14 Day 14 (June 22)

14.1 Announcements

  • Project proposals due friday
  • No office hours today

14.2 Confidence intervals for derived quantities

  • Demonstration of how not to do it.

    y <- c(63, 68, 61, 44, 103, 90, 107, 105, 76, 46, 60, 66, 58, 39, 64, 29, 37,
27, 38, 14, 38, 52, 84, 112, 112, 97, 131, 168, 70, 91, 52, 33, 33, 27,
18, 14, 5, 22, 31, 23, 14, 18, 23, 27, 44, 18, 19)
year <- 1965:2011
df <- data.frame(y = y, year = year)
plot(x = df$year, y = df$y, xlab = "Year", ylab = "Annual count", main = "",
col = "brown", pch = 20, xlim = c(1965, 2040))

    m1 <- lm(y~year,data=df)
coef(m1)
    ## (Intercept)        year 
##  2356.48797    -1.15784
    confint(m1,level=0.95) 
    ##                 2.5 %       97.5 %
## (Intercept) 929.80699 3783.1689540
## year         -1.87547   -0.4402103
    # This is ok!
beta0.hat <- as.numeric(coef(m1)[1])
beta1.hat <- as.numeric(coef(m1)[2])
extinct.date.hat <- -beta0.hat/beta1.hat
extinct.date.hat
    ## [1] 2035.245
    # This is not ok!

beta0.hat.li <- confint(m1)[1,1]
beta1.hat.li <- confint(m1)[2,1]
extinct.date.li <- -beta0.hat.li/beta1.hat.li
extinct.date.li
    ## [1] 495.7729
    beta0.hat.ui <- confint(m1)[1,2]
beta1.hat.ui <- confint(m1)[2,2]
extinct.date.ui <- -beta0.hat.ui/beta1.hat.ui
extinct.date.ui
    ## [1] 8594.004

  • The delta method

    library(msm)

y <- c(63, 68, 61, 44, 103, 90, 107, 105, 76, 46, 60, 66, 58, 39, 64, 29, 37,
   27, 38, 14, 38, 52, 84, 112, 112, 97, 131, 168, 70, 91, 52, 33, 33, 27,
   18, 14, 5, 22, 31, 23, 14, 18, 23, 27, 44, 18, 19)
year <- 1965:2011
df <- data.frame(y = y, year = year)
plot(x = df$year, y = df$y, xlab = "Year", ylab = "Annual count", main = "",
 col = "brown", pch = 20, xlim = c(1965, 2040))

    m1 <- lm(y~year,data=df)
coef(m1)
    ## (Intercept)        year 
##  2356.48797    -1.15784
    confint(m1,level=0.95)  
    ##                 2.5 %       97.5 %
## (Intercept) 929.80699 3783.1689540
## year         -1.87547   -0.4402103
    beta0.hat <- as.numeric(coef(m1)[1])
beta1.hat <- as.numeric(coef(m1)[2])
extinct.date.hat <- -beta0.hat/beta1.hat
extinct.date.hat
    ## [1] 2035.245
    extinct.date.se <- deltamethod(~-x1/x2, mean=coef(m1), cov=vcov(m1))

extinct.date.ci <- c(extinct.date.hat-1.96*extinct.date.se,
                 extinct.date.hat+1.96*extinct.date.se)
extinct.date.ci
    ## [1] 2005.598 2064.892

  • Live example

References

Powell, Larkin A. 2007. “Approximating Variance of Demographic Parameters Using the Delta Method: A Reference for Avian Biologists.” The Condor 109 (4): 949–54.
Ver Hoef, Jay M. 2012. “Who Invented the Delta Method?” The American Statistician 66 (2): 124–27.