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I have to make a statistical analysis of only four (n=4) data pairs. I have tested two specific warm-ups for 4000m Team Pursuit in track cycling, and used a 250 m. sprint test as a meassurement of the warm-ups effect. I only had four track riders available as test subject, så they underwent both types of warm-up followed by the sprint test, on seperate days (repeated meassures). That gives me the four data pairs that i would like to analyse. I first thought of a Wilcoxon signed-rank test, but found out that it required at least six data pairs. I haven't been able to find an alternative test. "Please help Obi-Wan Kanobi (read: stat experts at Cross Validated), you are my only hope!"

ps. all four data pairs show the same warm-up as the most effective.

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  • $\begingroup$ An exact, permutation-based significance can be done for every nonparametric test of a small sample size. $\endgroup$ – ttnphns May 20 '16 at 9:40
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    $\begingroup$ @ttnphns The problem is that with 4 pairs the smallest achievable two-tailed significance level is 0.125 ($2(\frac12)^4$). This applies equally to the permutation test as it does to the signed rank test. Similarly, when there are 6 pairs, the smallest achievable significance level for both tests is $2(\frac12)^6=\frac{1}{32}=0.03125$, which is the first one below 5% (this is the source of the "requires six pairs" in the question -- which is actually only true if you insist on a two-tailed test with a significance level of 5%). $\endgroup$ – Glen_b -Reinstate Monica May 20 '16 at 10:11
  • $\begingroup$ @Glen_b, thank you for reminding, and I'm aware of the problem. But could there be a solution for the OP, free of it? $\endgroup$ – ttnphns May 20 '16 at 10:22
  • $\begingroup$ Realistically a non-parametric test with a significance level of 0.05 is silly with n=4 and it would be hard to see how this could be a particularly compelling dataset - perhaps a little pilot trial that should not be overinterpreted? If there is previous data, then identifying a suitable parametric may be possible and may give some power. However, unless you are looking for enormous effect sizes, this is likely not a very conclusive experiment and ideally you should do a better experiment with a sample size suitable to detect realistically plausible effect sizes. $\endgroup$ – Björn May 20 '16 at 10:46
  • $\begingroup$ Hey folks. Thank you for taking your time to answer. As you might can tell I'm not a statistics expert of any kind, but your answers have been useful to me anyway. And Bjørn, you are right, this is a small trial, but smaller then planned to be. I had 8 subjects available to begin with, but injuries and sickness left me with only four and a statistical challenge I had no idea how to handle. Even though I can't say my results are statistical significant, they still show a tendensy which gives me something to elaborate on in my discussion chapter. $\endgroup$ – Thomas May 20 '16 at 11:29
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If you want to be able to use a significance level of 5% and you want a two tailed test, you're pretty much left with parametric tests.

However, this doesn't mean you're stuck with normal-theory tests.

For example you could treat times as (say) gamma-distributed and use a link-transformed sprint-time-for-first-warmup-type as an offset and an intercept-only model (a GLM). e.g. with a log-link you could use log-sprint-time-for-first-warmup-type as offset. Then a test of the intercept would be testing for change in sprint times. (Alternatively, generalized linear mixed-effect models might be used.)

There are numerous other possibilities (e.g. assume differences of log-times are normal).

You might even consider a normal theory test after all, and assume time differences are normal.

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  • $\begingroup$ (+1) for the overview of some possibilities. $\endgroup$ – ttnphns May 20 '16 at 10:48

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