A note on adapting propensity score matching and selection models to choice based samples

Abstract

The probability of selection into a treatment, also called the propensity score, plays a central role in classical selection models and in matching models (see, e.g., Heckman, 1980; Heckman and Navarro, 2004; Heckman and Vytlacil, 2007; Hirano et al., 2003; Rosenbaum and Rubin, 1983). 1 Heckman and Robb (1986, reprinted 2000), Heckman and Navarro (2004) and Heckman and Vytlacil (2007) show how the propensity score is used differently in matching and selection models. They also show that, given the propensity score, both matching and selection models are robust to choice-based sampling, which occurs when treatment group members are over- or under-represented relative to their frequency in the population. Choice-based sampling designs are frequently chosen in evaluation studies to reduce the costs of data collection and to obtain more observations on treated individuals. Given a consistent estimate of the propensity score, matching and classical selection methods are robust to choice-based sampling, because both are defined conditional on treatment and comparison group status. This note extends the analysis of Heckman and Robb (1985),Heckman and Robb (1986, reprinted 2000) to consider the case where population weights are unknown so that the propensity score cannot be consistently estimated. In evaluation settings, the population weights are often unknown or cannot easily be estimated.2 For example, for the National Supported Work training program studied in LaLonde (1986), Dehejia and Wahba (1999, 2002) and in Smith and Todd (2005), the population consists of all persons eligible for the program, which was targeted at drug addicts, ex-convicts, and welfare recipients. Few datasets have the information necessary to determine whether a person is eligible for the program, so it would be difficult to estimate the population weights needed to consistently estimate propensity scores. In this note, we establish that matching and selection procedures can still be applied when the propensity score is estimated on unweighted choice based samples. The idea is simple. To implement both matching and classical selection models, only a monotonic transformation of the propensity score is required. In choice based samples, the odds ratio of the propensity score estimated using misspecified weights is monotonically related to the odds ratio of the true propensity scores. Thus, selection and matching procedures can identify population treatment effects using misspecified estimates of propensity scores fit on choice-based samples.