1 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.

2 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])

3 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

4 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.

5 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.