Sign test

In statistics, the sign test can be used to test the hypothesis that the difference median is zero between the continuous distributions of two random variables X and Y, in the situation when we can draw paired samples from X and Y. It is a non-parametric test which makes very few assumptions about the nature of the distributions under test - this means that it has very general applicability but may lack the statistical power of other tests such as the paired-samples t-test^[1] or the Wilcoxon signed-rank test.^[2]

Method

Let p = Pr(X > Y), and then test the null hypothesis H₀: p = 0.50. In other words, the null hypothesis states that given a random pair of measurements (x_i, y_i), then x_i and y_i are equally likely to be larger than the other.

To test the null hypothesis, independent pairs of sample data are collected from the populations {(x₁, y₁), (x₂, y₂), . . ., (x_n, y_n)}. Pairs are omitted for which there is no difference so that there is a possibility of a reduced sample of m pairs.^[3]

Then let W be the number of pairs for which y_i − x_i > 0. Assuming that H₀ is true, then W follows a binomial distribution W ~ b(m, 0.5).

Assumptions

Let Z_i = Y_i – X_i for i = 1, ... , n.

1. The differences Z_i are assumed to be independent.
2. Each Z_i comes from the same continuous population.
3. The values X_i and Y_i represent are ordered (at least the ordinal scale), so the comparisons "greater than", "less than", and "equal to" are meaningful.

Significance testing

Since the test statistic is expected to follow a binomial distribution, the standard binomial test is used to calculate significance. The normal approximation to the binomial distribution can be used for large sample sizes, m>25.^[3]

The left-tail value is computed by Pr(W ≤ 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(W ≥ 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 smaller tail-value.

See also

• Wilcoxon signed-rank test - A more powerful variant of the sign test, but one which also assumes a symmetric distribution and interval data.
• Median test - An unpaired alternative to the sign test.

References

• Gibbons, J.D. and Chakraborti, S. (1992). Nonparametric Statistical Inference. Marcel Dekker Inc., New York.
• Kitchens, L.J.(2003). Basic Statistics and Data Analysis. Duxbury.
• Conover, W. J. (1980). Practical Nonparametric Statistics, 2nd ed. Wiley, New York.
• Lehmann, E. L. (1975). Nonparametrics: Statistical Methods Based on Ranks. Holden and Day, San Francisco.