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Inference in Case-Control Studies with Limited Auxilliary Information

Logistic Regression in Rare Events Data

We study rare events data, binary dependent variables with dozens to thousands of times fewer ones (events, such as wars, vetoes, cases of political activism, or epidemiological infections) than zeros (``nonevents''). In many literatures, these variables have proven difficult to explain and predict, a problem that seems to have at least two sources. First, popular statistical procedures, such as logistic regression, can sharply underestimate the probability of rare events. We recommend corrections that outperform existing methods and change the estimates of absolute and relative risks by as much as some estimated effects reported in the literature. Second, commonly used data collection strategies are grossly inefficient for rare events data. The fear of collecting data with too few events has led to data collections with huge numbers of observations but relatively few, and poorly measured, explanatory variables, such as in international conflict data with more than a quarter million dyads, only a few of which are at war. As it turns out, more efficient sampling designs exist for making valid inferences, such as sampling all available events (e.g., wars) and a tiny fraction of non-events (peace). This enables scholars to save as much as 99% of their (non-fixed) data collection costs, or to collect much more meaningful explanatory variables. We provide methods that link these two results, enabling both types of corrections to work simultaneously, and software that implements the methods developed.

Inference in Case-Control Studies With Limited Auxilliary Information

We address a disagreement between epidemiologists and econometricians (and also among several camps within the medical, epidemiological, and public health literatures) about inference from the simplest type of case-control samples. To estimate the conditional probability of disease, the relative risk, or the risk difference in these data, some assumption about the population fraction of ``cases'' is necessary. This population fraction is assumed to be effectively zero by epidemiologists and known exactly, but not necessarily zero, by econometricians. Since the population fraction is usually not known exactly, more recent econometric literature assumes complete ignorance. Methods based on this ignorance assumption produce bounds on the quantities of interest that, unfortunately, are often wide and always encompass a conclusion of no treatment effect (relative risks of one or risk differences of zero) no matter how strong the true effect is. We simplify the existing bounds for risk differences, making them easier to estimate, and then suggest a resolution of the disagreement by providing a method that allows researchers to include easily available information (e.g., that the fraction of cases in the population falls within [.2,.3]); this method avoids unrealistic assumptions and considerably narrows the bounds and hence confidence intervals on all quantities of interest. We also offer public-domain software for all methods introduced, and discuss implications for reporting standards in applied research.