I've got a data set consisting of a series of "broken stick" monthly case counts from a handful of sites. I'm trying to get a single summary estimate from two different techniques:
Technique 1: Fit a "broken stick" model with a Poisson GLM (in SAS) with a 0/1 indicator variable, and using one of three methods for controlling for trends in time: a simple "month" variable, "month^2" or a Loess smoother on month, depending on whether or not the more sophisticated methods change the estimate much.
The indicator variable's estimate and SE are then fed into either tlnise or just a simple method of moments approach to get a pooled estimate and variance. This is similar to what Peng and Dominici do with air pollution data, but with fewer sites (~a dozen).
Technique 2: Abandon some of the site-specific control for trends in time and use a linear mixed model. Particularly:
lmer(cases ~ indicator + (1+month+I(month^2)+I(month^3)|SiteID) + offset(log(p)), family="poisson", data=data)
My question involves the standard errors that come out of these estimates. Technique 1's standard error, which is actually using a weekly rather than monthly time set and thus should have more precision, has a standard error on the estimate of ~0.206 for the Method of Moments approach and ~0.306 for the tlnise.
The lmer method gives a standard error of ~0.09. The effect estimates are reasonably close, so it doesn't seem to be that they're just zeroing in on different summary estimates as much as the mixed model is vastly more efficient.
Is that something that's reasonable to expect? If so, why are mixed models so much more efficient?