In the studies on selection and population genetics of forest trees that include the analysis of genotype × environment interaction (GE), the use of biplot graphs is relatively rare. This article describes the models and analytic methods useful in the biplot graphs, which enable the analyses of mega-environments, selection of the testing environment, as well as the evaluation of genotype stability. The main method presented in the paper is the GGE biplot method (G - genotype effect, GE -genotype × environment interaction effect). At the same time, other methods have also been referred to, such as, SVD (singular value decomposition), PCA (principal component analysis), linear-bilinear SREG model (sites regression), linear-bilinear GREG model (genotypes regression) and AMMI (additive main effects multiplicative interaction). The potential of biplot method is presented based on the data on growth height of 20 European beech genotypes (Fagus sylvatica L.), generated from real data concerning selection trials and carried out in 5 different environments. The combined ANOVA was performed using fixed- -effects, as well as mixed-effects models, and significant interaction GE was shown. The GGE biplot graphs were constructed using PCA. The first principal component (GGE1) explained 54%, and the second (GGE2) explained more than 23% of the total variation. The similarity between environments was evaluated by means of the AEC method, which allowed us to determine one mega-environment that comprised of 4 environments. None of the tested environments represented the ideal one for trial on genotype selection. The GGE biplot graphs enabled: (a) the detection of a stable genotype in terms of tree height (high and low), (b) the genotype evaluation by ranking with respect to the height and genotype stability, (c) determination of an ideal genotype, (d) the comparison of genotypes in 2 chosen environments.
Balzarini M. 2002. Applications of mixed models in plant breeding. In: Quantitative genetics genomics and plant breeding (ed.: M.S. Kang). CABI Publishing UK 353-365.
Bradu D. Gabriel K.R. 1978. The biplot as a diagnostic tool for models of two-way tables. Technometrics 20 47-68. DOI: 10.1080/00401706.1978.10489617.
Cooper M. DeLacy I.H. 1994. Relationships among analytic methods used to study genotypic variation and genotype-by-environment interaction in plant breeding Multi-environment experiments. Theoretical and Applied Genetics 88 561-572. DOI: 10.1007/BF01240919.
Cornelius P.L. Crossa J. 1999. Prediction assessment of shrinkage estimators of multiplicative models for multi-environment cultivar trials. Crop Science 39 998-1009. DOI: 10.2135/cropsci-1999.0011183X003900040007x.
Cornelius P.L. Seyedsadr M. 1997. Estimation of general linear-bilinear models for two-way tables. Journal of Statistical Computation and Simulation 58 287-322. DOI: 10.1080/00949659708811837.
Cornelius P.L. Crossa J. Seyedsadr M. 1996. Statistical tests and estimators of multiplicative models for cultivar trials. In: Genotype- by-Environment Interaction (eds.: M.S. Kang H.G. Gauch Jr). CRC Press Boca Raton Florida 199-234.
Correia I. Alia R. Yan W. David T. Aguiar A. Almeida M.H. 2010. Genotype × environment interactions in Pinus pinaster at age 10 in a multi-environment trial in Portugal: a maximum likelihood approach. Annals of Forest Science 67 612p1-612p9. DOI: 10.1051/forest/2010025.
Crossa J. 2012. From genotype × environment interaction to gene × environment interaction. Current Genomics 13 (3) 225-244. Available at: http://ejournals.ebsco.com.prxy4.ursus.maine.edu/direct.asp?ArticleID=473EADE10B8406E6D70F.
Ding M. Tier B. Yan W. Wu H.X. Powell M.B. McRae T.A. 2008. Application of GGE biplot analysis to evaluate Genotype (G) Environment (E) a nd G×E interaction on Pinus radiata: a case study. New Zealand Journal of Forestry Science 38 (1) 132-142. Available at: http://www.scionresearch.com/__data/assets/pdf_file/0007/5596/NZJFS_38_12008_Ding_et_al_132-142.pdf.
Eberhart S.A. Russell W.A. 1966. Stability parameter for comparing varieties. Crop Science 6 36-40. Available at: http://www.sap.uchile.cl/descargas/fisiogenetica/Stability%20parameters%20for%20comparing%20varieties_Eberhart_Russell1966.pdf.
Finlay K.W. Wilkinson G.N. 1963. The analysis of adaptation in a plant breeding programme. Australian Journal of Agricultural Research 14 742-754. DOI: 10.1071/AR9630742.
Gabriel K.R. 1971. The biplot graphic display of matrices with application to principal component analysis. Biometrika 58 453-467. DOI: 10.1093/biomet/58.3.453.
Gabriel K.R. 1972. Analysis of meteorological data by means of canonical decompositions and biplots. Journal of Applied Meteorology 11 1071-1077. DOI: http://dx.doi.org/10.1175/1520-0450(1972)011<1071:AOMDBM>2.0.CO;2.
Gabriel K.R. 1978. Least squares approximation of matrices by additive and multiplicative models. Journal of the Royal Statistical Society Series B 40 186-196. Available at: http://www.jstor.org.prxy4.ursus.maine.edu/stable/2984752.
Gauch H.G. 1988. Model selection and validation for yield trials with interaction. Biometrics 44 705-715. DOI: 10.2307/2531585.
Gauch H.G 1992. Statistical Analysis of Regional Yield Trials: AMMI Analysis of Factorial Designs. Elsevier Amsterdam Netherlands.
Gauch G.H. Zobel R.W. 1997. Interpreting mega- -environments and targeting genotypes. Crop Science 37 311-326. DOI: 10.2135/cropsci1997 .0011183X003700020002x.
Golub G.H. Reinsch C. 1971. The singular value decomposition and least squares solutions. In: Handbook for Automatic Computation (eds.: J.H. Wilkinson C. Renisch). Springer-Verlag Berlin 134-151.
Hocking R.R. Speed F.M. 1975. A full-rank analysis of some linear model problems. Journal of the American Statistical Association 70 706-712. DOI: 10.2307/2285959.
Hotelling H. 1933. Analysis of a complex of statistical variables into principal components. Journal of Educational Psychology 24 417-441 498-520. DOI: 10.1037/h0071325.
Hotelling H. 1936. Relations between two sets of variates. Biometrika 27 321-377. DOI: 10.2307/2333955.
Jolliffe I.T. 1972. Discarding Variables in a Principal Component Analysis. I: Artifical Data. Applied Statistics 21 160-173. DOI: 10.2307/2346488.
Jolliffe I.T. 1973. Discarding Variables in a Principal Component Analysis. II: Real Data. Applied Statistics 22 21-31. DOI: 10.2307/2346300.
Khattree R. Naik D.N. 2000. Multivariate data reduction and discrimination with SAS software. SAS Institute Inc. Cary NC.
Kempton R.A. 1984. The use of biplots in interpreting variety by environment interactions. Journal of Agricultural Science 103 123-135. DOI: 10.1017/ S0021859600043392.
Kim I. Kwon H. Ryu K. Choi W.Y. 2008. Provenance by Site Interaction of Pinus densiflora in Korea. Silvae Genetica 57 (3) 131-139.
Littell R.C. Milliken G.A. Stroup W.W. Wolfinger R.D. 1996. SAS system for mixed models. SAS Institute Inc. Cary NC.
Liu G. Cornelius P.L. 2001. Simulations and derived approximations for the means and standard deviations of the characteristic roots of a Wishart matrix. Communications in Statistics - Simulation and Computation 30 963-989. DOI: 10.1081/SAC-100107791.
McCabe G.P. 1984. Principal Variables. Technometrics 26 (2) 137-144. DOI: 10.2307/1268108.
Murillo O. 2001. Genotype by environment interaction and genetic gain on unbalanced Pinus oocarpa provenances trials. Agronomia Costarricense 25 (1) 21-32.
Patterson H.D. Thompson R. 1971. Recovery of inter- -block information when block sizes are unequal. Biometrika 58 545-554. DOI: 10.2307/2334389.
Patterson H.D. Thompson R. 1975. Maximum likelihood estimation of components of variance. Proceedings of 8th International Biometric Conference 197-207.
Pearson K. 1901. On lines and planes of closest fit to systems of points in space. Philosophical Magazine 2 (11) 559-572. DOI: 10.1080/14786440109462720.
Piepho H.P. 1998. Empirical best linear unbiased prediction in cultivar trials using factor analytic variance- covariance structures. Theoretical and Applied Genetics 97 195-201.
Piepho H.P. Möhring J. 2006. Selection in cultivar trials - is it ignorable? Crop Sciences 46 192-201.
SAS Institute Inc. 2013. SAS/STAT® 13.1 User’s Guide. Cary NC:SAS Institute Inc.
Saxton A.M. 2004. Genetic analysis of complex traits using SAS. SAS Institute Inc. Cary N.C.
Sixto H. Gil P.M. Ciria P. Camps F. Cañellas I. Voltas J. 2015 Interpreting genotype-by-environment interaction for biomass production in hybrid poplars under short-rotation coppice in Mediterranean environments. GCB Bioenergy. DOI: 10.1111/ gcbb.12313.
Taibi K. 2014. Title of dissertation “Integrated approach for addressing assisted population migration programs in forest management to climate change: Out- -planting performance genotype by environment interactions physiological and molecular response”. The Polytechnic University of Valencia Spain.
Ukalska J. Kociuba W. 2013. Phenotypical diversity of winter triticale genotypes collected in the Polish Gene Bank between 1982 and 2008 with regard to major quantitative traits. Field Crops Research 149 203-212. DOI: 10.1016/j.fcr.2013.05.010.
Ukalski K. Śmiałowski T. Ukalska J. 2010a. Analysis of oat yield environments using graphical GGE method. Colloquium Biometricum 40 81-93.
Ukalski K. Śmiałowski T. Ukalska J. 2010b. Analiza plonowania i stabilności genotypów owsa za pomocą metody graficznej typu GGE. Żywność. Nauka. Technologia. Jakość 3 (70) 127-140.
Yan W. 1999. A study on the methodology of cultivar evaluation based on yield trial data with special reference to winter wheat in Ontario. Ph.D. thesis University of Guelph Guelph Ontario Canada.
Yan W. 2001. GGE biplot: a Windows application for graphical analysis of multi-environment trial data and other types of two-way data. Agronomy Journal 93 1111-1118. DOI: 10.2134/agronj2001.9351111x.
Yan W. 2002. Singular-value partitioning in biplot analysis of multienvironment trial data. Agronomy Journal 94 990-996. DOI: 10.2134/agronj2002.0990.
Yan W. Cornelius P.L. Crossa J. Hunt L.A. 2001. Two types of GGE biplots for analyzing multienvironment trial data. Crop Science 41 656-663. DOI: 10.2135/cropsci2001.413656x.
Yan W. Hunt L.A. 2001. Interpretation of genotype x environment interaction for winter wheat yield in Ontario. Crop Science 41 19-25. DOI: 10.2135/ cropsci2001.41119x.
Yan W. Hunt L.A. Sheng Q. Szlavnics Z. 2000. Cultivar evaluation and mega-environment investigation based on the GGE biplot. Crop Science 40 597-605. DOI: 10.2135/cropsci2000.403597x.
Yan W. Kang M.S. 2003. GGE biplot analysis: a graphical tool for breeders genetics and agronomists. CRC Press Boca Raton FL.
Yan W. Rajcan I.R. 2002. Biplot analysis of test sites and trait relations of soybean in Ontario. Canadian Journal of Plant Science 42 11-20. DOI: 10.2135/ cropsci2002.0011.
Yan W. Tinker N.A. 2005. An integrated biplot analysis system for displaying interpreting and exploring genotype-×environment interactions. Crop Science 45 1004-1016. DOI: 10.2135/cropsci2004.0076.
Yan W. Tinker N.A. 2006. Biplot analysis of multi-environment trial data: Principles and applications. Canadian Journal of Plant Science 86 623-645.
Zhao X. Xia H. Wang X. Wang C. Liang D. Li K. et al. 2016. Variance and stability analyses of growth characters in half-sib Betula platyphylla families at three different sites in China. Euphytica 208 173-186. DOI: 10.1007/s10681-015-1617-7.
Zobel R.W. Wright M.J. Gauch H.G. 1988. Statistical analysis of a yield trial. Agronomy Journal 80 388-393. DOI: 10.2134/agronj1988.000219620080 00030002x.