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 |R₁ - 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.