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What I Am Trying to Do?

Let's assume, I am working on a research problem where only one participant is present. I have a data set consisting that participant's respiration rate (The number of breaths a person takes per minute [1]) and average speed of running on each minute. There are 300 such instances and those data are that person's 5 hours data of a single day [5 hours is without any break]. I am trying to explore whether there remains statistically different respiration rate when the speed of running was high and when the speed of running was low. I found that there are 150 instances of respiration rate when the speed was high (Group 1) and there are 102 instances when the speed was low (Group 2) [Skipped describing the method, I used to group them].

There can be many methods to check the respiration rate difference of group1 and group 2. But I am interested to explore this using Permutation Test.

Then, What Is the Problem?

Here in Wikipedia, I found that

An important assumption behind a permutation test is that the observations are exchangeable under the null hypothesis

In answer of a question in SE, the author remarks that

... Under the null hypothesis of no difference between groups, the data is exchangeable ...

I am really confused whether these two groups of my data are exchangeable or not.

My null hypothesis is "There is no difference between respiration rate of group 1 and group 2". According to the second block quote, I suspect that Under the null hypothesis of no difference between groups, every data [i.e. not only in case of these data, but also in all other cases] are exchangeable. Then, permutation test will be applicable everywhere!

Therefore, My Questions

  1. Are these data of group 1 and group 2 exchangeable? If no, then, please explain why? If yes, is it okay to use permutation test in this scenario?
  2. If you disagree with "Under the null hypothesis of no difference between groups (let's say, two groups), all data are exchangeable", can you please provide an example when there are two groups of data, but those are not exchangeable? I will request you to provide more real life example, instead of using traditional coin, red/blue ball of urn.

Note: I know that having only one participant will create problem in generalization. But still, I am interested to explore this one. On the other hand, I like to mention that I have checked many questions (1, 2, 3, 4) of SE regarding ex-changeability, but, I did not find my answer.

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In principle, I think it is fine to use a permutation test with your data and hypothesis. It would just involve permuting the "group" labels between your respiration rates, and calculating a new average respiration rate for each "group" on each permutation. You then see whether your observed average rate falls outside the range of expected rates, or not.

The only note of caution I would have with your particular test, is whether your data points are truly independent. For example, it may be that the respiration rate is actually more affected by the running speed in the previous minute, rather than the concurrent minute. This is for you - the designer and analyst of your study - to decide if this matters for your hypothesis test or not.

There aren't really hard rules regarding exchangeability as it's dependent on your specific case. You just need to think carefully whether it makes sense to randomise your labels or not - for example, by permuting do you create 'impossible' combinations?

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