CROPSANDSOILSRESEARCHPAPER Applying the generalized ...ainfo.cnptia.embrapa.br/.../item/154434/1/Applying-the-generalized.pdf · Generalized linear models When a distribution is non-normal,

CROPS AND SOILS RESEARCH PAPER

Applying the generalized additive main effects and multiplicativeinteraction model to analysis of maize genotypes resistant to greyleaf spot

C. R. L. ACORSI1, T. A. GUEDES1, M. M. D. COAN2*, R. J. B. PINTO2, C. A. SCAPIM2,C. A. P. PACHECO3, P. E. O. GUIMARÃES3 AND C. R. CASELA3

1Departamento de Estatística (DES), Universidade Estadual de Maringá (UEM), Av. Colombo, 5·790 – Zip Code87020-900 Jd. Universitário, Maringá – Paraná, Brazil2Departamento de Agronomia (DAG), Universidade Estadual de Maringá (UEM), Av. Colombo, 5·790 – Zip Code87020-900 Jd. Universitário, Maringá – Paraná, Brazil3 Embrapa Milho e Sorgo, CNPMS, Rodovia MG 424 km 45, CP 285 – Zip Code 35701-970 – Sete Lagoas, MG, Brazil

(Received 27 May 2014; accepted 22 November 2016)

SUMMARY

Analysing the stability and adaptation of cultivars to different environments is always necessary before recom-mending them for planting on large areas. Additive main effects and multiplicative interaction (AMMI) modelshave been used to analyse genotype-by-environment interactions (G × E). AMMI models require data with homo-geneous variance, normal errors and additive effects. However, agronomic data do not always conform to thesestatistical assumptions. The objective of the present study was to analyse G × E interactions for severity and inci-dence of grey leaf spot, a foliar disease in maize caused by Cercospora zeae-maydis, using a generalized AMMImodel. Data were collected and evaluated for 36 maize cultivars from experiments carried out in nine Brazilianregions in 2010/11 by the Empresa Brasileira de Pesquisa Agropecuária (EMBRAPA –Milho e Sorgo). Only two ofthree stable genotypes defined by a quasi-likelihood model with a logistic link function could be recommendedfor their desirable agronomic characteristics. Four growing locations in which the genotypes were stable wereidentified, but in only one of these was stability associated with very severe grey leaf spot disease. Cultivarsadapted to specific locations with low percentage disease severity were also identified.

INTRODUCTION

Optimal maize production depends on genotype (G),environment (E) and both together when there is sig-nificant G × E interaction (Allard 1999). Efforts havebeen made to quantify, minimize or make use of theG × E interaction when making strategic decisionsregarding maize breeding (Cruz et al. 2006).The additive main effects and multiplicative inter-

action (AMMI) models developed by Kempton(1984); Gauch & Zobel (1988); Zobel et al. (1988)and Crossa et al. (1991) are important statisticalmethods for plant breeding. Although these modelsprovide easy and simple methods for interpreting

parametric estimators, they require normally distribu-ted data.

Kempton (1984) discusses the method of principalcomponents (PC) as a way to summarize the responseof a genotype to different environments. In thismethod, the matrix of estimated G × E interactioneffects from the classical analysis of variance(ANOVA) model is subjected to principal componentanalysis (PCA). The G × E interaction is thus decom-posed into a number of multiplicative terms. Thehypothesis is that most of the G × E interaction canbe explained by the first few terms of the PCA andthat these have some meaningful interpretations.

The AMMI model has been applied since the1990s to evaluate G × E interactions and allowbreeders to recommend stable cultivars adapted to

* To whom all correspondence should be addressed. Email:[email protected]

Journal of Agricultural Science, Page 1 of 15. © Cambridge University Press 2016doi:10.1017/S0021859616001015

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either broad or specific environments. However, theAMMI model can only be applied when the responsevariable Y follows a normal distribution with ahomogeneous variance. If these assumptions arenot met, a methodology based on a generalizedlinear model (GLM) is more appropriate.Algorithms for generalized additive main effectsand multiplicative interaction (GAMMI) developedby van Eeuwijk (1995) and Gabriel (1998) arebased on the basic concepts for AMMI expandedby the theories of GLM and the quasi-likelihoodmethod. These GAMMI models assume that theresponse variables have an exponential probabilitydistribution. Earlier, Wedderburn (1974) establishedthe quasi-likelihood method to accommodate awider range of possible distributions and variances(Agresti 2002). The quasi-likelihood method is ageneralization of the GLM (Paula 2004) thatassumes a single relationship between the meanand variance rather than an a priori distribution forY. Similar to the GLM, the quasi-likelihood modelassumes a link function that is a linear predictorinstead of a specific distribution of the response vari-able Y (McCullagh & Nelder 1989). It also assumesthat Var(Y) = ϕ Var(μ), where μ is the mean of Y, V(μ)is a new function of μ and ϕ is the dispersionparameter.

Grey leaf spot can severely affect susceptible culti-vars, resulting in crop losses of greater than 0·80. Inmaize, the symptoms of grey leaf spot include irregu-lar, rectangular grey spots that develop parallel to theleaf veins (Fantin et al. 2001; Fornasieri Filho 2007).According to Brito et al. (2007), the pathogen colo-nizes large areas of foliar tissue, reduces photosyn-thesis, induces early leaf senescence and decreasescrop yield. Wind or raindrops can disseminate thepathogen. Because the spores remain on the stoverafter harvest, a management strategy must beadopted to reduce recontamination (Bhatia &Munkvold 2002).

Thus, a maize variety carrying a large number ofresistance genes is likely to have better yield in envir-onments in which grey leaf spot is prevalent. Suchgenotypes must have stable, high yield with little vari-ation in different environments (Tarakanovas &Ruzgas 2006).

The objective of the present study was to evaluateand quantify the G × E interaction for response togrey leaf spot in maize using GAMMI models to iden-tify genotypes that are resistant to grey leaf spot,adapted to specific environments, or both.

MATERIALS AND METHODS

Data pertaining to grey leaf spot severity in Brazil werecollected from 36 maize cultivars evaluated in ninedifferent environments in 2010/11. The experimentaldesign in each environment was a randomized com-plete block with two replications. Plots consisted offour 5-m rows spaced 0·70 m apart with experimentalunits of 14 m2. Fertilization, liming and other culturalpractices were applied as required in each locationand experimental area. Grey leaf spot severity wasquantified as the percentage of diseased leaf areawithin each plot.

The locations in which the 36 genotypes (G1 to G36)were evaluated by EMBRAPA are shown in Table 1. Inthe first stage of the present study, the Shapiro–Wilkmultivariate normality test and the Bartlett test for homo-geneity of variance were used to determine whether toapply an AMMI model or a GAMMI model to analysethe G× E interactions for incidence and severity ofgrey leaf spot in maize in these environments.

Additive main effects and multiplicative interactionmodel

The AMMI model was composed of additive andmultiplicative components where Y represents avector of n independently distributed observationsthat can be predicted by the categorical variables forgenotypes and environments. The additive compo-nent, with fixed main effects for genotype (αi) andenvironment (βj), was assumed to be a fixed effect,and inferences were restricted to the grey leaf spotresponse variables, disease incidence and severity(Searle et al. 1992).

A least-squares method was used to estimate theseeffects via a two-way ANOVA using the meansmatrix for Y(gxe). The multiplicative component wasestimated by the singular value decomposition (SVD)of the residual matrix from the two-way ANOVA ofthe genotype–environment means Y(gxe). This matrixwill be denoted as R(gxe). Generally, the SVD of amatrix A is defined as the product of an orthogonalmatrix U by a diagonal matrix S and the transpose ofthe orthogonal matrix V; thus, Amn ¼ UmnSmnVTnn.Required conditions are that UTU = I and VTV = I; thecolumns of U are orthonormal eigenvectors of AAT;the columns of V are orthonormal eigenvectors ofATA; S is a diagonal matrix containing the squareroots of eigenvalues from U or V in descendingorder; and A = VS2VT (Ientilucci 2003).

2 C. R. L. Acorsi et al.


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Therefore, the model equation for the ith genotypein the jth environment in the rth block is (Gauch &Zobel 1988):

Yijr ¼ μþ αi þ βj þ ρrð jÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}additive terms

þXp

h¼1λhγihδ jh

|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}multiplicative terms

þ εijr

where Yijr is the phenotypic trait (i.e., the proportion ofplants affected by grey leaf spot) of genotype i in envir-onment j for replicate r; μ is the grand mean; αi is thefixed effect for genotype i, where i = 1, 2,…, g; βj is thefixed effect for the environment j, wherej ¼ 1; 2; :::; e; λh is the singular value for the inter-action principal component (IPC) axis k; γih and δjhare the IPC scores (i.e., the left and right singularvectors) for genotype and environment, respectively,for axis k; ρr(j) is the effect of the rth block in the jthenvironment; r is the number of blocks; p is the rankof the R(gxe) matrix that corresponds to the number ofmain effects from the interaction (PCI) retained bythe residual matrix, p¼minðg�1; e�1Þ; ðαβÞij ¼Pp

h¼1 λhγihδjh is the specific interaction of the ith geno-type with the jth environment and ɛijr is the experimen-tal error that is assumed to be independently andnormally distributed with a mean of zero and varianceσ2; εijr ∼ Nð0; σ2Þ: The decomposition of the residualmatrix into singular values (SVD) permits the partition-ing of the least squares from the elements of the R(gxe)matrix by reducing the number of axes, or K < p suchthat the model remains informative, where K is thenumber of axes or PC retained by the model, butwith fewer degrees of freedom. This partition is:

Xp

h¼1λhγihδ jh ¼

XK

h¼1λhγihδ jh þ

Xp

h¼1þKλhγihδ jh;

wherePp

h¼1þK λhγihδjh ¼ φijr quantifies the disturb-ance and φijr is the residual containing all of the multi-plicative terms not included in the model.

Therefore, using the least-squares approximation tothe R(gxe) matrix by the first n components of the SVD,the reduced model ηijr is estimated by:

Ŷijr ¼ μ̂þ α̂i þ β̂j þ ρ̂rð jÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}additive terms

þXK

h¼1λhγihδ jh

|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}multiplicative terms

The PCA permits the components from the interactionto capture the decreasing proportion of the variationthat is present in matrix GE, or λ21 � λ22 � � � � � λ2K . Asufficient number of components (K) to represent thetarget model can be identified using Gollob’s test(Table 2) (Gollob 1968).

Generalized linear models

When a distribution is non-normal, GLMs expand thepossibilities for statistical modelling. These modelsallow fitting of n random variables yi, where i = 1, 2,…,n, that are independently distributed with mean μi andan exponential probability density function. Theserandom variables are associated with the explanatoryvariables xj, j = 1, 2,…, p by means of a link function g(μi) designated as the linear predictor (ηi) that is mono-tonic and differentiable such that:

ηi ¼ gðμiÞ ¼Xp

j¼1xijψj

where ψj represents the coefficients of the linearpredictor.

The maximum-likelihood method is the most usefulmethod for estimating the vector of the unknown

Table 1. Codes and geographic coordinates for the locations in which maize genotypes were evaluated

Locations Codes Brazilian states* Latitude (S) Longitude (W)

Campo Mourão CM Paraná-PR 24°02′ 52°22′Goiânia GO Goiás-GO 16°40′ 49°15′Goianésia GS Goiás-GO 15°19′ 49°07′Jataí JT Goiás-GO 17°52′ 51°42′Londrina LD Paraná-PR 23°18′ 51°09′Ponta Grossa PG Paraná-PR 25°05′ 50°09′Planaltina PL Goiás-DF 15°27′ 47°36′Patos de Minas PM Minas Gerais-MG 18°34′ 46°31′São Sebastião do Paraíso SP Minas Gerais-MG 20°55′ 46°59′

* State name and state abbreviation.

GAMMI analysis of maize resistant to grey leaf spot 3



parameters of the linear predictor ψj. Cordeiro &Demétrio (2008) explained that the robust and fastGLM algorithm rarely fails to converge. However,when this does happen, the fitting procedure mustbe restarted using the current estimate as the startingvalue for another model.

The deviance function derived from the likelihoodratio statistic tests the significance of the coefficientsof the linear predictor. Therefore, in a sequence ofk nested models (which have the same probabilitydistribution and link function, but the linear compo-nent M0 is a special case of the general linear compo-nent M1) (Dobson 2002), tests of significance areperformed using an analysis of deviance (ANODEV)table. Thus the deviance function from the GLMs isanalogous to the residual squared sums from leastsquares. Standardized Pearson residuals, standar-dized deviance residuals, and Cook’s distance mea-sures were used to diagnoses in the quasi-likelihoodmodels.

Quasi-likelihood models

Quasi-likelihood has been used due to the character-istics of the data and the model. Although the GLMrepresents a great advance in statistical modellingbecause it allows the fitting of a large number ofmodels, in some instances the choice of an exponen-tial model is not adequate (McCullagh & Nelder1989), so Wedderburn (1974) proposed quasi-likelihood estimation. Assuming that Var(μ) is aknown function of the mean, and ϕ is the dispersionparameter, the quasi-likelihood function for every

observation is

Qi ¼ Qiðy; μÞ ¼ ∫μi

yi

yi � tf�VarðtÞdt; yi � t � μi

Inference in quasi-likelihood models is similar to thatin GLM because quasi-likelihood estimates maximizeQ or solve the following system of equations:

Xn

i¼1

ðyi � μiÞfVðμiÞ

∂μi∂ψj

¼ 0; j ¼ 1; . . . ;p

and

Xn

i¼1

ðyi � μiÞxijfVðμiÞ

∂μi∂ηi

¼ 0

In the first system, μi ¼ g�1ðηiÞ ¼ g�1ðzTi ψjÞ ¼ hðxTi ψjÞand this expression are based on the GLM theory. Thedispersion ϕ is estimated using the method of momentson the residual vector ðY � μ̂Þ:

f̂ ¼ 1n� p

Xn

i¼1

ðyi � μ̂iÞ2Vðμ̂iÞ

¼ χ2

n� pwhere χ2 is Pearson’s generalized chi-square statisticfor goodness-of-fit, n is the number of observations,and p is the number of parameters (ψj). The general-ized function can be estimated in a similar mannerto the deviance function, using the differencebetween the quasi-likelihood logarithm of thecurrent and the saturated models:

Dðy;μ̂Þ ¼2ffQðy;yÞ �Qðμ̂;yÞg¼ � 2ffQðμ̂;yÞ �Qðy;yÞg

Because the contribution from the saturated model iszero, then:

Dðy;μ̂Þ ¼ �2fQ ¼ �2fX

∫μi

yi

yi � tfVðtÞdt

¼ �2X

∫μi

yi

yi � tVðtÞ dt

Thus, the quasi-deviance function does not depend onthe dispersion parameter ϕ. The quasi-deviance func-tion Dðy;μ̂Þ=f is compared with the percentiles ofthe χ2 distribution with (n–p) degrees of freedom,although the null distribution of f�1Dðy;μ̂Þ is notusually known (Paula 2004).

Generalized additive main effects and multiplicativeinteraction

The GAMMI theory requires the same basic assump-tions as the GLMs. The response variable Y must be

Table 2. Complete variance analyses of the meansaccording to Gollob (1968)

Source of variation D.F. (Gollob)Deviance(Gollob)

Blocks/environment (E) e(j–1) Deviance (R|E)Genotype (G) g–1 DevianceGEnvironment (E) e–1 DevianceEG × E interaction (g–1)(e–1) DevianceGEAxis1 g + e–1–(2 × 1) λ

21

Axis2 g + e–1–(2 × 2) λ22

… … …

Axisk g + e–1–(2 × j) λ2k

Plot error e(g–1)(j–1) DeviancePlot error

Total gej–1 DevianceTotal




independently distributed and have a known expo-nential family distribution, and p associated explana-tory variables Xj, where j = 1, 2, …, p are determinedby a link function g(μi) that designates a linear pre-dictor ηi that is monotonic and differentiable such that:

ηi ¼ gðμiÞ ¼Xp

j¼1Xijψj

These linear predictors have been useful to estimatethe mean severity of grey leaf spot. Because the linkfunction was the logit in which η = g(μ) = log(μ/1− μ),the mean proportion of disease was estimated by therelationship

g�1ðηijrÞ ¼expðμþ αi þ βj þ ρrð jÞ þ

PKh¼1

λhγihδ jhÞ

1þ expðμþ αi þ βj þ ρrð jÞ þPKh¼1

λhγihδ jhÞ

where K is the number of axes considered.The GAMMI model is applied using van Eeuwijk’s

algorithm adapted from Sumertajaya (2007) in R soft-ware version 3.0·2 (R Development Core Team 2013)using the R package gnm (generalized nonlinearmodels) (Turner & Firth 2009). This algorithm uses itera-tive alternating generalized regression of rows andcolumns to estimate the parameters. The first step indetermining the appropriate model is to identify the dis-tribution and handling of the experimental data. Anerror plot should be used to visualize whether thedata have, for example, a Poisson or binomial distribu-tion instead of a normal distribution. The second step isto fit the GAMMI model, in which each regressionincludes a GLM class that is arrived at iteratively. Thisalgorithm involves convergence in row regression, incolumn regression, and in alternating regression (Hadiet al. 2010). If the model converges, then ANODEVmay then be performed. Finally, the data matrix isrepresented as a biplot. Figure 1 shows the algorithmnecessary for applying the GAMMI model.To determine the number of axes or the number

of multiplicative terms in a GAMMI model, a general-ization of the AMMI method via the tests describedbelow may be used. The F test does not require aspecial table and is easy to calculate. The statisticused is F = (Dev. restricted/D.F. sv restricted)−ðDev: full= D:F: fullÞ=f̂ , which approximates theF(D.F.source of variation; D.F.error) distribution. Where:

Dev: : deviance; f̂ is the dispersion parameter fromquasi-likelihood estimation, D.F..sv: degrees offreedom from source of variation that is being tested.

The test proposed by Gollob (1968) allocates(g− 1)(e− 1)− (2k− 1) = g + e− 1− 2k degrees offreedom to the eigenvalues associated with the kthaxis, where k= 1, 2,…, n and n =minimum (g–1, e–1),which corresponds to the difference between thenumber of parameters to be estimated and the numberof factors applied. Thus, the mean deviance is testedagainst the estimated error.

Stability is the maintenance or predictability of theresponse variable in various environments(Annicchiarico et al. 2005; Cruz et al. 2006). For theincidence or severity of disease, a genotype is consid-ered to be stable when its disease severity percentageis low and constant with respect to environmentalvariation under both specific and broad conditions.Stability is estimated by analysing the magnitude andsign of the biplot scores corresponding to the selectedGAMMI model. Genotypes and environments withlow (near zero) scores are considered stable, whichis expected for genotypes and environments thathave a small contribution to the overall interaction(Duarte & Vencovsky 1999).

The adaptability of a genotype indicates its ability totake advantage of environmental effects to ensure ahigh level of productivity. Adaptability is predictedas a function of the responses for each combinationof genotype and environment in the model selectedby GAMMI IPCAk (axis k: axis of interaction PCA).The correlation between cultivars and the environ-ment is based on the angles between vectors deter-mined by coordinates of the interaction (axis 1, axis 2)and the vertex. The cosine of the two vectors indicatesthe level of correlation between two corresponding vari-ables (Rencher 2002). Therefore, a small angle indicateshighly positively correlated variables, perpendicularvectors indicate non-correlated variables, and anangle greater than 90° indicates a negative correlation.

RESULTS

Box plots of grey leaf spot incidence showed strongevidence of asymmetric disease severity and discrep-ancies in the data for the distribution of disease bylocation and genotype (Fig. 2).

The results of the Shapiro–Wilk test (W) indicatedthat the data were not normally distributed (W =0·4974, P < 2·2 × 10−16). Similarly, the hypothesis ofhomogeneous variance was rejected based onresults of the Bartlett test, both for genotype (D.F. =35; χ2 = 520·30; P < 2·2 × 10−16) and location (D.F. =8; χ2 = 682·88; P < 2·2 × 10−16).




Thereafter, the first step in applying the GAMMImethodology was to determine the means, variancesand coefficients of variation (CV) for the severity ofgrey leaf spot. Some discrepant values for locationand genotype were detected. The highest diseaseseverity levels were detected in Campo Mourão(36·85%) and Patos de Minas (6·67%), and thelowest levels were detected in Goianésia (0·61%)and Londrina (1·45%). Campo Mourão had thelowest coefficient of variation (70·6%) for diseaseseverity, while those for Planaltina (329·5%) and SãoSebastião do Paraíso (260·9%) were very high.

Coefficient of variation values for other locationsranged from 83·4 to 250·7%. Moreover, there waslarge variability in disease severity among genotypes.Means for grey leaf spot severity ranged from 0·9 (G15and G10) to 34·5% (G29) and the CV values rangedfrom 81·6 to 283·9% (Table 3).

The models were fit using quasi-likelihood with thelogit link function. The first model (model 1) has thevariance function (μ) = μ(1 − μ) . The second model(model 2) was based on Wedderburn (1974), inwhich the variance function is equal to the square ofthe variance of the binomial distribution, Var(μ) = [μ

Fig. 1. van Eeuwijk’s algorithm for modelling GAMMI, adapted from Sumertajaya (2007). *Analysis of Deviance (ANODEV).




(1− μ)]2. The logit link function is the linear predictormodel.The quasi-likelihood models are depicted in Fig. 3.

Graphs of the standardized deviance residuals, linearpredictor, index and normal QQ plot for model 1 aredepicted in the first column, and those representingmodel 2 are depicted in the second column. Model2 fit the data better with a more normal distributionof residuals (Fig. 3).The ANODEV with the logit link function and vari-

ance function Var(μ) = [μ(1− μ)]2 was significant forgenotype and environment and also for the two firstaxes of the G × E interaction (Table 4). The relativecontribution of genotype and environment to the inter-action is shown in Fig. 4, and the genotypes with desir-able low mean disease severity are shown in Fig. 5.Figure 4 describes the variability associated with thefirst two axes and Fig. 5 shows the relationshipbetween the average severity of grey leaf spot andthe first term of the interaction.The first two components of the GAMMI graphic

that contain the average severity of grey leaf spotand the first term of the G × E interaction identified

Campo Mourão, Goianésia, Londrina and SãoSebastião do Paraíso as locations in which theaverage disease severity of genotypes is less variable.The scores from these environments are close to thevertex, which indicates minimal variation betweengenotypes within each environment. However, thedisease severity responses of these locations were dis-tinct. For example, genotypes in Campo Mourão hadlow variability and high severity of grey leaf spot(36·8%).

The contributions of Goianésia, Londrina and SãoSebastião do Paraíso to the G × E interaction wererelatively low, as indicated by average disease sever-ities of 0·6, 1·4 and 2·3%, respectively, in theseregions (Fig. 4 and Table 3). The genotypes G9, G1and G17 had average disease severities of 1·4, 1·9and 7·0%, respectively, which were close to thevertex. Therefore, these genotypes can be consideredstable because of their low variability for diseaseseverity.

Although genotype G17 is stable, it had greater greyleaf spot severity than did G9 and G1 (Fig. 5 andTable 3); thus, only G9 and G1 can be recommended

Fig. 2. Box plot of environments and genotypes showing the distribution of grey leaf spot severity. Locations: Campo Mourão(CM), Goiânia (GO), Goianésia (GS), Jataí (JT), Londrina (LD), Ponta Grossa (PG), Planaltina (PL), Patos de Minas (PM) and SãoSebastião do Paraíso (SP).




Table 3. Mean values of grey leaf spot severity estimated from two replications of 36 maize cultivars grown in nine locations during the 2010/11 growingseason

Genotypes

Locations

Mean Variance CV%CM GO GS JT LD PG PL PM SP

G1 0·150 0·000 0·005 0·000 0·000 0·010 0·000 0·005 0·005 0·019 0·0024 251·6G2 0·150 0·000 0·000 0·000 0·005 0·010 0·005 0·005 0·005 0·020 0·0024 244·2G3 0·250 0·005 0·005 0·000 0·005 0·010 0·000 0·055 0·000 0·037 0·0067 223·0G4 0·100 0·000 0·000 0·000 0·005 0·055 0·000 0·005 0·000 0·018 0·0013 192·5G5 0·200 0·000 0·005 0·000 0·000 0·055 0·000 0·055 0·000 0·035 0·0044 188·5G6 0·350 0·000 0·005 0·000 0·000 0·055 0·000 0·055 0·005 0·052 0·0130 218·0G7 0·150 0·005 0·005 0·005 0·010 0·010 0·000 0·055 0·010 0·028 0·0024 175·1G8 0·100 0·000 0·010 0·005 0·005 0·010 0·000 0·010 0·000 0·016 0·0010 204·8G9 0·100 0·000 0·000 0·000 0·005 0·010 0·000 0·010 0·005 0·014 0·0010 223·0G10 0·010 0·000 0·005 0·000 0·005 0·050 0·000 0·010 0·000 0·009 0·0003 179·1G11 0·100 0·005 0·005 0·000 0·005 0·005 0·000 0·010 0·000 0·014 0·0010 222·3G12 0·100 0·000 0·000 0·000 0·000 0·055 0·000 0·010 0·000 0·018 0·0013 192·9G13 0·600 0·000 0·010 0·055 0·055 0·100 0·005 0·100 0·010 0·104 0·0361 183·0G14 0·300 0·000 0·000 0·000 0·000 0·055 0·000 0·005 0·000 0·040 0·0098 247·2G15 0·055 0·000 0·005 0·005 0·000 0·010 0·000 0·005 0·000 0·009 0·0003 197·9G16 0·100 0·000 0·000 0·005 0·000 0·010 0·005 0·055 0·005 0·020 0·0012 173·1G17 0·600 0·005 0·005 0·010 0·000 0·005 0·000 0·005 0·000 0·070 0·0395 283·9G18 0·350 0·000 0·005 0·005 0·000 0·100 0·000 0·010 0·010 0·053 0·0134 216·7G19 0·700 0·000 0·055 0·010 0·005 0·010 0·150 0·250 0·005 0·132 0·0528 174·4G20 0·500 0·005 0·005 0·005 0·005 0·055 0·000 0·055 0·000 0·070 0·0265 232·6G21 0·350 0·000 0·005 0·010 0·000 0·055 0·000 0·010 0·005 0·048 0·0131 236·4G22 0·400 0·000 0·000 0·050 0·050 0·005 0·005 0·010 0·005 0·058 0·0168 222·0G23 0·700 0·000 0·005 0·055 0·005 0·055 0·005 0·005 0·005 0·093 0·0523 246·5G24 0·600 0·000 0·005 0·010 0·005 0·100 0·005 0·055 0·005 0·087 0·0381 223·8G25 0·250 0·055 0·000 0·005 0·005 0·010 0·000 0·010 0·010 0·038 0·0066 211·2G26 0·350 0·005 0·005 0·010 0·000 0·010 0·005 0·010 0·005 0·044 0·0131 257·6G27 0·600 0·005 0·010 0·005 0·005 0·005 0·010 0·010 0·005 0·073 0·0391 271·6G28 0·800 0·100 0·005 0·200 0·005 0·055 0·300 0·100 0·200 0·196 0·0609 125·8G29 0·950 0·350 0·005 0·500 0·100 0·100 0·400 0·400 0·300 0·345 0·0792 81·6G30 0·600 0·055 0·010 0·010 0·010 0·055 0·000 0·055 0·005 0·089 0·0373 217·2G31 0·700 0·200 0·005 0·005 0·050 0·055 0·010 0·105 0·055 0·132 0·0492 168·4G32 0·100 0·050 0·010 0·005 0·010 0·005 0·000 0·155 0·010 0·038 0·0030 141·9G33 0·500 0·050 0·000 0·200 0·005 0·005 0·005 0·100 0·005 0·097 0·0273 170·9

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for use in maize breeding programmes. The genotypesshown in Fig. 4 that appear in the upper or lowerquadrants on the left showed the lowest severity ofgrey leaf spot. The decreasing rank order of diseaseseverity for genotypes in the upper quadrant wasG12 (6ª) > G14 (17ª) > G4 (7ª) > G10 (1ª) > G5 (13ª) >G20 (23ª) > G3 (14ª) > G11 (3ª) > G6 (20ª) > G7 (12ª) >and G9 (4ª). The decreasing rank order of diseaseseverity for genotypes in the lower quadrant was G34(9ª) > G18 (21ª) > G21 (19ª) > G8 (5ª) > G15 (2ª) > G24(26ª) > G13 (30ª) > G1 (8ª) > and G17 (24ª). The geno-types G32 (15ª) > G31 (31ª) > G35 (35ª) > G30 (27ª) >G25 (16ª) > and G36 (33ª) in the upper right quadrantwere sequentially closest to the vertex of the 1° and2° axes and had the highest grey leaf spot severity.The genotypes G28 (34ª) > G29 (36ª) > G19 (32ª) > G23(28ª) > G33 (29ª) > G22 (22ª) > G2 (11ª) > G27 (25ª) >G26 (18ª) > and G16 (10ª) in the lower right quadranthad the highest grey leaf spot severity and are shownin decreasing order of disease severity.

Model 2 allowed detection of the variance asso-ciated with the G × E interaction (Fig. 4) and axis 1and axis 2 accounted for approximately 38·3 and23·0% of variance associated with this interaction,respectively.

The genotypes G9 and G1 were nearest to thevertex, which indicated that they were resistant togrey leaf spot and that this resistance was relativelyinsensitive to environmental effects due to minimalG × E interactions. However, the remaining genotypeswere sensitive to environmental effects in terms oftheir responses to grey leaf spot and exhibited largeG × E interactions.

Genotypes with specific adaptations to particularenvironments are generally chosen based on a posi-tive relationship between that genotype’s positionand the respective environment in the same vectorialdirection, such as for crop yield, for which avector with a small angle (coincident straight line)indicates a positive correlation between genotypeand environment. However, genotypes with specificadaptation for disease resistance can be identified bythe inverse orientation of the vectors for genotypeand environment. Thus, the best genotypes toselect for adaptation to environmental conditionsshould be those with the lowest average grey leafspot severity.

Genotypes with specific adaptations were thosewith a reverse vectorial orientation relative to theenvironment, according to the proposed model.Figures 4 and 5 show that genotypes G20 (0·1%) andT

able

3.(con

t)

G34

0·10

00·00

00·01

00·00

50·00

50·05

50·00

00·00

50·00

00·02

00·00

1217

3·1

G35

0·80

00·35

00·00

50·01

00·10

00·05

50·01

00·40

00·10

50·20

40·07

1613

1·2

G36

0·50

00·20

00·01

00·20

00·05

50·05

50·00

00·20

00·05

50·14

20·02

4811

1·2

Mean

0·36

80·04

00·00

60·03

80·01

40·03

80·02

60·06

70·02

30·06

9–

–

Varianc

e0·06

770·00

820·00

010·00

930·00

070·00

100·00

710·01

010·00

370·02

32–

–

CV%

70·6

226·1

147·8

251·0

182·2

83·4

330·4

150·7

261·3

220·9

––

CM,C

ampo

Mou

rão;

GO,G

oiân

ia;G

S,Goian

ésia;JT,

Jataí;LD

,Lon

drina;

PG,P

onta

Grossa;

PL,P

lana

ltina

;PM,P

atos

deMinas;S

P,SãoSeba

stiãodo

Paraíso.




G10 (0·0%) showed a specific adaptability to SãoSebastião do Paraíso. Similarly, genotype G35(35·0%), which had the same vectorial direction asthe environment and high average disease severity,could not be recommended for Goiania, while G24(0·0%) and G8 (0·0%) were adapted to Goiania.

The genotypes with higher specific adaptability forPonta Grossa are G16 and G19, with average diseaseseverities of 1%. Genotypes G3 and G11 had loweraverage disease severities (0·0%) and greater adapt-ability for Jataí. The most desirable genotypes for thePlanaltina region were G12 and G4, due to their

specific adaptability and disease severities of 0%.Because of their high disease severity, genotypesG29 (40·0%) and G28 (30·0%) should not be recom-mended for use in breeding cultivars to grow inPlanaltina. G33 (0·5%) and G26 (1%) are the mostappropriate genotypes to recommend for use inPatos de Mina. No genotype was particularly welladapted to the conditions of Campo Mourão. On theother hand, two genotypes could be highly recom-mended for use in Goianésia, G6 (0·5%) and G26(0·5%); the latter was highly adapted to thatenvironment.

Fig. 3. Graphical diagnoses for the quasi-likelihood models: Standardized deviance residuals/linear predictor, index, andNormal QQ plots; index (i), where i is the sequential order in which the values yi were measured (proportion orpercentage leaf area severity affected on plot for genotypes). (Ai) Model 1, link function logit and variance function V(μ) =μ(1− μ); (Bi) Model 2, logit link function and variance function V(μ) = [μ(1− μ)]2.




DISCUSSION

Because the current statistical approach is not rou-tinely applied for the analysis of disease severitydata in maize under field conditions, the data distribu-tion had to first be characterized, then the model thatbest fitted the data had to be determined. The suitabil-ity of the present data for the proposed model can beseen in Fig. 3. Note the random distribution of

residuals around zero, which suggests a lack of correl-ation between the errors; the influence of error wasminor, as can be seen in the Normal QQ plot(Fig. 3); points far out of alignment were not observed.Thus, the Wedderburn model with a logistic functionwas appropriate to describe the data set.

The ANODEV was significant for genotype andenvironment as well as significant for the two firstaxes of the interaction (Table 4). In this decomposition,the singular value represents the level of associationbetween these factors. Because the variable response

Table 4. Analysis of deviance (ANODEV) for proportion of grey leaf spot severity, using model 2 with logit linkfunction and variance function Var(μ) = [μ(1−μ)]2]

Source of variation D.F. Qdev. Qdev. mean Quasi-residuals F P > F D.F. Gollob FGollob P > F

Blocks/locations 9 12·3 1·37 2298·1 0·71 0·1005 9 1·92 0·0483Locations (L) 8 1045·1 130·6 2310·4 68·5

was within the interval [0, 1] the logistic link functionwas used. Thus, the quasi-likelihood models wereevaluated with the logit link and the variance functionsVar(μ) = μ(1− μ) and Var(μ) = [μ(1− μ)]2. Therefore, theWedderburn model, model 2, more reliably describedthe data (Fig. 3).

Among the adjusted models, model 2 showed fewerdiscrepant values, did not violate the initial assump-tions, and presented significant coefficients, so it wasthe most suitable model to describe the responses inthese data. The cumulative proportion of quasi-devi-ance of the two first axes relative to the total quasi-deviance was high (61·3%) (Table 5). However, atleast 75% of the total variance could be attributed tothe first two PC axes (Ferreira 2008). This indicatesthat these components could replace the n originalvariables without excessive loss of information.These axes measured methodological efficiency, butthey could also be used to quantify the G × Einteraction.

Although the basic assumptions necessary to estimatethe stability and adaptability of genotypes in variousenvironments are usually violated, the GAMMImethod-ology is a step forward in detecting interaction effects.Previously, the required computational methods hin-dered application of the GAMMI method, but specificroutines are now available in R (these commands areshownAppendix A) to fit thesemultiplicative interactionmodels using van Eeuwijk’s algorithm (1995). The SVDof the residual matrix used to obtain the coefficients forthe main effects is shown in Table 6 together with theenvironment and genotype scores.

Duarte & Vencovsky (1999) stated that favourablecombinations of genotypes and environments havecoordinates with the same sign and are graphicallydistant from the vertex. The positive or negative inter-actions depicted by the biplot, principally those ofhigh magnitude, can be useful in plant breedingprogrammes. For disease severity, combinationswith opposite signs were of interest because they indi-cated genotypes suited to particular environments.Graphical representation as a biplot also permits thequick identification of more productive environmentswith scores of approximately zero that contribute lessto the G × E interaction. Such environments could alsobe favourable locations for the preliminary steps of aplant breeding programme (Pacheco et al. 2003).Therefore, genotypes and environments with lowscores for the interaction axes contribute less tomodel variance and are considered stable. These gen-otypes could be recommended for growing on large

crop acreage due to their high mean crop yields anddisease resistance.

In the analyses of the stability and adaptability ofgenotypes using multiplicative models, the interactioneffects can be evaluated using graphical representa-tions that approximate the SVD residual matrix ofthe model with another low rank matrix. The biplotfacilitates identification and understanding of thecomponents of the G × E interaction. Rencher (2002)defined the biplot as a two-dimensional representationof the data matrix that defines the SVD produced bythe SVD method. Here, the data matrix is the R(gxe),which identifies an element for every g vector ofobservations (g lines in the R(gxe) matrix, or genotypes)simultaneously with an element for every e variable(e columns in the R(gxe) matrix, or locations).Therefore, with this technique, one can readily identifyproductive genotypes with wide adaptability for mega-environments, limit genotypes with specific adaptabil-ity to determined agronomic zones, and identify theenvironments that should be tested (Kempton 1984;Gauch & Zobel 1996; Ferreira et al. 2006).

The graphic interpretation in Fig. 5 depicts the vari-ation caused by the main additive effects of genotypeand environment and the multiplicative effect of theG × E interaction (Gauch & Zobel 1996; Smith et al.2005). The abscissa represents the main effects (i.e.,the overall averages of the variables for the genotypesevaluated) and the ordinate is the first interaction axis(axis 1). In this case, the lower the absolute value ofaxis 1, the lower its contribution to the G× E interaction;therefore, the more stable the genotype. The ideal geno-type is one with high productivity and an axis 1 valuenear zero. An undesirable genotype has low stabilityassociated with low productivity (Kempton 1984;Gauch & Zobel 1996; Ferreira et al. 2006).

In the biplot analysis shown in Fig. 4, the cosine ofthe angle between a vector and an axis indicates thecontribution of that variable to the axis dimension.Also, the cosine of the angle between the vectors for

Table 5. Quasi-deviance proportion in relation to theproposed axes for the mean values of grey leaf spotseverity

Axes Quasi-devianceQuasi-devianceproportion

Cumulativeproportion

Axis 1 429·6 0·383 0·383Axis 2 258·3 0·230 0·613Residuals 433·9 0·387 1·000




two environments approximates their correlation.Therefore, when vectors are perpendicular, thecosine of the angles between them equals zero andthe variables are independent. But if the vectors fortwo variables are at very close angles or at a 180°angle, they are highly positively or negatively corre-lated (Gower 1995; Kroonenberg 1997). The anglesbetween the vectors for sites and genotypes, and thepositions of the vectors, permitted us to identify geno-types positively or negatively correlated with particu-lar environments (Table 7).

The negative correlation between cultivar and loca-tion has helped to identify genotypes with specificadaptations. Genotypes with a highly negative correl-ation within an environment had the lowest diseaseseverities (Fig. 4 and Fig. 5), and should therefore berecommended for use in those locations.

CONCLUSIONS

The GAMMI method efficiently described the dataregarding stability and adaptability of genotypes togrey leaf spot incidence in various locations in Brazilusing available theories and the computationalresources outlined in the present paper. A pattern ofdifferential responses to grey leaf spot in differentenvironments was found, and the GAMMI methodcould explain 61·3% of the variance due to the G × Einteraction with only two PC. The two-dimensionalanalysis detected the presence of a strong interactionbetween genotype and environment.

The GAMMI model could efficiently identify andquantify the G × E interactions, even though the datawere not normally distributed and variances were het-erocedastic. The present analyses indicated that thegenotypes G9 and G1 could be recommendedbecause of their high stability and low severity of

Table 6. GAMMI coefficients for main effects and thescores from environments and genotypes

Locations andgenotypes

Estimates ofcoefficients Axis 1 Axis 2

Intercept −1·642 – –CM – 0·138 −0·135GO −5·840 2·270 2·171GS −5·377 −1·064 0·383JT −4·224 1·278 −0·458LD −4·983 0·856 0·971PG −2·934 −1·054 0·053PL −6·075 1·022 −2·323PM −2·478 0·112 0·099SP −5·039 1·610 −1·198CM:rep2 −0·109 – –GO:rep2 0·648 – –GS:rep2 0·696 – –JT:rep2 0·164 – –LD:rep2 0·928 – –PG:rep2 −0·102 – –PL:rep2 −0·167 – –PM:rep2 −0·353 – –SP:rep2 0·248 – –G1 – −0·423 −0·570G2 −0·084 0·229 −0·813G3 0·530 −0·437 0·892G4 0·020 −1·103 0·356G5 0·442 −1·506 0·272G6 0·789 −0·758 −0·535G7 0·849 0·326 0·276G8 0·640 −0·710 0·499G9 −0·318 0·143 −0·258G10 0·049 −1·158 0·381G11 −0·113 −0·237 0·798G12 −0·288 −1·175 0·162G13 2·402 −0·180 −0·248G14 −0·103 −1·210 0·155G15 −0·203 −0·619 −0·091G16 0·258 −0·009 −1·060G17 0·571 −0·077 0·559G18 1·139 −0·684 −0·817G19 2·771 −0·751 −1·411G20 1·259 −0·564 0·980G21 0·972 −0·630 −0·707G22 0·907 1·524 0·095G23 1·808 −0·286 −0·803G24 1·640 −0·732 −0·814G25 0·251 1·040 0·628G26 0·913 0·473 −0·193G27 1·538 0·110 −0·366G28 2·938 1·234 −0·830G29 3·840 1·476 −0·546G30 1·723 0·175 1·055

Table 6. (Cont.)

Locations andgenotypes

Estimates ofcoefficients Axis 1 Axis 2

G31 2·501 0·973 0·322G32 1·413 0·355 0·817G33 1·125 1·599 −0·198G34 0·652 −1·029 0·254G35 2·963 1·043 0·445G36 2·566 0·926 0·635

CM, Campo Mourão; GO, Goiânia; GS, Goianésia; JT, Jataí;LD, Londrina; PG, Ponta Grossa; PL, Planaltina; PM, Patosde Minas; SP, São Sebastião do Paraíso; Rep2, Repetition 2.




grey leaf spot. Campo Mourão, Goianésia, Londrina,and São Sebastião do Paraíso were locations inwhich average disease severity was more stable, indi-cating that these locations made a minor contributionto the G × E interaction. The scores from these envir-onments had values close to the vertex in thefigures, which indicated less variability among geno-types for disease severity. However, the responses todisease in these locations were distinct. Forexample, Campo Mourão exhibited low variabilityand high severity of grey leaf spot (36·8%).Genotypes with specific adaptability and low severityof grey leaf spot for specific locations were G26 forGoianésia, G24 and G8 for Goiânia, G3 and G11for Jataí, G19 and G16 for Ponta Grossa, G12 and G4for Planaltina, G26 and G33 for Patos de Minas, andG10 and G20 for São Sebastião do Paraíso. Theseresults will be useful to guide recommendations of cul-tivars with stable resistance to grey leaf spot and highyield in particular environments.

REFERENCES

AGRESTI, A. (2002). Categorical Data Analysis. New Jersey,USA: John Wiley & Sons.

ALLARD, R.W. (1999). Principles of Plant Breeding.New York, USA: John Wiley & Sons.

ANNICCHIARICO, P., BELLAH, F. & CHIARI, T. (2005). Defining sub-regions and estimating benefits for a specific-adaptationstrategy by breeding programs: a case study. Crop Science45, 1741–1749.

BHATIA, A. & MUNKVOLD, G. P. (2002). Relationships of envir-onmental and cultural factors with severity of gray leafspot in maize. Plant Disease 86, 1127–1133.

BRITO, A. H., VON PINHO, R. G., POZZA, E. A., PEREIRA, J. L. A. R.& FARIA FILHO, E. M. (2007). Efeito da Cercosporiose no

rendimento de híbridos comerciais de milho.Fitopatologia Brasileira 32, 472–479.

CORDEIRO, G.M. & DEMÉTRIO, C. G. B. (2008). ModelosLineares Generalizados e Extensões. Piracicaba, Brazil:Escola Superior de Agricultura “Luiz de Queiroz” –Universidade de São Paulo.

CROSSA, J., FOX, P. N., PFEIFFER, W. H., RAJARAM, S. & GAUCH, H.G., Jr (1991). AMMI adjustment for statistical analysis ofan international wheat yield trial. Theoretical andApplied Genetics 81, 27–37.

CRUZ, C. D., REGAZZI, A. J. & CARNEIRO, P. C. S. (2006).Modelos Biométricos Aplicados ao MelhoramentoGenético. Viçosa, Brazil: Universidade Federal de Viçosa.

DOBSON, A. J. A. (2002). Introduction to Generalized LinearModels. New York, USA: Chapman & Hall CRC.

DUARTE, J. B. & VENCOVSKY, R. (1999). Interação Genótipos xAmbientes uma introdução à análize “AMMI”. MonographSeries, 9. Ribeirão Preto, Brazil: Sociedade Brasileira deGenética.

FANTIN, G.M., BRUNELLI, K. R., RESENDE, I. C. & DUARTE, A. P.(2001). A mancha de Cercospora do milho. Campinas,Brazil: Instituto Agronômico de Campinas.

FORNASIERI FILHO, D. (2007). Manual da Cultura do Milho.Jaboticabal, Brazil: Funep.

FERREIRA, D. F. (2008). Estatística Multivariada. Lavras, Brazil:Universidade Federal de Lavras.

FERREIRA, D. F., DEMÉTRIO, C. G. B., MANLY, B. F. J., MACHADO, A.A., VENCOVSKY, R. (2006). Statistical models in agriculture:biometrical methods for evaluating phenotypic stability inplant breeding. Cerne 12, 373–388.

GABRIEL, R. (1998). Generalized bilinear regression.Biometrika 85, 689–700.

GAUCH, H. G. & ZOBEL, R.W. (1988). Predictive and postdic-tive success of statistical analyses of yield trials.Theoretical and Applied Genetics 76, 1–10.

GAUCH, H. G. & ZOBEL, R.W. (1996). AMMI analysis of yieldtrials. In Genotype by Environment Interaction (Eds M. S.Kang & H. G. Gauch), pp. 85–122. New York, USA:Boca Raton CRC Press.

GOLLOB, H. F. (1968). A statistical model which combinesfeatures of factor analytic and analysis of variance techni-ques. Psychometrika 33, 73–115.

GOWER, J. C. (1995). A general theory of biplots. In RecentAdvances in Descriptive Multivariate Analysis (Ed. W. J.Krzanowski), pp. 283–303. Royal Statistical SocietyLecture Notes, 2. Oxford: Clarendon Press.

HADI, A. F., MATTJIK, A. A. & SUMERTAJAYA, I. M. (2010).Generalized AMMI models for assessing theendurance of soybean to leaf pest. Jurnal Ilmu Dasar11, 151–159.

IENTILUCCI, E. J. (2003). Using the Singular ValueDecomposition. New York, USA: Chester F. CarlsonCenter for Imaging Science, Rochester Institute ofTechnology. Available from: http://www.cis.rit.edu/~ejipci/Reports/svd.pdf (verified 27 October 2016).

KEMPTON, R. A. (1984). The use of biplots in interpretingvariety by environment interactions. Journal ofAgricultural Science, Cambridge 103, 123–135.

KROONENBERG, P. M. (1997). Introduction to Biplots for G × ETables. Research Report #51. Leiden: Leiden University.

Table 7. Genotypes positively or negatively associatedwith specific environments

Locations

High positivecorrelation Genotypesunfavourable

High negativecorrelation Genotypesfavourable

GS G6 G26PM – G22SP G26 G6 and G10PG G14 G16 and G19GO G35 G24 and G8JT G22, G23 and G33 G3 and G11PL G2, G29 and G28 G12 and G4

GS, Goianésia; PM, Patos de Minas; SP, São Sebastião doParaíso; PG, Ponta Grossa; GO, Goiânia; JT, Jataí; LD,Londrina; PL, Planaltina.



http://www.cis.rit.edu/~ejipci/Reports/svd.pdfhttp://www.cis.rit.edu/~ejipci/Reports/svd.pdfhttp://www.cis.rit.edu/~ejipci/Reports/svd.pdfhttps://doi.org/10.1017/S0021859616001015https:/www.cambridge.org/corehttps:/www.cambridge.org/core/terms

MCCULLAGH, P. & NELDER, J. A. (1989). Generalized LinearModels. London, UK: Chapman & Hall.

PACHECO, R. M., DUARTE, J. B., ASSUNÇÃO, M. S., JUNIOR, J. N. &CHAVES, A. A. P. (2003). Zoneamento e adaptação produ-tiva de genótipos de soja de ciclo médio de maturaçãopara Goiás. Pesquisa Agropecuária Tropical 33, 23–27.

PAULA, G. A. (2004). Modelos de Regressão com ApoioComputacional. São Paulo, Brazil: Instituto de Matemáticae Estatística da Universidade de São Paulo (IME-USP)..

R Development Core Team (2013). R: A Language andEnvironment for Statistical Computing. Vienna, Austria:R Foundation for Statistical Computing. Available from:http://www.R-project.org/

RENCHER, A. C. (2002).Methods of Multivariate Analysis, 2ndedn. New York, USA: John Wiley & Sons.

SEARLE, S. R., CASELLA, G. &MCCULLOCH, C. E. (1992). VarianceComponents. New York, USA: John Wiley & Sons.

SMITH, A. B., CULLIS, B. R. & THOMPSON, R. (2005). The analysisof crop cultivar breeding and evaluation trials: an

overview of current mixed model approaches. Journal ofAgricultural Science, Cambridge 143, 449–462.

SUMERTAJAYA, I. M. (2007). Analisis Statistik Interaksi GenotipeDengan Lingkungan. Bogor, Indonesia: DepartemenStatistik, Fakultas Matematika dan IPA, IPB (Abstract inEnglish).

TARAKANOVAS, P. & RUZGAS, V. (2006). Additive main effect andmultiplicative interaction analysis of grain yield of wheatvarieties in Lithuania. Agronomy Research 4, 91–98.

TURNER, H. & FIRTH, D. (2009). Generalized Nonlinear Modelsin R: An Overview of the gnm Package (R Package Version0.10–0). Warwick, UK: University of Warwick.

VAN EEUWIJK, F. A. (1995). Multiplicative interaction in gener-alized linear models. Biometrics 51, 1017–1032.

WEDDERBURN, R.W.M. (1974). Quasi-likelihood functions,generalized linear models and the Gauss-Newtonmethod. Biometrika 61, 439–447.

ZOBEL, R.W., WRIGHT, M. J. & GAUCH, H. G. (1988). Statisticalanalysis of a yield trial. Agronomy Journal 80, 388–393.

APPENDIX A

The multiplicative term of this model was estimatedin R software with the generalized nonlinear modelsgnm function using the Mult (factor1, factor2, inst =…) command, which specifies the multiplicativeinteractions that are linear or nonlinear predictors.The subscripts 1 and 2 represent the multiplicativefactors of the interaction and inst is an integer thatspecifies the number of interactions (Turner & Firth

2009). The function residSVD (model, fac1, andfac2, d =…) performed the SVD of the residualmatrix. This residSVD function uses the first dcomponents of the SVD to approximate a residualvector from the model by adding d multiplicativeterms (Turner & Firth 2009). Finally, the model isre-adjusted by the update command (object,formula … evaluate = TRUE), which assumes thecoefficients from the previous model as startingvalues for the new model.



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Applying the generalized additive main effects and multiplicative interaction model to analysis of maize genotypes resistant to grey leaf spotINTRODUCTIONMATERIALS AND METHODSAdditive main effects and multiplicative interaction modelGeneralized linear modelsQuasi-likelihood modelsGeneralized additive main effects and multiplicative interaction

RESULTSDISCUSSIONCONCLUSIONSREFERENCES

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