Trial of wheat with nitrogen fertilizer in two fertility zones

Usage

data("bachmaier.nitrogen")

Format

A data frame with 88 observations on the following 3 variables.

nitro: nitrogen fertilizer, kg/ha
yield: wheat yield, Mg/ha
zone: fertility zone

Details

Data from a wheat fertilizer experiment in Germany in two yield zones. In each zone, the design was an RCB with 4 blocks and 11 nitrogen levels. The yield of each plot was measured.

Electronic data originally downloaded from http://www.tec.wzw.tum.de/bachmaier/vino.zip (no longer available).

Source

Bachmaier, Martin. 2009. A Confidence Set for That X-Coordinate Where a Quadratic Regression Model Has a Given Gradient. Statistical Papers 50: 649–60. https://doi.org/10.1007/s00362-007-0104-1.

References

Bachmaier, Martin. Test and confidence set for the difference of the x-coordinates of the vertices of two quadratic regression models. Stat Papers (2010) 51:285–296, https://doi.org/10.1007/s00362-008-0159-7

Examples


library(agridat)
data(bachmaier.nitrogen)
dat <- bachmaier.nitrogen

# Fit a quadratic model for the low-fertility zone
dlow <- subset(dat, zone=="low")
m1 <- lm(yield ~ nitro + I(nitro^2), dlow)

# Slope of tangent line for economic optimum
m <- .005454 # = (N 0.60 euro/kg) / (wheat 110 euro/Mg)
# x-value of tangent point
b1 <- coef(m1)[2]
b2 <- coef(m1)[3]
opt.bach <- (m-b1)/(2*b2)
round(opt.bach, 0)
#> nitro 
#>   199 

# conf int for x value of tangent point
round(vcovs <- vcov(m1), 7)
#>             (Intercept)      nitro I(nitro^2)
#> (Intercept)   0.1346512 -0.0016680    4.7e-06
#> nitro        -0.0016680  0.0000295   -1.0e-07
#> I(nitro^2)    0.0000047 -0.0000001    0.0e+00
b1b1 <- vcovs[2,2] # estimated var of b1
b1b2 <- vcovs[2,3] # estimated cov of b1,b2
b2b2 <- vcovs[3,3]
tval <- qt(1 - 0.05/2, nrow(dlow)-3)
A <- b2^2 - b2b2 * tval^2
B <- (b1-m)*b2 - b1b2 * tval^2
C <- ((b1-m)^2 - b1b1 * tval^2)/4
D <- B^2 - 4*A*C
x.lo <- -2*C / (B-sqrt(B^2-4*A*C))
x.hi <- (-B + sqrt(B^2-4*A*C))/(2*A)
ci.bach <- c(x.lo, x.hi)
round(ci.bach,0) # 95% CI 173,260 Matches Bachmaier
#> nitro nitro 
#>   173   260 

# Plot raw data, fitted quadratic, optimum, conf int
plot(yield~nitro, dlow)
p1 <- data.frame(nitro=seq(0,260, by=1))
p1$pred <- predict(m1, new=p1)
lines(pred~nitro, p1)
abline(v=opt.bach, col="blue")
abline(v=ci.bach, col="skyblue")
title("Economic optimum with 95 pct confidence interval")