Introduction

Consider a population each of whose members is described by a vector of covariates z and a binary response y. A common problem of empirical research is to infer the conditional response probabilities P(y|z) when the population is divided into response strata and random samples are drawn from one or both strata. This sampling process is known to epidemiologists studying the incidence of disease as case-control, case-referent, or retrospective sampling, and has been prominent in epidemiological research since the work of Cornfield (1951). The same sampling process in known to economists studying individual behavior as choice-based sampling (Manski and Lerman 1977) or as responsebased sampling (Manski 1986). The final synonym will be used here.

Sampling from the stratum with y = 1 reveals the distribution P(z | y = 1) of covariates within this stratum. Sampling from the stratum with y = 0 reveals P(z | y = 0). So response-based sampling raises this basic inferential question: What does knowledge of P(z | y = 1) and/or P(z | y = 0) reveal about P(y | z)?

Analysis of response-based sampling has concentrated on situations in which the empirical researcher is able to draw random samples from both response strata, and thus learns both P(z | y = 1) and P(z | y = 0). The epidemiological and econometrics literatures have emphasized the identifying power of auxiliary information on the distribution of response and covariates. These literatures have, however, differed in important ways.