R Function for Additive Interaction Measures

To the Editor

Interaction analyses are commonplace in the epidemiology literature. Predominantly, investigators assess for interaction on the multiplicative scale when the outcome of interest is binary.¹ For example, the standard exponentiated logistic regression coefficient corresponding to the product of two exposures represents the multiplicative ratio by which the joint effect (on the relative risk or odds ratio scale) of both exposures exceeds their individual contributions. Yet multiplicative measures alone are insufficient to fully assess the public health relevance of exposure interactions. For example, they can mislead strategies to target interventions to subgroups, thus reducing net benefit in the population.²

To overcome these limitations, several measures of interaction on the additive scale have been proposed^3,4, and editorial policy for the journal Epidemiology advocates their reporting as common practice. Additive measures assess the difference, rather than the ratio, by which the joint effect exceeds the individual contributions by the two exposures; common examples for a binary outcome and binary exposures appear in the Table (rows 1–3).

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	$\frac{\mathit{RERI}}{R{R}_{11}}$	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	$\frac{\mathit{RERI}}{R{R}_{11}-1}$	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 $R{R}_{ab}=\frac{P(D=1\mid E_1=a,E_2=b)}{P(D=1\mid E_1=0,E_2=0)}$ 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.

Such measures accurately identify subgroups to which interventions should be targeted² and furthermore can be used to assess for “mechanistic interactions” based on the sufficient cause framework.⁵ Examples of mechanistic interactions include sufficient cause synergism and compositional epistasis (Table, rows 4–5). Under the assumption that there is no unmeasured confounding of either exposure-outcome relationship, then a true relative excess risk due to interaction surpassing specific thresholds is sufficient to guarantee the existence of synergism and compositional epistasis. These thresholds on the relative excess risk due to interaction can be relaxed if one or both of the exposures can be assumed to have monotonic effects; that is, if the direction of the exposure’s causal effect would be the same direction for all individuals in the population.⁵

Existing software for additive interaction analyses in SAS and STATA⁶ allows the user to compute these measures, including the proportion of the effects attributable to interaction,^5,6 along with confidence intervals. The present software provides a similar implementation in R with more options and flexibility than other R implementations, and it does not require recoding of exposures. It also more directly allows for the assessment of mechanistic interaction. The user-friendly R function additive_interactions addresses all these topics. The code is publicly available (https://osf.io/7ccpp/), along with documentation and usage examples. We briefly describe the functionality here.

The user passes a standard model object from a logistic regression (fit via R’s glm) of the outcome on both binary exposures and their interaction. The linear predictor can include confounders of arbitrary specification. Using fitted coefficients and their estimated variance-covariance matrix, the function computes estimates of the measures listed in the Table along with confidence intervals and p-values based on the delta method.⁵ To test for mechanistic interactions, additive_interactions allows the user specify whether zero, one, or two of the exposures are assumed to have monotonic effects. Appropriate hypothesis tests are then conducted for both sufficient-cause interaction and compositional epistasis. All output is returned in the form of a dataframe. In the online documentation, we demonstrate application of the function to simulated data.

Additive interaction measures are typically conceptualized for settings in which both exposures are positively associated with the outcome (RR₁₀ > 1 and RR₀₁ > 1, denoting the joint relative risk $R{R}_{ab}=\frac{P(D=1\mid E_1=a,E_2=b)}{P(D=1\mid E_1=0,E_2=0)}$, where a, b ε{0,1}). Our function additive_interactions handles other cases by giving the option of automatically recoding of one or both exposures against new reference levels defined by the joint category with the lowest overall risk.⁷

We hope that the availability of a general, user-friendly R function may reduce a potential barrier to widespread reporting of additive interaction measures.