http://www.randomservices.org/random/data/Challenger2.txt # 23 previous space shuttle launches before the Challenger disaster # T is the temperature in Fahrenheit, I is the O-ring damage index oring=read.table("http://www.randomservices.org/random/data/Challenger2.txt",header=T) attach(oring) #note: masking T=TRUE plot(T,I) oring.lm=lm(I~T) summary(oring.lm) # add fitted line to scatterplot lines(T,fitted(oring.lm)) # 95% posterior interval for the slope -0.24337 - 0.06349*qt(.975,21) -0.24337 + 0.06349*qt(.975,21) # note that these are the same as the frequentist confidence intervals # the Challenger launch was at 31 degrees Fahrenheit # how much o-ring damage would we predict? # y-hat 18.36508-0.24337*31 coef(oring.lm) coef(oring.lm)[1] + coef(oring.lm)[2]*31 # posterior prediction interval (same as frequentist) predict(oring.lm,data.frame(T=31),interval="predict") 10.82052-2.102*qt(.975,21)*sqrt(1+1/23+((31-mean(T))^2/22/var(T))) # posterior probability that damage index is greater than zero 1-pt((0-10.82052)/(2.102*sqrt(1+1/23+((31-mean(T))^2/22/var(T)))),21) http://www.randomservices.org/random/data/Galton.txt # Galton's seminal data on predicting the height of children from the # heights of the parents, all in inches heights=read.table("http://www.randomservices.org/random/data/Galton.txt",header=T) attach(heights) names(heights) pairs(heights) summary(lm(Height~Father+Mother+Gender+Kids)) summary(lm(Height~Father+Mother+Gender)) heights.lm=lm(Height~Father+Mother+Gender) # each extra inch taller a father is is correlated with 0.4 inch extra height in the child # each extra inch taller a mother is is correlated with 0.3 inch extra height in the child # a male child is on average 5.2 inches taller than a female child # 95% posterior interval for the the difference in height by gender 5.226 - 0.144*qt(.975,894) 5.226 + 0.144*qt(.975,894) # posterior prediction interval (same as frequentist) predict(heights.lm,data.frame(Father=68,Mother=64,Gender="M"),interval="predict") predict(heights.lm,data.frame(Father=68,Mother=64,Gender="F"),interval="predict")