\[
\newcommand{\mY}{\mathbf{Y}}
\newcommand{\mT}{\mathbf{T}}
\newcommand{\mS}{\mathbf{S}}
\newcommand{\mR}{\mathbf{R}}
\newcommand{\vg}{\mathbf{g}}
\newcommand{\E}{\mathrm{E}}
\newcommand{\Var}{\mathrm{Var}}
\newcommand{\Cov}{\mathrm{Cov}}
\]
Abstract
A diallel grown in several environments shows whether combining ability is stable. GriffingEnv() pools Griffing’s analyses over environments and splits each line into its main effect and its interaction with environments. The biplot of the cross means shows the best general combiners and the best crosses on one graph.
A diallel over several environments
GriffingEnv() pools Griffing’s analyses of the same diallel in several environments. Each line of the analysis is a projection of the table of cross means, so the sum over environments of a line’s sum of squares splits into the line itself and its interaction with environments. The example simulates six parents of Method 2 in three environments with gca \(\times\) environment effects:
d <-
DiallelSim(
Parents = 6
, Method = 2
, Rep = 3
, Env = 3
, Mean = 50
, gca = 4
, sca = 2
, rec = 0
, gcaEnv = 1.5
, scaEnv = 1
, EnvSD = 2
, Block = 1
, Error = 3
, seed = 1
)
fe <-
GriffingEnv(
y = y
, Rep = Rep
, Cross1 = Cross1
, Cross2 = Cross2
, Env = Env
, data = d
, Method = 2
, Model = 2
)
fe$ANOVA
Df Sum Sq Mean Sq F value Pr(>F)
Env 2 81.91 40.95 9.1418 0.015084 *
Rep(Env) 6 26.88 4.48 1.5694 0.161861
gca 5 2073.78 414.76 48.4875 1.088e-06 ***
sca 15 200.96 13.40 5.1141 7.288e-05 ***
gca:Env 10 85.54 8.55 2.9968 0.002076 **
sca:Env 30 78.59 2.62 0.9178 0.593124
Error 120 342.52 2.85
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Components
gca 16.4760483
sca 3.5924821
gca:Env 0.7417799
sca:Env -0.2347162
Error 2.8543549
gca/sca 4.5862576
In Model 2 gca and sca are tested against their interactions with environments. The gca effects in each environment show which parents combine well everywhere:
plot(x = fe, which = c("gca", "gcaenv", "sca", "residuals", "qq")[2])
A biplot of the crosses
plot(fit, which = "biplot") draws the table of cross means as a biplot (Yan and Hunt 2002): the table of a cross and its reciprocal averaged, the selfs left out and filled by a rank-two fit, centred on the tester (column) means, and its first two principal components with the singular values split between parents and testers. Parents far along the average-tester axis (dashed) combine well in general; a parent close to a tester’s vector combines well with that tester.
g4 <-
Griffing(
y = Yield
, Rep = Rep
, Cross1 = Cross1
, Cross2 = Cross2
, data = GriffingData4
, Method = 4
, Model = 1
, Reciprocal = c("combined", "split")[1]
)
plot(x = g4, which = c("gca", "sca", "biplot", "residuals", "qq")[3])
When specific combining ability is large the two axes show mainly the sca pattern; the parent with the largest gca effect is printed in bold.
Summary
The gca \(\times\) environment and sca \(\times\) environment lines show whether combining ability holds across environments; the biplot shows which parents and crosses stand out.
References
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.
Yan, W. and Hunt, L. A. (2002). Biplot analysis of diallel data. Crop Science, 42(1), 21–30.
Zhang, Y. and Kang, M. S. (1997). DIALLEL-SAS: a SAS program for Griffing’s diallel analyses. Agronomy Journal, 89(2), 176–182.
