On identification and estimation for sufficient cause interaction through a quasi-instrumental variable

Mechanistic interaction concerns how exposures affect the outcome. When investigating mechanisms, synergism is the most mentioned type in the fields of genetic study and pharmacology. Synergism is defined under the framework of sufficient component cause model, which is difficult to be quantified directly. Sufficient cause interaction (SCI) is the only alternative metric to imply the existence of synergism. VanderWeele and Robins provided empirical tests for SCIs. However, this test only assesses the lower bound of SCIs rather than estimate SCIs directly due to the lack of the degree of freedom, which causes low power. To address this issue, in this study, we propose a novel method to estimate the probability of individual with SCI by introducing a new factor named quasi-instrumental variable, which is necessary for the background condition of SCI. We also develop a corresponding hypothesis test and show that it is more powerful than the existing empirical test. We demonstrate this method by applying it to estimate the synergistic effects between intestinal bacteria on the formation of Parkinson's disease.