How to normalize a Lmer model?

Question

lmer:

mixed.lmer6 <- lmer(Size ~ (Time+I(Time^2))*Country*STemperature +
   (1|Country:Locality)+ (1|Locality:Individual)+(1|Batch)+
   (1|Egg_masses), REML = FALSE, data = data_NoNA) 

residuals:

plot_model(mixed.lmer6, type = "diag")

Tried manual log,power, sqrt transformations in my formula but no improvement and I also can not find a suitable automatic transformation R function such as BoxCox (which does not work for LMER's)

Any help or tips would be appreciated

Answer 1

This might be better suited for CrossValidated ("what should I do?" is appropriate for CV; "how should I do it?" is best for Stack Overflow), but I'll take a crack.

• The QQ plot is generally the last /least important diagnostic you should look at (the order should be approximately (1) check for significant bias/missed patterns in the mean [fitted vs. residual, residual vs. covariates]; (2) check for outliers/influential points [leverage, Cook's distance]; (3) check for heteroscedasticity [scale-location plot]; (4) check distributional assumptions [QQ plot]). The reason is that any of the "upstream" failures (eg missed patterns) will show up in the QQ plot as well; resolving them will often resolve the apparent non-Normality.
• If you can fix the distributional assumptions by fixing something else about the model (adding covariates/adding interactions/adding polynomial or spline terms/removing outliers), then do that.
• you could code your own brute-force Box-Cox, something like

fitted_model <- lmer(..., data = mydata)
bcfun <- function(lambda, resp = "y") {
   y <- mydata[[resp]]
   mydata$newy <- if (lambda==0) log(y) else (y^lambda -1)/lambda
   ## https://stats.stackexchange.com/questions/261380/how-do-i-get-the-box-cox-log-likelihood-using-the-jacobian
   log_jac <- sum((lambda-1)*log(y))
   newfit <- update(fitted_model, newy ~ ., data = mydata)
   return(-2*(c(logLik(newfit))+ log_jac))
}
lambdavec <- seq(-2, 2, by = 0.2)
boxcox <- vapply(lambdavec, bcfun, FUN.VALUE = numeric(1))
plot(lambdavec, boxcox - min(boxcox))

(lightly tested! but feel free to let me know if it doesn't work)

• if you do need to fit a mixed model with a heavy-tailed residual distribution (eg Student t), the options are fairly limited. The brms package can fit such models (but takes you down the Bayesian/MCMC rabbit hole), and the heavy package (currently archived on CRAN) will work, but doesn't appear to handle crossed random effects.