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AMMI procedure

Allows exploratory analysis of genotype × environment interactions (M. Talbot, K. Brown & M.F. Smith).


PRINT = string tokens Results to be output (aovtable, genotype, environment, estimates, envtable, cluster, stability); default * i.e. none
NROOTS = scalar Number of IPCA scores required; default is to take as many roots as possible up to a maximum of 9
DIMENSIONS = scalars Two numbers specifying the dimensions to display in the biplots; default 1,2
PLOT = string tokens Types of biplot to display (mean, ipca); default * i.e. none
SCALING = string token Scaling to use for the ipca biplot (genotype, environment, symmetric); default envi


DATA = variates Provides the data to be analysed
GENOTYPES = factors Specifies the genotypes
ENVIRONMENTS = factors Specifies the environments
REPLICATES = factors Replication factor; this should be omitted if the data comprises just the genotype by environment means
GSCORES = pointers Pointer containing a set of variates (each of length equal to the number of genotypes) to save the genotype IPCA scores
ESCORES = pointers Pointer to a set of variates to save the environment IPCA scores
RESIDUALS = variates Saves the residuals from the AMMI model
FITTEDVALUES = variates Saves the fitted values from the AMMI model
TITLEPREFIX = texts Specifies a prefix to use for the titles of the plots
AOVTABLE = pointers Saves the analysis-of-variance table
STABILITY = variates Saves the AMMI stability values


AMMI is a procedure for fitting, to data classified by two factors, a model which involves the Additive Main effects of ANOVA along with the Multiplicative Interaction effects of principal components analysis (PCA). The method is used when analysing data from a series of trials with crop genotypes.

A principal components model is fitted to the residuals from the ANOVA and the resulting scores, called the I (for interaction) PCA are calculated for both the genotypes and the trials or environments.

The data to be analysed can be supplied in a variate using the DATA parameter. The associated genotype and environment factors are specified using the GENOTYPES and ENVIRONMENTS parameters, respectively. You can also use the REPLICATES parameter to specify a factor defining replicates within environments. When constructing the analysis-of-variance table, AMMI assumes that these replicates arise from the use of a randomized block design within each environment. No missing values are allowed, and there must be equal replication. If you have a more complicated structure, you can form the genotype × environment means (for example using ANOVA and AKEEP. or REML and VKEEP), and supply this instead. If the GENOTYPES and ENVIRONMENTS are not specified as well as the table, it is assumed that the rows of the table correspond to the genotypes, and the columns correspond to the environments.

The NROOTS option allows the number of roots (sets of scores) for the principal component analysis to be specified.

The PRINT option allows a choice of results to be requested by settings:

    aovtable analysis-of-variance table summarising the contribution of each component to the interaction term,
    genotype genotype means and scores and stability,
    environment environment means and scores,
    envtable table of environment means and variances,
    estimates genotype estimates for each environment,
    cluster hierarchical clustering of AMMI genotype estimates over environments (using the average link method and Euclidean test for the similarity matrix),
    stability AMMI stability values (Purchase, Hatting & van Deventer 2000).

The PLOT option controls the biplots that are displayed. The setting mean produces a biplot of the genotype and environment means against their corresponding IPCA scores. The setting ipca produces a biplot of the IPCA scores.

The scaling used for the ipca biplot is controlled by the SCALING option. The settings environment and genotype multiply the environment or genotype scores, respectively, by their corresponding eigenvalues. The symmetric multiplies both the environment and the genotype scores by the square roots of their corresponding eigenvalues.

By default, the plots are produced using the first two dimensions of IPCA scores, but you can specify other dimensions using the DIMENSIONS option.

The default titles for the plots are prefixed using the identifier of the DATA variate or table. However, you can supply an alternative prefix using the TITLEPREFIX parameter.

The genotype and environment IPCA scores can be saved within a pointer to a set of variates, using the GSCORES and ESCORES parameters respectively. The fitted values for the AMMI model can be saved using the FITTEDVALUES parameter, and the simple residuals can be saved using the RESIDUALS parameter.

The AOVTABLE parameter saves the analysis-of-variance table, in a pointer with elements labelled 'Source', 'd.f.', 's.s.', 'm.s.', 'v.r.' and 'F pr'.

The STABILITY parameter saves the AMMI stability values, defined as

√{ ( (IPCA1 scores) × (IPCA1 s.s.) / (IPCA2 s.s.) )2 + (IPCA2 scores)2 }

by Purchase, Hatting & van Deventer (2000). These are the distances of the genotypes from zero in the 2-dimensional plot of genotype scores, but with an additional weighting for the IPCA1 scores to take account of their larger contribution to the genotype-by-environment interaction.




The data are averaged over replicates, and the genotype by environment means are calculated. ANOVA is used to provide the main effects, sums of squares and degrees of freedom. The matrix of residuals from ANOVA are then decomposed by singular value decomposition to generate the AMMI analysis (see, for example, Gauch 1992).

Action with RESTRICT

If the DATA variate is restricted the analysis will involve only the units not excluded by the restriction.


Gauch, H.G. (1992). Statistical Analysis of Regional Yield Trials – AMMI analysis of factorial designs. Elsevier, Amsterdam.

Purchase, J.L., Hatting, H. & van Deventer, C.S. (2000). Genotype × environment interaction of winter wheat (Triticum aestivum L.) in South Africa: II Stability analysis of yield performance. S. Afr. J. Plant Soil, 17, 101-107.

See also


Commands for: REML analysis of linear mixed models.


CAPTION  'AMMI example'; STYLE=meta
VARIATE  [NVALUES=32; VALUES=2729,2662,2638,2680,2598,2908,2732,2747,\
         2432,2832,2528,2079,3218,2641,2823,2866,2543,3654,2875,2635] Yield
FACTOR   [NVALUES=32; LABELS=!T(G1,G2,G3,G4)] Genotype
GENERATE Env,Rep,Genotype
VARIATE  [NVALUES=4] Gsc[1, 2]
AMMI     [PRINT=aovtable,genotype,environment; NROOTS=2; PLOT=ipca,mean]\
Updated on March 11, 2019

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