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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.
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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
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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).
GAMMI analysis of maize resistant to grey leaf spot 5
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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).
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(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).
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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.
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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.
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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
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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.
GAMMI analysis of maize resistant to grey leaf spot 13
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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.
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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.
GAMMI analysis of maize resistant to grey leaf spot 15
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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