### Class examples from Week 8: Correlations
### Ken Benoit Mar 2010


## Height and Weight example from L&F p233-
# define the data from p233
child <- LETTERS[1:8]
x <- c(49, 50, 53, 55, 60, 55, 60, 50)
y <- c(81, 88, 87, 99, 91, 89, 95, 90)

# scatterplots from p244
plot(x, y, xlab="Height in inches", ylab="Weight in pounds", pch=19)
text(x, y, child, pos=4)
# add mean lines
abline(v=mean(x), h=mean(y), lty="dashed")


## computations from p245
xminusxbar <- x - mean(x)
yminusybar <- y - mean(y)
devproduct <- xminusxbar * yminusybar
data.frame(child=child, X=x, Y=y, xminusxbar, yminusybar, devproduct)
(SP <- sum(devproduct))
xminusxbar2 <- xminusxbar^2
yminusybar2 <- yminusybar^2
(SSx <- sum(xminusxbar2))
(SSy <- sum(yminusybar2))
data.frame(child=child, X=x, Y=y, xminusxbar, yminusybar,
           devproduct, xminusxbar2, yminusybar2)

## define function to compute r from p246
pearsonr <- function(x, y) {
  xdev <- x - mean(x)
  ydev <- y - mean(y)
  sum(xdev*ydev) / sqrt(sum(xdev^2)*sum(ydev^2))
}
pearsonr(x,y)

pearsonrcomp <- function(x, y) {
  N <- length(x)
  (sum(x*y) - N*mean(x)*mean(y)) /
    sqrt((sum(x^2)-N*mean(x)^2) * (sum(y^2)-N*mean(y)^2))
}
pearsonrcomp(x,y)


## Example L&F p247
x <- c(12,10,6,16,8,9,12,11)
y <- c(12,8,12,11,10,8,16,15)
# using our created function
pearsonr(x,y)
# using R's built-in correlation command
cor(x,y)

# compute the empirical t-value
(t.calc <- (.24*sqrt(8-2) / sqrt(1-.24^2)))
# compute the critical t-value
(t.crit <- qt(1-.05/2, 6))
# compare
t.calc > t.crit
# using R's built-in correlation significance test
cor.test(x,y)


## Final example from L&F from pp249-251
x <- c(10,3,12,11,6,8,14,9,10,2)
y <- c(1,7,2,3,5,4,1,2,3,10)
cor.test(x,y)
# make a picture of it
plot(x,y,pch=19)
respondent <- LETTERS[1:10]
text(x,y,respondent,pos=4)
abline(v=mean(x), lty="dashed")
abline(h=mean(y), lty="dashed")
abline(lm(y~x))  # "regression" line