Reference:

Siegel S,
Castellan NH. Nonparametric statistics
for the behavioral sciences. 2^{nd} ed New York: McGraw-Hill, Inc;
1988,

*post hoc* comparisons after Friedmans’
test . pages 180-183

*post hoc* comparisons after
Kruskal-Wallis test . pages 213-215

Tables: pages 319-321

The formulas 1a and 1b depict for *post
hoc* comparisons after Friedmans’ test .

In the formula Ia, |R_{v}
- R_{u}| represents difference in rank sums for a
particular pair, k represents the number of groups and the critical value for Z_{a/k(k-1) }is the
abscissa value from unit normal distribution above which lies a/k(k-1)which
is obtained from relevant statistical tables.
Similarly, in the formula Ib, the |R_{1} - R_{u}| is
the difference in rank sum between the control group and the group to be
compared, c is the number of comparisons i.e k-1. The critical values for q are obtained special table.

The formulas IIa and IIb depicted
below represent those for *post hoc* analysis after Kruskal-Wallis test
for all pair-wise comparisons and comparisons against a single control group.

In the formula IIa, the critical value
must exceed the difference in average ranks between a pair of groups to be
compared, N is the total sample size for all the groups, n_{u} and n_{v}
represent the sample size of the relevant groups to be compared. The critical value Z_{a/k(k-1)}
is the abscissa value from the unit normal distribution above which lies a/k(k-1)
percent of distribution. The values of
z can be obtained from relevant statistical tables.

In the formula IIb, the difference in average ranks between
control group and the group to be compared must exceed the critical value and n_{c}
represents the sample size of control group.