A common reason given for assessing interaction is to evaluate “whether the effect is larger in one group versus another”. It has long been known that the answer to this question is scale dependent: the “effect” may be larger for one subgroup on the difference scale, but smaller on the ratio scale. In this paper we show that if the relative magnitude of effects across subgroups is of interest then there exists an “interaction continuum” that characterizes the nature of these relationships. When both main effects are positive then the placement on the continuum depends upon the relative magnitude of the probability of the outcome in the doubly exposed group. For high probabilities of the outcome in the doubly exposed group the interaction may be positive-multiplicative positive-additive, the strongest form of positive interaction on the “interaction continuum”. As the probability of the outcome in the doubly exposed group goes down, the form of interaction descends through ranks we will call: positive-multiplicative positive-additive; no-multiplicative positive-additive, negative-multiplicative positive-additive, negative-multiplicative zero-additive, negative-multiplicative negative-additive, single pure interaction, single qualitative interaction, single-qualitative single-pure interaction, double qualitative interaction, perfect antagonism, inverted interaction. One can thus place a particular set of outcome probabilities into one of these states on the interaction continuum. Analogous results are also given when both exposures are protective, or when one is protective and one causative. The “interaction continuum” can allow for inquiries as to relative effects sizes, while also acknowledging the scale dependence of the notion of interaction itself.
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Funding: This research was funded by NIH grant R01CA222147.
The author reports no conflicts of interest.
Corresponding author: Tyler J. VanderWeele, Harvard T.H. Chan School of Public Health, Department of Epidemiology, 677 Huntington Avenue, Boston MA 02115, Phone: 617-432-7855, Fax: 617-432-1884, E-mail: email@example.com