Alternative to Wilcoxon signed-rank test with sample size < 6

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.

• An exact, permutation-based significance can be done for every nonparametric test of a small sample size. – ttnphns May 20 '16 at 9:40
• @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%). – Glen_b -Reinstate Monica May 20 '16 at 10:11
• @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? – ttnphns May 20 '16 at 10:22
• 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. – Björn May 20 '16 at 10:46
• 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. – Thomas May 20 '16 at 11:29