Should I chose a linear, a generalised or a mixed model? I have a dataset of n=3000 nested within 8 countries with approximately 200 or 400 responses in each country and planned to perform multilevel modelling with 4 dependent variables (DV) as fixed effects in SPSS.
The DV variables are responses in a scale of 1-100 and this kind of variables is treated as metric in psychology.
However, all my DV and the error terms are clearly skewed or clearly curtotic. My questions are:

*

*I have read that in some cases the size of the dataset or the number of nesting subgroups allow to use the general linear model. Does it make sense, however, to do so if the dataset clearly shows extreme tendencies? It looks to me like clearly different distributions, but I am not sure how to define them. Should I regard them as continuous distributions?

*Am I right to think that data transformation is not a good option if there is a different form of distribution?

*What would be the advantages and disadvantages of bootstrapping or simulation?

*What would be good reasons for using a generalized linear or a mixed model?

*Would it be appropriate to perform a factor analysis of the four DV and if not are there alternatives?

I would appreciate if someone can answer any of these questions or suggest some not very technical references !

 A: *

*When you have a small number of clusters then it may make sense to fit fixed effects for the cluster IDs. Since you have repeated measures (in countries) this needs to be accounted for and in a mixed effects or multilevel framework we usually fit random intercepts for this. There is no requirement for the dependent variables to have, or not have, any particular distribution. For certain inferential purposes we may like the residuals to be normally distributed.


*Transformation may be a good idea if we are fitting a linear model and the residuals are not plausibly from a normal distribution. A linear model has the form $Y = X\beta + \epsilon$ where $X\beta$ is the linear predictor and $\epsilon$ is typically normally distributed. If the residuals from the model do not plausibly follow a normal distribution then transformation of $Y$ and/or the $X$ variables might help.


*There are some of advantages and disadvantages to bootstrapping and simulation but I don't see why that is relevant here.


*A generalised model may be appropriate when we want to model the mean response using a link function and specify the residual variation according to some distribution (ie a GLM can be written as $g(Y) = X\beta + \epsilon$ where $g$ is the link function, $\epsilon$ follows some distribution, and $X\beta$ is, as with the linear model, the linear predictor). This is often useful when we have a response variable that is a count, a proportion, or cetagorical, for example. A mixed model may be appropriate when you have repeated measures or some other kind of clustering or nested structure, in which case the linear predictor is of the form $X\beta + Zu$ where $X\beta$ is the fixed part and $Zu$ is the random part ($X$ and $Z$ being the respective model matrices, and $\beta$ and $u$ being the fixed and random effects, respectively). So a linear mixed model (LMM) would be of the form $Y =  X\beta + Zu + \epsilon$. These can be combined in a generalised linear mixed model (GLMM), which would then be of the form $g(Y) = X\beta + Zu + \epsilon$


*Factor analysis can be useful especially when the variables in question are measuring the same thing.
In your particular case, from the information given I would suggest starting with a linear mixed model, fitting random intercepts for countries.
