Does the interpretation of fixed effects depend on random-effects specification? (example provided) Imagine a dataset where there are 3 nested grouping variables: study $>$ group $>$ outcome (read outcome is nested in group which in turn is nested in study.)
We have two predictors. ktype is a categorical factor (0=direct,1=indirect) that can vary between studys, between groups, and between outcomes. And, treat is a continuous variable only varying between studys.
Question: If I fit 3 models (see below) that only differ in their random-effects specification, then, how does the interpretation or meaning of any of the 3 fixed-effect coefficients (A, B, C) for each model change?
         Estimate
ktype0     A
ktype1     B
treat      C

# Syntax in R's `lme4` package (DON'T RUN)
1) lmer(y ~ 0 + ktype + treat + (ktype |study))

2) lmer(y ~ 0 + ktype + treat + (ktype |study) + (treat |study))

3) lmer(y ~ 0 + ktype + treat + (ktype |study/group/outcome) + (treat |study))

# STRUCTURE OF GROUPING VARIABLES:
"
study  group outcome
1       1    1
1       1    2
1       1    1
1       1    2
1       1    1
1       1    2
1       2    1
1       2    2
1       2    1
1       2    2
1       2    1
1       2    2
2       1    1
2       1    2
2       2    1
2       2    2"
```


In my experience, fixed effects are usually well-estimated when the number of clusters is large enough so that the assumption of multivariate normal random effects can be justified, even when the random structure is mis-specified.
When the number of clusters is small, then parameter estimates can be biased. The extent of the bias is hard to know in advance, since it depends very much on the structure of the data and th underlying data generation process. Simulation studies are the best way to approach this, using specific data structures and a range of values for various options:

*

*number of levels of each grouping factor

*correlations among covariates

*types of covariates

*values for the fixed effects in the model

*variance of random intercepts for each grouping variable

*variance of random slopes where applicable

*correlations between random intercepts and slopes

The idea would be to simulate data for such a model, and then run a Monte-Carlo simulation to assess whether the fitted values are biased. I have posted code on here which could be easily adapted for this. eg:
Why is this linear mixed model singular?
If I consider the fixed factor as a random slope, the p-value changes from p<0,05 to p>0,05
