W = 1.00, Q = 9.00 ~ X^2, DoF = 3, p <= (this p is not reliable)
The distributions of ranks within rows are unrelated between rows.
This is a distribution free test, so there are no strong assumptions.
Ordinal within rows.
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.
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.
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.
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.