Normal distribution necessary for linear-mixed effects? (R) This is my first post on this site. I'm a linguistics graduate student who is struggling to grasp the basics of statistics.
I've run a questionnaire in which participants had to rate sentences from 1 (totally unacceptable) to 7 (fully acceptable). I had two different factors with two levels each (a 2x2 design).
Following previous papers whose authors used the same design, I have log-transformed the ratings and then I have calculated z-scores by subject:
dat$rating.log <- log(dat$rating)
dat$z.score.rating2 <- ave(dat$rating.log, dat$subject, FUN=scale)

After that, I've considered ratings above and below 2.5 standard deviations from the mean as outliers and I've removed them (also following previous studies).
I report here the histogram for the cleaned data:

And these are the histograms per condition:

As you can see, the data is far from normal. My question is the following: does this matter if I want to conduct a linear-mixed effects model? If it does, how can I normalize the data?
Thank you very much!
 A: As per the comment by @Roland, there is no requirement for the response variable itself to be normally distributed in a linear mixed model (LMM). It is the distribution of the response, conditional on the random effects, that is assumed to be normally distributed.  This means that the residuals should be normally distributed. Therefore, you can proceed with fitting an LMM and then check the residuals to see if they are normally distributed. Treating likert item responses as continuous data is a contentious topic - for example see here:
Parametric tests and Likert Scales (Ordinal data) - Two different views
This simulation study plays down the concerns. Clearly, with fewer levels in the likert scale there is going to be more of a problem. This presentation from one of the authors of the lme4 package for R seems to suggest that 10 or more levels is OK. 
So with a 7 point scale, there is a good chance that the residuals will not be normally distributed, in which case you can look at fitting a generalised linear mixed model for ordinal data - two such packages  which fit these models in R are ordinal and MCMCglmm
A: If you use something like a generalized linear mixed model, then the response variables don't have to be gaussians.  This fact is the key differentiator from GLMM and LMM.
