13.3 : Introduction to the Sign Test

The sign test is an important tool in nonparametric statistics, offering a straightforward yet effective method for analyzing matched pairs, nominal data, or hypotheses concerning the median of a population. It transforms data points into positive or negative signs, avoiding the need for assumptions about data distribution and instead focusing on the direction of change. It is particularly valuable when data does not conform to the normal distribution requirements of many parametric tests. For instance, researchers might employ the sign test to evaluate pre- and post-treatment effects in a medical study, determining whether the treatment correlates with an improvement (positive sign) or deterioration (negative sign) in patient outcomes. The null hypothesis proposes no difference in medians between two populations, while a predominance of one sign over the other may suggest a statistically significant effect.

As the name suggests, the sign test compares data in terms of the signs of their differences. For each pair of observations, we compare their values and:

1. If the value in sample A is greater than in sample B, assign a "+" sign.
2. If the value in sample A is less than in sample B, assign a "−" sign.
3. If the values are equal, discard the pair (no sign is assigned).

With this, we can count the number of positive and negative signs and follow on with the test. In analyzing small datasets with up to 25 observations, the test statistic (x) represents the count of the less frequent sign. A z-score is computed for larger datasets, facilitating a comparison against critical values from standard statistical tables. If the test statistic is less than or equal to these critical values, the null hypothesis is rejected, indicating a significant difference. If the opposite happens, the null hypothesis cannot be dismissed, reflecting insufficient evidence of a significant effect.