# Fundamentals of Nonparametric Methods

Here we continue to present some basic tools (permutation test and sign test) and principles (order statistics, ranks, and efficiency) that will be useful moving forward.

## Permutation (Randomization) test

Consider the following pretty artificial scenario in which four out of nine subjects are selected at random to receive a new drug, and the other five get a placebo. After some time, all nine subjects are assessed on some outcome and are ranked from “best” ($rank = 1$) to “worst” ($rank = 9$). We assume no ties - for now.

Here’s our question: if the new drug has no beneficial effect, what is the probability that the subjects who got it were ranked $(1, 2, 3, 4)$, i.e. best responses?

First, we need to think about what it means when we say that the four were selected “at random”. If the drug really has no effect, it means that the labels “new drug” and “placebo” are essentially meaningless, and any subject is equally likely to be ranked low, medium, or high after the treatment. The ranks are essentially assigned at random.

There are in total $\binom{9}{4} = 126$ ways of picking four people out of the nine to receive the new drug. If the drug has no effect, then the set of ranks belonging to the chosen four is equally likely to be any of the 126 possible sets of ranks. Here are some of the possibilities:

ABCD
1234$\rightarrow$ most favorable to the new drug
1235
1236
1237

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