# Three-tier Multi-Level Model: violation of the assumptions of normality and heteroscedasticity

I am working with educational data. To do so, I am using the classic three-level hierarchical linear model (student, class and school). I am using the R software lmer package and the stata software. When I perform the residual analysis, the assumptions of homoscedasticity and normality are not met. Here is the adjusted model: m33 <- lmer(pt_ex ~ gen_alun + rep_alun + comp_alun + educ_ee + gen_prof + idad_prof + nro_alun_turm + sase_esc + reg_esc + area_esc + (1|id_esc) + (1|id_turm), REML = T, date = data)

1. The dependent variable pt_ex (Exam Score) despite being continuous, has only discrete values ​​(0 to 100).
2. Regarding the independent variables, with the exception of the variable nro_alun_turm (Number of students in the class), these are nominal/binary categorical.

I thought of using a GLM, namely a Poisson or Negative Binomial multilevel model, but these have infinite support. So, could you try a Gamma, since the two assumptions of the Gaussian model were not met?

[1]: https://i.sstatic.net/nffjA.png


• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
– Community Bot
Commented Aug 4, 2022 at 12:56
• close enough to a normal
– user318514
Commented Aug 4, 2022 at 13:20
• I disagree with @Germania. The distribution is not close to Normal as it's not symmetric. For some perspectives on the effect of skewness see here. Commented Aug 7, 2022 at 22:10

With values ranging from zero to 100, such a response variable can often be considered continuous. You mentioned a gamma model, but the support for the gamma distribution is $$(0, \infty)$$. A good starting point would be a beta model - since the beta distribution has support on $$[0, 1]$$.
(1|id_esc) + (1|id_turm)

The diagnostic plots that you show are not really appropriate for a GLMM. Instead you should use the excellent DHARMa package, and the following thread discusses how to assess a beta GLMM: