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The distinction between fixed effects and random effects seems intuitively clear to me. A factor is a fixed effect if the set of possible levels for the factors is fixed. A fixed effect factor would e.g. be Treatment with the levels treatment and no treatment. Random effect factors on the other hand do not have their set of possible levels fixed. Rather they are usually sampled from a larger population. The prototypical random effect would probably be the subjects in an experiment. Running the experiment again usually involves sampling again and using different subjects.

I find things less clear when there are continuous predictors involved in a mixed-effect experiment. An easy case would be an experiment where I am interested in sleep deprivation and my predictor is the number of consecutive days subjects only get to sleep a certain amount of time. Let us say our predictor days has a range from $0$ to $14$ days. Clearly, modelling the predictor as a factor does not make much sense. The better option would be to model it as continuous. And here it actually does make sense to say that although continuous we have experimentally manipulated the predictor. But what about experiments where I did not experimentally manipulate the predictor but rather only observed it and the predictor is continuous; maybe not even all possible values have been realized. What would be the argument for treating it as a fixed-effect factor and not as a random factor?

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