Partial diallel between two groups

\[ \newcommand{\mY}{\mathbf{Y}} \newcommand{\mT}{\mathbf{T}} \newcommand{\mS}{\mathbf{S}} \newcommand{\mR}{\mathbf{R}} \newcommand{\vg}{\mathbf{g}} \newcommand{\E}{\mathrm{E}} \newcommand{\Var}{\mathrm{Var}} \newcommand{\Cov}{\mathrm{Cov}} \]

1 Abstract

When parents fall into two groups (for example lines and testers, or two heterotic groups) and each parent of one group is crossed with each parent of the other, PartialDiallel() gives the combining ability of each group and the specific combining ability of the crosses.

2 The data

head(PartialDiallelData)

  Cross1 Cross2 Rep     y
1      1     11   1 18.18
2      1     12   1 16.40
3      1     13   1 13.12
4      1     14   1 17.71
5      2     11   1 17.55
6      2     12   1 18.05

with(data = PartialDiallelData, expr = table(Cross1, Cross2))[1:3, ]

      Cross2
Cross1 11 12 13 14
     1  4  4  4  4
     2  4  4  4  4
     3  4  4  4  4

Ten parents (1 to 10) are crossed with four parents (11 to 14), in four replicates.

3 The analysis

p <-
  PartialDiallel(
      y      = y
    , Rep    = Rep
    , Cross1 = Cross1
    , Cross2 = Cross2
    , data   = PartialDiallelData
    )
knitr::kable(x = p$ANOVA, digits = 4)

	Df	Sum Sq	Mean Sq	F value	Pr(>F)
Treat	39	144.7748	3.7122	2.7460	0.0000
GCA I	9	58.5418	6.5046	4.8116	0.0000
GCA II	3	43.0531	14.3510	10.6158	0.0000
SCA	27	43.1800	1.5993	1.1830	0.2654
Error	117	158.1672	1.3519	NA	NA

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

	Components	Percent
GCA I	0.3066	0.4044
GCA II	0.3188	0.2974
SCA	0.0619	0.2983

With \(L\) and \(M\) parents in the two groups, \(\sigma^2_{g^I} = (MS_{g^I} - MS_s)/(BM)\), \(\sigma^2_{g^{II}} = (MS_{g^{II}} - MS_s)/(BL)\) and \(\sigma^2_s = (MS_s - MS_E)/B\). Percent is each source’s share of the treatment sum of squares.

4 Effects

round(x = p$General, digits = 3)

     1      2      3      4      5      6      7      8      9     10     11 
 0.338  0.571  0.712 -0.343 -0.567  0.005  0.275 -1.138  0.741 -0.594  0.895 
    12     13     14 
-0.329 -0.339 -0.227

plot(p)

plot(p, which = "sca")

5 Summary

Both groups show significant general combining ability; the specific combining ability of the crosses is small.

6 References

Kempthorne, O. and Curnow, R. N. (1961). The partial diallel cross. Biometrics, 17(2), 229–250.

Kempthorne, O. (1957). An Introduction to Genetic Statistics. John Wiley and Sons, New York.

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