*Example:*

**
W = 1.00,
Q = 9.00 ~ X^2,
DoF = 3,
p <=
(this p is not reliable)
**

The

The test parameter

Rank ordering is done within rows and each row is used independent of the others. That is, only the

For two columns the

*H0:*

The distributions of ranks within rows are unrelated between rows.

*Assumptions:*

This is a distribution free test, so there are no strong assumptions.

*Scale:*

Ordinal within rows.

*Procedure:*

Start with *n* rows and *k* columns. Rank order the entries of each
row independently of the other rows. Sum the ranks for each column
(*Rct*) and sum the squared column totals ( Sum(*Rct^2*) ).

The test statistics are:

*W* = Sum(*Rct^2*) * 12 / (*n*^2 * *k* * ( *k^2* - 1 ) ) -
3 * ( *k* + 1 ) / ( *k* - 1 )

*Q* = *n* * ( *k* - 1 ) * *W*

*Degrees of Freedom (DoF)* = ( *k* - 1 )

Furthermore, the mean
Spearman Rank Correlation Coeficient (*Rsm*)
between all the rows is:

*Rsm* = (*n * W* - 1) / ( *n* - 1 ) ( =
1 for the example above)

*Level of Significance:*

Consult a table of *Q* values for *k* = 3 and *n* <= 15,
or *k* = 4 and *n* <= 8. Otherwise, use the *Chi-square*
approximation below.

*Approximation:*

The probability distribution of *Q* under *H0* can be approximated with
a *X^2* distribution with ( *k* - 1 ) degrees of freedom if the number of
columns, *k*, > 4 or the number of rows *n*, > 15.

*Remarks:*

The *level of significance, p <= *, given above is based on an
approximation of the *Chi-square* distribution. Consult a **table** for the exact
*p* value that belongs to this *X^2 = 9.00* value.

*Note: A X^2 approximation of Q is not reliable for n=3 and k=4.*

Limitations

If there are ties, the *p* values are incorrect. There is a correction of Q necessary that can be found here. Thanks to Eben Goodale, University of
Massachusetts, Amherst, for pointing this out to me and finding the correction.

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