For simpler random effect models the residuals for the fixed effect and corresponding random effects model would be the same.  The main difference in analysis is that observations(residuals) are assumed independent in a fixed effects analysis and have a correlation structure in the random effects model.  This correlation affects how inference on the fixed effects are performed.

Thank you for your answer.

I am sorry for the lateness of my answer, I wanted to check some articles which confirm this hypothesis.

I don't understaind what is the point in defining a covariable "random" instead of "fixed" effect when you want to analyse residuals of a dependant variable.

When I do this, my residuals are different in fixed and random effects.

To me, a residual is the difference between the predicted value of the subject i and the value of the subject i. When we predict a value with a random variable, the value of each predicted group is the same but shouldn't be the residuals different ?

I understood what you showed me but I'm a bit confused, excuse me.

The definition of a residual is not well defined in random effect models and especially between different packages.   One definition of residual (called structural residuals in some packages) would be based on only the fixed effects portion of the model.  The other set of residuals would be those based on all factors in the model (fixed and random) essentially treating the random effects as fixed.    When working with random effect models I generally treat the mean structure to be defined by the fixed effects in the model and the correlation structure defined by the random terms in the model.  This is the framework used by the "MIXED" procedure in SAS.

Oh so it could depend on the package.

That explains a lot, I'm going to dig why they have a different formula.

Thank you for your answer.