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Hayman’s analysis

Muhammad Yaseen

2026-11-03

Source: vignettes/DiallelAnalysisR_06_hayman.qmd

1 Abstract

Hayman’s approach analyses a complete diallel (parents, \(F_1\) and reciprocals) for gene action: an analysis of variance that separates additive, dominance, maternal and reciprocal effects; the variances \(V_r\) and covariances \(W_r\) of the arrays and their graph; the components \(D\), \(H_1\), \(H_2\), \(F\), \(h^2\) and \(E\); and tests of the additive-dominance model.

2 The analysis 

h <-
  Hayman(
      y      = Yield
    , Rep    = Rep
    , Cross1 = Cross1
    , Cross2 = Cross2
    , data   = HaymanData
    )
knitr::kable(x = h$ANOVA, digits = 3)
Df Sum Sq Mean Sq F value Pr(>F)
Total 255 127716.027 NA NA NA
Rep 3 1036.192 345.397 3.001 0.032
Treatment 63 104923.873 1665.458 14.468 0.000
Additive 7 15218.571 2174.082 18.887 0.000
Non-Additive 28 88243.154 3151.541 27.378 0.000
b1 1 1367.594 1367.594 11.881 0.001
b2 7 13530.165 1932.881 16.791 0.000
b3 20 73345.396 3667.270 31.859 0.000
Maternal 7 310.737 44.391 0.386 0.910
Reciprocal 21 1151.411 54.829 0.476 0.976
Error 189 21755.963 115.111 NA NA

Line \(b_1\) tests the mean dominance deviation of the crosses from the parents (directional dominance), \(b_2\) whether this deviation differs among parents, \(b_3\) the remaining dominance. Maternal and reciprocal effects are not significant here.

3 Arrays: \(V_r\) and \(W_r\)

After averaging reciprocal crosses, array \(r\) (the progeny of parent \(r\)) has variance \(V_r\) and covariance \(W_r\) with the parents:

knitr::kable(x = h$Wr.Vr.Table[, c("Wr", "Vr", "Wr.Minus.Vr", "Wr.Plus.Vr", "Yr")], digits = 2)
Wr Vr Wr.Minus.Vr Wr.Plus.Vr Yr
-124.09 546.87 -670.97 422.78 85.65
-98.41 129.31 -227.72 30.90 98.26
97.33 171.95 -74.62 269.28 74.07
39.07 549.50 -510.43 588.57 91.64
155.78 317.59 -161.81 473.37 54.10
43.12 455.95 -412.83 499.08 100.39
44.36 428.02 -383.67 472.38 90.96
-160.30 824.10 -984.40 663.80 82.00

4 The \(W_r\)-\(V_r\) graph 

plot(h)

Under the additive-dominance model the points lie on a line of unit slope (dashed) inside the limiting parabola \(W_r^2 = V_r V_{0L0}\). Parents near the origin carry mostly dominant genes, parents far from it mostly recessive genes.

5 Tests of the model 

knitr::kable(x = h$Tests, digits = 4)
Statistic Df p.value
Uniformity of Wr - Vr (t^2) 4.2485 6 0.0849
b = 0 (t) -1.3293 6 0.2321
b = 1 (t) 6.8569 6 0.0005

A significant \(t^2\) or a slope that differs from 1 indicates epistasis or correlated gene distribution; the components below then need care. In this data set the slope is negative and differs from 1, so the additive-dominance model is not adequate.

6 Components of variation 

knitr::kable(x = h$Components.of.Variation, digits = 3)
Estimate StdErr t.value
E 29.677 74.009 0.401
D 195.287 222.026 0.880
F 406.984 524.628 0.776
H1 1856.573 510.406 3.637
H2 1516.416 444.053 3.415
h2 136.597 313.481 0.436 
knitr::kable(x = h$Other.Parameters, digits = 3)
Other Parameters
Mean degree of dominance sqrt(H1/D) 3.083
Proportion of genes H2/(4H1) 0.204
Dominant to recessive genes 2.021
r(Wr + Vr, Yr) -0.155
r^2(Wr + Vr, Yr) 0.024
Gene groups h2/H2 0.090
Narrow-sense heritability 0.136

\(\sqrt{H_1/D}\) above 1 indicates overdominance; \(H_2/(4H_1)\) below 0.25 an unequal distribution of positive and negative alleles among the parents.

7 Summary

Hayman’s analysis asks how genes act rather than which parents combine well; read the components only when the tests support the additive-dominance model.

8 References

Hayman, B. I. (1954). The theory and analysis of diallel crosses. Genetics, 39(6), 789–809.

Hayman, B. I. (1954). The analysis of variance of diallel tables. Biometrics, 10(2), 235–244.

Mather, K. and Jinks, J. L. (1982). Biometrical Genetics, 3rd edition. Chapman and Hall, London.

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