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I have a study design with a between subjects factor (treatment: verum vs placebo) and a within subjects factor (time: before vs after). Subjects were entered into the model as a random effect.

So far this is straightforward to implement, however now a third factor comes into play. The subjects were recruited in two "waves", so the this factor can be named seasonality with two levels (winter vs spring). I am not sure how this factor can be introduced into the mixed model. Because I'm not interested to estimate differences in the levels of seasonality I would think this can be considered as an additional random effect? On the other side seasonality has only two levels and so it doesn't seem to make sense to consider it as a random effect.

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It is a fixed effect. Just because you are not interested in it does not mean it is random.

One way of thinking about it is to ask whether, if you were to do the study again, you would get the same levels of the variable. "Subject" is a random effect (in most cases) because if you did the study again you would get different subjects.

Seasonality seems like a regular covariate - you aren't interested in it, per se, but it might affect other things and it should be in the model (at least, it should be considered for entry into the model).

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