Not normally distributed raw data are compatible with a multiple linear regression? I have to analyse the predictive value of some neuropsychological data (ex time taken to complete a test, anxiety scores..) with a multiple linear regression model.If the raw data don't follow a normal distribution what I have to do? My prof told me that if there is a really big asimmetry I can transform them in z-scores...it is correct?
Thank you for your answers!
 A: A standard assumption in linear regression is that the residuals will be normal.$^{\dagger}$ We do not, however, care about the distribution of the response variable (e.g. hist(y)), and we definitely do not care about the distribution of the predictor variables.
The assumption of normal residuals is not essential to estimating the parameters, note. In particular, the legendary Gauss-Markov theorem that gives conditions where the estimated parameters have a nice statistical property called unbiasedness does not assume normal residuals. The normal residuals help with doing hypothesis testing and confidence intervals on the parameters, which may be of interest to you.
I think your professor's comment has to do with accounting for when variables are on wildly different scales, such as having one variable range from $0$ to $0.001$ and another range from $0$ to $987,654,321$. Transforming to a z-score could help with some issues related to doing calculations on a computer. However, that has nothing to do with the distribution of data being normal or not.
$^{\dagger}$Technically speaking, the assumption is about the error term, which we do not observe but estimate by the residuals. See Glen’s comments.
A: I would expect time taken and anxiety scores to be normally distributed. Check for outliers using Cooks Distance and z-scores that are greater than some large size (e.g. greater than 4.0). You might choose to do a transformation, in which case I'd suggest Box-Cox transformation.
