

The main advantage of RRM over the CSM and MTM is the ability to predict genotypic performance in environments where a genotype has not been evaluated (and this can increase the efficacy of breeding programs). Besides that, MTM tends to present problems in relation to convergence due to the large number of parameters estimated. Thus, the MTM became prohibitive when the number of environments is high.

2.3) “Also, as mentioned by the AE, what are the uncertainties around these parameters? Uncertainty in parameters is essential for a proper interpretation of the results. Should we observe a similar type of discrepancy between the three models if we were working with another population in different environments? In other words, the manuscript needs to go beyond reporting the technical details and expand on the biological implications.”

We added in Table 2 the standard errors of the variance component estimates. The results for another population evaluated in other environments can be different, but the statistical properties of the models will remain unchanged.

2.4) “As admitted in the main text of the manuscript, some of these models do not even have parameters that are capable of estimating GEI components directly (such as the MTM), despite that being the main goal of the manuscript.”

The MTM are the most complete models to deal with GEI. The fact that these models don’t be able to estimate the variance of the GEI interaction is not a limitation, because the GEI can be observed through the genotype ranking. Please see the text highlight in blue in the manuscript.

3) “Another issue that was not properly addressed was the use of figures.”

3.1) “Other than the addition of few red lines in one of the figures (Figure 1), there was no noticeable attempt to make the figures clearer. Why not highlight the crossing interactions (if that is the main argument being made) in Figure 1?”

Aiming to clarify the Fig 1, we split in A and B. In B, we reduced the number of reaction norms to make it easier to see the reaction norms and their intersections (what represent the presence of the GEI).

3.2) “Also, the figures came without caption this time. The figure with large matrices (which has no number) is very hard to read.”

We made some changes in the letter’s font, and we changed it to bold letter, to become it easier to read.

In short, the revised version of Coelho et al. remains a potentially interesting contribution to the field but falls short of making a stronger case at this point in time.

Thank you very much for the review and for the excellent contributions to improvement of this manuscript.

Attachment

Submitted filename: Response Letter_IFC.docx

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PLoS One. doi: 10.1371/journal.pone.0242705.r005

Decision Letter 2

Diogo Borges Provete

9 Nov 2020

Multiple-trait, random regression, and compound symmetry models for analyzing multi-environment trials in maize breeding

PONE-D-20-15130R2

Dear Dr. Bhering,

We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication.

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Kind regards,

Diogo Borges Provete, PhD

Academic Editor

PLOS ONE

Additional Editor Comments (optional):

Reviewers' comments:

PLoS One. doi: 10.1371/journal.pone.0242705.r006

Acceptance letter

Diogo Borges Provete

11 Nov 2020

PONE-D-20-15130R2

Multiple-trait, random regression, and compound symmetry models for analyzing multi-environment trials in maize breeding

Dear Dr. Bhering:

I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now with our production department.

If your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information please contact onepress@plos.org.

If we can help with anything else, please email us at plosone@plos.org.

Thank you for submitting your work to PLOS ONE and supporting open access.

Kind regards,

PLOS ONE Editorial Office Staff

on behalf of

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Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Supplementary Materials

S1 Table. Location of the four environments (E1, E2, E3, and E4) with their respective geographic coordinates and altitudes.

(DOCX)

Click here for additional data file.^{(13.3KB, docx)}

(DOCX)

Click here for additional data file.^{(32.9KB, docx)}

S3 Table. Genotype ranking in each environment (E1, E2, E3, and E4), based on the compound symmetry (CSM), multiple-trait (MTM), and random regression (RRM) models.

(DOCX)

Click here for additional data file.^{(34.1KB, docx)}

Attachment

Submitted filename: Response letter.docx

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Attachment

Submitted filename: PONE-D-20-15130_R1.pdf

Click here for additional data file.^{(2.5MB, pdf)}

Attachment

Submitted filename: Response Letter_IFC.docx

Click here for additional data file.^{(24KB, docx)}

Data Availability Statement

All relevant data are within the paper and its Supporting Information files.

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PERMALINK

Multiple-trait, random regression, and compound symmetry models for analyzing multi-environment trials in maize breeding

Igor Ferreira Coelho

Marco Antônio Peixoto

Jeniffer Santana Pinto Coelho Evangelista

Rodrigo Silva Alves

Suellen Sales

Marcos Deon Vilela de Resende

Jefferson Fernando Naves Pinto

Edésio Fialho dos Reis

Leonardo Lopes Bhering

Roles

Abstract

Introduction

Material and methods

Experimental data

Statistical analyses

(i) Compound symmetry models

(ii) Multiple-trait models

(iii) Random regression models

Model selection

Variance components and genetic parameters

Genetic selection

Software

Results

Model selection

Table 1. Model, Akaike information criterion (AIC), difference among AIC values (ΔAIC), and likelihood ratio test (LRT) for genotypic effects and GxE effects (in parentheses), for grain yield (GY) evaluated for 84 maize hybrids in four environments.

Variance components and genetic parameters

Table 2. Variance components and their standard errors (in parentheses) and genetic parameters considering the compound symmetry (CSM), multiple-trait (MTM), and random regression (RRM) models for grain yield (GY) evaluated for 84 maize hybrids in four environments.

Reaction norms

Fig 1. Reaction norms for grain yield (GY) evaluated for maize hybrids in four environments.

Genetic selection

Table 3. Predicted selection gains in percentage by the compound symmetry (CSM), multiple-trait (MTM) and random regression (RRM) models for grain yield (GY) evaluated for 84 maize hybrids in four environments.

Fig 2. Coincidence index for all environments in all models.

Discussion

Model selection

Variance components and genetic parameters

Reaction norms

Genetic selection

Conclusion

Supporting information

Data Availability

Funding Statement

References

Decision Letter 0

Diogo Borges Provete

Roles

Author response to Decision Letter 0

Decision Letter 1

Diogo Borges Provete

Roles

Author response to Decision Letter 1

Decision Letter 2

Diogo Borges Provete

Roles

Acceptance letter

Diogo Borges Provete

Roles

Associated Data

Supplementary Materials

Data Availability Statement

ACTIONS

PERMALINK

RESOURCES

Similar articles

Cited by other articles

Links to NCBI Databases

Table 1. Model, Akaike information criterion (AIC), difference among AIC values (Δ_AIC), and likelihood ratio test (LRT) for genotypic effects and GxE effects (in parentheses), for grain yield (GY) evaluated for 84 maize hybrids in four environments.