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Alternative Definitions of “Proportion Eliminated”

Suzuki, Etsuji; Mitsuhashi, Toshiharu; Tsuda, Toshihide; Yamamoto, Eiji

doi: 10.1097/EDE.0000000000000050

Department of Epidemiology, Graduate School of Medicine, Dentistry and Pharmaceutical Sciences, Okayama University, Okayama, Japan,

Center for Innovative Clinical Medicine, Okayama University Hospital, Okayama University, Okayama, Japan

Department of Human Ecology, Graduate School of Environmental and Life Science, Okayama University, Okayama, Japan

Department of Information Science, Faculty of Informatics, Okayama University of Science, Okayama, Japan

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To the Editor:

By using the counterfactual framework, the causal inference literature has made a considerable contribution to mediation analysis in providing definitions for direct and indirect effects.1,2 Of particular interest is that a total effect (TE) of exposure on outcome can be decomposed into direct and indirect effect components using the concepts of natural direct effect and natural indirect effect. By using this property, it is now a common practice to calculate the “proportion mediated” or “proportion explained” (ie, natural indirect effect divided by TE) to capture the importance of the pathway through the intermediate in explaining the actual operation of the effect of the exposure on the outcome3,4; this measure has also been discussed in the sufficient-component cause framework.5 However, neither the natural direct effect nor the natural indirect effect corresponds to actual interventions. These measures have been criticized because of their limited relevance from a policy perspective.6

Recently, VanderWeele3 proposed “proportion eliminated” as an alternative proportion measure, which may be of more interest in policy settings. To discuss this measure, we will let X denote a binary exposure of interest, Y a binary outcome, and M a potential mediator. We also let

denote the potential outcomes for individual ω if, possibly contrary to fact, there had been interventions to set X to x and to set M to m. The proportion eliminated was then introduced as “the proportion of the effect of the exposure on the outcome that could be eliminated by intervening to set the intermediate to some fixed level m.” By using the concept of controlled direct effect (CDE), which captures the effect of exposure on outcome, intervening to fix M to m, the proportion eliminated was defined as

In general, the numerator of the proportion eliminated, which has been sometimes referred to as “controlled indirect effect,” loses causal meaning unless there is no interaction between the effects of the exposure and the mediator on the outcome.2,4,7

We propose two alternative definitions of the proportion eliminated that may be of more policy relevance. To illustrate their relevance, we consider two possible situations in which the redefined proportions eliminated can be of particular use.

First, we consider a situation in which investigators cannot intervene on the exposure to eliminate its adverse health effects and instead aim to possibly attenuate the impact by intervening to set the intermediate to some fixed level m for the whole population. We may encounter this situation when the exposure is not manipulable or beyond human control (eg, radiation exposure). In this case, we propose

as a measure to assess the effectiveness of the alternative intervention on the mediator. This measure would be of greater relevance from the policy perspective because we consistently use

as a reference, unlike the “proportion eliminated” of VanderWeele.3 Note that, if

, these measures become identical, and they can be obtained from the formula

, where



Second, we consider a situation in which investigators can intervene on the exposure and the intermediate. They may then consider intervening on both the exposure and the inter mediate, hoping to eliminate adverse health effects as much as possible. In this case, we propose another alternative definition of the proportion eliminated as

=E[Y1Y0m]/E[Y1Y0], Note that, when

, this measure exceeds 1.0. (Both the proportion mediated and the proportion eliminated can exceed 1.0, and they are not strictly proportions.)

We fully agree that the proportion eliminated is attractive because it describes the estimated effect of actual policy intervention. We hope that its alternative definitions may be of use to estimate the public health burdenfor researchers as well as policy makers.8

Etsuji Suzuki

Department of Epidemiology

Graduate School of Medicine

Dentistry and Pharmaceutical Sciences

Okayama University

Okayama, Japan

Toshiharu Mitsuhashi

Center for Innovative Clinical Medicine

Okayama University Hospital

Okayama University

Okayama, Japan

Toshihide Tsuda

Department of Human Ecology

Graduate School of Environmental and

Life Science

Okayama University

Okayama, Japan

Eiji Yamamoto

Department of Information Science

Faculty of Informatics

Okayama University of Science

Okayama, Japan

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1. Robins JM, Greenland S. Identifiability and exchangeability for direct and indirect effects. Epidemiology. 1992;3:143–155
2. VanderWeele TJ, Vansteelandt S. Conceptual issues concerning mediation, interventions and composition. Stat Interface. 2009;2:457–468
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8. Suzuki E, Yamamoto E, Tsuda T. On the relations between excess fraction, attributable fraction, and etiologic fraction. Am J Epidemiol. 2012;175:567–575
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