Table.

Additive and Mechanistic Interactions Implemented in R Function

Definition

Interpretation

Assumptions

Additive interactions

RERI (relative excess risk due to interaction)

RR₁₁ − RR₀₁ − RR₁₀ + 1

Difference between the joint RR and the separate contributions by the two exposures

None when interpreted associationally; otherwise NUCA for one or both exposures

Attributable proportion

\frac{RERI}{R R_{11}}

Proportion of outcome risk in the doubly-exposed group attributable to interaction

None when interpreted associationally; otherwise NUCA for one or both exposures

Proportion of joint effect due to interaction

\frac{RERI}{R R_{11} - 1}

Proportion of the joint effects that is attributable to interaction

None when interpreted associationally; otherwise NUCA for one or both exposures

Mechanistic interactions

Synergy

There exists an individual with D₁₁ = 1 but D₀₁ = D₁₀ = 0.

Presence of a mechanism such that some individuals would experience the outcome under both exposures, but not under either exposure alone

NUCA; optionally monotonicity assumptions for less stringent tests

Compositional epistasis

There exists an individual with D₁₁ = 1 but D₀₁ = D₁₀ = D₀₀ = 0.

Presence of a mechanism such that some individuals would experience the outcome if and only if both exposures were present

NUCA; optionally monotonicity assumptions for less stringent tests

Other measures

Proportion of joint effect due to exposure 1

\frac{R R_{10}}{R R_{11} - 1}

None when interpreted associationally; otherwise NUCA for one or both exposures

Proportion of joint effect due to exposure 2

\frac{R R_{01}}{R R_{11} - 1}

None when interpreted associationally; otherwise NUCA for one or both exposures

RERI indicates relative excess risk due to interaction; D = binary outcome variable; E₁ and E₂ = binary exposure variables. NUCA = “no-unmeasured-confounding assumptions” for one or both exposure-outcome relationships.⁵ $R R_{a b} = \frac{P (D = 1 ∣ E_{1} = a, E_{2} = b)}{P (D = 1 ∣ E_{1} = 0, E_{2} = 0)}$ where a, b ε{0,1}, which can be replaced with an odds ratio as appropriate to study design. D_ab = potential outcome for D under an intervention setting E₁ = a and E₂ = b.