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Let's say I have a variable with 6 levels, for example, different tasks which are just a subset of all possible tasks. I am interested in differences between the tasks so should add it as a fixed effect, but I might also want to take into account random differences among the tasks.

Doesn't it make sense to include task as a fixed effect as well as a random effect? I suspect it doesn't, according to some posts I read in CV, but how can I account for random differences among the tasks that are not induced by any experimental manipulation. Is this even possible in mixed models?

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