Abstract
Supplemental Digital Content is available in the text.
To the Editor:
Mediation analysis serves as a crucial tool to perform causal inferences on the relationship among an exposure, a mediator, and an outcome. Methods for statistical inference of the parameter estimates of the mediation effects are limited and need improvement, relying on Bayesian, bootstrapping or Delta methods.
VanderWeele has developed methods1 and together with Valeri published a SAS macro for estimating the natural direct and indirect effects for time-to-event outcomes.2,3 This approach involves fitting two models to infer the relationship between an exposure and survival outcome with a continuous mediator when an interaction effect exists between the exposure and mediator.1 One is the linear regression model, 
, 
, where M, A, and C represent the mediator, exposure, and additional covariates, respectively.1 The other is the survival model, which could employ a Cox proportional hazard model, 
. Whereas the Cox model requires the assumption of a rare survival outcome, the alternative accelerated failure time model does not require this assumption.2 The natural direct effect can be estimated as 
.1 The Delta method was proposed as an approximation method to calculate the standard error of the natural direct effect estimate, which requires estimation of the covariance matrix of the list of estimated parameters, including variance of 
(
).2,3
While we agree to use mean squared error (MSE) to estimate 
, we noticed that the SAS code provided also used MSE (we name it 
) to estimate 
.2,3 We think there are severe limitations for 
, which is a biased estimator in most cases and does not converge to the same limit as 
does as 
approaches infinity, where 
, n is the number of observations, and 
is the number of the predictor variables in the linear regression model that correlates mediator with exposure.
Our objective is to improve estimation of the standard error for natural direct effect in causal mediation inference, and we propose a new estimator 
for 
. We have proved theoretically that it possesses three desired statistical properties (eAppendix; http://links.lww.com/EDE/B497). First, 
is an unbiased estimator. Second, it achieves the minimum variance. Third, it converges to the same limit as 
does as 
approaches infinity.
We also examined the evaluator MSE (MSEeval) for the two estimators. As 
becomes large, 
outperforms 
in a clear manner since MSEeval of 
converges to 0 and that of 
converges to 
(eAppendix; http://links.lww.com/EDE/B497). We acknowledge that the MSEeval of 
is smaller than that of 
for small 
, but 
is still preferred for its unbiasedness. Also, epidemiologic studies rarely have sample sizes <100.
We performed simulation studies under different scenarios and provide code for reproducing simulation results in the eAppendix; http://links.lww.com/EDE/B497. The standard error of the exponential part of the estimated natural direct effect using our proposed 
is consistently smaller than that from 
for large 
, thereby providing higher statistical power to detect mediation effects. The Table presents an example simulation result for 
and 
. We observed that the estimated natural direct effect is very close to the true value with the difference being < 0.01. The 95% confidence interval from 
(1.3, 3.9) successfully captures true natural direct effect and offers much improved precision over that from 
.
TABLE.
Simulation Study with σ2 = 1, k = 200, to Compare the Performance of Our Proposed Estimator Wk with Sk = MSE (True Natural Direct Effect = 2.3)
Unlike Bayesian and bootstrapping methods, the Delta method is an approximation approach to obtaining standard error without massive computations. Accurate statistical inference of natural direct effect in mediation analysis requires improvement. We proposed a valid way to improve the estimation of standard error for natural direct effect with the Delta method. Furthermore, our approach also contributes to statistical inference in epidemiology or biostatistics through providing effective estimation of the variance of MSE.
ACKNOWLEDGMENTS
X.G. is grateful to Ronald Christensen for his inspiration on minimum variance unbiased estimator. X.G. also gives special thanks to James Duncan and Victoria J. Gao for their encouragement when he was frustrated.
Supplementary Material
Xin Gao
Department of Mathematics and Statistics
College of Arts and Sciences
University of New Mexico
Albuquerque, NM
Li Luo
Division of Epidemiology, Biostatistics and Preventive Medicine
Department of Internal Medicine and UNM Comprehensive Cancer Center
University of New Mexico
Albuquerque, NM
LLuo@salud.unm.edu
Footnotes
Supported in part by the National Cancer Institute (P30CA118100 to L.L.).
The authors report no conflicts of interest.
Supplemental digital content is available through direct URL citations in the HTML and PDF versions of this article (www.epidem.com).
The data and computing code required to replicate the results reported in this work will be available upon request.
REFERENCES
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