2026-11-03
Source:vignettes/DiallelAnalysisR_01_getting_started.qmd
\[ \newcommand{\mY}{\mathbf{Y}} \newcommand{\mT}{\mathbf{T}} \newcommand{\mS}{\mathbf{S}} \newcommand{\mR}{\mathbf{R}} \newcommand{\vg}{\mathbf{g}} \newcommand{\E}{\mathrm{E}} \newcommand{\Var}{\mathrm{Var}} \newcommand{\Cov}{\mathrm{Cov}} \]
DiallelAnalysisR analyses diallel crosses laid out in randomized complete blocks. This vignette takes one data set, an eight-parent diallel with reciprocals, from the plot data to the analysis of variance, the combining-ability effects with their standard errors and the plots, in a few minutes of reading. Each later vignette takes one part further.
Each row is one plot: the female parent (Cross1), the male parent (Cross2), the replicate (Rep) and the yield.
head(GriffingData1) Cross1 Cross2 Rep Yield
1 1 1 1 104.86
2 1 2 1 88.66
3 1 3 1 109.76
4 1 4 1 128.10
5 1 5 1 128.36
6 1 6 1 74.40
dim(GriffingData1)[1] 256 4 Cross2
Cross1 1 2 3 4
1 4 4 4 4
2 4 4 4 4
3 4 4 4 4
4 4 4 4 4The 8 parents, their 28 crosses and the 28 reciprocal crosses (64 cells) are each grown in 4 replicates.
Griffing’s Method 1 (parents, \(F_1\) and reciprocals) with the fixed-effects model (Model 1):
fit <-
Griffing(
y = Yield
, Rep = Rep
, Cross1 = Cross1
, Cross2 = Cross2
, data = GriffingData1
, Method = 1
, Model = 1
, Reciprocal = c("combined", "split")[1]
)
fitGriffing's Method 1, Model 1 (fixed effects); 8 parents, 4 replicates
Analysis of variance
Df Sum Sq Mean Sq F value Pr(>F)
gca 7 3805 543.5 18.887 <2e-16 ***
sca 28 22061 787.9 27.378 <2e-16 ***
reciprocals 28 366 13.1 0.454 0.992
Error 189 5439 28.8
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Components
Components
gca 32.17142
sca 759.10758
Reciprocal -7.86142
gca/sca 0.04238
General combining ability effects (standard error 1.255)
1 2 3 4 5 6 7 8
0.06188 0.46800 9.02700 -3.43800 -2.32500 -1.93500 7.23800 -9.09600
All components: names(x); tests of the effects: summary(x); figures: plot(x)The result is the list that Griffing() has always returned, with a class that gives it methods:
summary(fit)Griffing's Method 1, Model 1 (fixed effects); 8 parents, 4 replicates
Analysis of variance
Df Sum Sq Mean Sq F value Pr(>F)
gca 7 3805 543.5 18.887 <2e-16 ***
sca 28 22061 787.9 27.378 <2e-16 ***
reciprocals 28 366 13.1 0.454 0.992
Error 189 5439 28.8
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Components
Components
gca 32.17142
sca 759.10758
Reciprocal -7.86142
gca/sca 0.04238
General combining ability effects
parent Estimate StdErr t.value p.value
<char> <num> <num> <num> <num>
1: 1 0.06188 1.255 0.04932 9.607e-01
2: 2 0.46797 1.255 0.37303 7.095e-01
3: 3 9.02672 1.255 7.19544 1.420e-11
4: 4 -3.43797 1.255 -2.74050 6.723e-03
5: 5 -2.32547 1.255 -1.85369 6.534e-02
6: 6 -1.93484 1.255 -1.54232 1.247e-01
7: 7 7.23750 1.255 5.76921 3.208e-08
8: 8 -9.09578 1.255 -7.25050 1.034e-11
Crosses with the largest specific combining ability effects
Cross1 Cross2 Estimate StdErr t.value p.value
<char> <char> <num> <num> <num> <num>
1: 7 8 32.83 3.353 9.792 1.431e-18
2: 1 4 29.66 3.353 8.846 6.417e-16
3: 1 5 27.30 3.353 8.144 5.072e-14
4: 3 8 25.89 3.353 7.721 6.517e-13
5: 6 7 22.59 3.353 6.737 1.892e-10
knitr::kable(x = fit$ANOVA, digits = 3)| Df | Sum Sq | Mean Sq | F value | Pr(>F) | |
|---|---|---|---|---|---|
| gca | 7 | 3804.643 | 543.520 | 18.887 | 0.000 |
| sca | 28 | 22060.789 | 787.885 | 27.378 | 0.000 |
| reciprocals | 28 | 365.537 | 13.055 | 0.454 | 0.992 |
| Error | 189 | 5438.991 | 28.778 | NA | NA |
General and specific combining ability are both highly significant; the reciprocal differences are not. The mean squares are on the scale of a cell mean, so the error line is \(MS_E/B\), the residual mean square of the block analysis divided by the number of replicates.
round(x = fit$GCA, digits = 3) 1 2 3 4 5 6 7 8
0.062 0.468 9.027 -3.438 -2.325 -1.935 7.238 -9.096
round(x = fit$StdErr, digits = 3) gi sii sij rij gi_gj sii_sjj sii_sij sii_sjk sij_sik sij_skl
1.255 4.694 3.353 3.793 1.897 6.570 6.290 5.690 5.018 4.646
rij_rkl
5.364 Parents 3 and 7 are the best general combiners, parent 8 the poorest. A difference between two gca effects larger than about twice gi_gj is significant at about the 5% level.
plot(fit)
plot(fit, which = "sca")
summary() adds the \(t\) tests of every gca effect and the crosses with the largest sca effects; anova(), coef(), vcov(), confint(), fitted(), residuals() and predict() work as for other models in R.
summary(object = fit, top = 5)Griffing's Method 1, Model 1 (fixed effects); 8 parents, 4 replicates
Analysis of variance
Df Sum Sq Mean Sq F value Pr(>F)
gca 7 3805 543.5 18.887 <2e-16 ***
sca 28 22061 787.9 27.378 <2e-16 ***
reciprocals 28 366 13.1 0.454 0.992
Error 189 5439 28.8
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Components
Components
gca 32.17142
sca 759.10758
Reciprocal -7.86142
gca/sca 0.04238
General combining ability effects
parent Estimate StdErr t.value p.value
<char> <num> <num> <num> <num>
1: 1 0.06188 1.255 0.04932 9.607e-01
2: 2 0.46797 1.255 0.37303 7.095e-01
3: 3 9.02672 1.255 7.19544 1.420e-11
4: 4 -3.43797 1.255 -2.74050 6.723e-03
5: 5 -2.32547 1.255 -1.85369 6.534e-02
6: 6 -1.93484 1.255 -1.54232 1.247e-01
7: 7 7.23750 1.255 5.76921 3.208e-08
8: 8 -9.09578 1.255 -7.25050 1.034e-11
Crosses with the largest specific combining ability effects
Cross1 Cross2 Estimate StdErr t.value p.value
<char> <char> <num> <num> <num> <num>
1: 7 8 32.83 3.353 9.792 1.431e-18
2: 1 4 29.66 3.353 8.846 6.417e-16
3: 1 5 27.30 3.353 8.144 5.072e-14
4: 3 8 25.89 3.353 7.721 6.517e-13
5: 6 7 22.59 3.353 6.737 1.892e-10 2.5 % 97.5 %
g_1 -2.412755 2.536505
g_2 -2.006661 2.942599
g_3 6.552089 11.501349
g_4 -5.912599 -0.963339
g_5 -4.800099 0.149161
g_6 -4.409474 0.539786
g_7 4.762870 9.712130
g_8 -11.570411 -6.621151
predict(
object = fit
, newdata = data.frame(Cross1 = c(3, 3), Cross2 = c(7, 8))
, type = c("gca", "full")[1]
) Cross1 Cross2 predicted
<char> <char> <num>
1: 3 7 107.01281
2: 3 8 90.67953The residuals of the block analysis check its assumptions:
Any component converts to a table, with the row names in a column term:
data.table::as.data.table(fit, component = "ANOVA") term Df Sum Sq Mean Sq F value Pr(>F)
<char> <num> <num> <num> <num> <num>
1: gca 7 3804.6427 543.52039 18.8868412 5.293328e-19
2: sca 28 22060.7885 787.88531 27.3783005 3.518704e-52
3: reciprocals 28 365.5369 13.05489 0.4536456 9.923291e-01
4: Error 189 5438.9907 28.77773 NA NAOne call of Griffing() gives the analysis of variance, the effects with their standard errors and the variance components; plot() draws the effects. Vignette 02 helps choose an analysis; vignettes 04 to 07 explain Griffing’s, Hayman’s and the partial diallel analyses, 08 to 11 heterosis, related designs, environments and planning, and 12 the tables for papers and the interactive application. Vignettes 13 to 15 cover validation, speed and extending the package.
Griffing, B. (1956). Concept of general and specific combining ability in relation to diallel crossing systems. Australian Journal of Biological Sciences, 9(4), 463–493.
Singh, R. K. and Chaudhary, B. D. (2004). Biometrical Methods in Quantitative Genetic Analysis. Kalyani Publishers, New Delhi.