How to properly diagnose a linear mixed model I am trying to study the relationship between the type of bariatric surgery and weight loss. This is longitudinal data with different BMI measurements per patient.
This the formula of the model: lmer.bmi<-lmer(BMI~Treatment+Time+Sex+age+(1+Treatment|ID),data =mydata)
And below:
The conditionnal studentized residuals vs fitted studentized.
The residual quantile plot for the error term
I don't know how to interpret it very well. But it seems to me that it does not respect the conditions of validity.
The scatter plot seems to be denser at the bottom of the horizontal line. Looking at the qqplot, the distribution of the residuals is not on the diagonal. The interpretation of these graphs remains relatively subjective. Can we say that the conditions of validity are sufficiently violated for me to revise my model. If so, what could I do as a transformation?


However, the random effects seem to have a fairly normal distribution. What do you think about this?
NB: Numero= ID, BP_demblee= Treatment


Are there more important conditions to be met than these?
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To answer your first question, there is some debate about whether or not you should transform data for lmer models (see Lo & Andrews, 2015). Others, like Brysbaert & Stevens, show that transformations can reduce error in the model if done correctly (in their case they use inverse Gaussian RT data as an example). The most important bit is that your residual plots look a bit off, which is an indicator that you may need a different function for the data.
If I'm not mistaken, BMI data is proportional (BMI = kg/m2), which may be part of the issue. Look at some density plots/histograms of the BMI data and see where the distribution lies. If the data is shifted somewhere away from the center of the plot or skewed in some way, you may need to fit a GLMM instead (you can simply switch to the glmer function and pick an appropriate distribution to model). From what I can gather from the residual plot, your data looks to possibly right skewed, as most of the data is centrally located on the left. The data is also densely packed below the line as you mentioned, whereas the top of the line is more broadly distributed.
There are other diagnostic checks you can also use to see if your model is predicting accurately (which is really the most important part), many of them you can use from the performance package. One way of achieving that is the posterior predictive check. The check_model function in R allows you to examine this fairly easily. As an example:
#### Load Libraries ####
library(performance)
library(lmerTest)

#### Make Model ####
iris.lmer <- lmer(Sepal.Length 
                  ~ Sepal.Width
                  + (1|Species),
                  data = iris)

#### PP Check ####
check <- check_model(iris.lmer)
plot(check$PP_CHECK) 

You can see here for example that the raw data (green line) is not fitting where the rest of the blue simulated lines are, indicating the linear model is not fitting properly around the real data.

This makes sense because the real data distribution of sepal length is not normally distributed:

This however alone doesn't really check lack of fit, its just simply a tool to take a peak at what your data and model are doing. There are other things you should check as well, such as linearity, collinearity, autocorrelation, outliers, and heteroscedascity.
