1. Lecture 17 Rank Correlation Coefficient
Outline of Today
Rank Correlation Coefficients
11/14/2018
SA3202, Lecture 17

2. Properties: 1. -1<= r <=1
Rank Correlation Coefficient
Recall that a measure of association (correlation) between two (numerical) random variables X and Y is Pearson’s correlation coefficient:
r=Cor(X,Y)
Properties: <= r <=1
2. X and Y is positively related when r>0
negatively when r<0
linearly uncorrelated when r=0.
Given a sample (Xi, Yi), i=1,2, ..,n, the sample correlation coefficient r can be used to test whether there is an association between X and Y. But the null distribution of r depends on the joint distribution of X and Y, usually assumed to be bivariate normal.
11/14/2018
SA3202, Lecture 17

3. The Definition
For a distribution-free test of association, we replace the observations with their ranks and then compute the sample correlation coefficient (based on ranks). This measure is known as Spearman’s rank correlation coefficient:
Where xi is the rank of Xi within the X’s, and yi is the rank of Yi within the Y’s. Ties are treated in a usual manner.
11/14/2018
SA3202, Lecture 17

4. Note that if there are no ties, we can show that
11/14/2018
SA3202, Lecture 17

5. Example
Eight elementary science teachers have been ranked by a judge according to their teaching ability, and all have taken a national teachers’ examination. Do the data suggest agreement between the judge’s ranking and the examination score?
Teacher
Judge’s Rank
Exam Score
For consistency, we ranked the examination scores from the highest from the lowest. We got the following table:
Total
Teacher
Judge’s Rank (xi)
Exam Score Rank (yi)
xi-yi
(xi-yi)^
rs=1-6x24/8/(64-1)=.714
This indicates agreement between the Judge’s ranking and the examination score.
11/14/2018
SA3202, Lecture 17

6. Hypothesis Test
The Spearman rank correlation coefficient may be used to test the hypothesis
H0: there is no association between the two variables.
The upper quantiles of the null distribution of rs is symmetric about 0; so the lower quantiles are just the negative of the upper quantiles.
Example For the above example, consider testing
H0: there is no association between the judge’s ranking and the exam scores
H1: the relationship is positively related.
For the 5% level, we reject H0 when rs> Since the observed value of rs is .714, H0 is rejected.
11/14/2018
SA3202, Lecture 17