agridat (version 1.8.1)

henderson.milkfat: Milk fat yields for a single cow

Description

Average daily fat yields (kg/day) from milk from a single cow for each of 35 weeks.

source

Charles McCulloch. Workshop on Generalized Linear Mixed Models. Used with permission of Charles McCulloch and Harold Henderson.

References

None.

Examples

dat <- henderson.milkfat

plot(yield~week, data=dat, cex = 0.8, ylim=c(0,.9), xlab = "Week",
     ylab = "Fat yield (kg/day)")

# Yield ~ a * t^b * exp(g*t) # where t is time
m1 <- nls(yield ~ alpha * week^beta * exp(gamma * week),
          data=dat,
          start=list(alpha=.1, beta=.1, gamma=.1))

# Or, take logs and fit a linear model
# log(yield) ~ log(alpha) + beta*log(t)  + gamma*t
m2 <- lm(log(yield) ~ 1 + log(week) + week, dat)

# Or, use glm and a link to do the transform
m3 <- glm(yield ~ 1 + log(week) + week, quasi(link = "log"), dat)

# Note: m2 has E[log(y)] = log(alpha) + beta*log(t)  + gamma*t
#  and m3 has  log(E[y]) = log(alpha) + beta*log(t)  + gamma*t

# Generalized additive models
require("mgcv")
m4 <- gam(log(yield) ~ s(week), gaussian, dat)
m5 <- gam(yield ~ s(week), quasi(link = "log"), dat)

# Model predictions
pdat <- data.frame(week = seq(1, 35, by = 0.1))
pdat <- transform(pdat, p1 = predict(m1, pdat),
                  p2 = exp(predict(m2, pdat)), # back transform
                  p3 = predict(m3, pdat, type="resp"), # response scale
                  p4 = exp(predict(m4, pdat)),
                  p5 = predict(m5, pdat, type="response"))

# Compare fits
with(pdat, {
  lines(week, p1)
  lines(week, p2, col = "red", lty="dotted")
  lines(week, p3, col = "red", lty="dashed")
  lines(week, p4, col = "blue", lty = "dashed")
  lines(week, p5, col = "blue")
})

legend("topright",
  c("nls", "lm, log-transformed", "glm, log-link",
    "gam, log-transformed", "gam, log-link"),
  lty = c("solid", "dotted", "dashed", "dashed", "solid"),
  col = c("black", "red", "red", "blue", "blue"),
  cex = 0.8, bty = "n")