Bayesian estimation of genotypic and phenotypic correlations from crop variety trials

Siraj Osman Omer Abdel Wahab H Abdalla Mohammed H. Mohammed Murari Singh About the authors

Abstract

Genotypic and phenotypic correlations are necessary for constructing indirect selection indices. Bayesian analysis, therefore, was applied to obtain posterior distributions of the correlations, and the estimates were compared with those under a frequentist approach. Three a priori distributions for standard deviation components based on uniform distribution, positive values from t- distribution, and positive values from normal distribution were examined, while a priori distribution for correlation was taken as a uniform distribution. The prior based on uniform was best found using the deviation information criterion. Data from sorghum genotypes evaluated in complete blocks in 2010-2011 in Northern Kordofan, Sudan, resulted in a posterior mean of 0.48 for genotypic correlation between seed yield and seed weight with posterior standard deviation of 0.24. Due to a wider inference base and the fact that it makes use of prior information, we recommend the Bayesian approach in estimation of genotypic correlations.

Key words:
Bayesian estimation; genotypic and phenotypic correlations; heritability; R2WinBUGS

INTRODUCTION

Genotypic and phenotypic correlations between plant traits are used as measures of their association (Ahmad et al. 2010Ahmad B, Khalli IH, Igbal M and Ur-Rahman H (2010) Genotypic and phenotypic correlation many yield components in bread wheat under normal and late planting. Sarhad Journal of Agriculture 26: 259-265.). Estimates of genotypic and phenotypic correlations between traits are useful in planning and evaluating breeding value (Desalegn et al. 2009Desalegn Z, Ratanadilok N and Kaveeta R (2009) Correlation and heritability for yield and fiber quality parameters of Ethiopian cotton (Gossypium hirsutum L.) estimated from 15 (diallel) crosses. Kasetsart Journal Natural Science 43: 1-11.). Knowledge of genotypic and phenotypic association among economically valuable traits can help plant breeders in identifying efficient breeding strategies for development of high yielding wheat cultivars (Abbasi et al. 2014Abbasi S, Baghizadeh A, Mohammadi-Nejad G and Nakhoda B (2014) Genetic analysis of grain yield and its components in bread wheat (Triticum aestivum L.). Annual Research & Review in Biology 24: 3636-3644.). Though estimation of genotypic correlations and phenotypic correlations is straightforward, evaluation of their precision in terms of standard errors and significance testing is quite cumbersome (Singh et al. 1997Singh M , Ceccarelli S and Grando S (1997) Precision of the genotypic correlation estimated from variety trials conducted in incomplete block designs. Theoretical and Applied Genetics 95: 1044-1048.). Over the course of experimentation, crop improvement programs gather information on genotypic and experimental error variability, which can be used in the Bayesian approach. In the Bayesian framework, one integrates prior information with the likelihood of current data and draws inferences in terms of conditional distribution of parameters of interest, given the data. In this process, an estimate of the parameter is assessed as posterior mean and precision as posterior standard deviation (Gelman et al. 2004Gelman A, Chew GL and Shnaidman M (2004) Bayesian analysis of serial dilution assays. Biometrics 60: 407-417.). In contrast, the commonly used frequentist approach does not make use of such information. Singh et al. (2015Singh M, Al-Yassin A and Omer S (2015) Bayesian estimation of genotypes means, precision and genetic gain due to selection from routinely used barley trials. Crop Science 55: 501-513. ) have presented a systematic approach for Bayesian analysis of trials conducted in complete or incomplete block designs. The priors discussed in their work have been incorporated in this study. This paper focuses on the Bayesian approach for estimation of genotypic and phenotypic correlations from crop variety trials and compares them with a frequentist approach.

The frequentist approach is normally based on estimation of variance components using a mixed model. The MIXED procedure in SAS software (SAS Institute 2011SAS Institute (2011) SAS/STAT(r) 9.2 User's guide. 2nd edn, SAS Institute, Cary, 7869p.) provides REML estimates of variance and covariance components among model factors and allows both fixed and random effects to be fitted in a mixed model analysis (Littell et al. 1998Littell RC, Henry PR and Ammerman CB (1998) Statistical analysis of repeated measures data using SAS procedures. Journal of Animal Science 76: 1216-1231.). Plant breeders have traditionally estimated genotypic and phenotypic correlations between traits using a multivariate analysis of variance (MANOVA) or a REML method (Hussain et al. 2012Hussain K, Khan IA, Sagat HA and Amjad M (2012) Genotypic and phenotypic correlation analysis of yield and fiber quality determining traits in upland cotton (Gossypim hirsutum). International Journal of Agricultural and Biology 12: 348-352. ). From the Bayesian perspective on genotypic and phenotypic correlations, posterior inference can be drawn using Markov Chain Monte Carlo (MCMC) methods (Tierney 1994Tierney L (1994) Markov chains for exploring posterior distributions (with discussion). Annual Statistics 22: 1701-1786. ). Schisterman et al. (2003Schisterman EF, Moysich KB, England LJ and Rao M (2003) Estimation of the correlation coefficient using the Bayesian approach and its applications for epidemiologic research. BMC Medical Research Methodology 3: 1-5.) investigated estimation of the correlation coefficient using the Bayesian approach and its applications in epidemiological research and found it useful for evaluating relationships between variables with measurement errors. More details on Bayesian estimation of correlation may be found in Liechty et al. (2004Liechty JC, Liechty MW and Muller P (2004) Bayesian correlation estimation. Biometrika 91: 1-14. ) for models providing a framework for representing and learning about dependence structures. The objective of this study is to estimate genotypic and phenotypic correlations and their standard errors using Bayesian and frequentist approaches when data on traits have been collected from a crop variety trial conducted in a randomized complete block design. The necessary computing codes are also provided using R2WinBUGS and R-packages.

MATERIAL AND METHODS

Experimental data

A set of 18 sorghum genotypes were evaluated in a randomized complete block design (RCBD) with four replications. The experiment was carried out in the 2010-2011 season at El Obeid Research Station, Agricultural Research Corporation (ARC), Northern Kordofan, Sudan. Plot-wise data on grain yield in kg ha-1 (GY) and 1000 seed weight in gm (SW) were recorded.

Estimation of genotypic and phenotypic correlation

Frequentist approach

In this approach, we consider estimation of genotypic correlation from a randomized complete block design (RCBD) data on two traits - X (for example, yield) and Y (for example, seed weight). The denotes the genotypic correlation between traits X and Y in a population of inbred lines. We consider v inbred lines are randomly selected from the population of interest and are evaluated in an RCBD with r replications in a single environment. The responses Xij and Yij from the plot of the genotype of the replicate are modeled as:

(1)

where for the two traits X and Y, and are general means, and are effects of the block, and are effects of the genotype sampled, and and are random errors, respectively (Singh and Hinkelmann 1992Singh M and Hinkelmann K (1992) Distribution of genotypic correlation coefficient and its transforms for non-normal populations. Sankhya Series B 54: 42-66.).

The parameter vector is assumed to be fixed. However, we make the following assumptions for the other vectors:

1- is bivariate normally distributed with mean vector, a variance-covariance matrix , and independent of for

2- is bivariate normally distributed with mean vector, a variance-covariance matrix, and independent of for

3- is bivariate normally distributed with mean vector , a variance-covariance matrix , and independent , where

4- The vectors, , and are pairwise independent of each other (Singh and Hinkelmann 1992Singh M and Hinkelmann K (1992) Distribution of genotypic correlation coefficient and its transforms for non-normal populations. Sankhya Series B 54: 42-66.).

Given the above background, the genotype correlation between traits X and Y is estimated as:

(2)

where is the estimated genotypic covariance between traits X and Y, is the estimated genotypic standard deviation for trait X, and is the estimated genotypic standard deviation for trait Y. Thus, the estimate of is obtained in terms of the estimates of the variance and covariance components , and . The variance components and can be estimated by using the residual (otherwise known as "restricted") maximum likelihood (REML) method (Patterson and Thompson 1971Patterson HD and Thompson R (1971) Recovery of inter-block information when block sizes are unequal. Biometrika 58: 545-554., Singh et al. 1997Singh M , Ceccarelli S and Grando S (1997) Precision of the genotypic correlation estimated from variety trials conducted in incomplete block designs. Theoretical and Applied Genetics 95: 1044-1048.). From the covariance obtained, we can construct a new variable Z with the plot-wise values as

(3)

where,

where

The genotypic variability of variable Z, denoted by , is expressed as:

, or (4)

Thus, the covariance component can be written in terms of variance components as

(5)

We now apply the REML method on values of Z to obtain an estimate of . Substituting the estimates of the three variance components in (5), we get an estimate where

Substituting the estimates of ,, and in (2), we obtain the estimate

In order to compute phenotypic correlation, we consider the additive model for the phenotypic value - phenotypic value = genotypic value + environmental effect. After ignoring the variation in controlled factors, if any, we can write the phenotypic variances and covariance as follows:

Using equation (3), the covariance can be obtained from the variance components ,and , where using

(6)

Thus, the phenotypic correlation and the environmental correlation between the traits X and Y are expressed as:

and (7)

Standard error of the estimates of phenotypic and environmental correlation can be obtained using Singh et al. (1997Singh M , Ceccarelli S and Grando S (1997) Precision of the genotypic correlation estimated from variety trials conducted in incomplete block designs. Theoretical and Applied Genetics 95: 1044-1048.) with the delta method. Similar approaches have been described by Miller et al. (1958Miller PA, Williams JC, Robinson HF and Comstock RE (1958) Estimates of genotypic and environmental variances and covariances in upland cotton and their implications in selection. Agronomy Journal 50: 126-131. ) using the corresponding variance and covariance components (Fikreselassie et al. 2012Fikreselassie M, Zeleke H and Alemayehu N (2012) Correlation and path analysis in Ethiopian fenugreek (Trigonella foenum-graecum L.) landraces. Crown Research in Education 3: 132-142.). The approach presented here is based on a univariate approach to variables X, Y, and Z=X+Y. An alternative approach is to use a multivariate formulation implemented in several software programs. In our experience, multivariate approaches more often resulted in non-convergence than the univariate approach (e.g., REML method) did. The variance components for X and Y were also used to estimate the broad-sense heritability of the traits on a mean basis, using the expression for trait X (as for trait Y), where r is the number of replications; see also Singh el al. (2015Singh M, Al-Yassin A and Omer S (2015) Bayesian estimation of genotypes means, precision and genetic gain due to selection from routinely used barley trials. Crop Science 55: 501-513. ). The estimation under the frequentist approach was carried out using Genstat software (Payne 2014Payne RW (2014) The guide to GenStat(r) command language (Release 17). Part 2: Statistics. VSN International, Hemel Hempstead, 1032p.).

Bayesian approach

Knowledge of a priori probability distribution of parameters of interest is required for making estimates under the Bayesian paradigm (Kizilkaya et al. 2002Kizilkaya K, Banks BD, Carnier P, Albera A, Bittante G and Tempelman RJ (2002) Bayesian inference strategies for the prediction of genetic merit using threshold models with an application to calving ease scores in Italian Piemontese cattle. Journal of Animal and Breeding Genetics 119: 209-220. ). To introduce the subject, consider the Bayesian approach for estimation of a single parameter using an observed data vector. One introduces a degree of belief in the parameter in terms of its probability distribution function, for example , called a priori distribution of , or simply a prior for . The inference about is obtained in terms of the probability distribution of given the data y and is expressed as and called the a posteriori, or simply a posterior, density function of,which is obtainable from the famous Bayes' Theorem available in standard texts (Ntzoufras 2002Ntzoufras I (2002) Gibbs variable selection using BUGS. Journal of Statistical Software 7: 1-19. , Rowe 2003Rowe DB (2003) Multivariate Bayesian statistics: Models for source separation and signal unmixing. Chapman & Hall/CRC, New York, 352p. , Gelman et al. 2004Gelman A, Chew GL and Shnaidman M (2004) Bayesian analysis of serial dilution assays. Biometrics 60: 407-417., Robert and Casella 2004Robert C and Casella G (2004) Monte Carlo statistical methods. 2nd edn, Springer-Verlag, New York, 645p. ). Using this a posteriori density, one can obtain the expected value of as an estimate of , standard error, and its Bayesian confidence intervals. The posterior distributions for each of,, and can be obtained using the following expression for the situation of a general case of s parameters . Let us denote the vector . Furthermore, let the bivariate data be generated on a pair of variables (X, Y) from the probability density function denoted by. The a posteriori distribution of (k = 1, 2..., s) based on an assumed joint a priori distribution of is given by:

The priors used include uniform, half normal, and gamma distributions for genotypic and phenotypic standard deviation components and uniform distribution for the correlations. Wong et al. (2003Wong F, Carter CK and Kohn R (2003) Efficient estimation of covariance selection models. Biometrika 90: 809-830.) proposed a prior probability model for the precision matrix in the case of multivariate responses. For responses from an RCBD, mixed linear models were used to estimate the variance components (Vargas et al. 2013Vargas M, Combs E, Alvarado G, Atlin G, Mathews K and Crossa J (2013) META: A suite of SAS programs to analyze multi-environment breeding trials. Agronomy Journal 105: 11-19.). In the present context, the parameters of model (1) are (the effects), (the standard deviations), and (the correlations). Priors are needed for standard deviations and correlations in the above. Following Gelman (2006Gelman A (2006) Prior distributions for variance parameters in hierarchical models. Bayesian Analysis 3: 515-533.), we used non-informative priors for scale parameters involved in these correlation parameters as uniform, positive half-t, and positive half-normal families of distributions (Crossa et al. 2010Crossa J, de los Campos G, Pérez P, Gianola D, Burgueño J, Araus JL, Makumbi D, Singh RP, Dreisigacker S, Yan J, Arief V, Banziger M and Braun HJ (2010) Prediction of genetic values of quantitative traits in plant breeding using pedigree and molecular markers. Genetics 186: 713-724. ). The following sets of prior distribution were considered.

P1: the priors for block, genotypic, plot-error standard deviations

and the priors for block, genotypic, and environmental correlations .

P2: the priors for block, genotypic, plot-error standard deviations

, and the priors for block, genotypic, and environmental correlations . Here the precision parameter is the inverse of the variance.

P3: the priors for block, genotypic, plot-error standard deviations

, and the priors for block, genotypic and environmental correlations. Here is a non-centrality parameter and is the degree of freedom of the t-distribution.

Since there are multiple priors, the best prior distribution was selected using a discrepancy criterion, the deviance information criterion (DIC), commonly considered for prior model selection (Gelman et al. 2004Gelman A, Chew GL and Shnaidman M (2004) Bayesian analysis of serial dilution assays. Biometrics 60: 407-417., Griffin and Brown 2012Griffin JE and Brown PJ (2012) Structuring shrinkage: some correlated priors for regression. Biometrika 99: 481-487.). The inference on the correlations was drawn using the best prior. We used the R2WinBUGS package and R- codes given in the Appendices (A and B). The number of iterations was set at 100,000 with three chains, and 5000 simulation values were taken for statistical summaries on the posteriors. Unlike the univariate approach in the frequentist method, here we used a multivariate (bivariate) framework in the Bayesian computations. In the bivariate case, the calculations were carried out by defining the priors at each element of the variance-covariance matrix. Alternatively, particularly with more than two traits, one may use Wishart distribution.

RESULTS AND DISCUSSION

Selection of priors

Choices of priors for Bayesian analysis were made from the statistics given in Table 1. Deviance information criteria (DIC) values were 1158.02 for P1, 1168.11 for P2, and 1631.9 for P3. However, the prior set P1 has the lowest numerical value of DIC (1158.02); we took P1 for estimation of the genetic parameters.

Genotypic and phenotypic variance components and heritability

Table 2 shows the frequentist estimates of the genotypic, phenotypic, and environmental variances and their estimated standard errors, as described in Singh and El-Bizri (1992Singh M and El-Bizri KS (1992) Phenotypic correlation: its estimation and test of significance. Biometrical Journal 43: 165-171.) and the asymptotic 95% confidence intervals. Bayesian estimates are based on the best priors set (P1) selected using the DIC. The posterior means of genotypic and environmental variance components were higher than the associated estimates in the frequentist version. Estimates of broad-sense heritability on a mean basis followed a similar trend, with Bayesian vs frequentist approach estimates as 0.94 vs. 0.95 for GY and 0.67 vs. 0.70 for SW.

Genotypic, phenotypic, and environmental correlations

For the frequentist approach, Table 3 presents estimates, estimated standard errors, and asymptotic confidence intervals of the genotypic, phenotypic, and environmental correlations between GY and SW, whereas for Bayesian and frequentist approaches, it presents their posterior means, standard deviations, and medians, along with credible and confidence intervals. Genotypic, phenotypic, and environmental correlations between GY and SW under the frequentist vs. Bayesian approach were 0.547 vs. 0.475, 0.377 vs. 0.328, and 0.226 vs. 0.216, respectively. A comparison between means and median showed that the Bayesian posterior distributions of these correlations are slightly skewed. The precision levels of various correlations were reasonably close for the two approaches.

Sorghum genotypes considered in the trial showed significant genetic variability for grain yield (GY) and 1000 seed weight (SW). The study makes use of prior information in terms of distributions of various variance components that may be made available from an ongoing series of crop variety trials. How the information can be utilized has been shown by the Bayesian approach, which integrates the prior information with the likelihood of the current datasets, so as to draw inferences on genotypic, phenotypic, and environmental correlations. Variable degrees of differences between the Bayesian and frequentist approaches have been found in the precision levels of the estimates of variance-component-based parameters in other studies (Singh et al. 2015Singh M, Al-Yassin A and Omer S (2015) Bayesian estimation of genotypes means, precision and genetic gain due to selection from routinely used barley trials. Crop Science 55: 501-513. ). In the case of the Bayesian approach, the precision associated with a parameter depends on the priors used. The merit of the Bayesian approach depends on the premise of its allowing for a realistic coverage of the distribution of various parameters used as priors. The Bayesian approach may not necessarily result in a lower posterior standard deviation of a parameter in comparison to the standard error of estimate of the parameter in the frequentist approach. Such investigations need to be carried out on other datasets to make an assessment of trends in the precision obtained by these two approaches. The most commonly used priors for variance components in terms of the standard deviation components have been used (Gelman 2006Gelman A (2006) Prior distributions for variance parameters in hierarchical models. Bayesian Analysis 3: 515-533.), but classes of other relevant priors (Crossa et al. 2010Crossa J, de los Campos G, Pérez P, Gianola D, Burgueño J, Araus JL, Makumbi D, Singh RP, Dreisigacker S, Yan J, Arief V, Banziger M and Braun HJ (2010) Prediction of genetic values of quantitative traits in plant breeding using pedigree and molecular markers. Genetics 186: 713-724. ) may also be included to examine support from data using the deviance information criterion. The simulation in the Bayesian approach using the R2WinBUGS software (Spiegelhalter et al. 2002Spiegelhalter DJ, Best NG, Garlin BA and Van der Linde A (2002) Bayesian measures of model complexity and fit (with discussion). Journal of the Royal Statistical Society 64: 583-639.) enables evaluation of the posterior distribution of the derived correlations in terms of variance and covariance components, unlike the frequentist methods where the simplification of the distribution is commonly made as asymptotic approximation (Singh and El- Bizri 1992Singh M and Hinkelmann K (1992) Distribution of genotypic correlation coefficient and its transforms for non-normal populations. Sankhya Series B 54: 42-66.). The R2WinBUGS software facilitated summaries in terms of posterior mean and median to make inferences regarding the symmetry of the distributions and the percentiles in reporting the credible intervals. Bayesian computation can also use the information from the experimental units that have data on additional units for only a single trait (broken samples) to estimate the genotypic and phenotypic correlations and the variance components for those traits. Furthermore, study in Bayesian estimation should be extended to multivariate cases (with more than two traits) in future investigations in plant breeding. Accordingly, heterogeneity in environmental variances and in genotype variances should also be the aspect of a future study by considering suitable models for heterogeneity of variances.

In summary, this study presents the Bayesian approach for estimation of genotypic and phenotypic correlations between traits from crop variety trials using the priors on standard deviation components and correlations obtainable from a series of previously conducted trials. The R2WinBUGS software was used for Bayesian estimates of genotypic and phenotypic correlations using experimental design data. Uniform distribution based on the priors set was found to be best, which led to precision similar to the frequentist approach. Due to its sound inference base, the Bayesian approach with WinBUGS and R codes is recommended for use in estimation of genotypic correlation in plant breeding trials.

ACKNOWLEDGMENTS

The authors are thankful to the reviewer for his/her suggestions which led to substantial improvement of the earlier version of the manuscript. The first author is grateful to ICARDA and the Arab Fund for Economic and Social Development (AFESD) for granting a fellowship for carrying out the research study.

REFERENCES

  • Abbasi S, Baghizadeh A, Mohammadi-Nejad G and Nakhoda B (2014) Genetic analysis of grain yield and its components in bread wheat (Triticum aestivum L.). Annual Research & Review in Biology 24: 3636-3644.
  • Ahmad B, Khalli IH, Igbal M and Ur-Rahman H (2010) Genotypic and phenotypic correlation many yield components in bread wheat under normal and late planting. Sarhad Journal of Agriculture 26: 259-265.
  • Crossa J, de los Campos G, Pérez P, Gianola D, Burgueño J, Araus JL, Makumbi D, Singh RP, Dreisigacker S, Yan J, Arief V, Banziger M and Braun HJ (2010) Prediction of genetic values of quantitative traits in plant breeding using pedigree and molecular markers. Genetics 186: 713-724.
  • Desalegn Z, Ratanadilok N and Kaveeta R (2009) Correlation and heritability for yield and fiber quality parameters of Ethiopian cotton (Gossypium hirsutum L.) estimated from 15 (diallel) crosses. Kasetsart Journal Natural Science 43: 1-11.
  • Fikreselassie M, Zeleke H and Alemayehu N (2012) Correlation and path analysis in Ethiopian fenugreek (Trigonella foenum-graecum L.) landraces. Crown Research in Education 3: 132-142.
  • Gelman A, Chew GL and Shnaidman M (2004) Bayesian analysis of serial dilution assays. Biometrics 60: 407-417.
  • Gelman A (2006) Prior distributions for variance parameters in hierarchical models. Bayesian Analysis 3: 515-533.
  • Griffin JE and Brown PJ (2012) Structuring shrinkage: some correlated priors for regression. Biometrika 99: 481-487.
  • Hussain K, Khan IA, Sagat HA and Amjad M (2012) Genotypic and phenotypic correlation analysis of yield and fiber quality determining traits in upland cotton (Gossypim hirsutum). International Journal of Agricultural and Biology 12: 348-352.
  • Kizilkaya K, Banks BD, Carnier P, Albera A, Bittante G and Tempelman RJ (2002) Bayesian inference strategies for the prediction of genetic merit using threshold models with an application to calving ease scores in Italian Piemontese cattle. Journal of Animal and Breeding Genetics 119: 209-220.
  • Liechty JC, Liechty MW and Muller P (2004) Bayesian correlation estimation. Biometrika 91: 1-14.
  • Littell RC, Henry PR and Ammerman CB (1998) Statistical analysis of repeated measures data using SAS procedures. Journal of Animal Science 76: 1216-1231.
  • Miller PA, Williams JC, Robinson HF and Comstock RE (1958) Estimates of genotypic and environmental variances and covariances in upland cotton and their implications in selection. Agronomy Journal 50: 126-131.
  • Ntzoufras I (2002) Gibbs variable selection using BUGS. Journal of Statistical Software 7: 1-19.
  • Patterson HD and Thompson R (1971) Recovery of inter-block information when block sizes are unequal. Biometrika 58: 545-554.
  • Payne RW (2014) The guide to GenStat(r) command language (Release 17). Part 2: Statistics. VSN International, Hemel Hempstead, 1032p.
  • Robert C and Casella G (2004) Monte Carlo statistical methods. 2nd edn, Springer-Verlag, New York, 645p.
  • Rowe DB (2003) Multivariate Bayesian statistics: Models for source separation and signal unmixing. Chapman & Hall/CRC, New York, 352p.
  • SAS Institute (2011) SAS/STAT(r) 9.2 User's guide. 2nd edn, SAS Institute, Cary, 7869p.
  • Schisterman EF, Moysich KB, England LJ and Rao M (2003) Estimation of the correlation coefficient using the Bayesian approach and its applications for epidemiologic research. BMC Medical Research Methodology 3: 1-5.
  • Singh M, Al-Yassin A and Omer S (2015) Bayesian estimation of genotypes means, precision and genetic gain due to selection from routinely used barley trials. Crop Science 55: 501-513.
  • Singh M and El-Bizri KS (1992) Phenotypic correlation: its estimation and test of significance. Biometrical Journal 43: 165-171.
  • Singh M , Ceccarelli S and Grando S (1997) Precision of the genotypic correlation estimated from variety trials conducted in incomplete block designs. Theoretical and Applied Genetics 95: 1044-1048.
  • Singh M and Hinkelmann K (1992) Distribution of genotypic correlation coefficient and its transforms for non-normal populations. Sankhya Series B 54: 42-66.
  • Spiegelhalter DJ, Best NG, Garlin BA and Van der Linde A (2002) Bayesian measures of model complexity and fit (with discussion). Journal of the Royal Statistical Society 64: 583-639.
  • Tierney L (1994) Markov chains for exploring posterior distributions (with discussion). Annual Statistics 22: 1701-1786.
  • Vargas M, Combs E, Alvarado G, Atlin G, Mathews K and Crossa J (2013) META: A suite of SAS programs to analyze multi-environment breeding trials. Agronomy Journal 105: 11-19.
  • Wong F, Carter CK and Kohn R (2003) Efficient estimation of covariance selection models. Biometrika 90: 809-830.

Appendix A   R-codes for Bayesian analysis of genotypic, phenotypic, and environmental correlations

Appendix B   WinBUGS codes for Bayesian analysis of genotypic correlation

Publication Dates

  • Publication in this collection
    Mar 2016

History

  • Received
    27 Aug 2014
  • Accepted
    04 July 2015
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