Suppose that we have samples $x$ and $y$, where $x_i$ and $y_i$ are the $i$-th subject's responses to the first and the second treatment respectively.

Then, we perform a two-sided statistical test, trying to reject the null hypothesis that both treatments have the same effect. For example, the Wilcoxon signed ranked test can be used.

Then, we find the neutral value $s_0$ of statistic $s$ that used in the test (the value that one would get if $x_i = y_i$ for all $i$). Let's assume that $s > s_0$ means that the first treatment is better than the second.

Suppose that the obtained value $s_c = s(x, y)$ is greater than $s_0$ and the null hypothesis was rejected.

I wonder whether I can conclude that the first treatment is better than the second, since the negation of the null hypothesis in our case is that the treatments have different effect.

  • 1
    $\begingroup$ See the 2nd paragraph of my answer here. Less formally: if you can reject the null, you might feel entitled to cast about for better hypotheses among those comprising the (composite) alternative. $\endgroup$ May 4, 2017 at 11:24
  • $\begingroup$ @Scortchi A. We can perfom a two-sided test and hope that $p< \alpha$, or we can perform two one-sided tests and hope that $p< 2\alpha$. It the latter case, we have to use some kind of multiple-hypothesis testing correction, e.g. Bonferroni correction, so we hope that $2p< 2\alpha$, but this condition is already satisfied. Thank you. $\endgroup$
    – Antoine
    May 4, 2017 at 11:59
  • $\begingroup$ Yes. There's a little more to say about discrete test statistics, but that's the idea. $\endgroup$ May 4, 2017 at 12:18


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