We study the identification of individual-level associations when only aggregate data are available. We characterize the biases of, and relationships among, canonical ecological inference (EI) estimators. We use these results to develop a partial identification approach: monotone EI. The approach exploits information about one or both of the following conditional associations: (1) outcome differences between groups within the same neighborhood and (2) outcome differences within the same group between neighborhoods with different group compositions. We show how assumptions about the sign of these conditional associations, whether individually or in relation to one another, can yield informative sharp bounds. We illustrate our results using county-level data to study differences in COVID-19 vaccination rates among Republicans and Democrats in the United States.
