## Calculating Bounds for Principal Causal Effects: Interactive Analysis Tool

### What is principal stratification?

With the proliferation of randomized trials in education, researchers are asking ever more sophisticated questions about program impacts. Collectively, the field is evolving from first-order questions about "what works overall" to more nuanced questions about what works, for whom, when, and under what circumstances. Today, analyses that examine variation in treatment effects based on pre-randomization characteristics are common. In contrast to examining variation in effects by such baseline characteristics, principal stratification (Frangakis and Rubin 2002) allows researchers to answer questions related to post-randomization actions. More specifically, principal stratification provides a framework to specify different strata defined by the combination of experimental subjects' observed and counterfactual post-randomization actions or behaviors.

For example, Feller *et al,* 2016 examined the impact of the offer of Head Start enrollment on children vocabulary skills for who, had they not received this offer would have otherwise enrolled at a non-Head Start childcare center versus children who would otherwise have been cared for in a home setting. Miratrix *et al,* 2017 have looked at how the impacts of Early College High Schools (ECHSs) vary by the quality of the school lottery winners would have attended absent the opportunity to enroll in an ECHS.

Principal stratification is a useful framework for identifying subgroups and defining principal causal effects of interest, which are the subgroup-specific intent-to-treat effects in a randomized control study. However, estimation of principal causal effects is challenging and requires a slew of scientific and substantive assumptions. Alternatively, one solution is to generate bounds for the principal causal effects in lieu of solving for point estimates.

This tool illustrates how to calculate the bounds for principal causal effects. The bounds identify the range of possible values of the principal causal effects that are consistent with the observed data. While the calculations needed to solve for the bounds are relatively straightforward, researchers must make assumptions using scientific or substantive knowledge to enable the process. This tool will guide you through the necessary framing, assumptions and calculations to estimate the bounds on stratum-specific treatment effects.

With this website, researchers can

- Determine whether a principal stratification framework is appropriate for their data and research questions
- Calculate bounds for the casual effects of treatment within the specified strata