43. The model for the data from a randomized block experiment for comparing I treatments was Xij  m 

aibjeij, where the a s are treatment effects, the b s are block effects, and the e s were assumed normal with mean 0 and variance s2. We now replace normality by the assumption that the e s have the same continuous distribution. A distribution-free test of the null hypothesis of no treatment effects, called Friedman’s test, involves rst ranking the observations in each block separately from 1 to I. The rank average is then calculated for each of the I treatments. If H0 is true, the expected value of each rank average is (I  1)/2. The test statistic is Fr 

12J I1I  12 a aRi 

I  1 2

b 2

For even moderate values of J, the test statistic has approximately a chi-squared distribution with I  1 df when H0 is true.
The article Physiological Effects During Hypnotically Requested Emotions (Psychosomatic Med., 1963: 334—343) reports the following data (xij) on skin potential in millivolts when the emotions of fear, happiness, depression, and calmness were requested from each of eight subjects.
Blocks (Subjects)
1 2 3 4 Fear 23.1 57.6 10.5 23.6 Happiness 22.7 53.2 9.7 19.6 Depression 22.5 53.7 10.8 21.1 Calmness 22.6 53.1 8.3 21.6 5 6 7 8 Fear 11.9 54.6 21.0 20.3 Happiness 13.8 47.1 13.6 23.6 Depression 13.7 39.2 13.7 16.3 Calmness 13.3 37.0 14.8 14.8 Use Friedman s test to decide whether emotion has an effect on skin potential.