  Detecting complex interactions among risk factors in case-control studies is
a fundamental task in clinical and population research. However, though
hypothesis testing using logistic regression (LR) is a convenient solution, the
LR framework is poorly powered and ill-suited under several common
circumstances in practice including missing or unmeasured risk factors,
imperfectly correlated "surrogates", and multiple disease sub-types. The
weakness of LR in these settings is related to the way in which the null
hypothesis is defined. Here we propose the Asymmetric Independence Model (AIM)
as a biologically-inspired alternative to LR, based on the key observation that
the mechanisms associated with acquiring a "disease" versus maintaining
"health" are asymmetric. We prove mathematically that, unlike LR, AIM is a
robust model under the abovementioned confounding scenarios. Further, we
provide a mathematical definition of a "synergistic" interaction, and prove
that theoretically AIM has better power than LR for such interactions. We then
experimentally show the superior performance of AIM as compared to LR on both
simulations and four real datasets. While the principal application here
involves genetic or environmental variables in the life sciences, our
methodology is readily applied to other types of measurements and inferences,
e.g. in the social sciences.
