A Fundamental Equivalence between Randomized Experiments and Observational Studies

Abstract A fundamental probabilistic equivalence between randomized experiments and observational studies is presented. Given a detailed scenario, the reader is asked to consider which of two possible study designs provides more information regarding the expected difference in an outcome due to a time-fixed treatment. A general solution is described, and a particular worked example is also provided. A mathematical proof is given in the appendix. The demonstrated equivalence helps to clarify common ground between randomized experiments and observational studies, and to provide a foundation for considering both the design and interpretation of studies.