\[
\newcommand{\mY}{\mathbf{Y}}
\newcommand{\mT}{\mathbf{T}}
\newcommand{\mS}{\mathbf{S}}
\newcommand{\mR}{\mathbf{R}}
\newcommand{\vg}{\mathbf{g}}
\newcommand{\E}{\mathrm{E}}
\newcommand{\Var}{\mathrm{Var}}
\newcommand{\Cov}{\mathrm{Cov}}
\]
Abstract
When the parents are grown beside their crosses, the crosses can be compared with the parents: the heterosis of each cross, and Gardner and Eberhart’s split of the crosses into variety effects and heterosis. In a diallel with reciprocals, the reciprocal effects split into maternal and non-maternal parts. This vignette takes these three in turn.
Heterosis
With the parents grown beside the crosses, Heterosis() gives for each cross the heterosis over the mid-parent \(MP = (P_i + P_j)/2\), the better parent and, if a check variety is named, the standard variety, in percent, each with a \(t\) test. With \(B\) replicates the standard errors of the differences are \(\sqrt{3MS_E/(2B)}\) for \(F_{ij} - MP\) and \(\sqrt{2MS_E/B}\) for the other two.
het <-
Heterosis(
y = Yield
, Rep = Rep
, Cross1 = Cross1
, Cross2 = Cross2
, data = GriffingData2
, Check = 3
, Better = c("higher", "lower")[1]
)
het$StdErr
F1_MP F1_BP
6.471663 7.472833
head(x = het$Heterosis[order(-het$Heterosis$MP)], n = 6)
Cross1 Cross2 F1 MidParent BetterParent MP t_MP p_MP
<char> <char> <num> <num> <num> <num> <num> <num>
1: 1 5 120.290 69.87375 85.645 72.15335 7.790308 5.025436e-12
2: 3 8 117.735 78.03500 82.000 50.87461 6.134435 1.531526e-08
3: 3 5 94.285 64.08625 74.070 47.12204 4.666304 9.085783e-06
4: 7 8 125.270 86.48000 90.960 44.85430 5.993822 2.926609e-08
5: 5 8 94.820 68.05125 82.000 39.33616 4.136301 7.130931e-05
6: 3 7 111.540 82.51500 90.960 35.17542 4.484937 1.870084e-05
BP t_BP p_BP Check SH t_SH p_SH
<num> <num> <num> <num> <num> <num> <num>
1: 40.45187 4.636127 1.025726e-05 74.07 62.40043 6.185071 1.211061e-08
2: 43.57927 4.781988 5.683737e-06 74.07 58.95099 5.843165 5.813966e-08
3: 27.29175 2.705132 7.968289e-03 74.07 27.29175 2.705132 7.968289e-03
4: 37.71988 4.591298 1.227161e-05 74.07 69.12380 6.851485 5.139696e-10
5: 15.63415 1.715547 8.919434e-02 74.07 28.01404 2.776725 6.503853e-03
6: 22.62533 2.753976 6.940153e-03 74.07 50.58728 5.014163 2.174202e-06
plot(x = het, which = c("MP", "BP", "SH", "residuals", "qq")[1])
For traits where a lower value is better (days to flowering, disease score), Better = "lower" takes the lower parent as the better one.
Maternal and non-maternal reciprocal effects
Cockerham and Weir (1977) write the reciprocal effect of Methods 1 and 3 as \(r_{ij} = m_i - m_j + n_{ij}\): a maternal effect of each parent and a non-maternal remainder. Griffing(..., Reciprocal = "split") splits the reciprocal line of the analysis of variance accordingly, on \(P - 1\) and \((P - 1)(P - 2)/2\) degrees of freedom:
rs <-
Griffing(
y = Yield
, Rep = Rep
, Cross1 = Cross1
, Cross2 = Cross2
, data = GriffingData1
, Method = 1
, Model = 1
, Reciprocal = c("combined", "split")[2]
)
rs$ANOVA
Df Sum Sq Mean Sq F value Pr(>F)
gca 7 3804.6 543.52 18.8868 <2e-16 ***
sca 28 22060.8 787.89 27.3783 <2e-16 ***
maternal 7 77.7 11.10 0.3856 0.9100
non-maternal 21 287.9 13.71 0.4763 0.9758
Error 189 5439.0 28.78
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
round(x = rs$Maternal, digits = 3)
1 2 3 4 5 6 7 8
0.087 0.681 0.968 0.830 -0.818 0.196 -1.257 -0.686
round(x = rs$StdErr[c("mi", "mi_mj", "nij")], digits = 3)
mi mi_mj nij
1.255 1.897 3.285
Here neither part is significant: the reciprocal differences of this diallel are small and show no maternal pattern.
Gardner and Eberhart’s Analyses II and III
For parents and one set of crosses, Gardner and Eberhart (1966) model the parents by variety effects \(v_j\) and the crosses by the mean of their parents plus heterosis. Analysis II splits the heterosis into average heterosis \(\bar{h}\), variety heterosis \(h_j\) and specific heterosis \(s_{jj'}\):
ge2 <-
GardnerEberhart(
y = Yield
, Rep = Rep
, Cross1 = Cross1
, Cross2 = Cross2
, data = GriffingData2
, Analysis = 2
, Model = 1
)
ge2$ANOVA
Df Sum Sq Mean Sq F value Pr(>F)
Varieties 7 1850.9 264.41 9.4698 4.546e-09 ***
Heterosis 28 12529.1 447.47 16.0259 < 2.2e-16 ***
Average heterosis 1 373.8 373.81 13.3879 0.000398 ***
Variety heterosis 7 2850.7 407.25 14.5854 3.807e-13 ***
Specific heterosis 20 9304.6 465.23 16.6619 < 2.2e-16 ***
Error 105 2931.8 27.92
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Estimate Std. Error t value Pr(>|t|)
mu_v 84.633437 1.868208 45.301929 9.626013e-71
hbar 7.750938 2.118349 3.658952 3.980309e-04
plot(x = ge2, which = c("effects", "specific", "residuals", "qq")[1])
Analysis III keeps the variety effects for the parents and analyses the crosses by general and specific combining ability, with average heterosis between the two; its gca and sca lines are Griffing’s Method 4:
ge3 <-
GardnerEberhart(
y = Yield
, Rep = Rep
, Cross1 = Cross1
, Cross2 = Cross2
, data = GriffingData2
, Analysis = 3
, Model = 1
)
ge3$ANOVA
Df Sum Sq Mean Sq F value Pr(>F)
Varieties 7 1574.8 224.96 8.057 8.099e-08 ***
Average heterosis 1 373.8 373.81 13.388 0.000398 ***
gca 7 3126.9 446.70 15.998 3.672e-14 ***
sca 20 9304.6 465.23 16.662 < 2.2e-16 ***
Error 105 2931.8 27.92
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Summary
Heterosis() compares each cross with its parents and a check; Griffing(..., Reciprocal = "split") asks whether the reciprocal differences come from the mother; GardnerEberhart() describes parents and crosses with one model of variety effects and heterosis.
References
Cockerham, C. C. and Weir, B. S. (1977). Quadratic analyses of reciprocal crosses. Biometrics, 33(1), 187–203.
Gardner, C. O. and Eberhart, S. A. (1966). Analysis and interpretation of the variety cross diallel and related populations. Biometrics, 22(3), 439–452.
Murray, L. W., Ray, I. M., Dong, H. and Segovia-Lerma, A. (2003). Clarification and reevaluation of population-based diallel analyses: Gardner and Eberhart Analyses II and III revisited. Crop Science, 43(6), 1930–1937.
Singh, R. K. and Chaudhary, B. D. (2004). Biometrical Methods in Quantitative Genetic Analysis. Kalyani Publishers, New Delhi.
