How valid is it to run one-way ANOVA on other means? For background, I have a set of data where I asked two sets of  50 people to repeat a task 12 times, which generates a single number. Each 1x12 vector represents a single vector of data. I then group the data together in a 50x12 matrix.
After this, I compute the arithmetic mean typically and get a 50x1 vector. I'll repeat for the second set and now I have two 50x1 vectors.
I'll group them together as a 50x1 and run one way ANOVA on it to measure the differences between the two sets of people.
My question is: Is it valid for me to use the geometric mean or the harmonic mean on the data and run ANOVA on that? What about if I were to run one-way ANOVA on the logarithm of the geometric or harmonic means, do I still have validity? I went searching for anwwers on this but can't find anything; let me know what you guys think. Statistics nooblet here ...
 A: If the statistic corresponds to a population quantity of interest, for which a comparison is meaningful, and under the null the arrangement of group labels would be arbitrary (i.e. there's nothing to distinguish the groups), then you can use permutation tests. These allow you to choose a very wide variety of statistics without necessarily needing specific distributional assumptions.
So for example, you could compare several groups on their geometric mean if you could come up with a suitable statistic that measured deviation of any groups from a common geometric mean ... but with positive data it might make more sense to take logs of the data (making the geometric mean an arithmetic mean of the logs) and apply a straight ANOVA on that scale ... if the distribution of the original values might not be pretty close to lognormal, you can still perform a permutation test. So for example, if the original data might be say, from some gamma distribution, the logs will be left-skew, perhaps enough to make a normal-theory ANOVA not perform as you would like. A permutation test on the logs would still work just fine.
Rejection of equal means of logs would imply rejection of equal geometric means on the original data.
(Similarly, one could consider comparing harmonic means by either coming up with a suitable statistic for deviation from equal harmonic means, or by taking inverses and performing an ANOVA on the inverses - unequal mean-inverses implies unequal harmonic means.)
