Improving the Power to Detect Indirect Effects in Mediation Analysis

Abstract

Causal mediation analysis seeks to determine whether an independent variable affects a response variable directly or whether it does so indirectly, by way of a mediator. The existing statistical tests to determine the existence of an indirect effect are overly conservative or have inflated type I error. In this article, we consider the principle of intersection–union tests and a method called the S-test. This method increases power but is not appropriate for statistical tests as small significance levels may cause the test to reject a null hypothesis, but larger significance levels will not reject the same hypothesis. We propose two new methods that provide increased power over existing methods while controlling type I error. We demonstrate through extensive simulation that the S-test and proposed methods control type I error and increase power over existing methods, and that while the proposed methods do not have the same problems, they provide similar power to the S-test. Finally, we provide an application to a large proteomic study.

Appendices

Appendix A

Potential Inflation of the Type I Error for the PS-Test

Due to the symmetric nature of the rejection region, the potential inflation of the Type I error for the PS-test is the same when $\gamma = 0$ and $\beta \ne 0$ versus when $\beta = 0$ and $\gamma \ne 0$. Thus, we assume that $\gamma = 0$ and $\beta \ne 0$, in which case $U_{\gamma }$ is standard uniform. The probability that $(U_{\beta }, U_{\gamma })$ lies within the rejection region depends on the value of $\alpha$ (just like the original S-test) and on the value of $U_{\beta }$ (unlike the original S-test). Let $g(u_{\beta })$ denote the probability that $H_0$ is rejected for the chosen $\alpha$ when $U_{\beta }=u_{\beta }$, and let $f_{U_{\beta }}(u_\beta )$ denote the density function of $U_{\beta }$. The probability of making a type I error at the significance level $\alpha$ equals $\int _0^1 g(x) f_{U_{\beta }}(x) dx$. We can determine the noncentrality parameter of $T_{\beta }$ that causes the largest type I error for any value of $\alpha$ and then determine the maximum inflation of the type I error over all possible values of $\alpha$.

We use numerical integration to calculate the type I error. We consider both small-sample and asymptotic scenarios, using a noncentral t-distribution with five degrees of freedom and a normal distribution with mean equal to the noncentrality parameter and unit variance. In the small-sample scenario, the maximum possible type I error rate occurs when $\alpha \approx 0.002$ and is approximately 1.0001 times $\alpha$. In the asymptotic case, the maximum type I error occurs when $\alpha \approx 0.028$ and is approximately 1.0084 times $\alpha$. In each case, the increase in the type I error for the PS-test is less than 1% of $\alpha$, and more common choices of $\alpha$ have even lower inflation of the type I error.