*Example:*

**R = 0.7667, N = 9, p <= 0 (Z = 2.1685)**

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.

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