Genetic parameters and selection of maize cultivars using Bayesian inference in a multi-trait linear model

ABSTRACT

Variance components must be obtained to estimate genetic parameters and predict breeding values. In studies which take many traits into account, it is reasonable to use the Bayesian approach for the estimation of genetic parameters. The main goal of the present research was not only to consider the genetic correlations of the examined traits, but above all to estimate unknown genetic parameters and to gain profits from the selection. Bayesian inference was also useful for the selection of the best maize varieties. It was applied to predict genetic values in the multi-traits linear model. Thirteen maize cultivars representing the traits of our interest were studied by means of Bayesian inference. The traits are the number of plants before harvest, the grain yield, the length of the ears, the mass of leaves and the number of ears. The experiment involved a randomised block design with four replications and ten plants per plot. The highest correlation estimates were found between the number of plants before harvest and the number of ears, jointly with the grain yield and the number of ears. Lower correlation estimates were found between the length of the ears and the number of ears as well as the grain yield and the length of the ears. The research confirms that the best varieties to be grown are: Clarica, NK Cooler, Drim and PR 39K13. The Bayesian approach proved to be useful in selection studies, which can further be used to improve the studied genotypes.

KEYWORDS:

• Breeding progress
• correlations
• genetic parameters
• integrated control
• modelling
• Zea mays L

Introduction

From 1 January 2014, pursuant to the 21 October 2009 Directive of the European Parliament and of the European Council establishing the framework for community action with regard to the use of pesticides (2009/128/CE), all professional users are obliged to observe the rules of integrated plant protection. The rules and guidelines for integrated plant protection specified in Annex III ‘General rules for integrated plant protection’ emphasise the application of all possible and available methods aimed at reducing the development of harmful organisms to a level of harmlessness. Integrated plant control emphasises the importance of research on plant diseases and breeding for resistance, as well as more extensive use of varieties resistant to diseases in production conditions (Piesik et al. 2007).

Maize (Zea mays L.), jointly with wheat and rice, is one of the main grain crops grown in the world. It is a widely used plant species and it forms the basis for agriculture in the European Union. As a valuable source of raw material for food, alcohol and chemical industries, it is also an excellent feed for animals (Aslam et al. 2011). Its future use may also include energy production. Furthermore, maize may be a remedy for climate change. It may prevent these changes thanks to its ability to intensively bind CO₂ and to form yield using some amount of heat (Kim et al. 2007). Due to its origin, maize belongs to thermophilic plants (Szulc et al. 2017). Therefore, in the period of its growth and development, apart from appropriate agrotechnology, it requires more heat, compared to other cereal plants (Bos et al. 2000; Kim et al. 2007). The productivity of plants depends on the so-called photosynthetic production, that is, the amount of assimilates produced in the process of photosynthesis. It determines the biological yield of plants to a large extent, especially the potential to produce fresh or dry matter per unit area. In addition to the standard photosynthesis, characteristic of most species in our climatic zone, at higher air temperatures (above 25°C) and high light intensity, the maize launches the C4 assimilation track, doubling the photosynthetic efficiency. In one cycle, it connects 2 carbon dioxide molecules (Pengelly et al. 2011). Therefore, C4 plants, including maize, have an adaptive advantage over C3 plants, because the photodream loss in C3 plants increases with increasing temperature (Tausta et al. 2002). In other words, the maize uses light radiation and high air temperature very effectively, which translates into large increments of the leaf surface (the main photosynthetic apparatus). Varieties characterised by a high intensity of photosynthesis very often give lower yields than varieties showing a lower intensity of the process. It is due to the fact that two factors play an important role in the production of biomass: the intensity of photosynthesis and the size of the assimilation surface (leaves), in particular life expectancy and the period of capacity of assimilation bodies for effective photosynthesis (Szulc et al. 2012). Due to the unpredictability of weather conditions, one of the main challenges facing growers is to create maize hybrids capable of withstanding in environments featuring limited water resources. One solution is to breed plants characterised by greater health in a later period of vegetation or the ‘stay-green’ trait. This feature is an indicator of high plant health (Costa et al. 2002), reduced progressive aging and tolerance to drought after flowering. Greater durability (greenness of the leaf) or ‘stay-green’ trait are positively correlated with the yielding of plants (Osaki et al. 1993).

Finding the estimators of the variance components, predicting the breeding values and calculating the genetic parameters are the necessary conditions to propose new breeding strategies for maize (Gonçalves-Vidigal et al. 2008; Fritsche-Neto et al. 2010; Peiffer et al. 2013; Oliveira et al. 2014). In standard tests in which one feature is considered, the first step is to estimate unknown components of the variance. To achieve it (in a standard case), the least squares method or the standard maximum likelihood method is usually used. If the experiment is carried out with additional restrictions, the restricted maximum likelihood (REML) method is appropriate. In a situation where many traits are examined and many factors are considered, it is recommended to use Bayesian methods. Such methods allow not only for the calculation of breeding values and genetic parameters, such as genotypic variability coefficients and residual variance, heritability but also for the determination of the reliability intervals of the variance components and the genotypic correlation. In general, two or more traits are considered in maize studies. In such analyses, in order to find relationships between the tested traits, multivariate methods based on multifactorial models are used. In such a situation, Bayesian methods are very useful. Bayesian multi-trait models have become useful statistics method for plant and animal genetic evaluations. Many authors have shown that this model is more flexible and effective than the least squares method since it is not only based on the likelihood function, but it also allows a priori knowledge assumption when defining prior distribution (Hayashi and Iwata 2013). The Bayesian methods are often used to estimation of genetic parameters in different plants: barley (Li and Sillanpää 2012), Eucalyptus bosistoana (Davies et al. 2017), Eucalyptus cladocalyx (Cané-Retamales et al. 2011), jatropha (Junqueira et al. 2016), maize (Cuevas et al. 2017; Torres et al. 2018), soybean (Arriagada et al. 2012), wheat (Gianola et al. 2011; Cuevas et al. 2017).

The aim of the research presented in this paper was to apply Bayesian approach, involving a multi-trait linear model, to obtain estimates of the genetic parameters, to assess the genetic correlation between traits, to predict breeding values, and to select the best maize cultivars.

Material and methods

The experiment was carried out at the Department of Agronomy, the Poznań University of Life Sciences, in 2016 in the fields of the Agricultural Experimental Station in Swadzim (52°26′20″ N; 16°44′58″ E). Thirteen cultivars (NK Cooler, Delitop, Gazele, NK Ravello, ES Palazzo, ES Paroli, Clarica, PR 39 G 12, SY Cooky, Drim, SY Mascotte, ES Fortran and PR 39K 13) of fodder maize, grown for grain and purchased from different seed production companies, were compared. The selection of cultivars for the tests was not accidental. The cultivars were chosen mainly on the basis of the popularity of a given farm on the domestic market and the market share of breeders depending on the direction of cultivation. In the case of maize grain cultivation, the largest share in the sowing structure in 2018 was characterised by Pioneer varieties (20%), followed by Syngenty (12%) and Euralis (10%). The trend has been maintained for a long time in Poland. Therefore, selecting 13 maize cultivars coming from three breeding and seeding companies, which are the leaders on the domestic market in grain cultivation, should be considered as one of the novel elements of the study. The experiment was established in a complete randomised block design with four field replications and ten plants per plot. It was carried out in a typical grey-brown podzolic soil, composed of coarse sandy soil, shallowly deposited on light loam. The same NPK fertilisation was applied throughout the experimental field: 100 kg N ha⁻¹ in the form of urea, 80 kg P₂O₅·ha⁻¹ in the form of NPK-fertiliser (6% N), and 120 kg K₂O·ha⁻¹ in the form of 60% potassium salt. All the agricultural and cultivation measures were performed, following the guidelines of good agricultural practices. In each plant, the number of plants before harvest, the grain yield, the length of the ears, the mass of leaves and the number of ears were evaluated.

Statistical analyses were carried out using the plot mean. In this paper, we consider the multi-trait mixed model offered by Henderson and Quaas (1976). Let vector y_t represent the n_t observations for trait t. Then, the multi-trait mixed linear model for n traits can be written as follows:$y_i=X_i{\beta }_i+Z_i{u}_i+e_i,i=1,2,\dots ,n$where: X_i denotes the incidence matrix of the block effects associated with trait i, β_i denotes the vector of fixed blocks effects summed to the overall mean associated with trait i, Z_i denotes the incidence matrix of the genetic effect of each genotype for trait i, u_i denotes is the vector of random effects of genotypes associated with trait i, e_i denotes the vector of random residual effects associated with trait i. The assumptions on the distribution of y, u, and e are described as:$\left[\begin{array}{c}\tilde{y}\\ \tilde{u}\\ \tilde{e}\end{array}\right]\sim NMV\left[\left[\begin{array}{c}X\tilde{\beta }\\ \tilde{0}\\ \tilde{0}\end{array}\right]\left[\begin{array}{ccc}ZG{Z}^{\mathrm{\prime }}+R& ZG& R\\ G{Z}^{\mathrm{\prime }}& G& \mathrm{\Phi }\\ R& \mathrm{\Phi }& R\end{array}\right]\right],$here, R and G denote covariance matrices associated with the vector e of residuals and vector u of random effects. If R0 (of order 5×5) is the residual covariance for the five traits, then R can be calculated as R = R0⊗I (‘⊗’ – the Kronecker product of two matrices, I – the identity matrix). Similarly, the genetic covariance matrix G can be calculated as G = G0⊗I. In the mixed model used, β is considered the vector of solutions for the systemic effects; however, from the Bayesian point of view, it is a vector of random effects in which the initial distribution values have uninformative priors; thus, they do not provide much information about the parameter and, therefore, have a uniform probability distribution (Everling et al. 2014).

Bayesian inference was used for estimating the variance components and predicting the breeding values of the accessions. The data analysis was carried out using Markov Chain Monte Carlo (MCMC) methods, through the Gibbs sampling algorithm. A Markov chain with 6,000,000 cycles was generated. It was considered a burn in of 300,000 cycles and a thinning interval of 3000 cycles, which ensured that the serial correlations were zero or very low. As a check criterion for convergence, the p-value of the Geweke test (Geweke 1992) was analysed. From the posterior distribution, the following parameters were calculated: the genetic variation coefficients: $CVg={\displaystyle \frac{\sqrt{{\hat{\sigma }}_g^2}}{{\hat{m}}_g}}$, the residual coefficients of variation: $CVe={\displaystyle \frac{\sqrt{{\hat{\sigma }}_e^2}}{{\hat{m}}_e}}$, the variation index: I_v = CVg/CVe, the genotypic correlation coefficients: $r={\displaystyle \frac{\sqrt{\widehat{Co{v}_g\left(X,Y\right)}}}{\sqrt{{\hat{\sigma }}_g^2\left(X\right){\hat{\sigma }}_g^2\left(Y\right)}}}$, the heritability in the broad sense: $h^2=\left({\hat{\sigma }}_g^2/\left({\hat{\sigma }}_g^2+{\hat{\sigma }}_e^2\right)\right)$, and the gain with selection: $Gs=h^2\left({\hat{m}}_s-{\hat{m}}_g\right)/{\hat{m}}_g$, where ${\hat{\sigma }}_g^2$ denotes the genetic variance, ${\hat{\sigma }}_e^2$ – the residual variance, $\widehat{Co{v}_g}$ – the genetic covariance obtained by the multi-trait model, ${\hat{m}}_g$ – the trait overall mean, and ${\hat{m}}_s$ – the mean of the top 30% of individuals. For the variance components, e.g. genetic parameters and breeding values, we calculated the mean, median, mode and interval of higher density of posterior distribution (HPD). All the analyses were conducted using the GenStat v. 18 statistical software package.

Results

The p-value of the Geweke test was used as an indicator of convergence with the Markov Chains (Tables 1–6). All the p-value estimates obtained in the Geweke test were lower than .05. Values closer to the mean, median and mode for the marginal posterior distribution of the parameters obtained were found. The highest heritability estimates were found for the mass of leaves and the number of ears (Tables 1–6). The grain yield had the lowest heritability estimates. However, these estimates did not differ significantly from the credibility interval. The grain yield also had the lowest estimate for the genetic variation coefficient. This estimate did not significantly differ from that calculated for the number of plants before harvest. The number of ears and the mass of leaves presented the highest coefficient of genetic variation. The mode for the posterior distribution of the coefficient of environmental variation ranged from 8.79% for grain yield to 14.51% for the number of ears (Table 1). Only the grain yield and the number of ears showed significant differences in the estimates of the coefficient of environmental variation. Moreover, only in the case of the grain yield was the mode of the posterior distribution of the coefficient of relative variation lower than 1.00 (0.866). However, its estimates were not significantly lower than 1.00 according to the credibility interval (0.445, 1.416). The mass of leaves turned out the highest coefficient of relative variation, which differs significantly from the estimates found for the grain yield and the length of the ears (Table 1).

Table 1. Mean, median, mode, higher posterior density intervals (HPD), and p-value of the Geweke test for the posterior distribution of the estimates of heritability, coefficient of genotypic variation (%), coefficient of residual variation (%), relative variation index, and genotypic correlation in maize cultivars.

Traits	Descriptive analysis			HDP (95%)		p-value (Geweke)
	Mean	Median	Mode	Infimum	Supremum
Heritability
Number of plants before harvest (NPH)	0.823	0.843	0.856	0.684	0.949	.005
Grain yield (GY)	0.721	0.723	0.758	0.509	0.907	.009
Length of the ears (LE)	0.879	0.907	0.922	0.799	0.976	.038
Mass of leaves (ML)	0.959	0.977	0.977	0.935	0.987	.035
Number of ears (NE)	0.924	0.935	0.948	0.887	0.979	.021
Coefficient of genetic variation (%)
NPH	19.71	19.28	20.55	12.12	27.87	.046
GY	10.48	10.14	9.09	5.59	15.09	.001
LE	29.99	28.99	27.87	19.19	41.92	.048
ML	44.28	42.87	37.52	30.07	64.01	.043
NE	47.21	44.33	45.01	30.01	58.48	.008
Coefficient of residual variation (%)
NPH	14.66	14.45	14.97	11.19	18.53	.018
GY	11.01	11.07	10.93	8.79	13.39	.047
LE	17.08	16.68	15.45	12.49	22.41	.042
ML	13.05	12.69	13.69	9.51	17.04	.004
NE	20.05	20.06	20.27	14.51	26.32	.037
Relative variation index
NPH	1.374	1.357	1.987	0.705	2.022	.014
GY	0.958	0.928	0.866	0.445	1.416	.035
LE	1.798	1.758	1.601	1.036	2.558	.015
ML	3.418	3.455	3.502	2.099	5.008	.005
NE	2.357	2.299	2.301	1.488	3.328	.007
Genotypic correlation
NPH – GY	0.621	0.654	0.668	0.557	0.786	.024
NPH – LE	−0.26	−0.274	−0.28	−0.378	−0.007	.014
NPH – ML	0.591	0.599	0.602	0.488	0.701	.021
NPH – NE	0.801	0.814	0.826	0.731	0.945	.028
GY – LE	−0.110	−0.145	−0.170	−0.211	0.073	.027
GY – ML	0.293	0.302	0.319	0.184	0.394	.023
GY – NE	0.729	0.739	0.755	0.701	0.821	.039
LE – ML	0.381	0.399	0.411	0.318	0.455	.047
LE – NE	−0.060	−0.066	−0.100	−0.118	0.094	.012
ML – NE	0.487	0.491	0.503	0.417	0.568	.001

Table 2. Mean, median, mode, higher posterior density intervals (HPD), and p-value of the Geweke test for the posterior distribution of the estimates of heritability for individual genotypes.

Traits	Cultivar	Descriptive analysis			HDP (95%)		p-value (Geweke)
		Mean	Median	Mode	Infimum	Supremum
NPH	Clarica	0.787	0.794	0.801	0.437	0.987	.034
	Delitop	0.699	0.706	0.707	0.329	0.999	.024
	Drim	0.977	0.977	0.978	0.607	0.977	.017
	ES Fortran	0.854	0.885	0.887	0.414	0.914	.048
	ES Palazzo	0.584	0.589	0.594	0.264	0.884	.047
	ES Paroli	0.794	0.801	0.811	0.354	0.847	.016
	Gazele	0.784	0.799	0.806	0.401	0.941	.033
	NK Cooler	0.801	0.811	0.809	0.457	0.914	.001
	NK Ravello	0.914	0.915	0.917	0.514	0.957	<.001
	PR 39 G 12	0.887	0.897	0.891	0.542	0.924	.002
	PR 39K 13	0.833	0.841	0.839	0.483	0.964	.009
	SY Cooky	0.897	0.899	0.902	0.521	0.994	.007
	SY Mascotte	0.854	0.865	0.851	0.511	0.915	.004
GY	Clarica	0.907	0.909	0.911	0.557	0.992	<.001
	Delitop	0.841	0.844	0.847	0.492	0.966	.007
	Drim	0.541	0.558	0.551	0.212	0.784	.029
	ES Fortran	0.801	0.807	0.806	0.432	0.899	.001
	ES Palazzo	0.273	0.277	0.272	0.067	0.573	.049
	ES Paroli	0.812	0.821	0.822	0.465	0.978	.017
	Gazele	0.554	0.568	0.564	0.201	0.854	.027
	NK Cooler	0.761	0.779	0.772	0.411	0.892	.038
	NK Ravello	0.806	0.822	0.813	0.464	0.982	.019
	PR 39 G 12	0.824	0.821	0.826	0.494	0.993	.009
	PR 39K 13	0.809	0.814	0.822	0.468	0.886	.003
	SY Cooky	0.707	0.727	0.721	0.365	0.847	.029
	SY Mascotte	0.791	0.792	0.804	0.454	0.901	.009
LE	Clarica	0.814	0.821	0.817	0.471	0.964	.002
	Delitop	0.905	0.907	0.911	0.561	0.995	.001
	Drim	0.854	0.849	0.864	0.512	0.974	.008
	ES Fortran	0.866	0.867	0.869	0.527	0.966	.004
	ES Palazzo	0.921	0.901	0.922	0.582	0.998	.003
	ES Paroli	0.954	0.845	0.958	0.604	0.997	.002
	Gazele	0.598	0.601	0.599	0.248	0.706	.041
	NK Cooler	0.864	0.855	0.869	0.522	0.969	.025
	NK Ravello	0.799	0.801	0.822	0.451	0.976	.036
	PR 39 G 12	0.897	0.884	0.901	0.554	0.994	.004
	PR 39K 13	0.691	0.704	0.724	0.356	0.991	.042
	SY Cooky	0.785	0.788	0.797	0.449	0.891	.039
	SY Mascotte	0.781	0.791	0.817	0.443	0.947	.027
ML	Clarica	0.892	0.901	0.914	0.558	0.969	.002
	Delitop	0.857	0.866	0.859	0.508	0.995	.017
	Drim	0.901	0.891	0.904	0.896	0.994	.002
	ES Fortran	0.944	0.922	0.955	0.799	0.999	.049
	ES Palazzo	0.975	0.957	0.965	0.801	0.997	.039
	ES Paroli	0.981	0.983	0.984	0.802	0.994	.025
	Gazele	0.631	0.699	0.744	0.298	0.709	.048
	NK Cooler	0.983	0.981	0.978	0.799	0.998	<.001
	NK Ravello	0.973	0.974	0.979	0.632	0.995	<.001
	PR 39 G 12	0.966	0.968	0.967	0.806	0.999	.001
	PR 39K 13	0.979	0.968	0.951	0.845	0.998	<.001
	SY Cooky	0.758	0.766	0.754	0.412	0.865	.017
	SY Mascotte	0.904	0.905	0.922	0.561	0.966	.005
NE	Clarica	0.921	0.934	0.939	0.855	0.997	.006
	Delitop	0.937	0.941	0.954	0.601	0.999	.001
	Drim	0.707	0.722	0.727	0.557	0.891	.027
	ES Fortran	0.699	0.706	0.709	0.362	0.999	.042
	ES Palazzo	0.508	0.524	0.534	0.161	0.808	.049
	ES Paroli	0.957	0.958	0.966	0.61	0.994	.006
	Gazele	0.964	0.965	0.964	0.621	0.996	.004
	NK Cooler	0.947	0.955	0.949	0.619	0.995	.009
	NK Ravello	0.904	0.909	0.911	0.561	0.995	.009
	PR 39 G 12	0.971	0.972	0.974	0.808	0.999	.01
	PR 39K 13	0.943	0.951	0.955	0.817	0.995	.011
	SY Cooky	0.931	0.947	0.942	0.593	0.993	.003
	SY Mascotte	0.894	0.899	0.904	0.554	0.995	.009

Table 3. Mean, median, mode, higher posterior density intervals (HPD), and p-value of the Geweke test for the posterior distribution of the estimates of coefficient of genotypic variation (%) for individual genotypes.

Traits	Cultivar	Descriptive analysis			HDP (95%)		p-value (Geweke)
		Mean	Median	Mode	Infimum	Supremum
NPH	Clarica	21.48	19.98	21.87	16.23	28.28	.003
	Delitop	21.94	20.41	22.31	17.21	27.84	.012
	Drim	23.09	23.12	23.42	18.46	28.79	.015
	ES Fortran	21.87	21.47	22.23	17.31	27.69	.009
	ES Palazzo	14.21	13.79	14.62	9.73	20.02	.007
	ES Paroli	18.94	18.44	19.39	14.41	24.75	.026
	Gazele	16.97	16.55	17.07	12.39	23.06	.003
	NK Cooler	17.84	17.43	18.26	13.31	23.73	.011
	NK Ravello	19.84	19.76	20.21	15.1	25.99	.032
	PR 39 G 12	20.91	20.85	21.23	16.23	27.88	.036
	PR 39K 13	20.17	19.99	20.49	15.48	26.57	.017
	SY Cooky	20.02	19.87	21.28	15.41	26.32	.016
	SY Mascotte	21.84	21.98	22.01	17.02	28.43	.043
GY	Clarica	17.54	17.21	18.02	13.01	23.48	.021
	Delitop	16.91	16.99	17.32	12.01	23.01	.015
	Drim	11.09	11.57	12.26	6.17	16.91	.048
	ES Fortran	7.04	6.98	7.21	2.41	12.89	.033
	ES Palazzo	6.51	6.62	6.83	2.02	12.55	.014
	ES Paroli	7.84	7.94	8.17	3.23	13.69	.037
	Gazele	13.21	12.99	13.45	8.55	19.52	.009
	NK Cooler	8.88	8.82	9.05	4.18	15.01	.007
	NK Ravello	12.45	11.98	12.99	7.77	18.95	.001
	PR 39 G 12	15.47	14.94	15.46	10.17	21.87	.001
	PR 39K 13	9.84	9.81	9.97	5.23	15.77	.024
	SY Cooky	14.09	13.9	14.21	9.44	20.03	.021
	SY Mascotte	10.07	9.92	10.07	5.32	15.88	.024
LE	Clarica	37.14	36.91	37.24	32.52	43.08	.016
	Delitop	34.08	33.97	34.37	29.51	39.81	.027
	Drim	20.17	19.75	20.21	15.06	25.92	.001
	ES Fortran	30.17	29.79	31.01	25.07	35.97	.015
	ES Palazzo	35.81	35.82	35.92	30.31	42.05	.032
	ES Paroli	38.91	38.98	38.95	32.11	44.21	.049
	Gazele	19.42	18.99	19.71	14.17	25.34	.038
	NK Cooler	24.07	24.57	24.36	19.22	29.89	.004
	NK Ravello	32.47	33.97	32.78	27.03	38.33	.001
	PR 39 G 12	21.48	22.08	21.73	14.88	27.97	<.001
	PR 39K 13	22.19	23.61	22.17	17.12	28.08	.029
	SY Cooky	33.74	33	34.16	27.11	39.91	.031
	SY Mascotte	36.84	36.91	37.07	32.03	45.01	.035
ML	Clarica	34.06	34.56	34.26	23.06	39.99	.001
	Delitop	55.81	55.92	56.24	47.33	62.62	.01
	Drim	36.91	36.55	37.39	31.01	43.12	.032
	ES Fortran	53.97	53.82	54.29	45.54	59.89	.017
	ES Palazzo	44.47	44.97	45.01	39.02	51.02	.025
	ES Paroli	49.81	50.32	50.14	45.04	55.77	.021
	Gazele	31.36	32.86	31.77	25.81	37.35	.048
	NK Cooler	56.78	57.18	57.23	50.27	62.99	.003
	NK Ravello	39.41	39.93	39.79	34.04	45.55	.012
	PR 39 G 12	30.17	29.99	30.66	25.11	36	.003
	PR 39K 13	47.01	48.02	47.61	39.21	52.8	.017
	SY Cooky	32.79	32.91	33.18	27.21	39.08	.033
	SY Mascotte	51.79	51.93	52.21	45.23	57.51	.014
NE	Clarica	55.17	55.66	55.47	50.01	61	.001
	Delitop	38.49	37.99	38.78	33.17	44.56	.036
	Drim	40.57	41.07	40.93	33.06	46.21	.049
	ES Fortran	42.06	42.56	42.44	37.06	47.65	.041
	ES Palazzo	36.83	36.99	37.07	29.39	42.44	.029
	ES Paroli	43.97	43.71	44.02	38.43	50.12	.021
	Gazele	51.36	51.86	51.75	46.81	57.09	.028
	NK Cooler	41.34	42.04	41.71	36.04	46.97	.026
	NK Ravello	38.46	37.91	38.81	33.06	44.15	.001
	PR 39 G 12	57.01	56.55	57.39	51.61	63.02	.027
	PR 39K 13	49.34	49.43	49.71	44.14	55.01	.026
	SY Cooky	42.48	42.92	42.83	37.18	48.33	.012
	SY Mascotte	53.47	54.91	53.81	48.07	59.21	.011

Table 4. Mean, median, mode, higher posterior density intervals (HPD), and p-value of the Geweke test for the posterior distribution of the estimates of coefficient of residual variation (%) for individual genotypes.

Traits	Cultivar	Descriptive analysis			HDP (95%)		p-value (Geweke)
		Mean	Median	Mode	Infimum	Supremum
NPH	Clarica	17.08	17.31	17.38	13.4	21.09	.043
	Delitop	18.31	18.51	18.61	14.62	22.33	.012
	Drim	19.01	19.18	19.29	15.32	23.08	.014
	ES Fortran	15.4	15.66	15.59	11.71	19.47	.034
	ES Palazzo	9.54	9.87	9.84	5.86	13.58	.049
	ES Paroli	11.24	11.36	11.28	7.59	15.35	.005
	Gazele	14.31	14.54	14.61	10.62	18.36	.024
	NK Cooler	16.24	16.46	16.54	12.54	20.29	.003
	NK Ravello	10.04	10.22	10.23	6.33	14.08	.001
	PR 39 G 12	17	17.28	17.19	13.3	21.11	.019
	PR 39K 13	12.45	12.66	12.55	8.77	16.47	.003
	SY Cooky	14.64	14.58	14.49	10.94	18.61	.016
	SY Mascotte	13.54	13.49	13.73	9.73	17.53	.032
GY	Clarica	17.57	17.79	17.75	13.91	21.49	.012
	Delitop	6.57	6.84	6.85	2.94	10.68	.017
	Drim	14.67	14.92	14.88	11.01	18.62	.025
	ES Fortran	4.67	4.95	4.91	1.99	8.78	.035
	ES Palazzo	3.54	3.79	3.77	0.19	7.77	.026
	ES Paroli	5.14	5.18	5.21	1.49	9.18	.026
	Gazele	15.08	15.31	15.31	11.33	19.19	.001
	NK Cooler	12.99	13.03	13.17	9.31	17.16	.045
	NK Ravello	11.05	11.19	11.22	7.36	15.02	.025
	PR 39 G 12	7.73	7.92	8.07	4.16	11.73	.011
	PR 39K 13	8.16	8.49	8.46	4.36	12.28	.016
	SY Cooky	10.38	10.64	10.63	6.66	14.4	.003
	SY Mascotte	16.47	16.77	16.71	12.81	20.43	.015
LE	Clarica	12.74	12.77	13.02	9.17	16.73	.028
	Delitop	23.54	23.68	23.73	19.83	27.62	.031
	Drim	24.54	24.58	24.73	20.68	28.57	.015
	ES Fortran	24.21	24.25	24.51	20.52	28.34	.016
	ES Palazzo	25.57	25.31	25.56	21.91	29.59	.041
	ES Paroli	27.01	27.05	27.38	23.43	32.01	.001
	Gazele	11.07	11.33	11.35	9.41	14.18	.003
	NK Cooler	19.33	19.55	19.61	15.65	24.35	.034
	NK Ravello	18.45	18.61	18.54	14.79	22.96	.005
	PR 39 G 12	16.94	17.12	17.22	13.25	21.35	.015
	PR 39K 13	15.48	15.73	15.74	11.73	19.48	.019
	SY Cooky	21.54	21.74	21.79	17.17	25.52	.028
	SY Mascotte	14.03	14.25	14.32	10.16	19.01	.009
ML	Clarica	6.94	7.02	7.12	3.06	12.16	.024
	Delitop	9.57	9.82	9.87	5.89	13.98	.007
	Drim	10.04	10.11	10.38	6.29	16.01	.014
	ES Fortran	15.57	15.15	15.83	11.89	20.51	.031
	ES Palazzo	17.44	17.29	17.73	13.65	21.65	.026
	ES Paroli	19.01	19.15	19.29	15.32	23.72	.017
	Gazele	6.57	6.41	6.85	2.91	10.79	.047
	NK Cooler	19.87	20.02	20.12	16.22	23.81	.011
	NK Ravello	11.64	11.85	11.83	7.95	15.77	.022
	PR 39 G 12	10.17	10.45	10.46	6.59	14.39	.037
	PR 39K 13	16.17	16.42	16.45	12.57	20.47	.007
	SY Cooky	8.45	8.33	8.66	4.67	12.96	.006
	SY Mascotte	9.67	9.74	9.87	6.66	13.54	.014
NE	Clarica	16.87	17.01	17.17	13.21	21.18	.012
	Delitop	26.54	26.65	26.82	22.83	31.05	.031
	Drim	22.14	22.22	22.41	18.42	26.46	.005
	ES Fortran	14.24	14.15	14.52	10.55	18.77	.036
	ES Palazzo	10.87	11.09	11.06	7.21	15.22	.041
	ES Paroli	12.47	12.31	12.73	8.78	16.84	.016
	Gazele	20.17	20.11	20.36	16.55	24.81	.023
	NK Cooler	18.94	19.04	19.13	15.28	24.07	.018
	NK Ravello	18.45	18.45	18.54	14.71	22.64	.021
	PR 39 G 12	30.87	30.11	31.06	27.22	36.01	.015
	PR 39K 13	28.45	27.69	28.44	24.66	32.64	.001
	SY Cooky	19.54	19.32	19.48	15.73	24.31	.016
	SY Mascotte	24.67	24.51	24.67	21.02	29.03	.024

Table 5. Mean, median, mode, higher posterior density intervals (HPD), and p-value of the Geweke test for the posterior distribution of the estimates of relative variation index for individual genotypes.

Traits	Cultivar	Descriptive analysis			HDP (95%)		p-value (Geweke)
		Mean	Median	Mode	Infimum	Supremum
NPH	Clarica	1.624	1.642	1.641	0.872	2.194	.021
	Delitop	1.701	1.718	1.717	0.949	2.272	.023
	Drim	1.789	1.811	1.802	1.038	2.361	.005
	ES Fortran	0.993	1.021	1.009	0.242	1.562	.004
	ES Palazzo	0.954	0.962	0.971	0.201	1.526	.031
	ES Paroli	0.995	1.011	1.007	0.242	1.56	.014
	Gazele	1.406	1.403	1.414	0.651	1.961	.021
	NK Cooler	1.125	1.112	1.141	0.373	1.707	.027
	NK Ravello	1.294	1.307	1.312	0.539	1.806	.023
	PR 39 G 12	1.514	1.532	1.535	0.766	2.084	.009
	PR 39K 13	1.414	1.421	1.431	0.666	1.993	.023
	SY Cooky	1.234	1.249	1.252	0.481	1.814	.006
	SY Mascotte	1.124	1.138	1.128	0.372	1.695	.002
GY	Clarica	1.457	1.47	1.463	0.709	2.033	.002
	Delitop	0.846	0.863	0.851	0.099	1.411	.021
	Drim	0.914	0.927	0.921	0.162	1.482	.022
	ES Fortran	0.694	0.708	0.711	0.056	1.269	.018
	ES Palazzo	0.671	0.682	0.685	0.079	1.244	.023
	ES Paroli	0.706	0.719	0.721	0.044	1.376	.047
	Gazele	0.735	0.742	0.748	0.015	1.305	.023
	NK Cooler	0.727	0.734	0.743	0.023	1.272	.041
	NK Ravello	1.184	1.199	1.201	0.534	1.759	.015
	PR 39 G 12	1.006	1.017	1.019	0.356	1.589	.019
	PR 39K 13	1.102	1.109	1.108	0.392	1.672	.011
	SY Cooky	1.299	1.317	1.305	0.519	1.873	.014
	SY Mascotte	1.406	1.422	1.402	0.566	1.977	.009
LE	Clarica	2.125	2.141	2.123	1.775	2.699	.017
	Delitop	1.841	1.852	1.852	1.491	2.411	.029
	Drim	2.287	2.299	2.301	1.937	2.851	.014
	ES Fortran	2.008	2.025	2.022	1.367	2.582	.026
	ES Palazzo	2.415	2.431	2.422	1.775	2.983	.032
	ES Paroli	2.574	2.588	2.579	1.994	3.166	.021
	Gazele	1.157	1.178	1.143	0.721	1.727	.009
	NK Cooler	1.578	1.591	1.591	0.898	2.148	.029
	NK Ravello	1.945	1.961	1.959	1.495	2.509	.011
	PR 39 G 12	1.254	1.263	1.269	0.594	1.823	.007
	PR 39K 13	1.408	1.415	1.421	0.671	1.979	.009
	SY Cooky	2.001	2.014	2.015	1.512	2.569	.017
	SY Mascotte	1.947	1.961	1.954	1.313	2.517	.011
ML	Clarica	4.037	4.044	4.022	3.281	4.601	.021
	Delitop	2.456	2.473	2.454	1.799	3.028	.016
	Drim	2.945	2.952	2.939	2.206	3.519	.009
	ES Fortran	3.224	3.238	3.224	2.704	3.801	.028
	ES Palazzo	3.254	3.262	3.257	2.593	3.833	.021
	ES Paroli	3.145	3.157	3.159	2.584	3.715	.01
	Gazele	1.446	1.459	1.462	0.961	2.021	.036
	NK Cooler	5.208	5.222	5.221	4.584	5.78	.008
	NK Ravello	3.25	3.261	3.256	2.509	3.827	.007
	PR 39 G 12	3.547	3.554	3.561	2.799	4.119	.026
	PR 39K 13	4.519	4.506	4.515	3.869	5.093	.015
	SY Cooky	3.047	3.044	3.033	2.504	3.618	.012
	SY Mascotte	4.467	4.444	4.476	3.783	5.039	.009
NE	Clarica	2.301	2.308	2.307	1.595	2.892	.036
	Delitop	2.494	2.501	2.502	1.924	3.063	.037
	Drim	2.348	2.365	2.344	1.948	2.922	.022
	ES Fortran	1.647	1.659	1.639	0.954	2.216	.029
	ES Palazzo	1.502	1.509	1.511	0.794	2.073	.037
	ES Paroli	1.729	1.738	1.742	0.991	2.403	.009
	Gazele	3.778	3.783	3.779	2.903	4.348	.014
	NK Cooler	2.016	2.027	2.031	1.457	2.581	.011
	NK Ravello	2.017	2.031	2.023	1.507	2.582	.019
	PR 39 G 12	4.073	4.089	4.079	3.723	4.648	.008
	PR 39K 13	3.941	3.935	3.939	3.292	4.533	.004
	SY Cooky	2.047	2.021	2.053	1.299	2.629	.007
	SY Mascotte	3.847	3.853	3.843	3.196	4.431	.017

Table 6. Mean, median, mode, higher posterior density intervals (HPD), and p-value of the Geweke test for the posterior distribution of the estimates of genotypic correlation for individual genotypes.

Traits	Cultivar	Descriptive analysis			HDP (95%)		p-value (Geweke)
		Mean	Median	Mode	Infimum	Supremum
NPH – GY	Clarica	0.601	0.643	0.636	0.551	0.691	.024
	Delitop	0.597	0.621	0.612	0.491	0.704	.015
	Drim	0.632	0.629	0.623	0.549	0.768	.009
	ES Fortran	0.619	0.616	0.607	0.501	0.714	.034
	ES Palazzo	0.622	0.626	0.615	0.597	0.694	.041
	ES Paroli	0.547	0.641	0.65	0.443	0.641	.027
	Gazele	0.599	0.587	0.693	0.501	0.734	.024
	NK Cooler	0.584	0.599	0.682	0.357	0.748	.008
	NK Ravello	0.608	0.612	0.607	0.534	0.781	.012
	PR 39 G 12	0.601	0.623	0.617	0.501	0.777	.023
	PR 39K 13	0.678	0.651	0.662	0.529	0.801	.017
	SY Cooky	0.666	0.659	0.667	0.533	0.727	.016
	SY Mascotte	0.694	0.685	0.679	0.624	0.814	.025
NPH – LE	Clarica	0.021	0.019	0.022	−0.239	0.108	.047
	Delitop	−0.254	−0.273	−0.281	−0.301	−0.016	.015
	Drim	−0.197	−0.201	−0.199	−0.294	0.001	.021
	ES Fortran	−0.241	−0.254	−0.244	−0.357	−0.011	.016
	ES Palazzo	−0.273	−0.272	−0.267	−0.401	−0.084	.037
	ES Paroli	−0.282	−0.279	−0.281	−0.337	−0.026	.009
	Gazele	−0.301	−0.312	−0.317	−0.371	−0.287	<.001
	NK Cooler	−0.299	−0.281	−0.301	−0.399	0.011	.022
	NK Ravello	−0.254	−0.266	−0.259	−0.297	−0.002	.008
	PR 39 G 12	−0.407	−0.383	−0.394	−0.501	−0.127	.014
	PR 39K 13	−0.259	−0.255	−0.261	−0.343	−0.018	.026
	SY Cooky	−0.255	−0.269	−0.259	−0.374	0.016	.017
	SY Mascotte	−0.261	−0.264	−0.27	−0.346	0.022	.003
NPH – ML	Clarica	0.602	0.587	0.601	0.473	0.712	.016
	Delitop	0.584	0.579	0.589	0.403	0.691	.021
	Drim	0.588	0.567	0.573	0.427	0.747	.033
	ES Fortran	0.561	0.551	0.557	0.339	0.659	.017
	ES Palazzo	0.704	0.692	0.703	0.501	0.917	.026
	ES Paroli	0.624	0.631	0.628	0.418	0.734	.017
	Gazele	0.578	0.583	0.671	0.499	0.657	.018
	NK Cooler	0.582	0.579	0.585	0.507	0.714	.006
	NK Ravello	0.594	0.582	0.591	0.484	0.728	.007
	PR 39 G 12	0.577	0.578	0.573	0.436	0.655	.015
	PR 39K 13	0.589	0.571	0.582	0.491	0.809	.036
	SY Cooky	0.547	0.546	0.55	0.401	0.678	.024
	SY Mascotte	0.491	0.501	0.495	0.367	0.602	.009
NPH – NE	Clarica	0.782	0.789	0.773	0.657	0.901	.017
	Delitop	0.822	0.831	0.819	0.704	0.914	.003
	Drim	0.791	0.802	0.795	0.597	0.899	.014
	ES Fortran	0.818	0.821	0.815	0.699	0.953	.028
	ES Palazzo	0.782	0.789	0.786	0.658	0.952	.017
	ES Paroli	0.911	0.907	0.909	0.822	0.998	.001
	Gazele	0.809	0.791	0.794	0.658	0.936	.012
	NK Cooler	0.754	0.766	0.751	0.587	0.879	.015
	NK Ravello	0.829	0.82	0.818	0.709	0.947	.002
	PR 39 G 12	0.658	0.667	0.661	0.487	0.793	.046
	PR 39K 13	0.691	0.683	0.684	0.361	0.773	.039
	SY Cooky	0.814	0.807	0.808	0.659	0.901	.019
	SY Mascotte	0.794	0.788	0.792	0.667	0.892	.024
GY – LE	Clarica	−0.099	−0.102	−0.095	−0.203	0.184	.026
	Delitop	−0.117	−0.109	−0.115	−0.228	0.025	.034
	Drim	−0.124	−0.123	−0.117	−0.294	0.088	.038
	ES Fortran	−0.097	−0.106	−0.101	−0.198	0.106	.049
	ES Palazzo	−0.104	−0.109	−0.099	−0.239	0.001	.041
	ES Paroli	−0.222	−0.219	−0.224	−0.417	−0.055	.016
	Gazele	−0.112	−0.107	−0.118	−0.295	0.037	.049
	NK Cooler	−0.109	−0.114	−0.093	−0.306	0.124	.045
	NK Ravello	−0.096	−0.107	−0.102	−0.203	0.204	.049
	PR 39 G 12	−0.114	−0.123	−0.107	−0.323	0.071	.036
	PR 39K 13	−0.097	−0.095	−0.105	−0.191	0.229	.045
	SY Cooky	−0.173	−0.17	−0.167	−0.404	0.158	.027
	SY Mascotte	−0.118	−0.109	−0.123	−0.339	0.204	.031
GY – ML	Clarica	0.286	0.291	0.283	0.181	0.557	.036
	Delitop	0.302	0.299	0.301	0.174	0.561	.039
	Drim	0.289	0.297	0.288	0.134	0.492	.015
	ES Fortran	0.357	0.351	0.349	0.205	0.618	.017
	ES Palazzo	0.291	0.284	0.286	0.169	0.579	.031
	ES Paroli	0.297	0.302	0.289	0.174	0.492	.015
	Gazele	0.273	0.279	0.269	0.099	0.407	.027
	NK Cooler	0.305	0.291	0.296	0.167	0.573	.005
	NK Ravello	0.278	0.271	0.284	0.149	0.357	.011
	PR 39 G 12	0.012	0.027	0.017	−0.155	0.148	.038
	PR 39K 13	0.173	0.168	0.17	−0.094	0.297	.018
	SY Cooky	0.289	0.278	0.284	0.085	0.618	.024
	SY Mascotte	0.276	0.282	0.279	0.076	0.553	.036
GY – NE	Clarica	0.731	0.745	0.729	0.697	0.834	.001
	Delitop	0.687	0.671	0.691	0.628	0.792	.007
	Drim	0.314	0.309	0.316	0.216	0.501	.049
	ES Fortran	0.729	0.731	0.724	0.658	0.851	.001
	ES Palazzo	0.681	0.669	0.672	0.521	0.793	.023
	ES Paroli	0.844	0.835	0.841	0.792	0.922	<.001
	Gazele	0.703	0.692	0.705	0.591	0.857	.014
	NK Cooler	0.657	0.661	0.65	0.487	0.801	.027
	NK Ravello	0.522	0.501	0.511	0.395	0.658	.019
	PR 39 G 12	0.241	0.198	0.227	0.094	0.417	.027
	PR 39K 13	0.309	0.273	0.286	0.147	0.492	.04
	SY Cooky	0.682	0.669	0.683	0.548	0.801	.016
	SY Mascotte	0.771	0.782	0.765	0.662	0.82	.021
LE – ML	Clarica	0.352	0.341	0.359	0.298	0.467	.009
	Delitop	0.367	0.378	0.351	0.301	0.494	.012
	Drim	0.392	0.401	0.386	0.314	0.438	.025
	ES Fortran	0.297	0.327	0.317	0.248	0.367	.034
	ES Palazzo	0.384	0.378	0.381	0.307	0.478	.014
	ES Paroli	0.383	0.384	0.382	0.298	0.451	.001
	Gazele	0.278	0.281	0.267	0.195	0.358	.042
	NK Cooler	0.374	0.384	0.377	0.284	0.447	.017
	NK Ravello	0.329	0.317	0.327	0.197	0.494	.008
	PR 39 G 12	0.127	0.117	0.131	0.008	0.257	.033
	PR 39K 13	0.098	0.101	0.087	−0.032	0.193	.048
	SY Cooky	0.378	0.386	0.364	0.258	0.504	.026
	SY Mascotte	0.072	0.069	0.077	−0.173	0.173	.037
LE – NE	Clarica	−0.051	−0.049	−0.05	−0.125	0.085	.011
	Delitop	−0.049	−0.042	−0.047	−0.121	0.093	.034
	Drim	−0.034	−0.029	−0.036	−0.097	0.101	.033
	ES Fortran	−0.147	−0.129	−0.138	−0.237	0.007	.027
	ES Palazzo	−0.109	−0.099	−0.102	−0.287	0.091	.017
	ES Paroli	−0.201	−0.187	−0.198	−0.358	−0.143	.008
	Gazele	−0.047	−0.041	−0.039	−0.157	0.102	.049
	NK Cooler	−0.007	0.001	−0.005	−0.057	0.073	.041
	NK Ravello	0.004	0.011	0.007	−0.033	0.057	.032
	PR 39 G 12	0.012	0.009	0.015	−0.069	0.093	.047
	PR 39K 13	0.028	0.015	0.017	−0.147	0.117	.045
	SY Cooky	−0.047	−0.064	−0.055	−0.107	0.046	.039
	SY Mascotte	−0.053	−0.048	−0.06	−0.144	0.077	.037
ML – NE	Clarica	0.476	0.467	0.481	0.421	0.501	.026
	Delitop	0.404	0.395	0.406	0.357	0.459	.014
	Drim	0.433	0.429	0.447	0.369	0.488	.018
	ES Fortran	0.602	0.597	0.614	0.552	0.638	.009
	ES Palazzo	0.558	0.552	0.561	0.514	0.601	.017
	ES Paroli	0.781	0.792	0.782	0.747	0.817	<.001
	Gazele	0.447	0.433	0.448	0.386	0.485	.015
	NK Cooler	0.364	0.354	0.369	0.308	0.402	.025
	NK Ravello	0.397	0.401	0.399	0.327	0.448	.021
	PR 39 G 12	0.124	0.117	0.127	−0.002	0.201	.036
	PR 39K 13	0.188	0.192	0.179	0.117	0.257	.047
	SY Cooky	0.453	0.45	0.459	0.407	0.502	.007
	SY Mascotte	0.433	0.429	0.441	0.373	0.494	.008

The highest correlation estimates were found between (1) the number of plants before harvest and the number of ears (0.826), and (2) the grain yield and the number of ears (0.755) (Table 1). Lower correlation estimates were found between the length of the ears and the number of ears (−0.100) as well as the grain yield and the length of the ears (−0.170), which were not significantly different from zero.

The cultivars Clarica and Drim presented the highest stem diameters, which were significantly different only from the ES Palazzo and ES Paroli, according to the credibility interval. Cultivar ES Palazzo had the lowest grain yield, which was not significantly different from other cultivars. The cultivar with the lowest length of the ears was Gazele, and it did not significantly differ from the Clarica and PR 39K 13 cultivars. For the mass of leaves, the best results were found for the NK Cooler cultivars. The highest number of ears was observed for PR 39 G 12, and this value was significantly different from the number of ears for ES Palazzo, ES Paroli and Gazele cultivars. The genetic values for individual cultivars were presented in Tables 2–5: estimates of heritability (Table 2), coefficient of genotypic variation (Table 3), coefficient of residual variation (Table 4), relative variation index (Table 5), and genotypic correlation (Table 6).

The grain yield presented the lowest gain with selection estimates. For this trait, the mode of the posterior distribution was −15%. The other traits turned out higher estimates for gain with the selection. The number of ears, the mass of leaves and the length of the ears had higher estimates of gain with the selection.

Discussion

The obtained asymmetric credible intervals for the variance components, breeding values and genetics parameters are a peculiarity of Bayesian inference. They make this approach notably informative (Mathew et al. 2012) and facilitative of hypothesis testing. Asymmetric credibility intervals obtained using posterior distribution, and shown in Tables 1–6, make the conclusions more realistic than those based on symmetric confidence intervals of frequentist statistics (Apiolaza et al. 2011). Additionally, Bayesian inference allows the evaluation of unbalanced experiments and the study of more complex statistical models (Bink et al. 2008). Consequently, its use has become more common among breeders not only for the analysis of molecular data but also for phenotypic data (Kärkkäinen and Sillanpää 2012; Taylor and Friesen 2012). The results of the Geweke test presented in Tables 1–6 showed the reliability of the results for all analysed parameters. Therefore, it is reasonable to posit that the samples are truly representative in the underlying stationary distribution of the Markov chain (Cowles and Carlin 1996).

As has been shown in Tables 1–6, the lowest heritability estimate found for the grain yield is the consequence of the lowest genotypic variation because the residual variation was low when compared with the other traits (Table 1). The grain yield also turned out the lowest estimate of genetic variation, which indicates a difficulty for the improvement of grain yield in this set of cultivars. However, there was an overlap of the credibility intervals between the traits with regard to the heritability and the coefficient of relative variation (Tables 1–6). This outcome indicates that these estimates were not significantly different. The number of plants before harvest is not only influenced by genetic factors but also by environment effects, such as the weather conditions of the growing season (Synowiec et al. 2017). The reduction in the length of the ears is advantageous because it reduces the need for cultural practices. From a commercial point of view, another advantage of a lower length of the ears is the reduction of the potential for vegetative propagation of the plant, which can increase the continuous sale of seeds.

The correlation study turned out favourable associations for the breeding process between (1) the number of plants before harvest and the grain yield, (2) the number of plants before harvest and the mass of leaves as well as (3) the number of plants before harvest and the number of ears. This result indicates that the selection of genotypes characterised by a larger number of plants before harvest can provide a reduction in grain yield and increase the mass of leaves and the number of ears. The selection of genotypes with a lower grain yield may facilitate an increase in the mass of leaves and the number of ears, which is also desirable. However, the selection taking the lowest length of the ears into account can cause an increase in grain yield and reduce the mass of leaves and the number of ears, which is undesirable. According to Falconer and Mackay (1996), the correlations between traits can be a result of pleiotropy or genetic linkage. Thus, if the undesirable correlations are caused by genetic linkage, these associations can be broken through recombination caused by crossing or self-fertilisation; consequently, these factors do not necessarily become major impediments to breeding programmes (Szulc and Bocianowski 2011b; Liersch et al. 2016; Radkowski et al. 2017). The increase in the maize yield is the result of a greater use of nitrogen fertilisers. On the other hand, lack of dose optimisation may have a huge negative impact on the natural environment. Therefore, it is necessary to have knowledge of the effectiveness of the fertiliser, with a simultaneous application of a crop control system (Erley et al. 2005). The improvement of agrotechnical applications of nitrogen fertilisers should always aim at increasing the use of the mineral fertiliser component (Szulc et al. 2016). On the other hand, breeding new varieties which more efficiently use nitrogen and, as a consequence, who feature a greater capability to absorb microelements from the soil and to manage the nitrogen collected, in synergy constitute a complementary strategy for managing the nutrient in the modern agriculture. Knowledge of genetics, physiology, as well as information about nitrogen collecting and utilisation, is indispensable in the process of breeding new varieties which are to be characterised by an effective nitrogen economy. The high population variability of maize, according to Paponov et al. (2005), enhances chances of finding potent genotypes, thanks to which it is and will be possible to obtain varieties which produce efficiently, also under conditions of moderate stress of nitrogen deficiency. The observed large variation among maize genotypes in terms of nitrogen utilisation indicates that this characteristic is genetically determined and can be improved in the process of breeding. New crop varieties use the nitrogen better because their physiological features associated with the creation of the crop are comprehensively improved and corrected. The way in which plants use the accumulated nitrogen has a significant impact on the size of grain yield (Szulc and Bocianowski 2011a). It has been shown that a well-developed root system of maize is an important factor in the nitrogen uptake throughout the vegetation period of the plant. Therefore, the early creation of a large root system is crucial for increasing the efficiency of nitrogen uptake (Niu et al. 2007). Complementary action strategy comprises improving the efficiency of the use of nitrogen already collected by the plant, operationalised as using smaller amounts of nitrogen to produce as large a mass of assimilates as possible. When compared to hybrids which use nitrogen inefficiently, a hybrid well-managed with the component of nitrogen utilisation is characterised by a higher rate of photosynthesis during the pouring of kernels, even when the concentration of nitrogen in vegetative organs is comparable (Chen et al. 2006). The only possible explanation for this phenomenon is the increase in the efficiency of nitrogen transformation in the genotype. According to Hirel et al. (2007), increasing the absorption of solar radiation by plants grown on less soils indirectly increases the use of nitrogen. Furthermore, stay-green hybrids have a larger assimilation area than traditional varieties do (Subedi and Ma 2005). Therefore, it is possible indirectly to explain the greater use of nitrogen by the type of hybrids.

In addition to the commercial cultivars, the following genotypes are recommended to integrate into breeding programmes: Clarica because it is one of the genotypes with the largest grain yield; NK Cooler because it is one of the genotypes with the largest mass of leaves; Drim because it is one of the cultivars with the highest production of plants before harvest; and PR 39K 13 because it is one of the cultivars with the highest production of ears. The estimates of gain with selection found in this study indicate the potential of selection for the genetic improvement of the genotypes evaluated. The rules for integrated plant protection emphasise the application of all possible and available methods reducing the development of a population of harmful organisms to a level of harmlessness. Guidelines for integrated protection indicate the priority of using non-chemical methods in pests and diseases control before applying chemical treatments. One of the easiest methods of protecting plants from pests and diseases may be using resistant plant varieties, therefore understanding resistance and adaptation in crop varieties is of great importance. All disturbances in the system of environmental factors cause an imbalance in the plant life activities, becoming the cause of stress. Maize yields are limited by a number of abiotic and biotic factors. Hence, crop production may fail to meet the objectives without introducing technological solutions required by the new varieties. The main reasons for the low level of maize yield are due to too acidic soils, insufficient supply of nutrients, as well as maladjustment of the fertilisation system to quantitative needs, and especially the dynamics of nutrient uptake by plants throughout the growing season. Periodic shortages of water in the soil (drought) during the growing season, limit the yield potential of new maize varieties. Due to unpredictable weather conditions, one of the main challenges for breeders is to create hybrids with features that give an advantage in an environment with limited water resources. Breeding works should be directed towards greater plant health in the later growing season or/and work on increasing the greenness of the leaf, known as ‘stay-green’. This feature is an indicator of high plant health, slower aging rate and tolerance to drought after flowering. The ‘stay-green’ feature shows a positive correlation with the yielding of plants. The conducted studies clearly showed that stay-green hybrids are definitely better compared to traditional varieties.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes on contributors

Jan Bocianowski has an M.Sc. in mathematics awarded in 1998; he achieved a Ph.D. in agriculture six years later in 2004. He is an interdisciplinary researcher, specialising in biometry and statistics for agriculture, plant breeding, biology and genetics. Jan has authored or co-authored over 300 scientific publications, of which around 150 were published in journals listed in Journal Citation Reports by Thompson Reuters.

Kamila Nowosad achieved a Ph.D. in agriculture in 2009. Her main research area is plant genetics, agrobiotechnology and plant breeding.

Piotr Szulc obtained his master's degree in 1998. He obtained the Ph.D. degree in 2003. Since 2017 he has held the position of associate professor at the Department of Agronomy of the Poznan, University of Life Sciences. His research interests mainly cover agrotechnics of agricultural plants with particular emphasis on cereal plants. He is the author of many scientific publications on national and international coverage.

Anna Tratwal achieved a Ph.D. in agriculture in 2006, since 2015 she has held the position of associate professor of the Institute of Plant Protection – National Research Institute. She is the author of about 170 scientific publications on national and international coverage. Within the framework of her professional activity she participate in seminars and post-graduate studies organised by IPP – NRI for the employees of National Health and Seed Inspection Service, Agricultural Advisory Center, agricultural producers and other parties interested in issues of signalling, forecasting and methods of monitoring of pests of cultivated plants.

Ewa Bakinowska has an M.Sc. in mathematics awarded in 1995; she achieved a Ph.D. in agriculture nine years later in 2004. She is an interdisciplinary researcher, specialising in statistics and biometry for agriculture, plant breeding and soil science. In the area of her scientific activities are, in particular, statistical analyses of categorical data using the logistic models. She is the author of many scientific publications on national and international coverage.

Dariusz Piesik graduated master's degree in 1995. He obtained the Ph.D. degree in 2000 and D.Sc. in 2009. Since 2018 he received the title of full professor. His research interests mainly cover pests of agricultural plants, volatile organic compounds released as consequence of abiotic and biotic stress, communication between plants. He is an author of many scientific publications on a global scale.

ORCID

Jan Bocianowski http://orcid.org/0000-0002-0102-0084

Kamila Nowosad http://orcid.org/0000-0001-6837-7806

Piotr Szulc http://orcid.org/0000-0002-9670-3231

Anna Tratwal http://orcid.org/0000-0001-9611-8799

Ewa Bakinowska http://orcid.org/0000-0002-1211-5046

Dariusz Piesik http://orcid.org/0000-0003-4192-9783