\[
\newcommand{\mY}{\mathbf{Y}}
\newcommand{\mT}{\mathbf{T}}
\newcommand{\mS}{\mathbf{S}}
\newcommand{\mR}{\mathbf{R}}
\newcommand{\vg}{\mathbf{g}}
\newcommand{\E}{\mathrm{E}}
\newcommand{\Var}{\mathrm{Var}}
\newcommand{\Cov}{\mathrm{Cov}}
\]
Abstract
Not every crossing scheme is a diallel. In a line \(\times\) tester experiment each of \(L\) lines is crossed with each of \(T\) testers; the North Carolina designs of Comstock and Robinson estimate additive and dominance variance from half-sib and full-sib families. Both are analysed from the plot data of a randomized complete block experiment, with the same result objects, plots and tables as the diallel analyses.
Line \(\times\) tester analysis
LineTester() gives Kempthorne’s analysis of \(L\) lines crossed with \(T\) testers: the split of the crosses into lines, testers and their interaction with the proportional contribution of each, the gca and sca effects with standard errors, and the genetic variances; Inbreeding is the inbreeding coefficient \(F\) of the parents.
lt <-
LineTester(
y = Yield
, Rep = Rep
, Line = Line
, Tester = Tester
, data = LineTesterData
, Inbreeding = 1
)
lt
Line x tester analysis: 8 lines, 3 testers, 3 replicates
Df Sum Sq Mean Sq F value Pr(>F)
Rep 2 427 213.3 14.894 1.03e-05 ***
Crosses 23 5469 237.8 16.607 2.76e-15 ***
Lines 7 3476 496.6 34.685 2.70e-16 ***
Testers 2 1038 519.0 36.244 3.54e-10 ***
Lines:Testers 14 955 68.2 4.762 2.76e-05 ***
Error 46 659 14.3
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Proportional contributions (%)
Lines Testers Lines:Testers
63.57 18.98 17.45
Genetic components
Components
gca_lines 47.6060
gca_testers 18.7824
gca 35.1418
sca 17.9554
gca/sca 1.9572
Additive 70.2835
Dominance 17.9554
Dominance/Additive 0.2555
Degree of dominance 0.7148
round(x = lt$GCA.Lines, digits = 2)
L1 L2 L3 L4 L5 L6 L7 L8
2.92 -9.28 -9.77 -1.63 9.74 -3.76 8.64 3.13
round(x = lt$StdErr, digits = 3)
gca_line gca_tester sca gi_gj_line gi_gj_tester sij_skl
1.261 0.772 2.185 1.784 1.092 3.090
plot(x = lt, which = c("gca", "sca", "contribution", "biplot", "residuals", "qq")[1])
North Carolina designs
NCDesign() analyses the three designs of Comstock and Robinson: females nested in males (Design I), males crossed with females (Design II), and \(F_2\) males backcrossed to both inbred parents (Design III). Each gives the additive and dominance variances and the average degree of dominance \(\sqrt{2\sigma^2_D/\sigma^2_A}\).
nc <- lapply(X = 1:3, FUN = function(k) {
NCDesign(
y = Yield
, Rep = Rep
, Male = Male
, Female = Female
, data = NCData[NCData$Design == k, ]
, Design = k
, Inbreeding = 0
)
})
nc[[2]]$ANOVA
Df Sum Sq Mean Sq F value Pr(>F)
Rep 2 280.67 140.333 13.6687 3.371e-05 ***
Males 4 250.42 62.605 2.9919 0.062947 .
Females 3 517.45 172.482 8.2428 0.003026 **
Males:Females 12 251.10 20.925 2.0382 0.047573 *
Error 38 390.13 10.267
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
round(
x = sapply(
X = nc
, FUN = function(z) z$Genetic.Components[c("Additive", "Dominance", "Degree of dominance"),
"Components"]
)
, digits = 3
)
[,1] [,2] [,3]
[1,] 11.867 27.154 58.248
[2,] 56.203 14.211 14.806
[3,] 3.078 1.023 0.713
Summary
LineTester() gives combining ability and the contribution of lines, testers and their interaction; NCDesign() gives the additive and dominance variances of Designs I, II and III.
References
Comstock, R. E. and Robinson, H. F. (1948). The components of genetic variance in populations of biparental progenies and their use in estimating the average degree of dominance. Biometrics, 4(4), 254–266.
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.
