  This paper considers small-area estimation with lung cancer mortality data,
and discusses the choice of upper-level model for the variation over areas.
Inference about the random effects for the areas may depend strongly on the
choice of this model, but this choice is not a straightforward matter. We give
a general methodology for both evaluating the data evidence for different
models and averaging over plausible models to give robust area effect
distributions. We reanalyze the data of Tsutakawa [Biometrics 41 (1985) 69--79]
on lung cancer mortality rates in Missouri cities, and show the differences in
conclusions about the city rates from this methodology.
