Our variables and residuals are not normally distributed. What we found is that regressions are usually quite robust against violations of normality. But we don't know to which degree, because our sample size is not that big either (71). We found some options how to deal with this: (1) use a non-parametric alternative (but we were not able to find one for regressions), (2) transform the data to be more normally distributed, but what are the implications?, (3) using a more conservative p-value to assess significance (i.e. 0.01 instead of 0.05). How do you deal with non-normality in this case? Is the sample size big enough to just assume robustness against normality or should we go for one of the 3 options?
(Please, provide references if they apply)
Regression analysis does not assume normality per se. Normality is required only to get certain properties that relate mainly to the multivariate normal distribution. Without doing special modifications: The assumptions are that the observations are i.i.d and what-ever is random it has finite variance. Everything else is bonus. Except for the infinite variance, every other violation has some reasonable remedies. But the exact "remedy" depends on the specific violation. E.g, you can use WLS to overcome non-constant variance, non-linear regression to overcome parametric non-linearity, certain non-parametric regressions (e.g., Kruskal-Wallis) to overcome asymmetry etc.