How can the significance of Spearman’s rank correlation be tested?

Let us assume

Ho:ρs=0  vs H1=ρs≠0 

The test statistics is

t=[r_s*SQRT(n-2)] / SQRT(1- r_s²)

t has (n-2) degrees of freedom.

The formula for r_s is based on the usual Pearsonian formula for the correlation coefficient. Here it is assumed that no ties occur in the ranks within a sample.

The value of rs lies between -1 and +1

If the ties occur within yhe samples, how the formula for rank correlation is amended?

Often the measurements are taken on individuals and then ranked. In such a situation, sometimes ties do occur.

If so, then a correlation has to be made in rank correlation. If the number of tied observations in X for a particular rank is tX. There can be more than one tX in X samples.

Let’s calculate the significance of Spearman’s Rank Correlation based on two judges’ ratings.

Following are the rank awarded to seven debaters in a competition by two judges.

Judge1: 3,2,1,6,7,4,5

Judge2: 5,6,3,7,4,2,1

Let’s calculate the rank correlation and test the significance.

First, we need to calculate the difference d’s which are:

d: -2, -4, -2, -1, 3, 2, 4

Also Sum of d square=54

rs=1-[ (6*54)/(7*48)]

rs=0.036

Significance Testing:

To test Ho: ρs=0  vs H1=ρs≠0 

t= 0.036*sqrt(7-2)/(sqrt(1-(0.036*0.036)

t=0.080/0.993

t=0.080

Calculated value of t=0.080<2.571,

Conclusion

The Calculated value less than the tabled value. It means that there is a dissociation between the ranks awarded by two judges.