Choosing an analysis • DiallelAnalysisR: Diallel Analysis, Heterosis and Related Mating Designs

1 Abstract

Which function, method and model to use depends on the crosses grown and on the question asked. This vignette gives a decision table and shows how to check that a data set has the crosses an analysis needs.

2 Decision table

Crosses grown	Question	Function
parents, \(F_1\) and reciprocals	combining ability, reciprocal effects	`Griffing(Method = 1)`
parents and one set of \(F_1\)	combining ability	`Griffing(Method = 2)`
\(F_1\) and reciprocals, no parents	combining ability, reciprocal effects	`Griffing(Method = 3)`
one set of \(F_1\) only	combining ability	`Griffing(Method = 4)`
parents, \(F_1\) and reciprocals	gene action: additive and dominance components, \(W_r\)-\(V_r\) graph	`Hayman()`
parents, \(F_1\) and reciprocals	maternal and non-maternal reciprocal effects	`Griffing(Method = 1, Reciprocal = "split")`
a published table of cross means	combining ability from the means and the published error	`GriffingMeans()`
any of Griffing’s layouts in several environments	stability of gca and sca: interactions with environments	`GriffingEnv()`
parents and crosses (a check variety optional)	heterosis over the mid-parent, better parent, check	`Heterosis()`
parents and one set of \(F_1\)	variety effects and average, variety and specific heterosis	`GardnerEberhart()`
two groups of parents crossed with each other	combining ability of each group	`PartialDiallel()`
lines crossed with testers	combining ability, contributions of lines and testers	`LineTester()`
North Carolina Design I, II or III	additive and dominance variance	`NCDesign()`
not yet grown	number of replicates for the tests	`DiallelPower()`, `DiallelSim()`

Model. Choose Model 1 (fixed effects) when the parents are of interest in themselves (the usual case in a breeding programme): it gives effects and standard errors. Choose Model 2 (random effects) when the parents are a random sample of a population and the aim is to estimate variance components (additive and dominance variance).

3 Which method does a data set fit?

cells <- function(d) {
  i <- as.numeric(as.character(d$Cross1))
  j <- as.numeric(as.character(d$Cross2))
  u <- unique(data.frame(i = i, j = j))
  c(parents = length(unique(c(i, j))), selfs = sum(u$i == u$j), above = sum(u$i < u$j), below = sum(u$i > u$j))
}
rbind(
    Data1 = cells(d = GriffingData1)
  , Data2 = cells(d = GriffingData2)
  , Data3 = cells(d = GriffingData3)
  , Data4 = cells(d = GriffingData4)
  )

      parents selfs above below
Data1       8     8    28    28
Data2       8     8    28     0
Data3       8     0    28    28
Data4       8     0    28     0

With \(P\) parents: Method 1 needs \(P\) selfs and \(P(P-1)\) crosses, Method 2 \(P\) selfs and \(P(P-1)/2\) crosses (either way round), Method 3 \(P(P-1)\) crosses and no selfs, Method 4 \(P(P-1)/2\) crosses and no selfs.

4 What happens when a plot or a cross is missing

A missing plot is allowed in Griffing(), GriffingEnv(), Heterosis(), GardnerEberhart(), LineTester() and NCDesign(): the cross means are then least-squares means adjusted for blocks, the number of replicates is the harmonic mean of the plot numbers, and the label of the result says so. Hayman() and PartialDiallel() need every plot. Every cross of the method must be present, though; a missing cross is reported with its name:

d <- subset(x = GriffingData1, subset = !(Cross1 == 1 & Cross2 == 2))
tryCatch(
    expr  = Griffing(
        y          = Yield
      , Rep        = Rep
      , Cross1     = Cross1
      , Cross2     = Cross2
      , data       = d
      , Method     = 1
      , Model      = 1
      , Reciprocal = c("combined", "split")[1]
      )
  , error = function(e) conditionMessage(e)
  )

[1] "Griffing(): 1 cross(es) of Method 1 are missing from `data` (for example 1 x 2). The analysis needs every cross; for incomplete diallels fit a linear mixed model."

Incomplete diallels need a linear mixed model with the gca, sca and reciprocal effects as random terms.

5 Column names

The arguments take a column by name, as a string, or a vector:

d <- GriffingData1
names(d) <- c("Female", "Male", "Block", "GrainYield")
fit <-
  Griffing(
      y          = "GrainYield"
    , Rep        = Block
    , Cross1     = Female
    , Cross2     = Male
    , data       = d
    , Method     = 1
    , Model      = 1
    , Reciprocal = c("combined", "split")[1]
    )
fit$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 ***
reciprocals  28   365.5   13.05  0.4536 0.9923
Error       189  5439.0   28.78
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

6 Summary

The crosses grown fix the analysis and the method; the purpose fixes the model. Check the cells before the analysis; every analysis reports what is missing.

7 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.

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.