Sample size and power calculations for causal mediation analysis: A Tutorial and Shiny App

Abstract

When designing a study for causal mediation analysis, it is crucial to conduct a power analysis to determine the sample size required to detect the causal mediation effects with sufficient power. However, the development of power analysis methods for causal mediation analysis has lagged far behind. To fill the knowledge gap, I proposed a simulation-based method and an easy-to-use web application (https://xuqin.shinyapps.io/CausalMediationPowerAnalysis/) for power and sample size calculations for regression-based causal mediation analysis. By repeatedly drawing samples of a specific size from a population predefined with hypothesized models and parameter values, the method calculates the power to detect a causal mediation effect based on the proportion of the replications with a significant test result. The Monte Carlo confidence interval method is used for testing so that the sampling distributions of causal effect estimates are allowed to be asymmetric, and the power analysis runs faster than if the bootstrapping method is adopted. This also guarantees that the proposed power analysis tool is compatible with the widely used R package for causal mediation analysis, mediation, which is built upon the same estimation and inference method. In addition, users can determine the sample size required for achieving sufficient power based on power values calculated from a range of sample sizes. The method is applicable to a randomized or nonrandomized treatment, a mediator, and an outcome that can be either binary or continuous. I also provided sample size suggestions under various scenarios and a detailed guideline of app implementation to facilitate study designs.

Appendices

Appendix A. Proofs of identification of causal effects

As a supplement to the identification section, this appendix proves the identification results of the causal mediation effects.

As proved by Imai et al. (2010b) and VanderWeele and Vansteelandt (2009),

$$E\left[{Y}_i\left(t,{M}_i\left({t}^{\prime}\right)\right)\right]=\int E\left[{Y}_i\left(t,{M}_i\left({t}^{\prime}\right)\right)|{\textbf{X}}_i\right] dF\left({\textbf{X}}_i\right)=\iint E\left[{Y}_i\left(t,m\right)|{\textbf{X}}_i,{M}_i\left({t}^{\prime}\right)=m\right] dF\left({M}_i\left({t}^{\prime}\right)=m|{\textbf{X}}_i\right) dF\left({\textbf{X}}_i\right)=\iint E\left[{Y}_i\left(t,m\right)|{\textbf{X}}_i,{M}_i\left({t}^{\prime}\right)=m,{T}_i={t}^{\prime}\right] dF\left({M}_i\left({t}^{\prime}\right)=m|{\textbf{X}}_i\right) dF\left({\textbf{X}}_i\right)=\iint E\left[{Y}_i\left(t,m\right)|{\textbf{X}}_i,{T}_i={t}^{\prime}\right] dF\left({M}_i\left({t}^{\prime}\right)=m|{\textbf{X}}_i\right) dF\left({\textbf{X}}_i\right)=\iint E\left[{Y}_i\left(t,m\right)|{\textbf{X}}_i,{T}_i=t\right] dF\left({M}_i\left({t}^{\prime}\right)=m|{\textbf{X}}_i,{T}_i={t}^{\prime}\right) dF\left({\textbf{X}}_i\right)=\iint E\left[{Y}_i\left(t,m\right)|{\textbf{X}}_i,{T}_i=t,{M}_i(t)=m\right] dF\left({M}_i\left({t}^{\prime}\right)=m|{\textbf{X}}_i,{T}_i={t}^{\prime}\right) dF\left({\textbf{X}}_i\right)=\iint E\left[{Y}_i|{\textbf{X}}_i,{T}_i=t,{M}_i=m\right] dF\left({M}_i=m|{\textbf{X}}_i,{T}_i={t}^{\prime}\right) dF\left({\textbf{X}}_i\right),$$

where t and t^′ are any two different levels of the treatment. The third and fifth equalities hold based on Assumptions 1 and 2. The fourth and sixth equalities hold based on Assumptions 3 and 4. The same idea applies to the identifications of E[Y_i(t^′, M_i(t))], E[Y_i(t, M_i(t))], and E[Y_i(t^′, M_i(t^′))]. By taking contrasts of expected potential outcomes, we can identify the causal mediation effects defined in Table 1.

Continuous mediator and continuous outcome

Based on Eq. 1, we can identify that

$$E\left[{Y}_i\left(t,{M}_i\left({t}^{\prime}\right)\right)\right]=E\left\{E\left[{Y}_i|{\textbf{X}}_i,T=t,{M}_i=E\left[{M}_i|{\textbf{X}}_i,T={t}^{\prime}\right]\right]\right\}={\beta}_0^y+{\beta}_t^yt+\left({\beta}_m^y+{\beta}_{tm}^yt\right)\left({\beta}_0^m+{\beta}_t^m{t}^{\prime }+{{\boldsymbol{\beta}}_x^m}^{\prime }E\left[\textbf{X}\right]\right)+{{\boldsymbol{\beta}}_x^y}^{\prime }E\left[\textbf{X}\right].$$

Similarly,

$$E\left[{Y}_i\left({t}^{\prime },{M}_i(t)\right)\right]={\beta}_0^y+{\beta}_t^y{t}^{\prime }+\left({\beta}_m^y+{\beta}_{tm}^y{t}^{\prime}\right)\left({\beta}_0^m+{\beta}_t^mt+{{\boldsymbol{\beta}}_x^m}^{\prime }E\left[\textbf{X}\right]\right)+{{\boldsymbol{\beta}}_x^y}^{\prime }E\left[\textbf{X}\right],$$

$$E\left[{Y}_i\left(t,{M}_i(t)\right)\right]={\beta}_0^y+{\beta}_t^yt+\left({\beta}_m^y+{\beta}_{tm}^yt\right)\left({\beta}_0^m+{\beta}_t^mt+{{\boldsymbol{\beta}}_x^m}^{\prime }E\left[\textbf{X}\right]\right)+{{\boldsymbol{\beta}}_x^y}^{\prime }E\left[\textbf{X}\right],$$

$$E\left[{Y}_i\left({t}^{\prime },{M}_i\left({t}^{\prime}\right)\right)\right]={\beta}_0^y+{\beta}_t^y{t}^{\prime }+\left({\beta}_m^y+{\beta}_{tm}^y{t}^{\prime}\right)\left({\beta}_0^m+{\beta}_t^m{t}^{\prime }+{{\boldsymbol{\beta}}_x^m}^{\prime }E\left[\textbf{X}\right]\right)+{{\boldsymbol{\beta}}_x^y}^{\prime }E\left[\textbf{X}\right].$$

Hence, based on the definitions in Table 1, we can finally identify the causal effects as

$$TIE=E\left[{Y}_i\left(t,{M}_i(t)\right)\right]-E\left[{Y}_i\left(t,{M}_i\left({t}^{\prime}\right)\right)\right]=\left({\beta}_m^y+{\beta}_{tm}^yt\right){\beta}_t^m\left(t-{t}^{\prime}\right),$$

$$PDE=E\left[{Y}_i\left(t,{M}_i\left({t}^{\prime}\right)\right)\right]-E\left[{Y}_i\left({t}^{\prime },{M}_i\left({t}^{\prime}\right)\right)\right]=\left({\beta}_t^y+{\beta}_{tm}^y\left({\beta}_0^m+{\beta}_t^m{t}^{\prime }+{{\boldsymbol{\beta}}_x^m}^{\prime }E\left[\textbf{X}\right]\right)\right)\left(t-{t}^{\prime}\right),$$

$$PIE=E\left[{Y}_i\left({t}^{\prime },{M}_i(t)\right)\right]-E\left[{Y}_i\left({t}^{\prime },{M}_i\left({t}^{\prime}\right)\right)\right]=\left({\beta}_m^y+{\beta}_{tm}^y{t}^{\prime}\right){\beta}_t^m\left(t-{t}^{\prime}\right),$$

$$TDE=E\left[{Y}_i\left(t,{M}_i(t)\right)\right]-E\left[{Y}_i\left({t}^{\prime },{M}_i(t)\right)\right]=\left({\beta}_t^y+{\beta}_{tm}^y\left({\beta}_0^m+{\beta}_t^mt+{{\boldsymbol{\beta}}_x^m}^{\prime }E\left[\textbf{X}\right]\right)\right)\left(t-{t}^{\prime}\right),$$

$$INT= TIE- PIE={\beta}_t^y{\beta}_{tm}^y{\left(t-{t}^{\prime}\right)}^2.$$

Continuous mediator and binary outcome

Based on Eq. 2, we can identify that

$$E\left[{Y}_i\left(t,{M}_i\left({t}^{\prime}\right)\right)|{\textbf{X}}_i\ \right]$$

$$=\int \Pr \left({Y}_i=1|{\textbf{X}}_i,{T}_i=t,{M}_i=m\right) dF\left({M}_i=m|{\textbf{X}}_i,{T}_i={t}^{\prime}\right)$$

$$=\int \Pr \left({Y}_i^{\ast }>0|{\textbf{X}}_i,{T}_i=t,{M}_i=m\right) dF\left({M}_i=m|{\textbf{X}}_i,{T}_i={t}^{\prime}\right)$$

$$=\int \Pr \left({\beta}_0^y+{\beta}_t^yt+{\beta}_m^ym+{\beta}_{tm}^y tm+{{\boldsymbol{\beta}}_x^y}^{\prime }{\textbf{X}}_i+{\varepsilon}_{y_i^{\ast }}>0\right) dF\left({M}_i=m|{\textbf{X}}_i,{T}_i={t}^{\prime}\right)$$

$$=\int \Pr \left({\beta}_0^y+{\beta}_t^yt+\left({\beta}_m^y+{\beta}_{tm}^yt\right)\left({\beta}_0^m+{\beta}_t^m{t}^{\prime }+{{\boldsymbol{\beta}}_x^m}^{\prime }{\textbf{X}}_i+{\varepsilon}_{mi}\right)+{{\boldsymbol{\beta}}_x^y}^{\prime }{\textbf{X}}_i+{\varepsilon}_{y_i^{\ast }}>0\right) dF\left({\varepsilon}_{mi}\right)$$

$$=\iint \textbf{1}\left\{{\beta}_0^y+{\beta}_t^yt+\left({\beta}_m^y+{\beta}_{tm}^yt\right)\left({\beta}_0^m+{\beta}_t^m{t}^{\prime }+{{\boldsymbol{\beta}}_x^m}^{\prime }{\textbf{X}}_i+{\varepsilon}_{mi}\right)+{{\boldsymbol{\beta}}_x^y}^{\prime }{\textbf{X}}_i+{\varepsilon}_{y_i^{\ast }}>0\right\} dF\left({\varepsilon}_{y_i^{\ast }}\right) dF\left({\varepsilon}_{mi}\right)$$

$$=\Pr \left({\beta}_0^y+{\beta}_t^yt+\left({\beta}_m^y+{\beta}_{tm}^yt\right)\left({\beta}_0^m+{\beta}_t^m{t}^{\prime }+{{\boldsymbol{\beta}}_x^m}^{\prime }{\textbf{X}}_i+{\varepsilon}_{mi}\right)+{{\boldsymbol{\beta}}_x^y}^{\prime }{\textbf{X}}_i+{\varepsilon}_{y_i^{\ast }}>0\right)$$

$$=\Pr \left(\left({\beta}_m^y+{\beta}_{tm}^yt\right){\varepsilon}_{mi}+{\varepsilon}_{y_i^{\ast }}>-\left({\beta}_0^y+{\beta}_t^yt+\left({\beta}_m^y+{\beta}_{tm}^yt\right)\left({\beta}_0^m+{\beta}_t^m{t}^{\prime }+{{\boldsymbol{\beta}}_x^m}^{\prime }{\textbf{X}}_i\right)+{{\boldsymbol{\beta}}_x^y}^{\prime }{\textbf{X}}_i\right)\right)$$

$$=\Pr \left(\left({\beta}_m^y+{\beta}_{tm}^yt\right){\varepsilon}_{mi}+{\varepsilon}_{y_i^{\ast }}<\left({\beta}_0^y+{\beta}_t^yt+\left({\beta}_m^y+{\beta}_{tm}^yt\right)\left({\beta}_0^m+{\beta}_t^m{t}^{\prime }+{{\boldsymbol{\beta}}_x^m}^{\prime }{\textbf{X}}_i\right)+{{\boldsymbol{\beta}}_x^y}^{\prime }{\textbf{X}}_i\right)\right)$$

$$=\Phi \left(\frac{\left({\beta}_0^y+{\beta}_t^yt+\left({\beta}_m^y+{\beta}_{tm}^yt\right)\left({\beta}_0^m+{\beta}_t^m{t}^{\prime }+{{\boldsymbol{\beta}}_x^m}^{\prime }{\textbf{X}}_i\right)\right)+{{\boldsymbol{\beta}}_x^y}^{\prime }{\textbf{X}}_i}{\sqrt{{\left({\beta}_m^y+{\beta}_{tm}^yt\right)}^2{\sigma}_m^2+1}}\right),$$

where Φ represents the cumulative density function of the standard normal distribution, and the last equation holds because ${\varepsilon}_{mi}\sim N\left(0,{\sigma}_m^2\right)$ and ${\varepsilon}_{y_i^{\ast }}\sim N\left(0,1\right)$. Hence,

$$E\left[{Y}_i\left(t,{M}_i\left({t}^{\prime}\right)\right)\right]=E\left[\Phi \left(\frac{\left({\beta}_0^y+{\beta}_t^yt+\left({\beta}_m^y+{\beta}_{tm}^yt\right)\left({\beta}_0^m+{\beta}_t^m{t}^{\prime }+{{\boldsymbol{\beta}}_x^m}^{\prime }{\textbf{X}}_i\right)\right)+{{\boldsymbol{\beta}}_x^y}^{\prime }{\textbf{X}}_i}{\sqrt{{\left({\beta}_m^y+{\beta}_{tm}^yt\right)}^2{\sigma}_m^2+1}}\right)\right].$$

The same applies to the identifications of E[Y_i(t^′, M_i(t))], E[Y_i(t, M_i(t))], and E[Y_i(t^′, M_i(t^′))]. We are then able to identify each causal effect by taking a contrast of expected potential outcomes.

Binary mediator and continuous outcome

Based on Eq. 3, we can identify that

$$E\left[{Y}_i\left(t,{M}_i\left({t}^{\prime}\right)\right)\right]=E\left[{\sum}_{m=0}^1E\left[{Y}_i|{\textbf{X}}_i,{T}_i=t,{M}_i=m\right]\times \Pr \left({M}_i=m|{\textbf{X}}_i,{T}_i={t}^{\prime}\right)\right]=E\left[\left({\beta}_0^y+{\beta}_t^yt+{\beta}_m^y+{\beta}_{tm}^yt+{{\boldsymbol{\beta}}_x^y}^{\prime }{\textbf{X}}_i\right)\times \Phi \left({\beta}_0^m+{\beta}_t^m{t}^{\prime }+{{\boldsymbol{\beta}}_x^m}^{\prime }{\textbf{X}}_i\right)+\left({\beta}_0^y+{\beta}_t^yt+{{\boldsymbol{\beta}}_x^y}^{\prime }{\textbf{X}}_i\right)\times \left(1-\Phi \left({\beta}_0^m+{\beta}_t^m{t}^{\prime }+{{\boldsymbol{\beta}}_x^m}^{\prime }{\textbf{X}}_i\right)\right)\right],$$

$$E\left[{Y}_i\left({t}^{\prime },{M}_i(t)\right)\right]=E\left[{\sum}_{m=0}^1E\left[{Y}_i|{\textbf{X}}_i=\textbf{x},{T}_i={t}^{\prime },{M}_i=m\right]\times \Pr \left({M}_i=m|{\textbf{X}}_i=\textbf{x},{T}_i=t\right)\right]=E\left[\left({\beta}_0^y+{\beta}_t^y{t}^{\prime }+{\beta}_m^y+{\beta}_{tm}^y{t}^{\prime }+{{\boldsymbol{\beta}}_x^y}^{\prime }{\textbf{X}}_i\right)\times \Phi \left({\beta}_0^m+{\beta}_t^mt+{{\boldsymbol{\beta}}_x^m}^{\prime}\right)+\left({\beta}_0^y+{\beta}_t^y{t}^{\prime }+{{\boldsymbol{\beta}}_x^y}^{\prime }{\textbf{X}}_i\right)\times \left(1-\Phi \left({\beta}_0^m+{\beta}_t^mt+{{\boldsymbol{\beta}}_x^m}^{\prime }{\textbf{X}}_i\right)\right)\right],$$

$$E\left[{Y}_i\left(t,{M}_i(t)\right)\right]=E\left[{\sum}_{m=0}^1E\left[{Y}_i|{\textbf{X}}_i=\textbf{x},{T}_i={t}^{\prime },{M}_i=m\right]\times \Pr \left({M}_i=m|{\textbf{X}}_i=\textbf{x},{T}_i=t\right)\right]=E\left[\left({\beta}_0^y+{\beta}_t^yt+{\beta}_m^y+{\beta}_{tm}^yt+{{\boldsymbol{\beta}}_x^y}^{\prime }{\textbf{X}}_i\right)\times \Phi \left({\beta}_0^m+{\beta}_t^mt+{{\boldsymbol{\beta}}_x^m}^{\prime }{\textbf{X}}_i\right)+\left({\beta}_0^y+{\beta}_t^yt+{{\boldsymbol{\beta}}_x^y}^{\prime }{\textbf{X}}_i\right)\times \left(1-\Phi \left({\beta}_0^m+{\beta}_t^mt+{{\boldsymbol{\beta}}_x^m}^{\prime }{\textbf{X}}_i\right)\right)\right],$$

$$E\left[{Y}_i\left({t}^{\prime },{M}_i\left({t}^{\prime}\right)\right)\right]=E\left[{\sum}_{m=0}^1E\left[{Y}_i|{\textbf{X}}_i=\textbf{x},{T}_i={t}^{\prime },{M}_i=m\right]\times \Pr \left({M}_i=m|{\textbf{X}}_i=\textbf{x},{T}_i=t\right)\right]=E\left[\left({\beta}_0^y+{\beta}_t^y{t}^{\prime }+{\beta}_m^y+{\beta}_{tm}^y{t}^{\prime }+{{\boldsymbol{\beta}}_x^y}^{\prime }{\textbf{X}}_i\right)\times \Phi \left({\beta}_0^m+{\beta}_t^m{t}^{\prime }+{{\boldsymbol{\beta}}_x^m}^{\prime }{\textbf{X}}_i\right)+\left({\beta}_0^y+{\beta}_t^y{t}^{\prime }+{{\boldsymbol{\beta}}_x^y}^{\prime }{\textbf{X}}_i\right)\times \left(1-\Phi \left({\beta}_0^m+{\beta}_t^m{t}^{\prime }+{{\boldsymbol{\beta}}_x^m}^{\prime }{\textbf{X}}_i\right)\right)\right].$$

Hence, based on the definitions in Table 1, we can finally identify the causal effects as

$$TIE=E\left[{Y}_i\left(t,{M}_i(t)\right)\right]-E\left[{Y}_i\left(t,{M}_i\left({t}^{\prime}\right)\right)\right]=E\left[\left({\beta}_m^y+{\beta}_{tm}^yt\right)\left(\Phi \left({\beta}_0^m+{\beta}_t^mt+{{\boldsymbol{\beta}}_x^m}^{\prime }{\textbf{X}}_i\right)-\Phi \left({\beta}_0^m+{\beta}_t^m{t}^{\prime }+{{\boldsymbol{\beta}}_x^m}^{\prime }{\textbf{X}}_i\right)\right)\right],$$

$$PDE=E\left[{Y}_i\left(t,{M}_i\left({t}^{\prime}\right)\right)\right]-E\left[{Y}_i\left({t}^{\prime },{M}_i\left({t}^{\prime}\right)\right)\right]=E\left[{\beta}_{tm}^y\left(t-{t}^{\prime}\right)\Phi \left({\beta}_0^m+{\beta}_t^m{t}^{\prime }+{{\boldsymbol{\beta}}_x^m}^{\prime }{\textbf{X}}_i\right)+{\beta}_t^y\left(t-{t}^{\prime}\right)\right],$$

$$PIE=E\left[{Y}_i\left({t}^{\prime },{M}_i(t)\right)\right]-E\left[{Y}_i\left({t}^{\prime },{M}_i\left({t}^{\prime}\right)\right)\right]=E\left[\left({\beta}_m^y+{\beta}_{tm}^y{t}^{\prime}\right)\left(\Phi \left({\beta}_0^m+{\beta}_t^mt+{{\boldsymbol{\beta}}_x^m}^{\prime }{\textbf{X}}_i\right)-\Phi \left({\beta}_0^m+{\beta}_t^m{t}^{\prime }+{{\boldsymbol{\beta}}_x^m}^{\prime }{\textbf{X}}_i\right)\right)\right],$$

$$TDE=E\left[{Y}_i\left(t,{M}_i(t)\right)\right]-E\left[{Y}_i\left({t}^{\prime },{M}_i(t)\right)\right]=E\left[{\beta}_{tm}^y\left(t-{t}^{\prime}\right)\Phi \left({\beta}_0^m+{\beta}_t^mt+{{\boldsymbol{\beta}}_x^m}^{\prime }{\textbf{X}}_i\right)+{\beta}_t^y\left(t-{t}^{\prime}\right)\right],$$

$$INT= TIE- PIE=E\left[{\beta}_{tm}^y\left(t-{t}^{\prime}\right)\left(\Phi \left({\beta}_0^m+{\beta}_t^mt+{{\boldsymbol{\beta}}_x^m}^{\prime }{\textbf{X}}_i\right)-\Phi \left({\beta}_0^m+{\beta}_t^m{t}^{\prime }+{{\boldsymbol{\beta}}_x^m}^{\prime }{\textbf{X}}_i\right)\right)\right].$$

Binary mediator and binary outcome

Based on Eq. 4, we can identify that

$$E\left[{Y}_i\left(t,{M}_i\left({t}^{\prime}\right)\right)\right]=E\left[{\sum}_{m=0}^1E\left[{Y}_i|{\textbf{X}}_i=\textbf{x},{T}_i=t,{M}_i=m\right]\times \Pr \left({M}_i=m|{\textbf{X}}_i=\textbf{x},{T}_i={t}^{\prime}\right)\right]=E\left[\Pr \left({\beta}_0^y+{\beta}_t^yt+{\beta}_m^y+{\beta}_{tm}^yt+{{\boldsymbol{\beta}}_x^y}^{\prime }{\textbf{X}}_i+{\varepsilon}_{y_i^{\ast }}>0\right)\times \Phi \left({\beta}_0^m+{\beta}_t^m{t}^{\prime }+{{\boldsymbol{\beta}}_x^m}^{\prime }{\textbf{X}}_i\right)+\Pr \left({\beta}_0^y+{\beta}_t^yt+{{\boldsymbol{\beta}}_x^y}^{\prime }{\textbf{X}}_i+{\varepsilon}_{y_i^{\ast }}>0\right)\times \left(1-\Phi \left({\beta}_0^m+{\beta}_t^m{t}^{\prime }+{{\boldsymbol{\beta}}_x^m}^{\prime }{\textbf{X}}_i\right)\right)\right],=E\left[\Phi \left({\beta}_0^y+{\beta}_t^yt+{\beta}_m^y+{\beta}_{tm}^yt+{{\boldsymbol{\beta}}_x^y}^{\prime }{\textbf{X}}_i\right)\times \Phi \left({\beta}_0^m+{\beta}_t^m{t}^{\prime }+{{\boldsymbol{\beta}}_x^m}^{\prime }{\textbf{X}}_i\right)+\Phi \left({\beta}_0^y+{\beta}_t^yt+{{\boldsymbol{\beta}}_x^y}^{\prime }{\textbf{X}}_i\right)\times \left(1-\Phi \left({\beta}_0^m+{\beta}_t^m{t}^{\prime }+{{\boldsymbol{\beta}}_x^m}^{\prime }{\textbf{X}}_i\right)\right)\right].$$

Similarly, we can identify [Y_i(t^′, M_i(t))], E[Y_i(t, M_i(t))], and E[Y_i(t^′, M_i(t^′))]. Each causal effect can be identified via a contrast of expected potential outcomes.

Appendix B. Derivations of ${\beta}_x^t,{\beta}_x^m,$ and ${\beta}_x^y$ for data generation

This appendix derives ${\boldsymbol{\beta}}_x^t,{\boldsymbol{\beta}}_x^m,$ and ${\boldsymbol{\beta}}_x^y$ based on the parameters that users are required to specify, ${\beta}_t^m,{\beta}_t^y,{\beta}_m^y,{\beta}_{tm}^y,{R}_{tx}^2,$ ${R}_{mx}^2,$ and ${R}_{yx}^2$, where ${R}_{tx}^2,$ ${R}_{mx}^2,$ and ${R}_{yx}^2$ are respectively the proportion of the variance in T (or T^∗ if T is binary) explained by X, the proportion of the variance in M (or M^∗ if M is binary) explained by X, and the proportion of the variance in Y (or Y^∗ if Y is binary) explained by X, without controlling for any variables.

I first focus on a single standard normal covariate X. If the treatment is not randomized, and the treatment, mediator, and outcome are all continuous, the following standardized regressions are used for data generation.

$${T}_i={\beta}_x^t{X}_i+{\varepsilon}_{ti},$$

$${M}_i={\beta}_t^m{T}_i+{\beta}_x^m{X}_i+{\varepsilon}_{mi},$$

$${Y}_i={\beta}_t^y{T}_i+{\beta}_m^y{M}_i+{\beta}_{tm}^y{T}_i{M}_i+{\beta}_x^y{X}_i+{\varepsilon}_{yi},$$

where ${\hat{\beta}}_x^t={R}_{tx}.$ In the mediator model, based on the ordinary least squares (OLS) estimator

$$\left(\begin{array}{c}{\hat{\beta}}_t^m\\ {}{\hat{\beta}}_x^m\end{array}\right)={\left[\left(\begin{array}{c}{T}^{\prime}\\ {}{X}^{\prime}\end{array}\right)\left(T\kern0.5em X\right)\right]}^{-1}\left[\left(\begin{array}{c}{T}^{\prime}\\ {}{X}^{\prime}\end{array}\right)M\right]={\left[\begin{array}{cc}1& {R}_{tx}\\ {}{R}_{tx}& 1\end{array}\right]}^{-1}\left[\begin{array}{c}{R}_{mt}\\ {}{R}_{mx}\end{array}\right],$$

where T = (T₁, …, T_n)^′, X = (X₁, …, X_n)^′, and M = (M₁, …, M_n)^′ for n individuals, and they are all standardized, we can calculate ${\beta}_x^m$ as

$${\beta}_x^m={R}_{mx}-{\beta}_t^m{R}_{tx},$$

(B.1)

which is consistent with Eq. 1 in Mauro (1990). Similarly, in the outcome model, based on the OLS estimator

$$\left(\begin{array}{c}{\hat{\beta}}_t^y\\ {}{\hat{\beta}}_m^y\\ {}{\hat{\beta}}_{tm}^y\\ {}{\hat{\beta}}_x^y\end{array}\right)={\left[\left(\begin{array}{c}{T}^{\prime}\\ {}{M}^{\prime}\\ {}{TM}^{\prime}\\ {}{X}^{\prime}\end{array}\right)\left(\begin{array}{cccc}T& M& TM& X\end{array}\right)\right]}^{-1}\left[\left(\begin{array}{c}{T}^{\prime}\\ {}{M}^{\prime}\\ {}{TM}^{\prime}\\ {}{X}^{\prime}\end{array}\right)Y\right]$$

$$={\left[\begin{array}{cccc}1& {R}_{mt}& {R}_{t, tm}& {R}_{tx}\\ {}{R}_{mt}& 1& {R}_{m, tm}& {R}_{mx}\\ {}{R}_{t, tm}& {R}_{m, tm}& 1& {R}_{x, tm}\\ {}{R}_{tx}& {R}_{mx}& {R}_{x, tm}& 1\end{array}\right]}^{-1}\left[\begin{array}{c}{R}_{yt}\\ {}{R}_{ym}\\ {}{R}_{y, tm}\\ {}{R}_{yx}\end{array}\right],$$

where TM = (T₁M₁, …, T_nM_n)^′ and Y = (Y₁, …, Y_n)^′ for n individuals, and they are all standardized, we can calculate ${\beta}_x^y$ as

$${\beta}_x^y={R}_{yx}-{\beta}_{tm}^y{R}_{x, tm}-{\beta}_m^y{R}_{mx}-{\beta}_t^y{R}_{tx},$$

(B.2)

where R_{x, tm} = 0 because ${R}_{x, tm}=\frac{\textrm{E}(XTM)-\textrm{E}\left(X\left)\textrm{E}\right( TM\right)}{\sqrt{\textrm{Var}\left(X\left)\textrm{Var}\right( TM\right)}}$, where E(X) = 0, Var(X) = 1,

Var(TM) = Var(T)Var(M) + Var(T)E(M)² + Var(M)E(T)² = 1, and

$$\textrm{E}\left( XT M\right)=E\left( XT\left({\beta}_t^mT+{\beta}_x^mX+{\varepsilon}_m\right)\right)$$

$$={\beta}_t^mE\left(X{T}^2\right)+{\beta}_x^mE\left({X}^2T\right)+E\left( XT{\varepsilon}_m\right)$$

$$={\beta}_t^mE\left(X{\left({\beta}_x^tX+{\varepsilon}_t\right)}^2\right)+{\beta}_x^mE\left({X}^2\left({\beta}_x^tX+{\varepsilon}_t\right)\right)+E\left(X\left({\beta}_x^tX+{\varepsilon}_t\right){\varepsilon}_m\right)$$

$$=\left({\beta}_t^m{\beta}_x^{t2}+{\beta}_x^m{\beta}_x^t\right)E\left({X}^3\right)+{\beta}_t^mE\left({\varepsilon}_t^2X\right)+\left(2{\beta}_t^m{\beta}_x^t+{\beta}_x^m\right)E\left({X}^2{\varepsilon}_t\right)+{\beta}_x^tE\left({X}^2{\varepsilon}_m\right)+E\left(X{\varepsilon}_t{\varepsilon}_m\right),$$

where E(X³) = 0, $E\left({\varepsilon}_t^2X\right)=E\left({\varepsilon}_t^2\right)E(X)=0$, E(X²ε_t) = E(X²)E(ε_t) = 0, E(X²ε_m) = E(X²)E(ε_m) = 0, and E(Xε_tε_m) = E(X)E(ε_t)E(ε_m) = 0.

Therefore,

$${\beta}_x^y={R}_{yx}-{\beta}_m^y{R}_{mx}-{\beta}_t^y{R}_{tx}.$$

(B.3)

If the number of covariates is p, p independent standard normal covariates are generated. I have verified through simulations that power depends on the number of covariates and ${R}_x^2$ contributed by the set of covariates as a whole but is not affected by how much ${R}_x^2$ each specific covariate contributes and how the multiple covariates are correlated. Therefore, based on the assumption that the covariates evenly partition each of ${R}_{tx}^2$, ${R}_{mx}^2$, and ${R}_{yx}^2$, the coefficient of each covariate in each model can be calculated as the previously derived coefficient of the single covariate divided by $\sqrt{p}$.

Above, I focus on a nonrandomized treatment and continuous treatment, mediator, and outcome. If the treatment is randomized, R_tx = 0. If any of the treatment, mediator, and outcome is binary, the corresponding model becomes a latent regression, and the corresponding ${R}_x^2$ is defined as the proportion of the variance in the latent index of the corresponding response explained by X, and thus the Eqs. (B.1) and (B.2) still apply. However, R_{x, tm} is different under different scenarios. Its derivation is particularly tricky if T and/or M are binary. In addition, when T (or M) is binary, R_tx in Eq. (B.1) and Eq. (B.2) (or R_mx in Eq. (B.2)) is the correlation between X and the predictor T (or M), which is on the original scale rather than the latent scale, and thus it needs to be recalculated. These difficulties can be greatly alleviated by calculating R_{x, tm}, R_tx, and R_mx with simulated data. The accuracy of the approach has been verified through simulations.

As shown above, the calculation of β_x’s depends on not only ${R}_x^2$’s but also the signs of R_x’s. Nevertheless, the latter plays a trivial role with respect to power. When both the mediator and outcome are continuous and X is standardized, the effect estimands do not depend on β_x, in which case power is not affected by the signs of R_x’s. When the mediator and/or outcome is binary and X is standardized, although β_x is involved in the effect estimands, I have verified through simulations that different signs of R_x’s make little difference in power under various scenarios. Therefore, I calculate each R_x as the square root of the corresponding ${R}_x^2$.

Appendix C. Simulations comparing the Monte Carlo method and the bootstrapping method when the mediator and outcome are nonnormal

I conducted Monte Carlo simulations to compare the performance of the Monte Carlo confidence interval method and the bootstrapping method in terms of bias, mean squared error (MSE), 95% confidence/credible interval (CI) coverage rate, type I error rate, and power for the estimation and inference of the causal mediation effects under various scenarios that vary by the sample size and the skewness and kurtosis of the distributions of the mediator and the outcome.

Data generation

Focusing on a randomized treatment, I generated the treatment T from a Bernoulli distribution with Pr(T) = 0.5 and generated the pretreatment covariate X from a standard normal distribution. I generated the continuous mediator and outcome based on the following models:

$${M}_i={\beta}_0^m+{\beta}_t^m{T}_i+{\beta}_x^m{\textrm{X}}_i+{\varepsilon}_{mi},$$

$${Y}_i={\beta}_0^y+{\beta}_t^y{T}_i+{\beta}_m^y{M}_i+{\beta}_{tm}^y{T}_i{M}_i+{\beta}_x^y{\textrm{X}}_i+{\varepsilon}_{yi},$$

where ${\beta}_{00}^m$ and ${\beta}_{00}^y$ are set to be 0 in all the scenarios; ε_M and ε_Y have a mean of 0, a variance of 1, and different values of skewness and kurtosis under different scenarios. I considered three different sets of parameter values. In set 1, all the parameters take the value of 0, so that type I error rates can be assessed. Sets 2 and 3 are selected from those considered in the section of sample size planning, as shown in Table 3. Set 2 is the smallest set of parameter values, in which ${\beta}_t^y=0.14$ (S), ${\beta}_m^y=0.14$ (S), ${\beta}_{tm}^y=0.1$ (HS), ${R}_{mx}^2={R}_{yx}^2=0.02$ (S), ${\beta}_t^m=0.14$ (S). Set 3 is the largest set of parameter values, in which ${\beta}_t^y=0.36$ (M), ${\beta}_m^y=0.27$ (HM), ${\beta}_{tm}^y=0.14$ (S), ${R}_{mx}^2={R}_{mx}^2=0.075$ (HM), ${\beta}_t^m=0.27$ (HM). In addition, we considered three different sample sizes, 30, 50, and 150, and three different sets of skewness and kurtosis of the distributions of the mediator and the outcome, (1, 1), (2, 3), and (3, 8). In total, there are 3 × 3 × 3 = 27 different scenarios. For each scenario, I generated 10,000 replications.

Software implementation

I ran the simulations in R. I generated ε_M and ε_Y with the R package mnonr (Qu & Zhang, 2020). To implement the bootstrapping algorithm, I used the boot function and generated 1000 bootstrapped samples for each replication. Confidence intervals are constructed using the bootstrap percentile interval.

Evaluation criteria

Given the three different sets of model parameters values, we obtain three sets of true values of PIE, TIE, PDE, TDE, and INT: (1)(0, 0, 0, 0, 0)^’, (2) (0.04, 0.08, 0.26, 0.30, 0.04)^’, (3) (0.14, 0.25, 0.67, 0.77, 0.10)^’. The true effects are calculated in the same way as noted below Table 2. Let δ denote each true effect and ${\hat{\delta}}_i$ be the estimate obtained from the ith replication. The following evaluation criteria are used to evaluate the performance of each method. (1) Bias $=\frac{\sum_{i=1}^{\textrm{10,000}}\left({\hat{\delta}}_i-\delta \right)}{\textrm{10,000}}$ for δ = 0, and relative bias $=\frac{\sum_{i=1}^{\textrm{10,000}}\left({\hat{\delta}}_i-\delta \right)}{\textrm{10,000}\delta }$ for δ ≠ 0. (2) MSE $=\frac{\sum_{i=1}^{\textrm{10,000}}{\left({\hat{\delta}}_i-\delta \right)}^2}{\textrm{10,000}}$. (3) 95% CI coverage rate $=\frac{\#\left({\hat{L}}_i\le \delta \le {\hat{U}}_i\right)}{\textrm{10,000}}$, where $\left[{\hat{L}}_i,{\hat{U}}_i\right]$ is the 95% CI of the ith replication, and $\#\left({\hat{L}}_i\le \delta \le {\hat{U}}_i\right)$ is the number of replications with CI that includes δ. (4) Rejection rate $=\frac{\#\left({\hat{L}}_i>0\ or\ {\hat{U}}_i<0\right)}{\textrm{10,000}}$, where $\#\left({\hat{L}}_i>0\ or\ {\hat{U}}_i<0\right)$ is the number of replications that reject the null hypothesis that δ = 0 with the significance level alpha at 0.05. The rejection rate is the power when δ ≠ 0 and is the type I error rate when δ = 0.

Simulation results

We present the simulation results for PIE, TIE, PDE, TDE, and INT in Appendix Tables 8, 9, 10, 11 and 12 respectively.

Under the three sets of parameter values, the 95% CI coverage rate of INT is always away from 95% when the sample size is small. To find a scenario when the 95% CI coverage rate gets close to 95%, I increase the true value of INT by increasing ${\beta}_{tm}^y$ from S to HM. Hence, in Appendix Table 12, I further incorporate the simulation results based on Set 4 of the parameter values, ${\beta}_t^y=0.36$ (M), ${\beta}_m^y=0.27$ (HM), ${\beta}_{tm}^y=0.27$ (HM), ${R}_{mx}^2={R}_{mx}^2=0.075$ (HM), ${\beta}_t^m=0.27$ (HM), which leads to a true natural treatment-by-mediator interaction effect of 0.20.

Table 8 Simulation results for the pure indirect effect

Table 9 Simulation results for the total indirect effect

Table 10 Simulation results for the pure direct effect

Table 11 Simulation results for the total direct effect

Table 12 Simulation results for the natural treatment-by-mediator interaction effect

Appendix D. Sample size planning when the number of covariates is 10

Table 13 Sample sizes for achieving a power of 0.8 when both M and Y are normal

Table 14 Sample sizes for achieving a power of 0.8 when M is normal and Y is binary

Table 15 Sample sizes for achieving a power of 0.8 when M is binary and Y is normal

Table 16 Sample sizes for achieving a power of 0.8 when both M and Y are binary