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.