Estimating the natural indirect effect and the mediation proportion via the product method

Abstract

Background The natural indirect effect (NIE) and mediation proportion (MP) are two measures of primary interest in mediation analysis. The standard approach for mediation analysis is through the product method, which involves a model for the outcome conditional on the mediator and exposure and another model describing the exposure–mediator relationship. The purpose of this article is to comprehensively develop and investigate the finite-sample performance of NIE and MP estimators via the product method. Methods With four common data types with a continuous/binary outcome and a continuous/binary mediator, we propose closed-form interval estimators for NIE and MP via the theory of multivariate delta method, and evaluate its empirical performance relative to the bootstrap approach. In addition, we have observed that the rare outcome assumption is frequently invoked to approximate the NIE and MP with a binary outcome, although this approximation may lead to non-negligible bias when the outcome is common. We therefore introduce the exact expressions for NIE and MP with a binary outcome without the rare outcome assumption and compare its performance with the approximate estimators. Results Simulation studies suggest that the proposed interval estimator provides satisfactory coverage when the sample size ≥500 for the scenarios with a continuous outcome and sample size ≥20,000 and number of cases ≥500 for the scenarios with a binary outcome. In the binary outcome scenarios, the approximate estimators based on the rare outcome assumption worked well when outcome prevalence less than 5% but could lead to substantial bias when the outcome is common; in contrast, the exact estimators always perform well under all outcome prevalences considered. Conclusions Under samples sizes commonly encountered in epidemiology and public health research, the proposed interval estimator is valid for constructing confidence interval. For a binary outcome, the exact estimator without the rare outcome assumption is more robust and stable to estimate NIE and MP. An R package is developed to implement the methods for point and variance estimation discussed in this paper.

An exact regression‐based approach for the estimation of natural direct and indirect effects with a binary outcome and a continuous mediator

“…7,42 Therefore, this macro also returns controlled direct effects estimated on the OR, RR, and RD scales at a user-defined mediator level and calculated according to the formulas presented in Samoilenko and Lefebvre. 14 As the Vander-Weele and Vansteelandt, 6 Gaynor et al, 5 and Cheng et al 16 approaches, our exact approach targets conditional natural effects. By default, our SAS macro bin_cont_exactmed evaluates the natural indirect, natural direct and controlled direct effects at the sample-specific mean values of the adjustment covariates, but it also allows to estimate these effects at user-defined covariates levels (that is, stratum-specific effects).…”

“…Recently, there has been a strong interest in the development of so‐called exact regression‐based estimators for natural effects, where the term exact refers to estimators that are developed without any theoretical simplifying assumptions and whose accuracy is defined, beyond sample size consideration, by the numeric precision of the software tools utilized and the default/user‐specified tolerance of the routines applied. However most effort has been concentrated on the case of a binary outcome and a binary mediator 13‐16 . Notably, Samoilenko and Lefebvre 14 proposed exact estimators for the natural effects without invoking the rareness or commonness of the outcome, thereby addressing the inherent difficulty in assessing the adequacy of the ROA in a mediation setting.…”

“…Samoilenko and Lefebvre's exact approach 14 was found well‐performing in simulation scenarios ranging from a rare to a common outcome, including a marginally but not conditionally rare outcome. In a very recent paper, Cheng, Spiegelman and Li 16 compared exact and approximate natural effects estimators 5‐7 on the (log) OR scale in different simulation scenarios when the mediator was either binary or continuous. For the case of a continuous mediator, Cheng et al 16 studied the performance and numerical stability of the proposed exact natural effects estimators through a simulation study without covariates in which scenarios varied as a function of the number of outcome cases and sample size; in all the scenarios considered, the outcome was rare marginally (maximal prevalence $$$ \approx $$$ 6.1%).…”

“…In a very recent paper, Cheng, Spiegelman and Li 16 compared exact and approximate natural effects estimators 5‐7 on the (log) OR scale in different simulation scenarios when the mediator was either binary or continuous. For the case of a continuous mediator, Cheng et al 16 studied the performance and numerical stability of the proposed exact natural effects estimators through a simulation study without covariates in which scenarios varied as a function of the number of outcome cases and sample size; in all the scenarios considered, the outcome was rare marginally (maximal prevalence $$$ \approx $$$ 6.1%). The authors globally concluded favorably regarding exact estimators, although for a continuous mediator, few differences between the exact and approximate estimators by VanderWeele and Vansteelandt 6 were observed in the simulation settings with a rare marginal outcome they considered.…”

“…However most effort has been concentrated on the case of a binary outcome and a binary mediator. 13 , 14 , 15 , 16 Notably, Samoilenko and Lefebvre 14 proposed exact estimators for the natural effects without invoking the rareness or commonness of the outcome, thereby addressing the inherent difficulty in assessing the adequacy of the ROA in a mediation setting. These natural effects estimators are based on the specification of a logistic model for both the outcome and mediator, and are expressed on the OR, risk ratio (RR), and risk difference (RD) scales.…”