Friedman Test

The Friedman Test tests the Nullhypothesis of identical populations for dependent data. It is an equivalent to the one factorial variance analysis with repeated measurement without making any assumptions on the distributions of the populations. It uses only the rank information of the data.

Requirements:

Data must be ordinal (rank-order) scaled.

Distribution is free

Hypothesis:

H0:      The treatments have identical effects

Idea:

The data of k subjects and p treatments are first displayed in a two dimensional table. Then the data of each person are brought into one rank order (treatment ranks). This procedure ensures that the dependency of the data is taken into account. Next for each treatment the sum of ranks T_i is computed. Whereas the total sum of ranks is:

with

For sufficiently large sample sizes (k>10 and p>4) the following value is approximately Chi-Square distributed with p-1 degrees of freedom:

if there are tied ranks Chi-Square is computed the following way:

wheras

Conover recommends the following F statistic, which has a more accurate approximation:

with

  degrees of freedom