Nonlinear maize yield-density model

Nonlinear maize yield-density model.

Format

A data frame with 32 observations on the following 3 variables.

gen

genotype/hybrid, 8 levels

pop

population (plants)

yield

yield, pounds per hill

Details

Eight single-cross hybrids were in the experiment--Hy2xOh7 and WF9xC103 were included because it was believed they had optimum yields at relatively high and low populations. Planted in 1963. Plots were thinned to 2, 4, 6, 8 plants per hill, giving densities 8, 16, 24, 32 thousand plants per acre. Hills were in rows 40 inches apart. One hill = 1/4000 acre. Split-plot design with 5 reps, density is main plot and subplot was hybrid.

Source

S G Carmer and J A Jackobs (1965). An Exponential Model for Predicting Optimum Plant Density and Maximum Corn Yield. Agronomy Journal, 57, 241--244. https://doi.org/10.2134/agronj1965.00021962005700030003x

Examples

# \dontrun{

library(agridat)
data(carmer.density)
dat <- carmer.density
dat$gen <- factor(dat$gen, levels=c('Hy2x0h7','WF9xC103','R61x187-2',
                             'WF9x38-11','WF9xB14','C103xB14',
                             '0h43xB37','WF9xH60'))

# Separate analysis for each hybrid
# Model: y = x * a * k^x.  Table 1 of Carmer and Jackobs.
out <- data.frame(a=rep(NA,8), k=NA)
preds <- NULL
rownames(out) <- levels(dat$gen)
newdat <- data.frame(pop=seq(2,8,by=.1))
for(i in levels(dat$gen)){
  print(i)
  dati <- subset(dat, gen==i)
  mi <- nls(yield ~ pop * a * k^pop, data=dati, start=list(a=10,k=1))
  out[i, ] <- mi$m$getPars()
  # Predicted values
  pi <- cbind(gen=i, newdat, pred= predict(mi, newdat=newdat))
  preds <- rbind(preds, pi)
}
#> [1] "Hy2x0h7"
#> [1] "WF9xC103"
#> [1] "R61x187-2"
#> [1] "WF9x38-11"
#> [1] "WF9xB14"
#> [1] "C103xB14"
#> [1] "0h43xB37"
#> [1] "WF9xH60"
# Optimum plant density is -1/log(k)
out$pop.opt <- -1/log(out$k)
round(out, 3)
#>               a     k pop.opt
#> Hy2x0h7   0.782 0.865   6.875
#> WF9xC103  1.039 0.825   5.192
#> R61x187-2 0.998 0.798   4.441
#> WF9x38-11 1.042 0.825   5.203
#> WF9xB14   1.067 0.806   4.647
#> C103xB14  0.813 0.860   6.653
#> 0h43xB37  0.673 0.862   6.740
#> WF9xH60   0.858 0.854   6.358
##               a     k pop.opt
## Hy2x0h7   0.782 0.865   6.875
## WF9xC103  1.039 0.825   5.192
## R61x187-2 0.998 0.798   4.441
## WF9x38-11 1.042 0.825   5.203
## WF9xB14   1.067 0.806   4.647
## C103xB14  0.813 0.860   6.653
## 0h43xB37  0.673 0.862   6.740
## WF9xH60   0.858 0.854   6.358


# Fit an overall fixed-effect with random deviations for each hybrid.
libs(nlme)
m1 <- nlme(yield ~ pop * a * k^pop,
           fixed = a + k ~ 1,
           random = a + k ~ 1|gen,
           data=dat, start=c(a=10,k=1))
# summary(m1) # Random effect for 'a' probably not needed


libs(latticeExtra)
# Plot Data, fixed-effect prediction, random-effect prediction.
pdat <- expand.grid(gen=levels(dat$gen), pop=seq(2,8,length=50))
pdat$pred <- predict(m1, pdat)
pdat$predf <- predict(m1, pdat, level=0)

xyplot(yield~pop|gen, dat, pch=16, as.table=TRUE,
       main="carmer.density models",
       key=simpleKey(text=c("Data", "Fixed effect","Random effect"),
         col=c("blue", "red","darkgreen"), columns=3, points=FALSE)) +
  xyplot(predf~pop|gen, pdat, type='l', as.table=TRUE, col="red") +
  xyplot(pred~pop|gen, pdat, type='l', col="darkgreen", lwd=2)

# }