In general, fixed effects variables are included in statistical models to account for factors that are constant or do not vary within each unit of observation. Therefore, if a variable does not vary within each participant (unit of observation), it would not typically be appropriate to include it as a fixed effect in a model.

Fixed effects are used to control for the effects of specific factors that are constant within each unit. They capture the average effect of these factors by estimating separate intercepts or coefficients for each unit. This is useful when there are unobserved or time-invariant factors that may influence the outcome variable.

If a variable does not vary within each participant, it does not contribute any variation to the model, and including it as a fixed effect would not be meaningful or appropriate. Instead, such variables can be considered as constant or control variables and included in the model as covariates. By including them as covariates, their constant effect can be controlled while examining the relationships between the other variables of interest.

In summary, fixed effects variables are included in models to capture factors that vary across the units of observation. If a variable does not vary within each participant, it is more suitable to include it as a covariate rather than a fixed effect in the model.

Ma'Mon Abu Hammad, thanks for your response. I confess I'm a bit puzzled by your answer. As far as I know, covariates are always added within the model as fixed effects, just like factors.

The use of terminology in this context is very frustrating. It is important to distinguish between factors and covariates, but, as far as I'm concerned, it has little to do with the difference between fixed and random effects.

In a nutshell (please correct me if I'm wrong): both factors (i.e., categorical variables) and covariates (ordinal or continuous variables) can be added within a model as fixed effects; whereas categorical variables can ALSO be added as random effects (typically in the case of cluster variables such as participants or stimuli in repeated-measures contexts).

Both factors and covariates are used as fixed effects to predict the dependent variable and share a similar relationship with it. In a linear model, such as regression, GLMM, or analysis of covariance, both the variance from factors and covariates are taken into account.

Indeed, when modelling on R (lme4), you just type something like:

DV ~ 1 + Categorical Variable + Continuous Variable + Interaction + (1 | Participant [random intercept, fixed slope]).

There's no difference between factors and covariates in the syntax.

It is crucial to note that a covariate is not simply an unrelated third variable; rather, it is a dimensional variable that contributes to the model just like a factor.

The statistical packages offer different options for factors and covariates just because they treat them differently. For instance, factors may allow for contrasts between groups, whereas covariates do not (moderation aside, with the classic -1SD, Mean, 1SD sample division).

What I need to know is if it makes sense to model as fixed effect a categorical or continuous variable that does not vary within each level of the cluster variable (i.e., random effect).

Please correct any information you deem wrong on imprecise.

And thank you again for your response.

Alessandro Ansani your understanding given above seems accurate, to my mind, particularly in the way you have distinguished factors and covariates. I too am a bit confused by statements given in the first response.

To your actual original question: Your are right to be concerned about when random slopes can be meaningfully defined for a given clustering variable. Briefly, if a given factor doesn't vary between the levels of a clustering variable, then it should not be used as a random slope within that cluster.

I'm going to block quote here from Barr et al.'s (2013) infamous "Keep it Maximal" paper (which, while I disagree with many of its conclusions, is an excellent intro to many of the theoretical concepts in MLMs):

"At this point, we might wish to go further and consider other models. For instance, we have considered a by-subject random slope; for consistency, why don’t we also consider a model with a by-item random slope, I1i? A little reflection reveals that a by-item random slope does not make sense for this design. Words are nested within word types—no word can be both type A and type B— so it is not sensible to ask whether words vary in their sensitivity to word type. No sample from this experiment could possibly give us the information needed to estimate random slope variance and random slope/intercept covariance parameters for such a model. A model like this is unidentifiable for the data it is applied to: there are (infinitely) many different values we could choose for its parameters which would describe the data equally well.4 Experiments with a within-item manipulation, such as a priming experiment in which target words are held constant across conditions but the prime word is varied, would call for by-item random slopes, but not the current experiment.

The above point also extends to designs where one independent variable is manipulated within- and another variable between- subjects (respectively items). In case of between-subject manipulations, the levels of the subject variable are nested within the levels of the experimental treatment variable (i.e. each subject belongs to one and only one of the experimental treatment groups), meaning that subject and treatment cannot interact—a model with a by-subject random slope term would be unidentifiable. In general, within-unit treatments require both the by-unit intercepts and slopes in the random effects specification, whereas between-unit treatments require only the by-unit random intercepts." (Grabbed from the version posted here: Article Random effects structure for confirmatory hypothesis testing...

My short answer to the initial question is yes, you can. I agree that factors that do not change within each participant may not contribute much to the variance in your dependent variable (if that's what you meant), which is attributable to within-subject differences but they can still contribute to the between-subject variance, so they can be included in the model as FEs. However, if I were the reviewer of a paper doing that, I would like to see first the results of the most conservative model and, as supplementary analyses, the results of the model with the FEs, specially if these variables are not central to your research question, I guess the FEs will not improve the model fit significantly, but let us know.