Classification of the effectiveness of combination treatments

According to FDA regulations, a combination drug is not efficacious unless each component contributes to the claimed effects. For a univariate endpoint, this implies that the combination at specific doses must be superior to each of its components at the same doses. More demanding is the property of synergy, in which the effect of the combination must be superior to the effect expected based on those of its components. If it is equal to those effects, it is additive, and if it is inferior, it is antagonistic. We give regions in the combination dose plane where these concepts are well defined. If the effect of the combination is greater than the greatest effect achievable by any of its components it is therapeutically synergistic. A combination can be antagonistic, yet its components can still contribute to the claimed effects. If it is additive, synergistic or therapeutically synergistic, its components must contribute to the claimed effects. We relate these concepts and provide designs and sequential procedures for determining whether a combination is therapeutically synergistic, synergistic, additive, antagonistic and contributing or antagonistic and non-contributing.