Sign test

In statistics, the sign test can be used to test the hypothesis that there is "no difference" between the continuous distributions of two random variables X and Y. Formally:

Let p = Pr(X > Y), and then test the null hypothesis H0: p = 0.50. This hypothesis implies that given a random pair of measurements (xi, yi), then xi and yi are equally likely to be larger than the other.

Independent pairs of sample data are collected from the populations {(x1, y1), (x2, y2),. . ., (xn, yn)}. Pairs are omitted for which there is no difference so that there is a possibility of a reduced sample of m pairs.

Then let w be the number of pairs for which yi &minus; xi > 0. Assuming that H0 is true, then W follows a binomial distribution W ~ b(m, 0.5).

The left-tail value is computed by Pr(W &le; w), which is the p-value for the alternative H1: p < 0.50. This alternative means that the X measurements tend to be higher.

The right-tail value is computed by Pr(B &ge; w), which is the p-value for the alternative H1: p > 0.50. This alternative means that the Y measurements tend to be higher.

For a two-sided alternative H1 the p-value is twice the smallest tail-value.