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I have some data where I've measured the response times of the same participants in two sessions and within 36 conditions (each participant took part in all 36 conditions). So the data are repeated measures across both IVs, however I've been informed that I can run it as though it was an independent-measures ANOVA because all the data comes from one participant.

Is that true? If so, why does having only one participant make it OK?

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    $\begingroup$ Are the 36 conditions matched across sessions? And did you do all this for ony one subject, or do you have more? (You did say "participants", plural.) If there are more, how many are there, and are the 36 conditions the same for every subject? $\endgroup$ Aug 28, 2013 at 6:44
  • $\begingroup$ Sorry, I should have made that clear. Yes, they are matched across conditions. At present I only have one participant, and I may not have any more. If I do get more the conditions will be the same. $\endgroup$ Aug 28, 2013 at 6:45
  • $\begingroup$ Further clarification - I most likely will have more like 10 participants in the end. At present I'm just doing a preliminary analysis. However, I am interested in what would be correct in hypothetical event that I only had one participant. $\endgroup$ Aug 28, 2013 at 6:52
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    $\begingroup$ If you did have only one subject, what would you want to test? $\endgroup$ Aug 28, 2013 at 6:55
  • $\begingroup$ I'd want to test whether the participant changed from session 1 to session 2, whether the participant's RTs were different across conditions, and I'd also want to investigate the possibility of interaction. $\endgroup$ Aug 28, 2013 at 7:39

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This is a rather a long answer but I hope it's instructive for thinking about similar problems in the future. Independence sounds like an absolute term but it's not. Whether samples are independent or not is relative. If what you're trying to get at is what I think it is, then yes, you can use independent measurement tests (I'm guessing by "conditions" you meant what is commonly called "trials). It's how case studies are done in performance work.

The trials are independent samples of that subject's performance. Just as the samples across people are independent samples of human's performance. While the former are correlated by virtue of being from one subject, the latter are all correlated by virtue of being measures of the same species, or same set of undergraduate students, etc. So, there's rarely (never?) any such thing as complete independence of samples. They need to be independent across the domain of generalization. If your domain of generalization is only the performance of your subject then the individual trials are independent measures.

To make it more concrete, consider your most basic statistics example; is this coin fair? Each flip of the coin is independent of the next flip as a sample of the coin. You can use them, and nothing else, to test if the coin is fair. However, you cannot use those samples alone to test if coins from a batch are fair, or coins from a mint are fair, or coins flipped by people in general are fair (assuming you made all of the initial flips yourself). They are not independent across those parameters.

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