Which test is suitable for related data when differences between paired observations cannot be assumed to be interval-level?

Which test is suitable for related data when differences between paired observations cannot be assumed to be interval-level?

• A. The T-test for dependent samples
• B. The (binomial) sign test for related data
• C. The Chi-square test
• D. The F-test

Answer: (B)

When you have related (paired) data and you can’t assume the differences are measured on an interval scale, you need a method that doesn’t rely on the size of the change. The sign test does just that: it only looks at the direction of change in each pair—whether the second observation is higher or lower than the first—ignoring how large the change is. Under no systematic effect, roughly half of the pairs will show a positive difference and half a negative one, so the number of positive differences follows a binomial distribution with p = 0.5. You then compare the observed count of positives to this expectation to obtain a p-value. This makes the sign test appropriate for ordinal data or any situation where the magnitude of differences isn’t reliably interval-level. By contrast, a paired t-test requires the differences to be interval (or ratio) and roughly normally distributed; the chi-square test is for categorical data in contingency tables, not paired differences; and the F-test is tied to variance comparisons in ANOVA or regression with its own assumptions.

When you have related (paired) data and you can’t assume the differences are measured on an interval scale, you need a method that doesn’t rely on the size of the change. The sign test does just that: it only looks at the direction of change in each pair—whether the second observation is higher or lower than the first—ignoring how large the change is. Under no systematic effect, roughly half of the pairs will show a positive difference and half a negative one, so the number of positive differences follows a binomial distribution with p = 0.5. You then compare the observed count of positives to this expectation to obtain a p-value. This makes the sign test appropriate for ordinal data or any situation where the magnitude of differences isn’t reliably interval-level.

By contrast, a paired t-test requires the differences to be interval (or ratio) and roughly normally distributed; the chi-square test is for categorical data in contingency tables, not paired differences; and the F-test is tied to variance comparisons in ANOVA or regression with its own assumptions.