/math-9b0b09d3b8160549d46206793139aacb.png?v=0 "\{x_i,y_i\}\,\!")
, /math-6b80d670336d4b43b32dbd006770a4f1.png?v=0 "i=1,2,\ldots ,n")
. The null hypothesis /math-806277203dedea2ed8321f6cbd465a54.png?v=0 "H_0\,\!")
iis that the medians of the paired samples are the same, while the alternative hypothesis /math-fabff59271b950125b7a360fba21de2c.png?v=0 "H_1\,\!")
can be one- or two-tailed (see below). We compute:
The paired sample sign test tests the median difference between pairs of scores from two matched samples.
For two match samples , . The null hypothesis iis that the medians of the paired samples are the same, while the alternative hypothesis can be one- or two-tailed (see below). We compute:
/math-9fe1e9487c4a538f98d385e390785b8d.png?v=0 "S\,\!")
, which is the number of pairs for which /math-855d732beda343c0b8146b8c5b8ff61e.png?v=0 "x_i<y_i\,\!")
;/math-58a5f6a471c6abd9f79b3186ef729fb3.png?v=0 "n_i\,\!")
of non-tied pairs /math-6454948bccaa3123019e0072cd7c7a60.png?v=0 "x_i=y_i\,\!")
;/math-c0f582773fdbd168bbab09a1e6159c46.png?v=0 "p\,\!")
corresponding to (adjusted to allow the complement /math-72b0884141d731e8d2d12fba2154b124.png?v=0 "1-p\,\!")
to be used in an upper tailed or a two-tailed test). is the probability of observing a value /math-7afeae060e44e606bc6dbe9715aab926.png?v=0 "\leq S\,\!")
if /math-98e2b266a1fbea2a909c4c68653f6aaf.png?v=0 "S\leq \frac 12n_1")
; or of observing a value /math-e1e7abf9ce1e57d462c82337a9485eb2.png?v=0 "<S\,\!")
if /math-380193fef07d6f8a658f13c175f50de0.png?v=0 "S> \frac 12n_1")
, given that /math-544f6d19ae2c031bad897e04e59a20a5.png?v=0 "H_0 \,\!")
is true. If /math-966df41aeb4cf87c7a1ea63e31df0914.png?v=0 "S=\frac 12n_1")
, then /math-9d2830381a5dff59871640fc0371df96.png?v=0 "p=0.5 \,\!")
.Suppose that the significance test of a chosen size /math-4bc6c42bbabe567d1f2516326e52b775.png?v=0 "\alpha \,\!")
is to be performed (i.e., is the probability of rejecting when is true; typically is a small quantity such as 0.05 or 0.01). The returned value of /math-0df16721fb6ef31805ae616372f0abd0.png?v=0 "p \,\!")
can be used to perform the significance test on the median difference, against various alternative hypothesis as follows.
: median of /math-e2729ae96cfef5940f6b120729b0f3eb.png?v=0 "x\neq")
median of /math-bfb6488d6c250ac5aeed1bbf139baaa5.png?v=0 "y\,\!")
. is rejected if /math-21cdaad16c062c0f476b34cb2f742a7b.png?v=0 "2\times \min (p,1-p)<\alpha")
.: median of /math-aa0866b5e83253281e0989ab5e13bf25.png?v=0 "x< \,\!")
median of . is rejected if /math-24e3e58d4b73eef66a0817de59678a3c.png?v=0 "1-p<\alpha \,\!")
: median of /math-6b8c8cbc8029d27893bce8ccd14ecc25.png?v=0 "x> \,\!")
median of . is rejected if /math-f17d940553818fee4f08170e90217c63.png?v=0 "p<\alpha\,\!")
For more details of the algorithm, please refer to: nag_sign_test (g08aac).