Who's Fixed?

When comparing a multilevel model to a fixed-level model, it’s important to consider how things are parameterized. For instance, let’s say you’re conducting comparisons between a no-pooling model and a partial pooling variance components model. In this case, we have:

$$ y= \Delta u + \epsilon$$

as the specification, where $\Delta$ is the dummy variable matrix, $y$ is the outcome of interest, and $\epsilon$ is the usual homoeskedastic error term for the responses. Note that the no pooling model contains no intercept and instead contains $J$ means, one for each of the $j$ groups.

But, let’s consider what happens when we look at the multilevel estimate for an analogous model:

$$ y = \Delta \theta + \epsilon$$

$$\theta = \mu + \zeta$$

where now, $\zeta$ is a zero-mean error term for our region-level model of $\theta$. Our $\theta$ term expresses the unique effects of each observation with respect to a higher-level mean, $\mu$, with its own estimating uncertainty.

So, what is the correct comparison between terms in these models? Well Gelman has his whole pooled/not pooled dichotomy, and I think that makes sense. Most of the multilevel literature compares $\zeta$ to $u$, since $\zeta$ reflects the fact that we have “separated” the uncertainty in estimating the global mean out, placed it in $\mu$ and now are only concerned with estimating the noise in the distinct contribution added by being in a group. Thus, again, for centered $y$, $u$ sould look like $\zeta$, but with slightly wider intervals and slightly more extreme point estimates, and $\mu$ should be around zero. This argument is quite persuasive, and grounds the interpretation of these models in the literature.

But, the “direct” comparison involves comparing $\theta$ to $\gamma$, which both serve the same direct function in the mean predictor of the model. Practically speaking, if we think this is the correct apples-to-apples comparison, then both the uncertainty in $\alpha_0$ and $\zeta$ are included in comparisons to the fixed effect model’s unique region estimate. An example of how this might be done is:

  1. run a sampler for the parameters
  2. at convergence, resampling $\mu$ and $\zeta$ from their posteriors and obtain one draw of $\theta$.

I showed this in my recent presentation at the Royal Statistical Society conference. The uncertainty in estimating the state-specific effect $\zeta$ obeys the usual behavior. But, if we actually simulate samples of $\theta$ from the posterior, we get estimate bands that look much more similar to the original bands.

I’m sympathetic to the arguments that you shouldn’t lump the uncertainty in $\mu$ into the uncertainty around $\zeta$, but I also think that, from the fixed-effects user perspective, the “right” intuitive comparison is between $\theta$ and $u$. Not sure what this means for applied work; personally, I nearly always want to make the Gelman-style comparison, but I’m wondering if there are cases where it’s more useful to think critically about what uncertainty around $\mu$ means in the multilevel specificaation. For my spatial statistical context, end of the day, our predictions for any specific data point will involve both the uncertainty in $\mu$ and $\zeta$ for any observation, so while we can separate them out in the regression model, we cannot really in the discussion of prediction.