# Rare Events

How to save 99% of your data collection costs; bias corrections for logistic regression in estimating probabilities and causal effects in rare events data; estimating base probabilities or any quantity from case-control data; automated coding of events.

## Bias Correction

Develops corrections for the biases in logistic regression that occur when predicting or explaining rare outcomes (such as when you have many more zeros than ones). Corrections developed for standard prospective studies, as well as case-control designs. How to use "case-control designs" to save 99% of your data collection costs. These articles overlap:
For general mathematical proofs and other technical material: Gary King and Langche Zeng. 2001. “Logistic Regression in Rare Events Data.” Political Analysis, 9, Pp. 137–163.Abstract
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 variable events (e.g., wars) and a tiny fraction of nonevents (peace). This enables scholars to save as much as 99% of their (nonfixed) 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.
An applied companion paper to the previous article that has more examples and pedagogical material but none of the mathematical proofs. Gary King and Langche Zeng. 2001. “Explaining Rare Events in International Relations.” International Organization, 55, Pp. 693–715.Abstract
Some of the most important phenomena in international conflict are coded s "rare events data," binary dependent variables with dozens to thousands of times fewer events, such as wars, coups, etc., than "nonevents". Unfortunately, rare events data are difficult to explain and predict, a problem that seems to have at least two sources. First, and most importantly, the data collection strategies used in international conflict are grossly inefficient. 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. 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. Second, logistic regression, and other commonly used statistical procedures, can underestimate the probability of rare events. We introduce some 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. We also provide easy-to-use methods and software that link these two results, enabling both types of corrections to work simultaneously.

## Hidden Region 1

Example of an analysis of case-control data. Also an independent evaluation of the U.S. State Failure Task Force, including improved methods of forecasting state failure and assessing its causes. Gary King and Langche Zeng. 2001. “Improving Forecasts of State Failure.” World Politics, 53, Pp. 623–658.Abstract

## Estimating Base Probabilities

A method to estimate base probabilities or any quantity of interest from case-control data, even with no (or partial) auxiliary information. Discusses problems with odds-ratios.
The original article: Gary King and Langche Zeng. 2002. “Estimating Risk and Rate Levels, Ratios, and Differences in Case-Control Studies.” Statistics in Medicine, 21, Pp. 1409–1427.Abstract
Classic (or "cumulative") case-control sampling designs do not admit inferences about quantities of interest other than risk ratios, and then only by making the rare events assumption. Probabilities, risk differences, and other quantities cannot be computed without knowledge of the population incidence fraction. Similarly, density (or "risk set") case-control sampling designs do not allow inferences about quantities other than the rate ratio. Rates, rate differences, cumulative rates, risks, and other quantities cannot be estimated unless auxiliary information about the underlying cohort such as the number of controls in each full risk set is available. Most scholars who have considered the issue recommend reporting more than just the relative risks and rates, but auxiliary population information needed to do this is not usually available. We address this problem by developing methods that allow valid inferences about all relevant quantities of interest from either type of case-control study when completely ignorant of or only partially knowledgeable about relevant auxiliary population information.
A revised and extended version of the previous article. Gary King and Langche Zeng. 2004. “Inference in Case-Control Studies.” In Encyclopedia of Biopharmaceutical Statistics, edited by Shein-Chung Chow, 2nd ed. New York: Marcel Dekker.Abstract

Classic (or "cumulative") case-control sampling designs do not admit inferences about quantities of interest other than risk ratios, and then only by making the rare events assumption. Probabilities, risk differences, and other quantities cannot be computed without knowledge of the population incidence fraction. Similarly, density (or "risk set") case-control sampling designs do not allow inferences about quantities other than the rate ratio. Rates, rate differences, cumulative rates, risks, and other quantities cannot be estimated unless auxiliary information about the underlying cohort such as the number of controls in each full risk set is available. Most scholars who have considered the issue recommend reporting more than just the relative risks and rates, but auxiliary population information needed to do this is not usually available. We address this problem by developing methods that allow valid inferences about all relevant quantities of interest from either type of case-control study when completely ignorant of or only partially knowledgeable about relevant auxiliary population information.

## Hidden Region 2

The first extensive empirical study of the probability of your vote changing the outcome of a U.S. presidential election? Most previous studies of the probability of a tied vote have involved theoretical calculation without data. Andrew Gelman, Gary King, and John Boscardin. 1998. “Estimating the Probability of Events that Have Never Occurred: When Is Your Vote Decisive?.” Journal of the American Statistical Association, 93, Pp. 1–9.Abstract
Researchers sometimes argue that statisticians have little to contribute when few realizations of the process being estimated are observed. We show that this argument is incorrect even in the extreme situation of estimating the probabilities of events so rare that they have never occurred. We show how statistical forecasting models allow us to use empirical data to improve inferences about the probabilities of these events. Our application is estimating the probability that your vote will be decisive in a U.S. presidential election, a problem that has been studied by political scientists for more than two decades. The exact value of this probability is of only minor interest, but the number has important implications for understanding the optimal allocation of campaign resources, whether states and voter groups receive their fair share of attention from prospective presidents, and how formal "rational choice" models of voter behavior might be able to explain why people vote at all. We show how the probability of a decisive vote can be estimated empirically from state-level forecasts of the presidential election and illustrate with the example of 1992. Based on generalizations of standard political science forecasting models, we estimate the (prospective) probability of a single vote being decisive as about 1 in 10 million for close national elections such as 1992, varying by about a factor of 10 among states. Our results support the argument that subjective probabilities of many types are best obtained through empirically based statistical prediction models rather than solely through mathematical reasoning. We discuss the implications of our findings for the types of decision analyses used in public choice studies.
James E Alt, Gary King, and Curtis Signorino. 2001. “Aggregation Among Binary, Count, and Duration Models: Estimating the Same Quantities from Different Levels of Data.” Political Analysis, 9, Pp. 21–44.Abstract
Binary, count and duration data all code discrete events occurring at points in time. Although a single data generation process can produce all of these three data types, the statistical literature is not very helpful in providing methods to estimate parameters of the same process from each. In fact, only single theoretical process exists for which know statistical methods can estimate the same parameters - and it is generally used only for count and duration data. The result is that seemingly trivial decisions abut which level of data to use can have important consequences for substantive interpretations. We describe the theoretical event process for which results exist, based on time independence. We also derive a set of models for a time-dependent process and compare their predictions to those of a commonly used model. Any hope of understanding and avoiding the more serious problems of aggregation bias in events data is contingent on first deriving a much wider arsenal of statistical models and theoretical processes that are not constrained by the particular forms of data that happen to be available. We discuss these issues and suggest an agenda for political methodologists interested in this very large class of aggregation problems.