A recent analysis of solid cancer incidence in the Life Span Study of atomic bomb survivors (Hiroshima and Nagasaki, Japan) found evidence of a nonlinear, upwardly curving radiation dose response among males but not among females. Further analysis of this new and unexpected finding was necessary. We used two approaches to investigate this finding. In one approach, we excluded individual cancer sites or groups of sites from all solid cancers. In the other approach, we used joint analysis to allow for heterogeneity in background-rate parameters across groups of cancers with dissimilar trends in background rates. Exclusion of a few sites led to the disappearance of curvature among males in the remaining collection of solid cancers; some of these influential sites have unique features in their background age-specific incidence that are not captured by a background-rate model fit to all solid cancers combined. Exclusion of a few sites also led to an appearance of curvature among females. Misspecification of background rates can cause bias in inference about the shape of the dose response, so heterogeneity of background rates might explain at least part of the all solid cancer dose-response difference in curvature between males and females. We conclude that analysis based on all solid cancers as a single outcome is not the optimal method to assess radiation risk for solid cancer in the Life Span Study; joint analysis with suitable choices of cancer groups might be preferable by allowing for background-rate heterogeneity across sites while providing greater power to assess radiation risk than analyses of individual sites.