Causal Inference for Case Studies in Behavioral Health
We present a framework for causal inference in behavioral health case studies -- and observational N=1 settings more generally -- under unmeasured confounding. The framework rests on a class of causal estimands, termed $\Omega$ estimands, defined as contrasts of functions of an outcome variable's support rather than its distribution. Because such estimands do not depend on how probability is distributed over supports, they are insensitive to the confounding that limits other methods. We prove that, in a structural causal model, the observational and interventional supports of an outcome coincide under a single assumption -- positivity -- without any requirement that confounders be known, measured, or adjusted for. Two optional conditions extend the framework: one licensing a client's recalled baseline as a stand-in for sparsely measured baseline periods, and one connecting support contrasts to conventional mean contrasts through an expected-average identity. We adopt a subjectivist (de Finetti) interpretation of probability and situate the framework within mandates for measurement-based care. A case study of cognitive behavioral therapy for anxiety illustrates an elementary approach a provider can use.