Factor Analytic Selection Tools: FAST and iClass Analysis

A unified implementation of the FAST (Factor Analytic Selection Tools; Smith & Cullis 2018) and iClass (interaction class; Smith et al. 2021) approaches for summarising variety performance from a Factor Analytic Linear Mixed Model fitted in ASReml-R V4.

Both methods build on the rotated FA loadings $\hat{\bm{\Lambda}}$ and score EBLUPs $\hat{\bm{f}}$ returned by ASExtras4::fa.asreml(). The Common Variety Effect (CVE) for genotype $g$ in environment $j$ is the FA regression prediction:

$$ \widehat{\text{CVE}}(g,j) = \sum_{r=1}^{k} \hat{\lambda}_{rj}\,\hat{f}_{rg} $$

and the Variety Effect (VE) adds the environment-specific variance: $\widehat{\text{VE}}(g,j) = \widehat{\text{CVE}}(g,j) + \hat{\psi}_j$.

Usage

fast(
  model,
  term = "fa(Site, 4):Genotype",
  type = c("all", "FAST", "iClass"),
  ic.num = 2L,
  ...
)

Arguments

model

An ASReml-R V4 model object containing a Factor Analytic random term.

term

Character string identifying the FA random term, written as "fa(<EnvFactor>, k):<GenotypeFactor>". The genotype factor may be wrapped in vm(...) for pedigree/genomic models. Default "fa(Site, 4):Genotype".

type

Analysis type. One of:

"all" (default): Compute both FAST and iClass metrics.
"FAST": Compute global Overall Performance and Stability only.
"iClass": Compute iClass labels, iClassOP, and iClassRMSD only.

ic.num

Integer. Number of factors used to form iClasses and compute iClassOP. Must be $\le k$. Only used when type includes iClass. Default 2.

...

Additional arguments forwarded to ASExtras4::fa.asreml().

Value

A data frame with one row per environment $\times$ genotype combination, containing:

<EnvFactor>: Environment labels.
<GenotypeFactor>: Genotype labels.
loads1, ..., loadsK: Rotated FA loadings per environment.
spec.var: Specific (residual) genetic variance per environment.
score1, ..., scoreK: Rotated FA score EBLUPs per genotype.
fitted1, ..., fittedK: Per-factor contributions to CVE: $\hat{\lambda}_{rj}\hat{f}_{rg}$.
CVE: Common Variety Effect (sum of fitted values).
VE: Total Variety Effect: CVE + spec.var.
OP: (FAST) Overall Performance – same value repeated for each environment row of a genotype.
dev: (FAST, $k > 1$) Residual from first-factor regression.
stab: (FAST, $k > 1$) RMSD stability.
iclass: (iClass) Sign-pattern iClass label.
iClassOP: (iClass) Within-iClass Overall Performance.
iClassRMSD: (iClass) Within-iClass RMSD.

FAST (`type = "FAST"`)

After rotation, the first factor captures the dominant non-crossover genotype-environment interaction (GEI) pattern and typically has all-positive loadings. FAST summarises each genotype by:

Overall Performance (OP): $\text{OP}(g) = \bar{\lambda}_1 \cdot \hat{f}_{1g}$, where $\bar{\lambda}_1 = t^{-1}\sum_j\hat{\lambda}_{1j}$ is the mean first-factor loading. This is on the same scale as the trait and represents the genotype's expected performance at the average environment.
Stability (stab): $\text{stab}(g) = \sqrt{t^{-1}\sum_j \text{dev}(g,j)^2}$, where $\text{dev}(g,j) = \widehat{\text{CVE}}(g,j) - \hat{\lambda}_{1j} \hat{f}_{1g}$ is the residual from the first-factor regression (i.e.\ the combined contribution of all higher-order bipolar factors). A small RMSD indicates broad adaptation.

iClass (`type = "iClass"`)

When non-trivial crossover GEI is present, a single global OP is misleading. iClass resolves this by grouping environments into interaction classes based on the sign pattern of their first ic.num rotated loadings:

$$ \text{iClass}(j) = \bigwedge_{r=1}^{k} \begin{cases} \text{p} & \hat{\lambda}_{rj} \ge 0 \\ \text{n} & \hat{\lambda}_{rj} < 0 \end{cases} $$

Within each iClass $\omega$:

iClassOP: $\text{iClassOP}(g,\omega) = \sum_{r=1}^{k} \bar{\lambda}_{r\omega} \cdot \hat{f}_{rg}$, where $\bar{\lambda}_{r\omega}$ is the mean of factor $r$'s loadings across environments in $\omega$. Equals the mean CVE across environments in $\omega$.
iClassRMSD: $\text{iClassRMSD}(g,\omega) = \sqrt{|\omega|^{-1} \sum_{j\in\omega} \text{dev}_{ic}(g,j)^2}$, where $\text{dev}_{ic}(g,j) = \widehat{\text{CVE}}(g,j) - \sum_{r=1}^{k} \hat{\lambda}_{rj}\hat{f}_{rg}$ measures residual crossover GEI within the class.

References

Smith, A.B. & Cullis, B.R. (2018). Plant breeding selection tools built on factor analytic mixed models for multi-environment trial data. Euphytica, 214, 143.

Smith, A.B., Norman, A., Kuchel, H. & Cullis, B.R. (2021). Plant variety selection using interaction classes derived from factor analytic linear mixed models: models with independent variety effects. Frontiers in Plant Science, 12, 737462.