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My sample size is in the test group=11 and control group=19. I'd like to compare if there is a difference in hair cortisol concentrations (continuous variable). My main question is whether to use t-test with log transformed values or u-test with initial values.

The data is highly positively skewed and log transformation makes the distributions normal in both groups. All t-test assumptions are ok with log-values, but I have read that the log transformation is complex in many ways. In almost every article in this research area it is said that "the cortisol was not normally distributed and was logarithmically transformed". It's quite difficult to find papers where the sample size has been as small as mine, and I'd like to know if that limits me using log transformation?

There is no statistical significant difference between the groups, whether I use U-test with initial data or t-test for with log transformed data. I was wondering which one is the right way to test my hypothesis?

I also need to explore if hair cortisol and several continuous variables like height, weight, hair color and dummy variables like sex are related. Again, if I use log transformed values the residuals look OK and I could use Pearson's correlation. But again, I'm worried about using the log transformed values and I was wondering if using non-parametric Sperman's correlation is 'more safe' option.

I would really appreciate your help!

Best regards, HappyZebra

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Your sample size is 11 + 19 = 30. It is too small for making the judgement on the distribution. So your sample cannot provide enough information on log-transformation or not, and you need to find the evidence from others' researches.

If there is strong evidence that hair cortisol concentrations follow log-normal distribution (log(hair cortisol concentrations) follow normal distribution), of course the best method is t-test on log(hair cortisol concentrations).

log transformation does not bring any complex in anyway.

Given your small sample size, maybe it will be fruitless to explore more complicated relationship.

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