Bayesian Credible Intervals for Maize Grain Yields of the Maintenance Varieties Evaluated in Sudan

Improved of maintenance crop varieties developed at agricultural research corporation, Wad median, Sudan is intended for seed foundation which is recently established. This study was undertaken to establish statistical investigation using Bayesian estimation for credible interval or posterior interval as a Bayesian strategy for a maintenance variety. Data on grain yield (kg/ha) on a maize crop variety were used. Bayesian posterior information can be annoying to investigate but are important in maintenance varieties that foundational claims are used to make general recommendations for practice. Half normal informative priors set were used. The heritability of yield (varieties)) was (h = 0.75). Predicted posterior means of varieties were shown with a Bayesian interval for scientific inference in the maintenance maize grain yield. Bayesian approach is useful for reducing uncertainty on decisions based on economic evaluation of new maize varieties in Sudan, the use of credible intervals for grain yield allow for early decisions.


Introduction
The maize crop plays an active role in the world economy and widely used for trade (Olson et al., 2012).FAO (2007) reported that the demand of Maize in sub-Saharan Africa between increasing by 50% as worldwide and by 93% up to 2020.Also maize remains an important source of food with great potential to improve the livelihoods of most poor farmers in developing countries (Smale et al., 2001).To maintain maize varieties will assess the farmers to be in level of adoption for determining the costs and technologies to be chosen and selected for variety development programs (Feder et al., 1981).Prior information or identification of farmers' constraints and preferences is required in order to plan more appropriate, acceptable and cost effective development intervention programs for effective agricultural research and technology development, and their adoption (Thijssen et al., 2008).
Nowadays, many crop yields can be affected by climatic factors such as precipitation, temperature and recent warming (Lobell and Burke, 2010).Also the impact of climate change may damage crop yields on a global scale

Research Article
and lead to decreases in crop production (Chen et al., 2014).In rain-fed agricultural regions, also weather conditions have substantial impacts on crop productivity (Yu, 2011).The maintenance of forms can be managed through three phases of seed multiplication: breeder's seed, foundation seed, and certified seed.Therefore, we found the breeder's seed field should show the minimum variation for morphological traits to help guide quality control during multiplication, but not impede seed production and distribution (Rajhandari et al., 2001).Maintenance of varieties is preferred to be developing in the regions of their natural adaptation to minimize developmental shifts (NASA, 2000).Therefore, Bayesian approach can be very useful in maintenance varieties, seed production, because Bayesian credible information can help to understand the probability distribution over the parameters through rich statistical inferences of means (Omer et al., 2015).Bayesian approach in analysis crop variety trials have been discussed by many authors (Gelman et al., 2002): Theobald et al. (2002); Theobald and Talbot (2002) Omer et al. (2015) and Singh et al. (2015).Bayesian credible intervals based on Bayesian theories are conceptually different ways to quantify parametric and predictive uncertainties, because are always numerically identical when consistent prior information is used (Lu et al., 2012).

Objectives of the study
The purpose of this study is to use Bayesian approach for estimating posterior interval for maize grain yield of maintenance varieties and to estimate the heritability and genetic gain due to selection of maize varieties.

Plant Material and Growth Conditions
The field experiment was carried out during seasons 2014-2015 at Gezira Research Station in agricultural research corporation, Wad Medani, Sudan.The trial was designed as a randomized complete block design (RCBD) with 3 replications and the 16 new maize entries during 2015.Data on grain yield was used.To meet the objectives of this of objective, priors from past dataset will used.The performance of the methods developed will be evaluated using real maintenance variety data.Bayesian analysis will provide results from two types of priors' information for comparison of a differentiating characteristic, with the results analyzed separately using REML analysis.Bayesian evidences such credible interval is appropriate statistical information in this case that prevented the normal distribution and/or confidence of the data.However, Bayesian posterior means of genotypes has been compared with the frequentist means.In this paper, Bayesian estimation will based commonly used statistical tools such as coefficient of variation (CV%), predicted means of genotypes, with standard errors and confidence intervals, average standard error of differences of predicted means, heritability, genetic gain due to selection, realized gain due to selection, and ranking of genotypes for selection.We ran the MCMC algorithm on three separate chains for 100000 iterations each, and we saved every fifth iteration among the last 5000 to form a posterior sample of size 1000 for each of the three chains.

Statistical Methods
Recently, Bayesian method has been widely applied in many different sciences for decision analyses, the feature Bayesian method is that priors are superior to frequentist approach (Lin, 2013).In agricultural experiments, the Bayesian analysis was carried out using data from previous trials (Besag et al., 1999).Some advantages of use Bayesian analysis are easy to interpret confidence intervals and p-values in direct way (Besag and Higdon, 1999).In other words, Bayesian interval has a 95% chance of containing the true parameter.The Benefits of Bayesian approaches that the prior information allows partial pooling of different data sources (Gelman, 2014).

Frequentist Confidence and Bayesian Credible Interval
The confidence interval provides a range that is highly likely (often 95% or 99%) to contain the true parameter that is being estimated.A confidence interval for the mean X from a normal population, can be written as Where t n−1 is the appropriate percentage point of the tdistribution with (n-1) degrees of freedom.The 95% confidence interval is commonly used, for which t-values are 2.2, 2,1 and 2.0 for 10, 20 and 30 degrees of freedom.So we can usually write that the 95% confidence interval for the mean is roughly: x ± 2 × s. e. (x) For example say variety-16 of maize crop gave x = 2221.5tons with s.e.= 92.47The 95% confidence interval for x is therefore about 2036.3 to 2406.19 kg/ha; and so we can say that that this range is likely to contain the population mean of maize grain yield.Therefore, exact 95% interval, which one can get from a statistical software package, is (2036.3 to 2406.19 kg/ha.).In other words, there is a 95% probability that the population mean is in the interval 2036.3 to 2406.19 kg/ha.More generally, for almost any estimate, whether it be a mean, or some other characteristic, and from almost any population distribution, we can write that the 95% confidence.A confidence interval does not contain 95% of the data in the sample that generated it; very approximately it called a prediction, or tolerance interval.In this an examples G16 mean with interval would say that most of the variety have yields in this range only.When the assumptions about the data may not be quite right, scientists may feel they ought to abandon the ordinary confidence interval and use some different procedure altogether.In frequentist approach, the confidence interval is expressing an area around the sample mean.Therefore, the interpretation is less desirable performing inference about based on a single interval (Yang, 2006).There are various ways, including the mean, median, mode of parameters posterior distribution, in which one can summarize the distribution, mostly point estimation is conducted through reporting the posterior mean.In Bayesian approach the interval estimate for parameters of interest is called credible interval or posterior distribution (Jaynes, 1976).The differences between the frequentist and Bayesian conference intervals are that in the Frequentist approach, the confidence interval can claim that the region covers the true parameter, reflecting uncertainty in sampling procedure or implies the interval covers the true parameter among 95% of the experiments (Baklizi et al., 1999).While Bayesian approach, credible Interval: a claim that the true parameter is inside the region with measurable probability (Box and Tiao, 1992).One can make a direct probability statement about parameters can implies the chance that the true parameter falls in (a, b) is 95% (Lin, 2013).A credible interval (or in general, a credible set) is the Bayesian analogue of a confidence interval.One advantage of the Bayesian procedures is the possibility of using the confidence limits (or credibility regions in Bayesian context) as an alternative estimation of genetic parameters (Rodovalho et al., 2014).Agresti and Min (2005) discussed Bayesian confidence intervals for association parameters.They argued that if one desires good coverage performance (in the frequentist sense) over the entire parameter space, it is best to use quite disuse priors.Even uniform priors are often too informative, and they recommended the Jeffrey's prior.In Bayesian approach the interval for parameters of interest (a) is called credible interval or posterior intervals.Any interval that has 95% posterior is called a 95% credible interval (CI).Given a posterior distribution intervalp(θ|y), A is a credible set for θ if p(θ ∈ A|y) = ∫ A p(θ|y)dθ.
For example the two sided 95% CI for: θ: p(θL <  < |data) = 0.95 or p(θ ≥ θU or θ ≤ θL|data) = 0.05, The credible can be construct using equal trails, where equal tai area interval divide the portability of the complements in two equal area: p(θ ≤ θ L |data) = p( θ ≥ θ U |data) = 0.025; another frequently used Bayesian credible set is called the Highest Posterior Density (HPD) Intervals (Lin, 2013).A HPD interval is a region that contains parameters with highest posterior density values (Hashemi et al., 1997).

Prior Distribution
A prior distribution of a parameter is the probability distribution that represents one's belief or associated uncertainty (Gelman, 2006).In Bayesian statistics, a prior distribution for µ with density p(µ) is chosen that reflects a researcher's knowledge and uncertainty about µ.For an example in a maize crop trial, suppose that a researcher has estimated that 5% of the genotypes are reliable and the researcher is 95% sure that the actual variation in genotypes means was no larger than 25% (Finley et al., 2011).In order to estimate the average of grain yield for maize crop based on a given sample size of 16 genotypes, the priors on means and variance of genotypic response may be collected using the data from past trials.A Bayesian statistician could then estimate that, given the prior knowledge of distribution of means, the mean response for the maize crop is normally distributed with a mean of, say 150 kg/ha and a standard deviation of, say 25 kg/ha.When the sample of say, 16 genotypes is collected, it is approximately normally distributed as the Bayesian inference or the posterior probability distribution considers both the prior probabilities and the data (Yi et al., 2013).

Results and Discussion
Bayesian information is presented here and found that there was a statistical significant of genotypes means.Bayesian estimation of predicted crop yields reflected the trend of the observed yields the estimated using classical approach, which Bayesian interference give wide inference of posterior.The choice of priors for Bayesian analysis was made from the statistics for RCBD models.The estimates of variance components and heritability and genetic gain using Bayesian approach are presented in Table 2.The posterior mean yield of the maintenances varieties, ranks, conference /creditable intervals are presented in Table 3 and  4.

Selection of Priors
The choices of priors for Bayesian analysis were made from the statistics given in Table 1 for maize data.The five prior sets of half normal distributions (P1 -P5) were used.The priors set derived from informative priors were used to build more suitable priors for maintenances varieties in the future.The values of DIC are different to each other for priors sets, however, the prior set P3 seems to have numerically lowest value of DIC (764.28).Therefore, the prior P3 was chosen for predicting performance of the genotypes, genetic parameters and credible intervals as presented in Table 3.
Table 2 shows the estimates of genetic gain (family selection) considering three selection rates (5% , 10% and 20%) calculated according to the predicted Bayesian values of popcorn families (posterior mode).Genetic gains were of 40%, 34% and 28% respectively for grain yield.The significant genetic variability among genotypes was 22.51% of the grain yield.Furthermore, the heritability value that we obtained was high expected for grain yield.Calculation of heritability and Genetic Gain showed that the heritability in the Genetic Gain with higher heritability tended to show higher prediction accuracies.Priors set are: P1: σ r , σ g and σ e independently ~ positive Half − normal (100, 500).

Maize Predicted Means under Bayesian Approach
The a posteriori means of the genotypes, 95% confidence intervals and their a posteriori ranks from alternative ways are presented in Table 3.The precision of the means under Bayesian approach is different than that of the frequentist approach (average SE of 0.375 vs 0.304.Correlation between the predicted values under the two approaches was 0.999 (p<0.001).The performance of the genotypes under Bayesian approach has been given also in terms of the percentiles of the predicted values and of their ranks.This has been possible due to availability of a large number of simulated values under the Bayesian model.Denoting the genotype numbered 1 to 16 by G1 to G16, the top 4 genotypes were G16, G1, G14, and G12 using Bayesian approach, which were also selected under frequentist approach.Selection based on quantiles of ranks of the genotypes appears quite desirable, instead of using only means, which is possible due to the availability of distribution of ranks from the computation process.When selecting the genotypes based on distribution of ranks, the top 4 genotypes using the a posteriori mean ranks and or median of the ranks were also G16, G1, G14, and G12.While the selection of the genotypes would be based on a posteriori mean rank or mean value, the distribution of ranks at the extreme quantiles also could be useful.The Fig. 1 indicates comparison of Bayesian and frequentist estimates of predicted mean of each genotypes, The correlation between frequentist and Bayesian approaches was highly significantly positive (0.99, P<0.001).The Bayesian and frequentist approaches for maize yield performance prediction were evaluated, which mostly differed in mean predicted and their ranks.These results were to explore the extent and pattern of confidante intervals of grain yield.The present investigation revealed that there was wide chance of containing the true means for the top majority of desired genotypes.Overall the results emphasize that investigating a maintenance varieties for adaptation using Bayesian approach should consider in future work.The necessary codes of the WinBUGS (Spiegelhalter et al., 2003) and R-package (R Development Core Team, 2009) will obtained from first author.

Conclusions
Bayesian analysis was used to assess the maintenance of varieties on the predication genotypes of interest and to be undertaken to adjust for any genotype estimation.The results of this experiment could be used for further statistical information for Bayesian inferences, the variance components will saved as prior information for more investigation for other varieties.Bayesian confidence / credible interval seem to be on of importance strategies for statistical information for seed and yield production based on maintenance of varieties.The additional information obtained in the proposed method, successfully evaluation rich statistical information like posterior information in detailed understanding of maintenance of varieties distributions.

Fig. 1 :
Fig. 1: Scatterplot of predicted means of grain yield (kg/ha) of the 16 maize genotypes for comparing frequentist mean, Bayesian posterior mean and Bayesian credible bands using 95% credible interval values for individual genotypes.

Table 1 :
Discrepancy statistics values for selection of the priors for data on maize maintainers varieties for grain yield based on half-normal distribution.

Table 3 :
Predicted values of the genotypes means, credible intervals and their ranks for a Bayesian approach for maintainers varieties for grain yield based on Half-normal Distribution.