What's an example of a situation in which it makes sense to assume random slopes but a fixed intercept?

Question

I'm referring to multilevel modelling. Field (2013) writes:

It’s worth noting that it would be unusual in reality to assume random slopes without also assuming random intercepts, because variability in the nature of the relationship (slopes) would normally create variability in the overall level of the outcome variable (intercepts). Therefore, if you assume that slopes are random you would normally also assume that intercepts are random.

Can anyone think of an example (preferably, but not necessarily, from an actual analysis) in which it was appropriate to assume random slopes but a fixed intercept? Why was it appropriate in that case?

Field, Andy. Discovering statistics using IBM SPSS statistics. Sage, 2013.

Answer 1

If you have random slopes in your model, then your random intercept is a function of how your variables that enter the model with the random slopes are centered. See page 54 of the GLVM book ("search inside" for "cluster-specific regression lines")... which is arguably a better source of information regarding multilevel models than the undergraduate book you refer to.

You may have some exceptional situations when the center or the zero of that variable has a very strong meaning for the model, e.g., when you have a growth that must start from zero (e.g., if a farm has no land, it cannot produce any crop). Otherwise, I would say that the model without the random intercept is weird at the very least, and most likely is not making sense.

Answer 2

The intercept variance and the intercept-slope covariance depend on the x variable origin, so that information should be generated from the data. If you fix the intercepts, then the relationships between the factors at x=0 might make sense for one factor, but not wrt its relationship with any other factors.