Spearman rank correlation

Suppose that a random sample of n pairs of observations is taken. If and are each ranked in ascending order and the sample correlation of these ranks is calculated, the resulting coefficient is called the Spearman rank correlation coefficient. If there are no tied ranks, an equivalent formula for computing this coefficient is

where the are the differences of the ranked pairs:

Remark: Spearman rank correlation shares the properties of that and that values near +1 indicate a tendency for the larger values of X to be paired with the larger values of Y.

The following test of the null hypothesis of no association in the population

and are independent

have significance level :

1. To test against the alternative of positive association, the decision rule is

Reject if

2. To test against the alternative of negative association, the decision rule is

Reject if

3. To test against the two sided alternative of some association, the decision rule is

Reject if or

The table of critical values of the Spearman rank correlation coefficient is given in the Table 5 of the Appendix.

Example:

Scores that 8 salesmen made on a test that measures their aggressiveness (x), and their sales in thousands of dollars for their second year with a certain company (y).

x (aggressiveness)	30 17 35 28 42 25 19 34
y (sales)	35 31 40 46 50 32 33 42

a) Find and interpret Spearman rank correlation

b) Test the null hypothesis that aggressiveness and sales are independent again the alternative that they are positively correlated. Take .

Solution:

a) First of all, let us rank separately x and y in ascending order. These two rank appear in third and fourth columns of the following table 3.2

Table 3.2

x	y	Rank	Rank
				-2 -1
sum

The differences between ranks and squared differences between ranks are shown in the last two columns of the table. Substituting the values and into formula for Spearman rank correlation, we obtain

It means that there exists strong positive correlation between aggressiveness and sales volume.

b) The null and alternative hypotheses are

x and y are independent

x and y are positively correlated

The decision rule is

Reject if

For a sample of size n= 8, and ,

Since 0.81>0.643 we reject , and accept the alternative hypothesis that x and y are positively correlated.

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