> ## Comparing two proportions ##
> 
> # Table 2.3, p. 27 (attack v. no attack)
> 
> x<-c(189,104) # placebo, aspirin, 
> n<-c(11034,11037)
> 
> # test H0:p1=p2 (equal probabilities of heart attack)
> prop.test(x,n,p=c(.5,.5))

        2-sample test for given proportions with continuity correction

data:  x out of n, null probabilities c(0.5, 0.5) 
X-squared = 20911.98, df = 2, p-value < 2.2e-16
alternative hypothesis: two.sided 
null values:
prop 1 prop 2 
   0.5    0.5 
sample estimates:
    prop 1     prop 2 
0.01712887 0.00942285 

> prop.test(x,n)

        2-sample test for equality of proportions with continuity
        correction

data:  x out of n 
X-squared = 24.4291, df = 1, p-value = 7.71e-07
alternative hypothesis: two.sided 
95 percent confidence interval:
 0.004597134 0.010814914       ## contains only positive values
sample estimates:              ## ==> taking aspirin appears to result in 
                                      a diminished risk of heart attack (MI).
    prop 1     prop 2 
0.01712887 0.00942285 

> # Test H0:p1=p2 v. H1:p1>p2
> prop.test(x,n,alt="greater")

        2-sample test for equality of proportions with continuity
        correction

data:  x out of n 
X-squared = 24.4291, df = 1, p-value = 3.855e-07
alternative hypothesis: greater 
95 percent confidence interval:
 0.005082393 1.000000000 
sample estimates:
    prop 1     prop 2 
0.01712887 0.00942285 

> 
> # sample difference of proportions:
> temp<-prop.test(x,n)  
> names(temp$estimate)<-NULL  # optional
> temp$estimate[[1]]-temp$estimate[[2]]
[1] 0.007706024     ## very small ==> makes it seem as if the two groups differ by a trivial amount.
> 
> # relative risk
> temp$estimate[[1]]/temp$estimate[[2]]
[1] 1.817802      ## The sample % of MI cases was 82% higher for the group taking placebo.
>                 ## ==> the difference may have important public health implications.
> # odds ratio
> x[1]*(n[2]-x[2])/(x[2]*(n[1]-x[1]))
[1] 1.832054      ## The estimated odds of MI for group taking placebo equal 1.83 times 
                     that for group taking aspirin                 
                  ==> the estimated odds of MI were 83% higher for the placebo group.

> # Table 2.4, p. 32 (Smoker v.s MI)
> 
> x<-c(172,173) # smoker, non-smoker
> n<-c(262, 519)
> 
> # test H0:p1=p2 (equal probabilities of ........ )
> prop.test(x,n,p=c(.5,.5))

        2-sample test for given proportions with continuity correction

data:  x out of n, null probabilities c(0.5, 0.5) 
X-squared = 82.0439, df = 2, p-value < 2.2e-16
alternative hypothesis: two.sided 
null values:
prop 1 prop 2 
   0.5    0.5 
sample estimates:
   prop 1    prop 2 
0.6564885 0.3333333 


> # Test H0:p1=p2 v. H1:p1>p2
> prop.test(x,n,alt="greater")$p.value
[1] 8.67911e-18
> 
> # odds ratio
> temp<-prop.test(x,n)  
> names(temp$estimate)<-NULL  # optional
> x[1]*(n[2]-x[2])/(x[2]*(n[1]-x[1]))
[1] 3.822222
>