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I am analysing a data set with a cross-classified structure, using a GLMM with a logit link. The unit of observation is clustered within two crossed hierarchies: one has three levels, the other has two levels. See the image below.

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Hierarchy 1 Level 3 (H1L3) only has four levels. H2L2 also only has four levels. Therefore, I have been recommended to include H1L3 and H2L2 as fixed effects.

I was wondering whether fitting H1L3 and H2L2 as fixed effects is appropriate, given that H1L1, H1L2 and H2L1 would be fitted as random effects. I have been unable to find examples where the top level of the hierarchy is fitted as a fixed effect, while lower levels are fitted as random effects.

Side note: when H1L3 and H2L2 are included as random effects instead of fixed effects, the variance of H1L3 is estimated as 0 and glmer reports a singular fit.

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