Multi-environment trial of wheat varieties with heteroskedastic yields

Wheat varieties with heteroskedastic yields

Format

A data frame with 52 observations on the following 3 variables.

env: environment, 13 levels
gen: genotype, 4 levels
yield: yield

Details

Yield of 4 varieties of wheat at 13 locations in Oklahoma, USA.

The data was used to explore variability between varieties.

Source

F. A. Graybill, 1954. Variance heterogeneity in a randomized block design, Biometrics, 10, 516-520.

References

Hans-Pieter Piepho, 1994. Missing observations in the analysis of stability. Heredity, 72, 141–145. https://doi.org/10.1038/hdy.1994.20

Examples

if (FALSE) { # \dontrun{

library(agridat)
data(graybill.heteroskedastic)
dat <- graybill.heteroskedastic

# Genotypes are obviously not homoscedastic
boxplot(yield ~ gen, dat, main="graybill.heteroskedastic")

# Shukla stability variance of each genotype, same as Grubbs' estimate
# Matches Piepho 1994 page 143.
# Do not do this!  Nowadays, use mixed models instead.
libs("reshape2")
datm <- acast(dat, gen~env)
w <- datm
w <- sweep(w, 1, rowMeans(datm))
w <- sweep(w, 2, colMeans(datm))
w <- w + mean(datm)
w <- rowSums(w^2)
k=4; n=13
sig2 <- k*w/((k-2)*(n-1)) - sum(w)/((k-1)*(k-2)*(n-1))
## sig2
##     G1     G2     G3     G4
## 145.98 -14.14  75.15  18.25

var.shukla <- function(x,N){
  # Estimate variance of shukla stability statistics
  # Piepho 1994 equation (5)
  K <- length(x) # num genotypes
  S <- outer(x,x)
  S1 <- diag(S)
  S2 <- rowSums(S) - S1
  S[!upper.tri(S)] <- 0 # Make S upper triangular
  # The ith element of S3 is the sum of the upper triangular elements of S,
  # excluding the ith row and ith column
  S3 <- sum(S) - rowSums(S) - colSums(S)
  var.si2 <- 2*S1/(N-1) + 4/( (N-1)*(K-1)^2 ) * ( S2 + S3/(K-2)^2 )
  return(var.si2)
}

# Set negative estimates to zero
sig2[sig2<0] <- 0

# Variance of shukla stat. Match Piepho 1994, table 5, example 1
var.shukla(sig2,13)
##        G1        G2        G3        G4 
## 4069.3296  138.9424 1423.0797  306.5270 

} # }