Mixed model with spatial variance–covariance structure for accommodating of local stationary trend and its influence on multi-environmental crop variety trial assessment
AbstractThe most common procedure for analyzing multi-environmental trials is based on the assumption that the residual error variance is homogenous across all locations considered. However, this may often be unrealistic, and therefore limit the accuracy of variety evaluation or the reliability of variety recommendations. The objectives of this study were to show the advantages of mixed models with spatial variance–covariance structures, and direct implications of model choice on the inference of varietal performance, ranking and testing based on two multi-environmental data sets from realistic national trials. A model comparison with a chi-square test for the trials in the two data sets (wheat data set BW00RVTI and barley data set BW01RVII) suggested that selected spatial variance-covariance structures fitted the data significantly better than the ANOVA model. The forms of optimally-fitted spatial variance-covariance, ranking and consistency ratio test were not the same from one trial (location) to the other. Linear mixed models with single stage analysis including spatial variance-covariance structure with a group factor of location on the random model also improved the real estimation of genotype effect and their ranking. The model also improved varietal performance estimation because of its capacity to handle additional sources of variation, location and genotype by location (environment) interaction variation and accommodating of local stationary trend.
Bartlett MS, 1978. Nearest neighbour models in the analysis of field experiments. J R Stat Soc B Method 40: 147-174.
Bozdogan H, 1987. Model selection and Akaike's information criterion (AIC): The general theory and its analytical extensions. Psychometrika 52: 345-370. http://dx.doi.org/10.1007/BF02294361
Brownie C, Gumpertz ML, 1997. Validity of spatial analyses for large field trials. J Agr Biol Environ Stat 2: 1-23. http://dx.doi.org/10.2307/1400638
Burnham KP, Anderson DR, 2002. Model selection and multi-model inference: a practical information-theoretic approach, Springer.
Casanoves F, Macchiavelli R, Balzarini M, 2005. Error variation in multienvironment peanut trials. Crop Sci 45: 1927-1933. http://dx.doi.org/10.2135/cropsci2004.0547
Casanoves F, Macchiavelli R, Balzarini M, 2013. Models for multi-environment yield trials with fixed and random block effects and homogeneous and heterogeneous residual variances. Journal of Agriculture of the University of Puerto Rico, 91: 117-131.
Clarke FR, Baker RJ, 1996. Spatial analysis improves precision of seed lot comparisons. Crop Sci 36: 1180-1184. http://dx.doi.org/10.2135/cropsci1996.0011183X003600050019x
Cullis BR, Gleeson AC, 1991. Spatial analysis of field experiments-an extension to two dimensions. Biometrics 47: 1449-1460. http://dx.doi.org/10.2307/2532398
Cullis B, Gogel B, Verbyla A, Thompson R, 1998. Spatial analysis of multi-environment early generation variety trials. Biometrics 54: 1-18. http://dx.doi.org/10.2307/2533991
Davidoff B, Selim HM, 1988. Correlation between spatially variable soil moisture content and soil temperature. Soil Sci 145: 1-10. http://dx.doi.org/10.1097/00010694-198801000-00001
Denis JB, Piepho HP, Eeuwijk FA, 1997. Modelling expectation and variance for genotype by environment data. Heredity 79: 162-171. http://dx.doi.org/10.1038/hdy.1997.139
Fahrmeir L, Hamerle A, 1984. Multivariate Statistische Verfahren. Walter de Gruyter, Berlin, NY.
Giesbrecht FG, Burns JC, 1985. Two-stage analysis based on a mixed model: large-sample asymptotic theory and small-sample simulation results. Biometrics 41: 477-486. http://dx.doi.org/10.2307/2530872
Gilmour AR, Cullis BR, Verbyla AP, 1997. Accounting for natural and extraneous variation in the analysis of field experiments. J Agr Biol Environ Stat 2: 269-293. http://dx.doi.org/10.2307/1400446
Gleeson AC, 1997. Statistical methods for plant variety evaluation (Kempton RA, Fox PN, eds). Chapman & Hall, London, Chapter 5, pp: 68-85. http://dx.doi.org/10.1007/978-94-009-1503-9_5
Greene WH, 2003. Econometric analysis, 5th edition. Pearson Educ Int, University Prentice Hall, NY
Guerin L, Stroup WW, 2000. A simulation study to evaluate PROC MIXED analysis of repeated measures data. Proc 12th Ann Conf on Applied Statistics in Agriculture. Kansas State Univ (Manhattan KS, ed.). pp: 170-203.
Handcock MS, 1994. An approach to statistical spatial-temporal modeling of meteorological fields: rejoinder. J Am Stat Assoc 89: 388-390.
Handcock MS, Stein ML, 1993. A Bayesian analysis of kriging. Technometrics 35: 403-410. http://dx.doi.org/10.1080/00401706.1993.10485354
Hartley HO, Rao CR, 1967. Maximum likelihood estimation for the mixed analysis of variance model. Biometrica 54: 93-108.
Harville DA, 1977. Maximum likelihood approaches to variance component estimation and to related problems. J Am Stat Assoc 72: 320-338. http://dx.doi.org/10.1080/01621459.1977.10480998
Hong N, White JG, Gumpertz ML, Weisz R, 2005. Spatial analysis of precision agriculture treatments in randomized complete blocks. Agron J 97: 1082-1096. http://dx.doi.org/10.2134/agronj2004.0130
Hrong-Tai Fai A, Cornelius PL, 1996. Approximate F-tests of multiple degree of freedom hypotheses in generalized least squares analyses of unbalanced split-plot experiments. J Stat Comput Sim 54: 363-378. http://dx.doi.org/10.1080/00949659608811740
Kackar RN, Harville DA, 1984. Approximations for standard errors of estimators of fixed and random effect in mixed linear models. J Am Stat Assoc 79: 853-862.
Kelly AM, Smith AB, Eccleston JA, Cullis BR, 2007. The accuracy of varietal selection using factor analytic models for multi-environment plant breeding trials. Crop Sci 47: 1063-1070. http://dx.doi.org/10.2135/cropsci2006.08.0540
Kempton R, 1984. The use of biplots in interpreting variety by environment interactions. J Agric Sci 103: 123-135. http://dx.doi.org/10.1017/S0021859600043392
Kenward MG, Roger JH, 1997. Small sample inference for fixed effects from restricted maximum likelihood. Biometrics 53: 983-997. http://dx.doi.org/10.2307/2533558
Littell RC, Milliken GA, Stroup WW, Wolfinger RD, Schabenberger O, 2006. SAS for mixed models. SAS Institute Inc., Cary, NC, USA.
Matérn B, 1986. Spatial variation, 2nd edition. In: Lecture notes in statistics. Springer-Verlag, NY. pp: 705-711.
Mo H, Si Y, 1986. Trend variation and it's control in field experiment. Acta Agr Sinnica 12(4): 233-240.
Möhring, J, Piepho HP, 2009. Comparison of weighting in two-stage analysis of plant breeding trials. Crop Sci 49: 1977-1988. http://dx.doi.org/10.2135/cropsci2009.02.0083
Oman SD, 1991. Multiplicative effects in mixed model analysis of variance. Biometrika 78: 729-739. http://dx.doi.org/10.1093/biomet/78.4.729
Patterson HD, Silvey V, 1980. Statutory and recommended list trials of crop varieties in the United Kingdom. J R Stat Soc A 143: 219-252. http://dx.doi.org/10.2307/2982128
Piepho HP, 1997. Analyzing genotype-environment data by mixed models with multiplicative terms. Biometrics 53: 761-766. http://dx.doi.org/10.2307/2533976
Piepho HP, 1998. Methods for comparing the yield stability of cropping systems. J Agr Crop Sci 180: 193-213. http://dx.doi.org/10.1111/j.1439-037X.1998.tb00526.x
Piepho HP, 1999a. Cultivar comparisons in three-way data based on mixed models with flexible variance-covariance structure. Biuletyn Oceny Odmian 30: 31-48.
Piepho HP, 1999b. Stability analysis using the SAS System. Agron J 91: 154-160. http://dx.doi.org/10.2134/agronj1999.00021962009100010024x
Piepho HP, Möhring J, 2010. Generation means analysis using mixed models. Crop Sci 50: 1674-1680. http://dx.doi.org/10.2135/cropsci2010.02.0093
Piepho HP, Richter C, Williams E, 2008. Nearest Neighbour adjustment and linear variance models in plant breeding trials. Biometrical J 50: 164-189. http://dx.doi.org/10.1002/bimj.200710414
Satterthwaite FE, 1941. Synthesis of variance. Psychometrika 6(5): 309-316. http://dx.doi.org/10.1007/BF02288586
Schaalje GB, McBride J, Fellingham G, 2002. Adequacy of approximations to distributions of test statistics in complex mixed linear models. J Agr Biol Environ Stat 7: 512-524. http://dx.doi.org/10.1198/108571102726
Schabenberger O, Pierce FJ, 2002. Contemporary statistical models for the plant and soil sciences. CRC Press, Boca Raton. 738 pp.
Scharf PC, Alley MM, 1993. Accounting for spatial yield variability in field experiments increases statistical power. Agron J 85: 1254-1256. http://dx.doi.org/10.2134/agronj1993.00021962008500060029x
Schwarz G, 1978. Estimating the dimension of a model. Ann Stat 6: 461-464. http://dx.doi.org/10.1214/aos/1176344136
Schwarzbach E, 1984. A new approach in the evaluation of field trials. The determination of the most likely genetic ranking of varieties. Vortrage Pflanzen 6: 249-259.
Smith A, Cullis B, Gilmour A, 2001. Applications: the analysis of crop variety evaluation data in Australia. Aust New Zeal J Stat 43: 129-145. http://dx.doi.org/10.1111/1467-842X.00163
Smith A, Cullis BR, Thompson R, 2005. The analysis of crop cultivar breeding and evaluation trials: an overview of current mixed model approaches. J Agr Sci 143: 449-462. http://dx.doi.org/10.1017/S0021859605005587
Spilke J, Piepho HP, Meyer U, 2004. Approximating the degrees of freedom for contrasts of genotypes laid out as subplots in an alpha-design in a split-plot experiment. Plant Breeding 123: 193-197. http://dx.doi.org/10.1046/j.1439-0523.2003.00964.x
Spilke J, Piepho HP, Hu X, 2005. A simulation study on tests of hypotheses and confidence intervals for fixed effects in mixed models for blocked experiments with missing data. J Agr Biol Environ Stat 10: 374-389. http://dx.doi.org/10.1198/108571105X58199
Stefanova KT, Buirchell B, 2010. Multiplicative mixed models for genetic gain assessment in lupin breeding. Crop Sci 50: 880-891. http://dx.doi.org/10.2135/cropsci2009.07.0402
Stroup WW, 2002. Power analysis based on spatial effects mixed models: A tool for comparing design and analysis strategies in the presence of spatial variability. J Agr Biol Environ Stat 7: 491-511. http://dx.doi.org/10.1198/108571102780
Welham SJ, Gogel BJ, Smith AB, Thompson R, Cullis BR, 2010. A comparison of anlysis methods for late-stage variety evaluation trials. Aust New Zeal J Stat 52: 125-149. http://dx.doi.org/10.1111/j.1467-842X.2010.00570.x
Wilkinson GN, Eckert SR, Hancock TW, Mayo O, 1983. Nearest neighbour (NN) analysis of field experiments. J R Stat Soc B 45: 151-211.
Williams ER, 1986. A neighbour model for field experiments. Biometrika 73: 279-287. http://dx.doi.org/10.1093/biomet/73.2.279
Wolfinger R, 1993. Covariance structure selection in general mixed models. Commun Stat-Simul C 22: 1079-1106. http://dx.doi.org/10.1080/03610919308813143
Wu T, Dutilleul P, 1999. Validity and efficiency of neighbor analyses in comparison with classical complete and incomplete block analyses of field experiments. Agron J 91: 721-731. http://dx.doi.org/10.2134/agronj1999.914721x
Yang RC, Ye TZ, Blade SF, Bandara M, 2004. Efficiency of spatial analyses of field pea variety trials. Crop Sci 44: 49-55. http://dx.doi.org/10.2135/cropsci2004.0049
Zimmerman DL, Harville DA, 1991. A random field approach to the analysis of field-plot experiments and other spatial experiments. Biometrics 47: 223-239. http://dx.doi.org/10.2307/2532508
© INIA. Manuscripts published are the property of the Instituto Nacional de Investigación y Tecnología Agraria y Alimentaria, and quoting this source is a requirement for any partial or full reproduction.
SJAR is an Open Access Journal. All articles are distributed under the terms of the Creative Commons Attribution 4.0 International (CC BY 4.0) License. You may read here the basic information and the legal text of the license. The indication of the license CC-by must be expressly stated in this way when necessary.