Ecological Inference

The ecological fallacy, named after Robinson's 1950 finding, is the error of assuming that a correlation observed across aggregate units holds for individuals — for instance, inferring that immigrants are more literate because districts with more immigrants had higher literacy. Ecological inference avoids the fallacy by not equating aggregate associations with individual behavior; instead it uses the deterministic accounting identity and bounds in each unit, plus a model of how within-unit rates vary, to estimate the individual-level rates directly. It treats the within-group rates as the explicit unknowns rather than reading them off an aggregate slope.

Goodman's regression assumes the within-group rates are constant across all units, which is efficient when true but yields nonsensical estimates outside the zero-to-one interval when rates actually vary with composition, and it ignores the deterministic bounds. King's model lets each unit have its own pair of rates drawn from a truncated bivariate normal distribution confined to the unit square, estimates that distribution by maximum likelihood, and combines it with the unit-specific bounds. This respects the logical constraints, allows heterogeneity, and produces unit-level estimates and honest uncertainty, though it adds distributional assumptions that must be scrutinized.

It is unreliable when aggregation bias is strong — that is, when within-unit rates are correlated with group composition in ways the model does not capture — because then no distributional assumption can be trusted. It also struggles when there are few units, when units are demographically homogeneous so the bounds are wide and uninformative, and when no individual-level data exist to validate the assumptions. In such cases the quantities of interest are only weakly identified, and analysts should report the bounds, conduct sensitivity analysis, and treat point estimates cautiously rather than as definitive.