4.5 Comparing Two Proportions (pp75-79).txt

#4.5 Comparing Two Proportions
#This makes use of Howard's (1998) dependent prior function howardprior
#See p.76

library("LearnBayes")
#The dependent prior has 5 parameters! - alpha, beta, gamma, delta, and sigma
#The first four parameters reflect one's prior belief about the locations of p1 and p2
#the first two are success and failure counts for sample 1 and 
#the last two are success and failure counts for sample 2.
#sigma indicates ptior belief in the dependence between two proportions

#suppose prior location parameter is believed to be 1.
#that is alpha=beta=gamma=delta=1.
sigma=c(2,1,.5,.25)
plo=.0001;phi=.9999
par(mfrow=c(2,2))
for (i in 1:4)
	mycontour(howardprior, c(plo, phi, plo, phi), c(1,1,1,1,sigma[i]),
		main=paste("sigma=", as.character(sigma[i])),
		xlab="p1", ylab="p2")

#suppose we observe counts y1 and y2 from two binomial samples.
#see Likelihood function in p. 77
#combining the prior with the likelihood we obtain a 
#posterior density with the same functional "dependent" form BUT with updated parameters

#(alpha+y1, beta+n1-y1, gamma+y2, delta+n2-y2, and sigma)
#since the posterior is of the same functional form as the prioir,
#we can still use howardprior function
#we use data from Pearson (1947) see page 77 table 4.3

sigma=c(2,1,.5,.25)
par(mfrow=c(2,2))
for (i in 1:4)
{
mycontour(howardprior, c(plo,phi,plo,phi),
	c(1+3, 1+15, 1+7, 1+5, sigma[i]),
	main=paste("sigma=", as.character(sigma[i])),
	xlab="p1", ylab="p2")
	lines(c(0,1),c(0,1))
}

#How to test hypothesis?
#Suppose hypothesis is Hi:p1>p2
#We simulate sample from the posterior and 
#find the proportion of simulated pairs where p1>p2
#we do this with sigma=2

s=simcontour(howardprior, c(plo, phi, plo, phi),
	c(1+3,1+15,1+7,1+5,2), 1000)
sum(s$>s$y)/1000