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My key question is, why is the ranked version superior in power by such a huge amount? I believed that I'll get approximately the same sample size, because it is known that the rank transformation provides a bridge to the non-parametric hypothesis tests, see e.g. Ref1, or Ref2.

My key question is, why is the ranked version superior in power by such a huge amount? I believed that I'll get approximately the same sample size, because it is known that the rank transformation provides a bridge to the non-parametric hypothesis tests, see e.g. Ref1, or Ref2.

If I use the numeric value, the sample size $n=8$ satisfies the $\beta$-risk condition. In contrast, if I transform the random numbers to ranks, I only need $n=4$. Thus, if I am willing to use a randomly generated dataset to analyse (rank) the experimental data, the power of the test increases significantly. This logic also applies if I sample from a location scaled version of the $t$-distribution. What am I missing? Why is the ranked version superior in power?

If I use the numeric value, the sample size $n=8$ satisfies the $\beta$-risk condition. In contrast, if I transform the random numbers to ranks, I only need $n=4$. Thus, if I am willing to use a randomly generated dataset to analyse (rank) the experimental data, the power of the test increases significantly. This logic also applies if I sample from a location scaled version of the $t$-distribution. What am I missing?

My key question is, why is the ranked version superior in power by such a huge amount? I believed that I'll get approximately the same sample size, because it is known that the rank transformation provides a bridge to the non-parametric hypothesis tests, see e.g. Ref1, or Ref2.

The numerical case can be checked by using the following code as:

The numerical case can be checked by using the following code: