17.5.7.2 Algorithms (Moods Median Test)

The procedure below draws on NAG algorithms.

The median test investigates the difference between K\,\! samples of sizes n_1,n_2,...,n_k\,\! denoted by

x_1,x_2,....x_{n_1};x_{n_1+1},x_{n_1+2},...,x_{n_1+n_2};...;x_{n_1+n_2+...+n_{i-1}+1},...x_{n_1+n_2+...+n_i}

If the median value is not given by the user, the combined data from all groups are sorted and the median is calculated:

md=(x_{(n/2)}+x_{(n/2+1)})/2\,\! ,if n is even;md=x_{((n+1)/2)}\,\!, if n is odd.

Where n= \sum_{i=1}^k n_i,x_{(1)},...,x_{(n)}\,\! is the ordered data of all observations from small to large.

The test proceeds by forming a frequency table, giving the number of scores in each sample above and below the median of the pooled sample:

Sample 1 Sample 2 …… Sample K Total
Score \le md n_{11}\,\! n_{12}\,\! n_{1k}\,\! R_{1}\,\!
Score > md\,\! n_{21}\,\!…… n_{22}\,\!…… n_{2k}\,\!…… R_{2}\,\!……
Total n_{1}\,\!…… n_{2}\,\! n_{k}\,\! n\,\!

The x^2\,\!statistic foe all nonempty samples is calculated as:

x^2=\sum_{j=1}^k\sum_{i=1}^2(n_{ij}-e_{ij})^2/e_{ij}\,\! where e_{ij}=R_in_j/n\,\!

The significance level is from the x^2\,\! distribution with k-1\,\! degrees of freedom, where k-1\,\! is the number of nonempty samples. A message is printed if any cell has an expected value less than one, or more than 20% of the cells have expected values less than five.

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