# R analysis of 2 by 2 tables

# Inference for the difference of two proportions (independent samples)

# We use the Christmas tree data given in class

## 95% CI for pi_1 - pi_2:

# We can specify the number of successes and the sample sizes for each sample:

prop.test(c(64,89), n=c(160,261), conf.level=0.95, correct=FALSE)$conf.int

# Or we can equivalently specify the number of successes and 
# number of failures for each sample:

prop.test(matrix(c(64,89,96,172),nrow=2,ncol=2,byrow=FALSE), conf.level=0.95, correct=FALSE)$conf.int

## Testing for a difference between pi_1 and pi_2:

# We can specify the number of successes and the sample sizes for each sample:

prop.test(c(64,89), n=c(160,261), alternative="two.sided", correct=FALSE)

# Or we can equivalently specify the number of successes and 
# number of failures for each sample:

prop.test(matrix(c(64,89,96,172),nrow=2,ncol=2,byrow=FALSE), alternative="two.sided", correct=FALSE)

#########################################################################

# Small-sample inference for the difference of two proportions (independent samples)

# Can use Fisher's exact test with the fisher.test() function in R:

fisher.test(matrix(c(1,4,5,2),nrow=2,ncol=2,byrow=FALSE), alternative = "less", conf.int = FALSE)

# P-value given as 0.1212.

# 95% CI for odds ratio:

fisher.test(matrix(c(1,4,5,2),nrow=2,ncol=2,byrow=FALSE), conf.int = TRUE, conf.level=0.95)