Suppose instead of running a t-test they had decided to run the corresponding Wilcoxon instead. What would you expect the p-value to be?

Question

BlackTom AI · Accepted Answer

A Wilcoxon test is a rank-based, nonparametric alternative to the t-test that discards numeric magnitudes in favor of ranks; when the t-test assumptions (particularly approximate normality and homoscedasticity) hold, the t-test uses more information and is typically more powerful.

1. Relevant concepts
- t-test is parametric and uses sample means and variances; under normality it is (asymptotically) the most efficient test for detecting a location shift.
- Wilcoxon (signed-rank or rank-sum) is nonparametric, uses ranks rather than raw values, and is robust to non-normality and outliers but generally loses some efficiency when data are truly normal.
- Power and p-value relationship: a less powerful test tends to produce larger p-values for the same effect and sample size.

2. Step-by-step reasoning
- For paired or two-sample differences with near-normal data, the t-statistic = (mean difference) / (standard error), which fully exploits magnitude information.
- Wilcoxon replaces observations by ranks, forming a sum of signed ranks W, then standardizes it to a z-like statistic. Ranking removes magnitude information (how far observations deviate), increasing variability of the test statistic under alternatives.
- Because the variance of the rank-based statistic under the alternative is effectively larger relative to its signal, the standardized statistic tends to be smaller in absolute value than the t-statistic for the same data, producing a larger p-value.

3. Intermediate/calculation note
- A formal measure: asymptotic relative efficiency (ARE) of Wilcoxon versus t under normality is near 0.95 (i.e., Wilcoxon needs about 5% more data to match t-test power). That quantitative inefficiency implies p-values tend to be higher for Wilcoxon under the normal-case.

4. Why the correct answer is correct
- Given that Wilcoxon uses less information (ranks only), it usually has less power under the conditions where the t-test is appropriate, so you would expect a higher p-value.

5. Why the other options are incorrect
- "The same" is wrong because the tests are different procedures and only coincide rarely.
- "Lower than the t-test" is wrong: lower p-value would imply more power; Wilcoxon is not generally more powerful under normal data.
- "Impossible to say" is too pessimistic: while specific data pathologies can reverse the pattern, the general expectation (given t-test was considered originally) is that Wilcoxon yields a higher p-value.

6. Conclusion
Therefore the expected result is: Higher than the t-test; a Wilcoxon has less power since it uses less information. This matches the provided answer.

Suppose instead of running a t-test they had decided to run the corresponding Wilcoxon instead. What would you expect the p-value to be?Single choice

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