Data Fusion for Partial Identification of Causal Effects

Data fusion techniques integrate information from heterogeneous data sources to improve learning, generalization, and decision-making across data sciences. In causal inference, these methods leverage rich observational data to improve causal effect estimation, while maintaining the trustworthiness of randomized controlled trials. Existing approaches often relax the strong "no unobserved confounding" assumption by instead assuming exchangeability of counterfactual outcomes across data sources. However, when both assumptions simultaneously fail-a common scenario in practice-current methods cannot identify or estimate causal effects. We address this limitation by proposing a novel partial identification framework that enables researchers to answer key questions such as: Is the causal effect positive/negative? and How severe must assumption violations be to overturn this conclusion? Our approach introduces interpretable sensitivity parameters that quantify assumption violations and derives corresponding causal effect bounds. We develop doubly robust estimators for these bounds and operationalize breakdown frontier analysis to understand how causal conclusions change as assumption violations increase. We apply our framework to the Project STAR study, which investigates the effect of classroom size on students' third-grade standardized test performance. Our analysis reveals that the Project STAR results are robust to simultaneous violations of key assumptions, both on average and across various subgroups of interest. This strengthens confidence in the study's conclusions despite potential unmeasured biases in the data. We assume the following standard conditions hold: Assumption A1. (Treatment Positivity). For s ∈ 0, 1 and all x, ∃c > 0 such that c < P (T = 1 | X = x, S = s) < 1 -c. Assumption A2. (Study Positivity). For all x, ∃c > 0 such that c < P However, we acknowledge the possibility of unobserved confounders that concurrently influence S, T , and Y . Due to such unobserved confounding, the following exchangeability assumptions, which are standard in the literature, may fail to hold: In this paper, we explicitly consider scenarios in which A4 and A5 assumptions are simultaneously violated, thereby challenging the point identifiability of τ and τ (x). Discussion of Assumptions. A1 is the standard treatment positivity assumption, ensuring overlap between treated and control groups. A3 and A4 are structurally equivalent, differing only in the sample subset (experimental vs. observational units). Internal validity in RCTs is generally accepted due to randomization, whereas NUC is stronger, as treatment may depend on unobserved confounders. Combine experimental and observational samples requires the additional A2 and A5 assumptions. A2 states that each unit must have a nonzero probability of being an experimental unit and is necessary to ensure overlap between the two study cohorts. A5 is the study exchangeability assumption, which states that, conditional on covariates, potential outcomes are exchangeable across studies. Like NUC, it can be a strong assumption, as study participation may depend on unobservables.