# Specifying multilevel model structure when random effects exhaust the population

I have been working with a dataset featuring observations at the county level for about 1300 of the ~3100 or so counties of the United States. These 1300 counties are drawn from every state in the union plus DC.

In my initial models, I estimated relatively simple linear regressions:

lm(y ~ x)

I eventually worked my way up to multilevel models where I allow the intercept and slope to vary across the states. In lmer-style syntax, this looks like this:

lmer(y ~ x + (x|state))

Using a cross-validation model comparison procedure, the latter model is preferred quite strongly. However, I've begun to rethink the wisdom of even estimating this model in the first place. I can think of a number of competing reasons for and against specifying this model:

In it's favor, the observations within states are correlated. The ICC for a random-intercept only version of the model described above is .23. So, the observations are not independent, given the basic linear model.

Also in it's favor, the cross-validation procedure prefers it! In some ways, I think this is the best evidence in favor of using this model.

The main point against it is that by specifying varying intercepts/slopes across states, we are making the assumption that these states are drawn from some population of states. The fixed effects are the expected association across this population (i.e. population effect). The problem is that there is no such population of states. There are 51 (including DC), and we have sampled all of them.

I would appreciate anyone pointing out either where my thinking is incorrect, or indicating some way of resolving these apparent inconsistencies.

## 1 Answer

The idea that the clusters are drawn from a population is one story we tell ourselves.

Another story is that the random effects approach simply allows us to estimate coefficients for each cluster in a parsimonous way (multivariate normal assumption on random slopes that also does some regularization) - I like this one.

Another story is that it corrects for lack of independence between observations.

Another story that sticks within the sampling error framework is that your 50 states are just one sample from the time continuum of those 50 states.

These are all stories we tell ourselves. None of them is wrong. And even if you do not believe the idea of a population drawn from the time continuum (a story that addresses OP's case), you can still see the benefits to taking a multilevel approach if you settle for one of the other stories mentioned above.

• Thanks. I like this view insofar as it's totally sensible and leaves the decision up to the analyst. Since the multilevel model does the best at predicting out-of-sample, and has a bunch of other nice features (e.g. cluster-level coefficients), I'm inclined to use that. I'm going to leave this question open for a while to see if anyone else chimes in, but will likely accept your answer. Commented Nov 2, 2018 at 13:40
• @triddle I've struggled with the question too in the past but I take this simple approach nowadays. Commented Nov 2, 2018 at 15:40