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Griffing’s methods with random effects

Muhammad Yaseen

2026-11-03

Source: vignettes/DiallelAnalysisR_05_griffing_random.qmd

1 Abstract

With random effects (Model 2), Griffing’s analysis estimates the variances of general and specific combining ability and, through them, the additive and dominance variances of the population the parents come from.

2 Variance components 

r1 <-
  Griffing(
      y          = Yield
    , Rep        = Rep
    , Cross1     = Cross1
    , Cross2     = Cross2
    , data       = GriffingData1
    , Method     = 1
    , Model      = 2
    , Reciprocal = c("combined", "split")[1]
    )
knitr::kable(x = r1$ANOVA, digits = 3)
Df Sum Sq Mean Sq F value Pr(>F)
gca 7 3804.643 543.520 0.690 0.680
sca 28 22060.789 787.885 27.378 0.000
reciprocals 28 365.537 13.055 0.454 0.992
Error 189 5438.991 28.778 NA NA 
knitr::kable(x = r1$Genetic.Components, digits = 3)
Components
gca -14.440
sca 426.166
Reciprocal -7.861
Error 28.778
Additive -28.881
Dominant 426.166
gca/sca -0.034

In the random model gca is tested against sca (the mean square that contains every term of \(\E(MS_g)\) except \(\sigma^2_g\)), and sca and reciprocals against the error.

3 All four methods 

comp <-
  sapply(
      X   = 1:4
    , FUN = function(m) {
        f <-
          Griffing(
              y          = Yield
            , Rep        = Rep
            , Cross1     = Cross1
            , Cross2     = Cross2
            , data       = get(paste0("GriffingData", m))
            , Method     = m
            , Model      = 2
            , Reciprocal = c("combined", "split")[1]
            )
        f$Genetic.Components[c("gca", "sca", "Error"), "Components"]
      }
    )
dimnames(comp) <- list(c("gca", "sca", "Error"), paste("Method", 1:4))
round(x = comp, digits = 2)
      Method 1 Method 2 Method 3 Method 4
gca     -14.44   -18.31    -9.59    -3.09
sca     426.17   419.55   443.24   434.04
Error    28.78    27.92    30.33    31.19

The variance of sca is far larger than that of gca in every method: in this material most of the variation among crosses is non-additive. A negative estimate (here gca) means that the true component is small; it is reported as computed so that the analysis of variance and the components stay consistent.

4 Additive and dominance variance

For non-inbred parents without epistasis \(\sigma^2_A = 2\sigma^2_g\) and \(\sigma^2_D = \sigma^2_s\); their ratio, and the ratio \(\sigma^2_g/\sigma^2_s\) (also called the predictability ratio), indicate how far selection on parental performance can predict the crosses.

r1$Genetic.Components[c("Additive", "Dominant", "gca/sca"), , drop = FALSE]
          Components
Additive -28.8809045
Dominant 426.1656570
gca/sca   -0.0338846

5 Summary

Model 2 answers a question about a population (how much additive and dominance variance) rather than about the parents tested; its components come from the expected mean squares of vignette 03.

6 References

Griffing, B. (1956). Concept of general and specific combining ability in relation to diallel crossing systems. Australian Journal of Biological Sciences, 9(4), 463–493.

Singh, R. K. and Chaudhary, B. D. (2004). Biometrical Methods in Quantitative Genetic Analysis. Kalyani Publishers, New Delhi.