I have more of a stats question regarding mixed models.
salary ~ years_experience + (years_experience|department)
Salary is the salary of university faculty based on their years of experience. In this model, we account for the fact that different departments (our random effect) may have different starting salaries (random intercept) and different increases in salary/year (random slopes). So each department will have a different intercept and a different slope.
What I do not understand is what happens when you have multiple continuous fixed effects? For example, if you had two continuous x fixed effects, your predictors would take the shape of a plane (see image).
Where would the random slopes and intercepts end up on this plane? Would your random slopes and intercepts have a third dimension? Would it then be possible to visualize this in 2D like in the single fixed effect example?