Least-squares AMMI model

ge_ammi() computes the least-squares estimates of the AMMI model with all interaction axes: grand mean, genotype and environment effects, singular values and vectors of the interaction, and the error variance. They are the starting values of bayes_ammi() and the classical estimates it is compared with.

Usage

ge_ammi(.data, .y, .gen, .env, .rep)

# Default S3 method
ge_ammi(.data, .y, .gen, .env, .rep)

Arguments

.data: a data frame with one row per plot.
.y, .gen, .env, .rep: the columns of the response, genotypes, environments and replicates: bare names, strings or vectors.

Value

A list (class c("bam_ge_ammi", "bam_result", "list")) with the components of the earlier versions:

g, e, Rep, k: numbers of genotypes, environments, replicates and axes.
ge_means: the result of ge_mean().
mu, sigma2, tau: grand mean, error variance and its reciprocal (precision).
tao, delta: genotype and environment effects (g_eff(), e_eff()).
lambdas, alphas, gammas: singular values and genotype and environment singular vectors of the interaction.

Details

With the cell means $\bar{y}_{ij}$ of $B$ replicates, $\hat{\mu} = \bar{y}$, $\hat{\tau}_i = \bar{y}_{i\cdot} - \bar{y}$, $\hat{\delta}_j = \bar{y}_{\cdot j} - \bar{y}$ and the singular value decomposition of the interaction effects (ge_eff()), $K = \min(G, E) - 1$ axes. The error variance is the residual variance of the model with random replicates within environments (ge_model()), in closed form for balanced data: $$\hat{\sigma}^2 = \frac{\mathrm{SSW} - \mathrm{SS}_{B(E)}}{E(B - 1)(G - 1)}$$ when the replicate mean square is not smaller, otherwise (the replicate variance component estimated as zero) $\mathrm{SSW}/(GE(B - 1))$; SSW is the within-cell and $\mathrm{SS}_{B(E)}$ the replicate-within-environment sum of squares.

Change from the earlier versions. The error variance came from lme4::lmer(); the closed form gives the same value for balanced data without needing lme4.

References

Crossa, J., Perez-Elizalde, S., Jarquin, D., Cotes, J. M., Viele, K., Liu, G. and Cornelius, P. L. (2011). Bayesian estimation of the additive main effects and multiplicative interaction model. \emphCrop Science, \strong51(4), 1458–1469. \doi10.2135/cropsci2010.06.0343

Gauch, H. G. (1992). Statistical Analysis of Regional Yield Trials: AMMI Analysis of Factorial Designs. Elsevier, Amsterdam.

Author

Muhammad Yaseen (myaseen208@gmail.com)

Examples

data(Maiz)
fit <-
  ge_ammi(
      .data = Maiz
    , .y    = y
    , .gen  = entry
    , .env  = site
    , .rep  = rep
    )
fit$lambdas
#> [1] 5922.8913 3070.4543 2552.4065 2320.2216 1758.6707 1313.4148  792.6324
#> [8]  757.0277
fit$sigma2
#> [1] 602844.9