# 17.5.3.2 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;
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 $ 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).