RE: "Synergism Between Obesity and Alcohol in Increasing the Risk of Hepatocellular Carcinoma: A Prospective Cohort Study"

We read with interest the article by Loomba et al (1) on their analysis of the interaction between obesity and alcohol consumption as a risk factor for hepatocellular carcinoma in a population in Taiwan. Although the authors’ conclusions appear broadly consistent with the data presented, their analysis and report are marred by 3 flaws that could be potentially misleading to readers interested in these tools. The authors wrote that, “To test whether the interaction is additive or multiplicative, we examined the combined impact of alcohol and obesity on [hepatocellular carcinoma] risk by relative excess risk due to interaction (RERI), attributable proportion (AP), and synergy index (SI) and their respective confidence intervals, as previously described” (1, p. 135). First, this obscures the fact that interaction, as used in epidemiology, refers to deviation from exactly additive or exactly multiplicative effects. The joint effects of 2 risk factors being considered could be, on the additive scale, less than additive (antagonistic), additive, or more than additive (synergistic). Similarly, on the multiplicative scale, the joint effects could be less than multiplicative, multiplicative, or more than multiplicative. The effect in persons exposed to both variables can thus be separately assessed on the additive and multiplicative scales. Deviation from exact additivity is referred to as interaction on the additive scale. Likewise, deviation from exact multiplicativity is referred to as interaction on the multiplicative scale. The 2 judgments are not in any way mutually exclusive. Second, the use of measures of additive interaction (RERI, AP, and SI) to make inferences about multiplicative interaction is misguided. RERI, AP, and SI provide no direct insight regarding multiplicative interaction (2, 3). In fact, given the proportional hazards model that was used by the authors, it is simply the estimated regression coefficient for the product interaction term that provides direct assessment of deviation from the multiplicative form of the model. The authors also stated that, “Based on prior studies, a multiplicative interaction is suggested by the following scores: a RERI >1.5; an AP >0.25; and an SI >1.5” (1, p.135). This statement is alarming because the reference provided does not provide any such cutoffs and such cutoffs would not be meaningful or sensible (4). Instead, RERI >0, AP >0, and SI >1 are indicative of a greater than additive interaction (2, 3). On the basis of simple computations, it is immediately obvious that such cutoffs would be erroneous for inference about deviations from multiplicativity. For example, consider a case in which the risk ratio for individuals singly exposed to the first factor only (RR10) is 3, the risk ratio for individuals singly exposed to the second factor only (RR01) is 2, and the observed risk ratio for individuals doubly exposed to both factors is 6. RERI would be computed as (6 – 3 – 2 + 1) = 2, AP as ((6 – 3 –2 + 1)/6) = 0.33, and SI as (6 − 1)/((3 − 1) + (2 − 1)) = 1.67. Yet, this is certainly a clear case of exactly multiplicative joint effects, with RR11 = (RR10 × RR01); that is, it is a case of no interaction on the multiplicative scale. Our concern with the article by Loomba et al (1) is not in the substantive conclusion per se, but rather in the methodology used in arriving at the conclusion. Scenarios in which measures of additive interaction are contrary to those of multiplicative interaction are common in practice. The RERI, AP, and SI measures are useful for making additive scale inferences from multiplicative models, such as the proportional hazards model used by these authors, and it is deviation from exactly additive joint effects that is most informative regarding biologic or mechanistic interaction (5–8).

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