FA.AMMI {ammistability}

Stability Measure Based on Fitted AMMI Model

Description

FA.AMMI computes the Stability Measure Based on Fitted AMMI Model (FA) (Raju 2002) considering all significant interaction principal components (IPCs) in the AMMI model. Using FA, the Simultaneous Selection Index for Yield and Stability (SSI) is also calculated according to the argument ssi.method.

Usage

FA.AMMI(model, n, alpha = 0.05, ssi.method = c("farshadfar", "rao"), a = 1)

Arguments

model

The AMMI model (An object of class AMMI generated by AMMI).

n

The number of principal components to be considered for computation. The default value is the number of significant IPCs.

alpha

Type I error probability (Significance level) to be considered to identify the number of significant IPCs.

ssi.method

The method for the computation of simultaneous selection index. Either "farshadfar" or "rao" (See SSI).

a

The ratio of the weights given to the stability components for computation of SSI when method = "rao" (See SSI).

Details

The Stability Measure Based on Fitted AMMI Model (\(FA\)) (Raju 2002) is computed as follows:

\[FA = \sum_{n=1}^{N'}\lambda_{n}^{2}\gamma_{in}^{2}\]

Where, \(N'\) is the number of significant IPCs (number of IPC that were retained in the AMMI model via F tests); \(\lambda_{n}\) is the singular value for \(n\)th IPC and correspondingly \(\lambda_{n}^{2}\) is its eigen value; and \(\gamma_{in}\) is the eigenvector value for \(i\)th genotype.

When \(N'\) is replaced by 1 (only first IPC axis is considered for computation), then the parameter \(FP\) can be estimated (Zali et al. 2012).

\[FP = \lambda_{1}^{2}\gamma_{i1}^{2}\]

When \(N'\) is replaced by 2 (only first two IPC axes are considered for computation), then the parameter \(B\) can be estimated (Zali et al. 2012).

\[B = \sum_{n=1}^{2}\lambda_{n}^{2}\gamma_{in}^{2}\]

When \(N'\) is replaced by \(N\) (All the IPC axes are considered for computation), then the parameter estimated is equivalent to Wricke's ecovalence (\(W_{(AMMI)}\)) (Wricke 1962; Zali et al. 2012).

\[W_{(AMMI)} = \sum_{n=1}^{N}\lambda_{n}^{2}\gamma_{in}^{2}\]

Value

A data frame with the following columns:

FA

The FA values.

SSI

The computed values of simultaneous selection index for yield and stability.

rFA

The ranks of FA values.

rY

The ranks of the mean yield of genotypes.

means

The mean yield of the genotypes.

The names of the genotypes are indicated as the row names of the data frame.

References

Raju BMK (2002). “A study on AMMI model and its biplots.” Journal of the Indian Society of Agricultural Statistics, 55(3), 297–322.

Wricke G (1962). “On a method of understanding the biological diversity in field research.” Zeitschrift fur Pflanzenzuchtung, 47, 92–146.

Zali H, Farshadfar E, Sabaghpour SH, Karimizadeh R (2012). “Evaluation of genotype × environment interaction in chickpea using measures of stability from AMMI model.” Annals of Biological Research, 3(7), 3126–3136.

See Also

AMMI, SSI

Examples

library(agricolae)
data(plrv)

# AMMI model
model <- with(plrv, AMMI(Locality, Genotype, Rep, Yield, console = FALSE))

# ANOVA
model$ANOVA

# IPC F test
model$analysis

# Mean yield and IPC scores
model$biplot

# G*E matrix (deviations from mean)
array(model$genXenv, dim(model$genXenv), dimnames(model$genXenv))

# With default n (N') and default ssi.method (farshadfar)
FA.AMMI(model)

# With n = 4 and default ssi.method (farshadfar)
FA.AMMI(model, n = 4)

# With default n (N') and ssi.method = "rao"
FA.AMMI(model, ssi.method = "rao")

# Changing the ratio of weights for Rao's SSI
FA.AMMI(model, ssi.method = "rao", a = 0.43)