Factorial experiment of bermuda grass, 4x4x4, N, P, K fertilizers

Factorial experiment of bermuda grass, 4x4x4, N, P, K fertilizers.

Format

A data frame with 64 observations on the following 4 variables.

n: nitrogen fertilizer, 4 levels
p: phosphorus, 4 levels
k: potassium, 4 levels
yield: yield of grass, tons/ac

Details

The experiment was conducted 1955, 1956, and 1957.

There were 3 treatment factors:

4 n nitrogen levels: 0, 100, 200, 400 pounds/acre

4 p phosphorous levels: 0, 22, 44, 88 pounds/acre

4 k potassium levels: 0, 42, 84, 168 pounds/acre

There were 3 blocks. The harvests were oven-dried. Each value is the mean for 3 years and 3 replications. In most cases, the yield increased with additions of the fertilizer nutrients.

Source

Welch, Louis Frederick and Adams, William Eugenius and Carmon, JL. (1963). Yield response surfaces, isoquants, and economic fertilizer optima for Coastal Bermudagrass. Agronomy Journal, 55, 63-67. Table 1. https://doi.org/10.2134/agronj1963.00021962005500010023x

References

Jim Albert. Bayesian Computation with R. Page 256.

Peter Congdon. Bayesian Statistical Modeling. Page 124-125.

P. McCullagh, John A. Nelder. Generalized Linear Models, 2nd ed. Page 382.

Examples

if (FALSE) { # \dontrun{

library(agridat)
data(welch.bermudagrass)
dat <- welch.bermudagrass
# Welch uses 100-pound units of n,p,k.
dat <- transform(dat, n=n/100, p=p/100, k=k/100)

libs(latticeExtra)
useOuterStrips(xyplot(yield~n|factor(p)*factor(k), data=dat, type='b',
                      main="welch.bermudagrass: yield for each P*K",
                      xlab="Nitro for each Phosphorous level",
                      ylab="Yield for each Potassim level"))


# Fit a quadratic model
m1 <- lm(yield ~ n + p + k + I(n^2) + I(p^2) + I(k^2) + n:p + n:k + p:k + n:p:k, data=dat)
signif(coef(m1),4) # These match the 3-yr coefficients of Welch, Table 2
## (Intercept)           n           p           k      I(n^2)      I(p^2)
##     1.94300     2.00700     1.47100     0.61880    -0.33150    -1.29500
##      I(k^2)         n:p         n:k         p:k       n:p:k
##    -0.37430     0.20780     0.18740     0.23480     0.02789

# Welch Fig 4.  Modeled response curves
d1 <- expand.grid(n=seq(0, 4, length=50), p=0, k=0)
d1$pred <- predict(m1, d1)
d2 <- expand.grid(n=0, p=0, k=seq(0, 1.68, length=50))
d2$pred <- predict(m1, d2)
d3 <- expand.grid(n=0, p=seq(0, .88, length=50), k=0)
d3$pred <- predict(m1, d3)

op <- par(mfrow=c(1,3), mar=c(5,3,4,1))
plot(pred~n, data=d1, type='l', ylim=c(0,6), xlab="N 100 lb/ac", ylab="")
plot(pred~k, data=d2, type='l', ylim=c(0,6), xlab="K 100 lb/ac", ylab="")
title("welch.bermudagrass - Predicted yield vs fertilizer", outer=TRUE, line= -3)
plot(pred~p, data=d3, type='l', ylim=c(0,6), xlab="P 100 lb/ac",
ylab="")
par(op)

# Brute-force grid-search optimization of fertilizer quantities, using
# $25/ton for grass, $.12/lb for N, $.18/lb for P, $.07/lb for K
# Similar to Example 5 in Table 4 of Welch
d4 <- expand.grid(n=seq(3,4,length=20), p=seq(.5, 1.5, length=20),
                  k=seq(.8, 1.8, length=20))
d4$pred <- predict(m1, newdata=d4)
d4 <- transform(d4, income = 25*pred - .12*n*100 + -.18*p*100 -.07*k*100)
d4[which.max(d4$income),] # Optimum at 300 lb N, 71 lb P, 148 lb K


# ----- JAGS -----
if(0){
  # Congdon (2007) p. 124, provides a Bayesian model based on a GLM
  # by McCullagh & Nelder.  We use JAGS and simplify the code.
  # y ~ gamma with shape = nu, scale = nu * eps_i
  # 1/eps = b0 + b1/(N+a1) + b2/(P+a2) + b3/(K+a3)
  # N,P,K are added fertilizer amounts, a1,a2,a3 are background
  # nutrient levels and b1,b2,b3 are growth parameters.

  libs(rjags)

  mod.bug =
  "model {
  for(i in 1:nobs) {
    yield[i] ~ dgamma(nu, mu[i])
    mu[i] <- nu * eta[i]
    eta[i] <- b0 + b1 / (N[i]+a1) + b2 / (P[i]+a2) + b3 / (K[i]+a3)
    yhat[i] <- 1 / eta[i]
  }

  # Hyperparameters
  nu ~ dgamma(0.01, 0.01)
  a1 ~ dnorm(40, 0.01) # Informative priors
  a2 ~ dnorm(22, 0.01)
  a3 ~ dnorm(32, 0.01)
  b0 ~ dnorm(0, 0.0001)
  b1 ~ dnorm(0, 0.0001) I(0,) # Keep b1 non-negative
  b2 ~ dnorm(0, 0.0001) I(0,)
  b3 ~ dnorm(0, 0.0001) I(0,)
}"

  jdat <- with(welch.bermudagrass,
               list(yield=yield, N=n, P=p, K=k, nobs=64))
  jinit = list(a1=40, a2=22, a3=32, b0=.1, b1=10, b2=1, b3=1)

  oo <- textConnection(mod.bug)
  j1 <- jags.model(oo, data=jdat, inits=jinit, n.chains=3)
  close(oo)
  
  c1 <- coda.samples(j1, c("b0","b1","b2","b3", "a1","a2","a3"),
                     n.iter=10000)

  # Results nearly identical go Congdon
  print(summary(c1)$statistics[,1:2],dig=1)
  # libs(lucid)
  # print(vc(c1),3)
  ##       Mean     SD
  ## a1  44.85  4.123
  ## a2  23.63  7.37
  ## a3  35.42  8.57
  ## b0   0.092 0.0076
  ## b1  13.23  1.34
  ## b2   1.186 0.47
  ## b3   1.50  0.48

  d2 <- coda.samples(j1, "yhat", n.iter=10000)
  dat$yhat <- summary(d2)$statistics[,1]
  with(dat, plot(yield, yield-yhat))
}

} # }