# The Sign Test

Example:

n+ , n- , p <=

n+ >
n- >

Characteristics:
Crudest and most insensitive test. It is also the most convincing and easiest to apply. The level of significance can almost be estimated without the help of a calculator or table. If the Sign-test indicates a significant difference, and another test does not, you should seriously rethink whether the other test is valid.

H0:
The median value of the distribution is m (generally m = 0), values larger (+) and smaller (-) than the median are equally likely.
When matched pairs are used, the probability of observing (A, B) is equal to that of observing (B, A) and the value of A-B has median value 0.

Assumptions:
None other than H0.

Scale:
Ordinal, but sometimes nominal when it is used as a binomial test

Procedure:
Label observations as +/-, positive/negative, greater/smaller, left/right, black/white, male/female, or whatever dichotomy you want to use and count the number of times each label is observed. n+ is the number of times a "+" is observed and n- the number of times a "-" is observed. Ignore all zero or equal observations.

Level of Significance:
n+ and n- are binomial distributed with p = q = 1/2 and N = (n+) + (n-).
If k is the smaller of (n+) and (n-) then:
p <= 2 * Sum (i=0 to k) {N!/(i!*(N-i)!)}/4
(with k! = k*(k-1)*(k-2)*...*1 is the factorial of k and 0! = 1)

Approximation:
If (n+) + (n-) = N > 25, then Z = (| n+ - n- | - 1)/sqrt( N ) can be approximated with a Standard Normal distribution. In our example, we calculate the exact probabilities upto N = 100.
For N > 30, the Student t-test can be used. However, the Wilcoxon Matched-Pairs Signed-Ranks test is a better alternative.

Remarks:
The Sign-test answers the question: How often?, whereas other tests answer the question How much?. It must be kept in mind that these two questions might have different answers.
When the problem concerns the sizes of the differences, the Wilcoxon Matched-Pairs Signed-Ranks Test should be preferred. This test is also distribution free, but it is much more sensitive than the Sign-test.
Finally, the Sign-test is an almost ideal quick-and-dirty test that can be used to browse datasets or to check the results of other tests (e.g., as in: "All six subjects show an increase, so why does your test insist that p > 0.05?").