#  This is S-plus code for creating 
#  bootstrapped confidence intervals
#  for a correlation coefficient.  It is 
#  stored in the file 
#
#             lawschl.ssc 
#
#  Any line preceded with a pound sign 
#  is a comment that is ignored by the
#  program.  The law school data are 
#  read from the file
#
#              lawschl.dat

#  Enter the law school data into a data frame

 
  laws <- read.table("c:/MyDocuments/courses/st415/lawschl.dat",
               col.names=c("School","LSAT","GPA"))
  laws
 

#  Plot the data

par(fin=c(7.0,7.0),pch=16,mkh=.15,mex=1.5)
plot(laws$LSAT,laws$GPA, type="p",
         xlab="LSAT score",ylab="GPA",
         main="Law School Data")


#  Compute the sample correlation matrix

  rr<-cor(laws$LSAT,laws$GPA)
  cat("Estimated correlation:  ", round(rr,5), fill=T)


# First test for zero correlation
 
   n <- length(laws$LSAT);
   tt<- sqrt(n-2)*rr/sqrt(1 - rr*rr)
   pval <- 1 - pt(tt,n-2)
   pval <- round(pval,digits=5)

   cat("t-test for zero correlation: ",
                 round(tt,4), fill=T)
   cat("p-values for the t-test for zero correlation: ",
                 pval, fill=T)

# Use Fisher's z-transformation to construct
# approximate confidence intervals.
# First set the level of confidence at 1-alpha.

   alpha <- .10
   z <- 0.5*log((1+rr)/(1-rr))
   zl <- z - qnorm(1-alpha/2)/sqrt(n-3)
   zu <- z + qnorm(1-alpha/2)/sqrt(n-3)
   rl <- round((exp(2*zl)-1)/(exp(2*zl)+1),digits=4)
   ru <- round((exp(2*zu)-1)/(exp(2*zu)+1),digits=4)

   per <- (1-alpha)*100;
   cat( per,"% confidence interval:  (",
                    rl,", ",ru,")",fill=T)


# Compute bootstrap confidence intervals.  
# Use B=5000 bootstrap samples.

  nboot <- 5000
  rboot <- bootstrap(data=laws,
                statistic=cor(GPA,LSAT),B=nboot)

#   Both sets of confidence intervals could
#   have been obtained from the summary(  )
#   function

  summary(rboot)   

# Make a histogram of the bootstrapped 
# correlations.

  hist(rboot$rep,nclass=50,
           xlab=" ",
           main="5000 Bootstrap Correlations",
           density=.0001)