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

### Options

`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` |

### Parameters

`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 |

### Description

`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

√{ ( (IPCA_{1} scores) × (IPCA_{1} s.s.) / (IPCA_{2} s.s.) )^{2} + (IPCA_{2} 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 IPCA_{1} scores to take account of their larger contribution to the genotype-by-environment interaction.

Options: `PRINT`

, `NROOTS`

, `DIMENSIONS`

, `PLOT`

, `SCALING`

.

Parameters: `DATA`

, `GENOTYPES`

, `ENVIRONMENTS`

, `REPLICATES`

, `GSCORES`

, `ESCORES`

, `RESIDUALS`

, `FITTEDVALUES`

, `TITLEPREFIX`

, `AOVTABLE`

, `STABILITY`

.

### Method

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.

### References

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

Procedures: `GESTABILITY`

, `GGEBIPLOT`

, `RFINLAYWILKINSON`

, `DBIPLOT`

.

Commands for: REML analysis of linear mixed models.

### Example

CAPTION 'AMMI example'; STYLE=meta VARIATE [NVALUES=32; VALUES=2729,2662,2638,2680,2598,2908,2732,2747,\ 2238,2425,2072,3036,3430,2951,2593,2454,2191,2994,3097,3265,\ 2432,2832,2528,2079,3218,2641,2823,2866,2543,3654,2875,2635] Yield FACTOR [NVALUES=32; LEVELS=2] Rep FACTOR [NVALUES=32; LABELS=!T(E1,E2,E3,E4)] Env 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]\ Yield; GENOTYPES=Genotype; ENVIRONMENTS=Env; REPLICATES=Rep