Example: H0: Assumptions: Scale: Procedure: Level of Significance: Approximation: Remarks:
Characteristics:
The Rank Correlation test is a distribution free test that determines whether
there is a monotonic relation between two variables ( x , y ).
A monotonic relation exists when any increase in one variable is invariably
associated with either an increase or a decrease in the
other variable. In equation form, for the pairs (X1, Y1) and
(X2, Y2):
If X2 > X1 then Y2 >= Y1 for a monotonic
increase
If X2 > X1 then Y2 <= Y1 for a monotonic
decrease
The monotonic relation is expressed using rank-order numbers instead of the
values. This also makes the Rank Correlation a test distribution free
test. Although the Rank Correlation coefficient can be interpreted as
indicating the "strength" of the monotonic association, quantifying this
strength is so complex that for all practical purposes this is a
non-parametric test.
There is no monotonic relation between the variables.
None realy
Ordinal
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 ))
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).
If N > 30, the distribution of Z = Rs * sqrt( N - 1 )
can be approximated by a
>
Standard Normal Distribution.
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