Sign test

The sign test is a non-parametric statistical test used to evaluate the difference between paired observations. It is particularly useful when the assumptions of the paired t-test are not met, such as when the data are not normally distributed or when the sample size is small. The sign test is based on the direction of the differences between pairs, rather than their magnitude.

Overview

The sign test is used to test the null hypothesis that the median of the differences between paired observations is zero. It is applicable in situations where the data are ordinal or when the assumptions of parametric tests are violated.

Procedure

The procedure for conducting a sign test is as follows:

1. Identify paired observations: Collect paired data from two related samples or repeated measurements on a single sample.
2. Calculate differences: For each pair, calculate the difference between the two observations.
3. Determine signs: Assign a positive sign (+) if the first observation is greater than the second, a negative sign (-) if the first observation is less than the second, and ignore pairs where the difference is zero.
4. Count signs: Count the number of positive and negative signs.
5. Test statistic: The test statistic is the smaller of the number of positive or negative signs.
6. Determine significance: Use the binomial distribution to determine the probability of observing the test statistic under the null hypothesis.

Example

Suppose a researcher wants to test the effectiveness of a new drug. Ten patients are measured for a specific health metric before and after taking the drug. The differences in measurements are recorded, and the sign test is applied to determine if there is a significant change.

Assumptions

The sign test makes the following assumptions:

• The data consist of paired observations.
• The differences between pairs are independent.
• The measurement scale is at least ordinal.

Advantages and Disadvantages

Advantages

• Does not require the assumption of normality.
• Simple to compute and interpret.
• Applicable to ordinal data.

Disadvantages

• Less powerful than parametric tests like the paired t-test when assumptions of those tests are met.
• Ignores the magnitude of differences, only considering their direction.

Also see

• Wilcoxon signed-rank test
• Paired t-test
• Non-parametric statistics
• Binomial test

References

• Conover, W. J. (1999). Practical Nonparametric Statistics. John Wiley & Sons.
• Hollander, M., Wolfe, D. A., & Chicken, E. (2013). Nonparametric Statistical Methods. John Wiley & Sons.