Sign test

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The sign test can be used to test the hypothesis that there is "no difference" between two continuous distributions X and Y. Formally:

Let p = P(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 both 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 - xi > 0. Assuming that Ho is true, then W follows a binomial distribution B ~ b(m, 0.5).

The left-tail value is computed by P(B <= 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 P(B >= 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.

[edit] References

Abdi, H. (2007).[1] Binomial Distribution: Binomial and Sign Tests. In N.J. Salkind (Ed.): Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage.