Bounding Treatment Effects In Controlled and Natural Experiments Subject to Post-Randomized Treatment Choice

In this paper, we explore a strategy for constructing non-parametric bounds on the effects of treatments actually received in experiments. While treatment statuses may be randomly assigned, experiments involving human subjects are potentially vulnerable to non-random post randomization treatment choice. In contrast to previous proposals for dealing with this problem, our approach focuses on what can be learned (identified) with data from a controlled or naturally occurring randomized experiment. Data from such experiments are not sufficient to achieve point identification of the effects of treatments actually received. Nonetheless, one can identify bounds on these effects without appealing to strong, and possibly untestable, assumptions. Our bounding strategy only requires that randomization itself does not alter behavior outside of the direct effect of assigning different levels of treatment. Much tighter bounds are possible when members of a control group are perfectly embargoed from obtaining the treatments available to the treatment group members. Other assumptions can tighten the bounds further. To assess how informative these bounds are likely to be in practice, we present a numerical example. This example shows that the bounds narrow as the variance of the outcome decreases or as the degree of post-randomization selection becomes small. The example also shows that for any given variance and degree of post-randomization selection the bounds we propose are narrower than bounds proposed elsewhere.