Encounted a problem that the treatment is confounded with the blocking structure. In this situation, it is not possible to assess the interaction of the treatment and genotypes.

Example

Here is an example of the trial:

set.seed(1234)
Rep <- rep(as.vector(replicate(2,sample(c("Rep1","Rep2","Rep3"),3))),each=5)
Gens <- paste0("Gen",1:5)
Geno <- as.vector(replicate(6, sample(Gens,5)))
TOS <- rep(c("TOS1","TOS2"),each=15)
Rows <- rep(1:5,6)
Cols <- rep(1:6,each=5)
dat <- data.frame(Rows,Cols,Rep,Geno,TOS)
ggplot(dat) + geom_tile(aes(Cols,Rows,fill=Rep))

ggplot(dat) + geom_tile(aes(Cols,Rows,fill=TOS))

ggplot(dat) + geom_tile(aes(Cols,Rows,fill=Geno))

dat$y <- 3+rnorm(length(Rep))
names(dat)
## [1] "Rows" "Cols" "Rep"  "Geno" "TOS"  "y"
nms <- c("Rows","Cols","Rep","Geno","TOS")
dat[nms] <- lapply(dat[nms], factor)

The researchers would like to assess the effects of different Time of Sowing, Genotypes and their interactions on a set of traits.

Model 1

With the trial, the interaction is not possibly being assessed because the treatment is confounded with the blocking structure.

mod1 <- asreml(y ~ TOS*Geno,
               random = ~ TOS:Rep,
               data = dat)
## Online License checked out Thu Mar  4 16:57:37 2021
## Model fitted using the gamma parameterization.
## ASReml 4.1.0 Thu Mar  4 16:57:37 2021
##           LogLik        Sigma2     DF     wall    cpu
##  1      -12.0709      0.654877     20 16:57:37    0.0
##  2      -12.0484      0.644739     20 16:57:37    0.0
##  3      -12.0259      0.624656     20 16:57:37    0.0
##  4      -12.0256      0.622321     20 16:57:37    0.0
plot(mod1)

wald(mod1,denDF = "default",ssType = "conditional")
## Model fitted using the gamma parameterization.
## ASReml 4.1.0 Thu Mar  4 16:57:37 2021
##           LogLik        Sigma2     DF     wall    cpu
##  1      -12.0256      0.622286     20 16:57:37    0.0
##  2      -12.0256      0.622286     20 16:57:37    0.0
##  3      -12.0256      0.622286     20 16:57:37    0.0
## $Wald
## 
## Wald tests for fixed effects.
## Response: y
## 
##             Df denDF   F.inc   F.con Margin      Pr
## (Intercept)  1     4 131.600 131.600        0.00033
## TOS          1     4   0.003   0.003      A 0.95740
## Geno         4    16   0.394   0.394      A 0.80986
## TOS:Geno     4    16   0.619   0.619      B 0.65513
## 
## $stratumVariances
##         df  Variance TOS:Rep units!R
## TOS:Rep  4 1.1778601       5       1
## units!R 16 0.6222861       0       1

Model 2

The two TOS blocks were treated as two different environments. As such, a MET analysis techniques were used assuming heterogeneous variance for both TOS.

mod2 <- asreml(y ~ TOS + Geno,
               random = ~ at(TOS):Rep,
               data = dat)
## Model fitted using the gamma parameterization.
## ASReml 4.1.0 Thu Mar  4 16:57:37 2021
##           LogLik        Sigma2     DF     wall    cpu
##  1      -12.3657      0.609972     24 16:57:37    0.0
##  2      -12.3039      0.599510     24 16:57:37    0.0
##  3      -12.2332      0.579837     24 16:57:37    0.0
##  4      -12.2288      0.575302     24 16:57:37    0.0
## Warning in asreml(y ~ TOS + Geno, random = ~at(TOS):Rep, data = dat): Some
## components changed by more than 1% on the last iteration.
plot(mod2)

wald(mod2,denDF = "default",ssType = "conditional")
## Model fitted using the gamma parameterization.
## ASReml 4.1.0 Thu Mar  4 16:57:38 2021
##           LogLik        Sigma2     DF     wall    cpu
##  1      -12.2287      0.574921     24 16:57:38    0.0
##  2      -12.2287      0.574920     24 16:57:38    0.0
##  3      -12.2287      0.574918     24 16:57:38    0.0
## $Wald
## 
## Wald tests for fixed effects.
## Response: y
## 
##             Df denDF   F.inc   F.con Margin      Pr
## (Intercept)  1   3.8 138.100 138.100        0.00039
## TOS          1   3.8   0.003   0.003      A 0.95753
## Geno         4  20.0   0.427   0.427      A 0.78766
## 
## $stratumVariances
##                   df  Variance at(TOS, TOS1):Rep at(TOS, TOS2):Rep units!R
## at(TOS, TOS1):Rep  2 1.4385847                 5     -2.297635e-17       1
## at(TOS, TOS2):Rep  2 0.9171355                 0      5.000000e+00       1
## units!R           20 0.5749179                 0      0.000000e+00       1

Hans-Peter has a paper discussing this issue and an example shows that there is complete confounding of area and treatment effects. So the model 1 in our case is not valid.

References