1 Abstract

Griffing’s four methods with the fixed-effects model (Model 1): the analysis of variance, the gca, sca and reciprocal effects, their standard errors and how to compare parents and crosses.

2 The four methods on one data set

The data sets for Methods 2 to 4 are subsets of the complete diallel, so the methods can be compared directly.

fits <-
  lapply(
      X   = 1:4
    , FUN = function(m) Griffing(
          y          = Yield
        , Rep        = Rep
        , Cross1     = Cross1
        , Cross2     = Cross2
        , data       = get(paste0("GriffingData", m))
        , Method     = m
        , Model      = 1
        , Reciprocal = c("combined", "split")[1]
        )
    )
names(fits) <- paste("Method", 1:4)
do.call(
    what = rbind
  , args = lapply(X = fits, FUN = function(f) f$ANOVA[c("gca", "sca"), "Mean Sq"])
  )

             [,1]     [,2]
Method 1 543.5204 787.8853
Method 2 264.4133 447.4681
Method 3 801.7759 916.8174
Method 4 446.6969 465.2278

3 Method 1: parents, \(F_1\) and reciprocals

With \(a_i = Y_{i\cdot} + Y_{\cdot i}\) and \(Y_{\cdot\cdot}\) the grand total, \[ SS_g = \frac{1}{2P}\sum_i a_i^2 - \frac{2}{P^2}Y_{\cdot\cdot}^2, \qquad \hat{g}_i = \frac{a_i}{2P} - \frac{Y_{\cdot\cdot}}{P^2}, \qquad \hat{R}_{ij} = \frac{Y_{ij} - Y_{ji}}{2} . \]

f1 <- fits[["Method 1"]]
knitr::kable(x = f1$ANOVA, digits = 3)

knitr::kable(x = f1$Genetic.Components, digits = 4)

The table Effects keeps the layout of earlier versions: gca effects on the diagonal, sca effects above it and reciprocal effects below it. The full matrices are in SCA and Reciprocal:

round(x = f1$Effects, digits = 2)

      1     2     3     4     5      6      7      8
1  0.06 -7.43 -3.49 29.66 27.30 -20.18  10.41 -31.03
2  3.16  0.47  7.82  0.96 12.52 -17.00  -5.72   2.28
3 -5.89  3.51  9.03  5.32 -6.17   2.39   2.98  25.89
4 -2.09 -0.57 -1.02 -3.44 -0.65  21.85 -33.50 -31.42
5  4.50 -1.48  3.01  0.95 -2.33  -0.43 -15.32  14.74
6 -0.14  0.98  0.65 -1.43 -1.91  -1.93  22.59 -22.73
7 -0.81  4.91  1.55  3.40  1.59   2.97   7.24  32.83
8  1.96  1.25  1.17  0.06  0.75  -3.25   3.55  -9.10

round(x = f1$SCA[1:4, 1:4], digits = 2)

      1     2      3     4
1 -5.23 -7.43  -3.49 29.66
2 -7.43  6.58   7.82  0.96
3 -3.49  7.82 -34.73  5.32
4 29.66  0.96   5.32  7.77

4 Standard errors and comparisons

StdErr gives the standard errors of single effects and of differences between effects; for Method 1 \(\Var(\hat{g}_i) = (P-1)\sigma^2_m/(2P^2)\) and \(\Var(\hat{g}_i - \hat{g}_j) = \sigma^2_m/P\).

round(x = f1$StdErr, digits = 3)

     gi     sii     sij     rij   gi_gj sii_sjj sii_sij sii_sjk sij_sik sij_skl 
  1.255   4.694   3.353   3.793   1.897   6.570   6.290   5.690   5.018   4.646 
rij_rkl 
  5.364 

t_gca <- f1$GCA / f1$StdErr[["gi"]]
round(x = t_gca, digits = 2)

    1     2     3     4     5     6     7     8 
 0.05  0.37  7.20 -2.74 -1.85 -1.54  5.77 -7.25 

A parent’s gca effect is significant at the 5% level when \(|t|\) exceeds the \(t\) quantile on the error degrees of freedom:

qt(p = 0.975, df = f1$ANOVA["Error", "Df"])

[1] 1.972595

5 Predicting crosses from general combining ability

Under the fixed model a cross is predicted by \(\hat{\mu} + \hat{g}_i + \hat{g}_j\) from the gca effects alone (the prediction for a cross not yet made) or by adding \(\hat{S}_{ij} + \hat{R}_{ij}\) (which reproduces the observed mean). The gap between the two is the specific combining ability of the cross:

pr_g <- predict(object = f1, newdata = NULL, type = c("gca", "full")[1])
pr_f <- predict(object = f1, newdata = NULL, type = c("gca", "full")[2])
pr <- merge(x = pr_g, y = pr_f, by = c("Cross1", "Cross2"), suffixes = c("_gca", "_full"))
head(pr[order(-predicted_full)], 8)

   Cross1 Cross2 predicted_gca predicted_full
   <char> <char>         <num>          <num>
1:      7      8      88.89031        125.270
2:      6      7      96.05125        121.610
3:      1      5      88.48500        120.290
4:      4      1      87.37250        119.115
5:      8      7      88.89031        118.170
6:      3      8      90.67953        117.735
7:      7      6      96.05125        115.670
8:      8      3      90.67953        115.400

6 Methods 2, 3 and 4

knitr::kable(x = fits[["Method 2"]]$ANOVA, digits = 3)

knitr::kable(x = fits[["Method 3"]]$ANOVA, digits = 3)

knitr::kable(x = fits[["Method 4"]]$ANOVA, digits = 3)

round(
    x      = rbind(
        Method1 = fits[[1]]$GCA
      , Method2 = fits[[2]]$GCA
      , Method3 = fits[[3]]$GCA
      , Method4 = fits[[4]]$GCA
      )
  , digits = 2
  )

            1     2     3     4     5     6     7      8
Method1  0.06  0.47  9.03 -3.44 -2.33 -1.93  7.24  -9.10
Method2 -0.31  2.38  5.93 -2.65 -4.71 -0.72  7.60  -7.52
Method3 -0.09 -1.65 13.80 -5.75  1.99 -5.21  8.60 -11.69
Method4 -0.86 -0.58 13.40 -6.76  2.33 -6.45 10.57 -11.66

The best general combiners (parents 3 and 7) and the poorest (parent 8) are the same in all four methods; including the parents themselves (Methods 1, 2) or the reciprocals (Methods 1, 3) changes the size of the effects and the order of the parents in between.

7 Plots

plot(fits[["Method 2"]])

8 Summary

Model 1 gives the effects of each parent and cross with their standard errors; the Effects table keeps the layout of earlier versions and GCA, SCA, Reciprocal give the effects in full.

9 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.