I've read that working with paired data requires other method than Mann-Whitney $U$ test. I am not sure about how formally accurate is this statement.
Reasons:
Suppose that there is a set of $n$ (for a very large $n$) systems $S=\{s_1, s_2, … , s_n\}$. In each system there is a property, and the exact value for the property in the system $s_i$ is a well defined and constant value $y_i$.
Suppose that there are two imperfect instrument $A$ and $B$ for the measurement of the property related to the $\{y_i\}$ values. The respective values of the measurements of $y_i$ are $x_{A,i}$ and $x_{B,i}$. And also suppose that the measurements errors of both instruments are independent of the $y_i$ value in the range of $y_i$ values found in $S$.
If I am right, in principle, we can take two randomly chosen subsets (of $m$ and $l$ elements, with $20 < m$, $l \ll n$) of $S$, $S_A$ and $S_B$, and analyze the distributions of the $\{x_{A,i}\}$ found in $S_A$ and the $\{x_{B, i}\}$ found in $S_B$ by comparing them with the Mann-Whitney $U$ test.
Depending of the chosen $S_A$ and $S_B$ we can get different results ($U$ and $p$-value). The key point here is: If we take $S_A = S_B$, we can pair data ($x_{A,i};x_{B,i}$) and use the Wilcoxon signed-rank test. If $m = l$, each randomly chosen $S_i$ has the same occurrence probability, and $S_A = S_B$ should be a valid choice.
If so, the Mann-Whitney $U$ test can be used for the comparison even if the data can be paired.
Questions:
- Is the reasoning above correct? If so, which is the advantage of using Wilcoxon signed-rank test instead of Mann-Whitney $U$ test? Is it related to confidence?
- If it is wrong, where is the mistake?
- What exactly does the $p$-value mean in the case of the Wilcoxon signed-rank test?
Best Answer