Algorithm (sign2)

The paired sample sign test tests the median difference between pairs of scores from two matched samples.

For two match samples \{x_i,y_i\}\,\!, i=1,2,\ldots ,n. The null hypothesis H_0\,\! iis that the medians of the paired samples are the same, while the alternative hypothesis H_1\,\!can be one- or two-tailed (see below). We compute:

  1. The test statistics S\,\! , which is the number of pairs for which x_i<y_i\,\!;
  2. The number n_i\,\! of non-tied pairs x_i=y_i\,\!;
  3. The lower tail probability p\,\! corresponding to S\,\! (adjusted to allow the complement 1-p\,\! to be used in an upper tailed or a two-tailed test). p\,\! is the probability of observing a value \leq S\,\! if S\leq \frac 12n_1; or of observing a value <S\,\! if S> \frac 12n_1, given that H_0 \,\! is true. If S=\frac 12n_1, then  p=0.5 \,\! .

Suppose that the significance test of a chosen size \alpha \,\! is to be performed (i.e., \alpha \,\! is the probability of rejecting H_0\,\!when H_0\,\! is true; typically \alpha \,\!is a small quantity such as 0.05 or 0.01). The returned value of p \,\!can be used to perform the significance test on the median difference, against various alternative hypothesis H_1\,\!as follows.

  1. H_1\,\!: median of  x\neq median of y\,\!. H_0\,\! is rejected if  2\times \min (p,1-p)<\alpha .
  2. H_1\,\!: median of  x< \,\! median of y\,\!. H_0\,\! is rejected if  1-p<\alpha \,\!
  3. H_1\,\!: median of  x> \,\! median of y\,\!. H_0\,\! is rejected if  p<\alpha\,\!

For more details of the algorithm, please refer to: nag_sign_test (g08aac).