*Example:*

The Rank Correlation test is a distribution free test that determines whether there is a monotonic relation between two variables (

If

If

The monotonic relation is expressed using rank-order numbers instead of the values. This also makes the Rank Correlation a test

*H0:*

There is *no* monotonic relation between the variables.

*Assumptions:*

None realy

*Scale:*

Ordinal

*Procedure:*

Rank order all *x* and *y* values seperately. Determine the
differences between the ranks of both variables
*V* = Rank(*x*) - Rank(*y*). Sum the squares of the differences
in rank order numbers (i.e., Sum( *V***2 ) ).

The Spearman Rank Correlation Coefficient is:

*Rs* = 1 - 6 * Sum( *V***2 ) / ( *N* * ( *N***2 - 1 ))

*Level of Significance:*

Look up the values of *Rs* and *N* in a table. The level of
significance is determined by checking all permutations of ranks in the
sample and counting the fraction for which the *Rs'* is more
extreme than the *Rs* found. As the number of permutations grows
proportional to *N!* (the factorial of *N*), this is not very
practical for large values of *N*. For *N > 10* this example
uses only an approximation (i.e., only a random subset of the permutations
is actualy checked).

*Approximation:*

If *N* > 30, the distribution of Z = *Rs* * sqrt( *N* - 1 )
can be approximated by a
>
Standard Normal Distribution.

*Remarks:*

This example uses the
>
Standard Normal approximation for *N > 30*. For *N < 11*
the exact value is calculated. For all other values of *10 < N < 31*,
*p* is calculated from a random subset of the possible permutations.
This latter value is not very exact.

As a statistical test to check whether a relation between two variables exists,
this test is better than the standard
>correlation coefficient
because the latter will *only* work when there is a *linear*
relation between the variables. In practical situations, assuming a linear
relation will very often be unrealistic.

This test is also usefull to check whether matched pairs are realy matched.
If they are, their rank correlation should be statistically significant.

Return to: Statistics