Generalized Inference for Mediation Analysis

Abstract

Mediation analysis has a long history and its use in applied research has been increasing. Studying mediators can help improve our understanding of mechanisms relating independent and dependent variables. The objective of this paper is to compare different methods to construct confidence intervals for the mediation effect for the one-mediator and two-mediator models. For the one-mediator model, we evaluated the generalized pivotal quantity (GPQ) method, the PRODCLIN method, bootstrap methods, the Sobel method, the Goodman method, and the Monte Carlo method. For the two-mediator model, we evaluated a new GPQ method, bootstrap methods, the Sobel method, the Goodman method, and the Monte Carlo method. Simulation studies compared the performance of the methods for sample sizes of 50, 100, and 200. The results of the simulation studies indicated that, for the simple traditional mediation models under consideration, the GPQ method performed well when compared with the other methods. Future work should consider the extension of the GPQ method to causal mediation analysis involving more complex models with multiple mediators.

1. Introduction

The use of mediation analysis in many research fields has increased because it provides a method to investigate the process by which an independent variable affects a dependent variable, see [1,2]. For example, the use of mediation analysis for treatment studies provides a way to investigate how a treatment achieved its effects. As mediation analysis is increasingly applied in research studies, the construction of confidence intervals for quantities of interest (e.g., mediated effects) with good coverage properties is becoming very important. Although there are many methods available to construct confidence intervals for the simplest case of a single-mediator model, their extensions to more complex cases involving multiple mediators are either difficult (e.g., the PRODCLIN method) or computationally intensive (e.g., bootstrap methods). Statistical methods that can be easily generalized to more complex mediation models while having reduced computational requirements are needed. The Monte Carlo method is simpler to use than other existing methods because it only requires the availability of estimates for the parameters of interest together with the corresponding estimated variance-covariance matrix. The Monte Carlo method for mediation was introduced in [3] and was called the empirical M-method. This Monte Carlo method, which involves simulations from the assumed joint multivariate normal distribution for the vector of parameter estimators, has been shown in simulation studies to perform better than the traditional methods that are based on the multivariate delta method [4]. The relatively good performance of the Monte Carlo method, together with the fact that it does not require the original data (only summary statistics), makes it a very competitive method for mediation analysis; see [5].

In this paper, we use the concept of a generalized pivotal quantity (GPQ) to construct confidence intervals for the mediated effect for the single-mediator model and the multiple-mediator model. While having the advantages of the Monte Carlo method mentioned above, and being comparable to it with respect to computational demands, the GPQ approach has the additional advantage of not being an ad hoc approach, but rather being based on the theory of generalized inference (see [6]), with well-established asymptotic properties for the resulting generalized confidence intervals (see [7]).

There is only one previous publication (that we are aware of) where generalized inference methods have been applied to construct confidence intervals for the mediated effect for the simplest case involving one-mediator and linear regression models [8]. A simulation study similar to the one from [3] was performed to compare the performance of the GPQ method with other methods: Sobel, Goodman, PRODCLIN, percentile bootstrap, and the bias-corrected and accelerated (BCa) bootstrap. The Sobel and Goodman methods are asymptotic approaches; the first one is based on the use of the multivariate delta method, while the second is based on the use of the second-order Taylor series and exact variance. Based on the results of their simulations, the authors concluded that the GPQ method and the percentile bootstrap method outperform the other methods for small and medium sample sizes, with the GPQ method also being more stable for small sample sizes. The proposed new GPQ method for the multiple-mediator models is the main contribution of this paper.

A brief description of the GPQ approach is provided in Appendix A of this paper. Since its introduction in [6], the GPQ methodology has been successfully used by a growing number of researchers to obtain improved confidence intervals for several parametric problems for which traditional approaches are either unavailable or provide unsatisfactory results [9]. In particular, the GPQ methodology has been applied for constructing confidence intervals for ratios of parameters, see [10], where the GPQ approach has been used successfully for the situation involving a ratio of regression parameters. The asymptotic accuracy of the GPQ methodology has been established in [7]; however, satisfactory small sample performance has been noted in many situations, for example in [10]. An added appeal of the GPQ methodology is that if a vector GPQ is available for a vector parameter, then a GPQ can be constructed for any function of the vector parameter by simply using the corresponding function of the vector GPQ. We will take advantage of this very useful property in this paper. For a detailed discussion of the GPQ methodology, along with numerous applications, we refer to the books by Weerahandi [9] and Meeker [11].

This paper is organized as follows. After briefly describing the single-mediator model, we use the GPQ for the linear regression model to construct a GPQ for the single-mediator model. We then describe an extension of the GPQ approach to the situation involving two mediators. We assess the performance of the GPQ-based confidence interval for the mediated effect for the single-mediator model using a simulation study similar to those employed in [3,8]. We then illustrate the application of the new GPQ method using a hypothetical data set, and assess the performance of the GPQ-based confidence interval for the mediated effect for the two-mediator model using a simulation study. We end with a discussion where we propose other extensions of the GPQ approach.

2. Materials and Methods

2.1. Single Mediator

2.1.1. The Model

Consider an independent variable Z, a dependent variable Y, and a mediator M of the effect of Z on Y. Then, one can consider the following two regression models (where for simplicity we drop the index used to identify each observation):

$$\begin{array}{ccc}\hfill M& =& {\delta }_1+\alpha Z+{ϵ}_1\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill Y& =& {\delta }_2+\tau Z+\beta M+{ϵ}_2,\hfill \end{array}$$

(2)

where ${ϵ}_i\sim N(0,{\sigma }_i^2)$, $i=1,2$ are independent error terms. The mediated (indirect) effect of Z mediated through M is given by $\alpha \beta$, while the total effect is $\tau +\alpha \beta$.

2.1.2. A GPQ for the Linear Regression Model

The following construction follows the approach described in [10].

Consider the linear regression model

$$y=X\gamma +ϵ,$$

where y is the $n\times 1$ dependent variable vector, X is an $n\times p$ design matrix (assumed of full column rank), and $ϵ\sim N(0,{\sigma }^2{I}_p)$. Using the usual estimators

$$\begin{array}{ccc}\hfill \widehat{\gamma }& =& {\left(X^{\prime }X\right)}^{-1}{X}^{\prime }y,\hfill \\ \hfill {\widehat{\sigma }}^2& =& {(y-X\widehat{\gamma })}^{\prime }(y-X\widehat{\gamma })/(n-p),\hfill \end{array}$$

(3)

one has

$$\begin{array}{ccc}\hfill Z& =& {\displaystyle \frac{{\left(X^{\prime }X\right)}^{1/2}}{\sigma }}(\widehat{\gamma }-\gamma )\sim N(0,I_p)\hfill \\ \hfill U^2& =& {\displaystyle \frac{(n-p){\widehat{\sigma }}^2}{{\sigma }^2}}\sim {\chi }_{(n-p)}^2\hfill \\ \hfill T& =& {\displaystyle \frac{\sqrt{n-p}}{U}}Z\sim t_{(n-p)},\hfill \end{array}$$

(4)

where ${\chi }_{(n-p)}^2$ denotes the chi-square distribution with $n-p$ degrees of freedom (df), and, Z and $U^2$ are independent. Let ${\widehat{\gamma }}_{\mathrm{obs}}$ and ${\widehat{\sigma }}_{\mathrm{obs}}^2$ denote the observed values of $\widehat{\gamma }$ and ${\widehat{\sigma }}^2$, respectively, and define

$$\begin{array}{ccc}\hfill T_{\gamma }& =& {\widehat{\gamma }}_{\mathrm{obs}}-{\widehat{\sigma }}_{\mathrm{obs}}{\left(X^{\prime }X\right)}^{-1/2}{\displaystyle \frac{\sigma }{\widehat{\sigma }}}{\displaystyle \frac{{\left(X^{\prime }X\right)}^{1/2}}{\sigma }}(\widehat{\gamma }-\gamma )\hfill \\ & =& {\widehat{\gamma }}_{\mathrm{obs}}-{\widehat{\sigma }}_{\mathrm{obs}}{\left(X^{\prime }X\right)}^{-1/2}{\displaystyle \frac{\sqrt{n-p}}{U}}Z\hfill \\ & =& {\widehat{\gamma }}_{\mathrm{obs}}-{\widehat{\sigma }}_{\mathrm{obs}}{\left(X^{\prime }X\right)}^{-1/2}T,\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill T_{{\sigma }^2}& =& {\displaystyle \frac{(n-p)·{\widehat{\sigma }}_{obs}^2}{U^2}},\hfill \end{array}$$

(6)

where Z, $U^2$, and T are defined in (4); ${\left(X^{\prime }X\right)}^{1/2}$ denotes the positive definite square root of $X^{\prime }X$; and ${\left(X^{\prime }X\right)}^{-1/2}$ is the inverse of ${\left(X^{\prime }X\right)}^{1/2}$. $T_{\gamma }$ is a GPQ for the entire vector $\gamma$, and therefore $g\left(T_{\gamma }\right)$ provides a GPQ for any function $g\left(\gamma \right)$, while $T_{{\sigma }^2}$ is a GPQ for ${\sigma }^2$.

2.1.3. A GPQ for the Mediated Effect for the Single-Mediator Model

Using (5), one can construct GPQs for the two regression models in (1) and (2). More precisely, if $T_{{\gamma }_1}$ denotes the GPQ for the parameters in (1), then a GPQ for $\alpha$ is given by $T_{{\gamma }_1}^{\left(2\right)}$, the second component of $T_{{\gamma }_1}$. Similarly, GPQs for $\tau$ and $\beta$ are given by $T_{{\gamma }_2}^{\left(2\right)}$ and $T_{{\gamma }_2}^{\left(3\right)}$, the second and the third components of $T_{{\gamma }_2}$, a GPQ for the parameters in (2).

Then, the GPQs for the mediated (indirect) effect and the total effect are given by

$$\begin{array}{ccc}\hfill T_{\alpha \beta }& =& T_{{\gamma }_1}^{\left(2\right)}{T}_{{\gamma }_2}^{\left(3\right)},\hfill \\ \hfill T_{\tau +\alpha \beta }& =& T_{{\gamma }_2}^{\left(2\right)}+T_{{\gamma }_1}^{\left(2\right)}{T}_{{\gamma }_2}^{\left(3\right)}.\hfill \end{array}$$

(7)

The algorithm used to construct the generalized confidence intervals is as follows:

(i)

Generate a random sample $T^1,\dots ,T^m$ in (5) from the t distribution with $n-p$ df for each GPQ.

(ii)

Construct $T_{\alpha \beta }^1,\dots ,T_{\alpha \beta }^m$.

(iii)

Choose the $\varphi$th quantile and $1-\varphi$th quantile of the distribution of $T_{\alpha \beta }$ to form the lower and upper bounds of $100(1-2\varphi )\%$.

A similar algorithm can be used to construct a GPQ-based confidence interval for the mediation proportion, the ratio between the mediated effect and the total effect.

2.2. Extension to Multiple Mediators

The extension described in this paper is the use of the GPQ method for situations involving multiple mediators. For simplicity, we only consider the case of two mediators. The linear regression models are

$$\begin{array}{ccc}\hfill M_1& =& {\delta }_1+{\alpha }_{1}Z+{ϵ}_1\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill M_2& =& {\delta }_2+{\alpha }_{2}Z+{ϵ}_2\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill Y& =& {\delta }_3+\tau Z+{\beta }_1{M}_1+{\beta }_2{M}_2+{ϵ}_3\hfill \end{array}$$

(10)

where ${ϵ}_i\sim N(0,{\sigma }_i^2)$, $i=1,\dots ,3$ are independent error terms.

Then, the effect of Z on Y mediated through $M_i$ is ${\alpha }_i{\beta }_i$, $i=1,2$; the (total) effect mediated through the two mediators is ${\alpha }_1{\beta }_1+{\alpha }_2{\beta }_2$; and the difference between the two mediated effects is ${\alpha }_1{\beta }_1-{\alpha }_2{\beta }_2$. The GPQs for all these quantities of interest can be obtained similarly to the one-mediator model case, by first constructing the GPQs for the regression parameters from models (8) to (10).

Using (5), if $T_{{\gamma }_1}$ denotes the GPQ for (8), then a GPQ for ${\alpha }_1$ is given by $T_{{\gamma }_1}^{\left(2\right)}$, the second component of $T_{{\gamma }_1}$. Similarly, a GPQ for ${\alpha }_2$ from (9) is given by $T_{{\gamma }_2}^{\left(2\right)}$; GPQs for $\tau$, ${\beta }_1$, and ${\beta }_2$ are given by $T_{{\gamma }_3}^{\left(2\right)}$, $T_{{\gamma }_3}^{\left(3\right)}$, and $T_{{\gamma }_3}^{\left(4\right)}$, the second, third, and fourth components, respectively, of $T_{{\gamma }_3}$, the GPQ from (10).

Then, the GPQs for the mediated effect and the total effect are given by

$$\begin{array}{ccc}\hfill T_{{\alpha }_1{\beta }_1+{\alpha }_2{\beta }_2}& =& T_{{\gamma }_1}^{\left(2\right)}{T}_{{\gamma }_3}^{\left(3\right)}+T_{{\gamma }_2}^{\left(2\right)}{T}_{{\gamma }_3}^{\left(4\right)},\hfill \\ \hfill T_{\tau +{\alpha }_1{\beta }_1+{\alpha }_2{\beta }_2}& =& T_{{\gamma }_4}^{\left(2\right)}+T_{{\gamma }_1}^{\left(2\right)}{T}_{{\gamma }_3}^{\left(3\right)}+T_{{\gamma }_2}^{\left(2\right)}{T}_{{\gamma }_3}^{\left(4\right)}.\hfill \end{array}$$

(11)

3. Results

3.1. Illustrative Example

As an illustrative example for the case of two mediators, we use data from a study of how two different ways of processing a list of words affect subsequent recall of the words (see [12] for more details on this study). The data set contains 44 observations. The independent variable Z is a random assignment of participants to either make images of words or to repeat the words over and over. The two mediators were participants’ self-reported imagery ($M_1$) and self-reported repetition of the words ($M_2$) when hearing the word list. The dependent variable, Y, was the number of words recalled. The Z variable was binary and $M_1$, $M_2$, and Y were treated as continuous variables for the sake of illustration. The research hypothesis is that the effect on word recall would be mediated through the use of imagery when memorizing the word list. The point estimates for the mediated effects are 2.447 for imagery and −0.875 for repetition, respectively. The 95% CI for the effect mediated through imagery is (0.655, 4.590), while the 95% CI for the effect mediated through repetition is (−2.775, 0.896). The 95% CI for the effect mediated through the two mediators is (−1.019, 4.298), while the 95% CI for the difference between the two mediated effects, namely ${\alpha }_1{\beta }_1-{\alpha }_2{\beta }_2$, is (0.764, 6.115). Figure 1 is a forest plot that shows the 95% CI for the effect mediated through the two mediators by using five different methods. We note that the point estimates are very close, and the GPQ-based 95% CI is the widest CI among the five CIs.

3.2. Simulation Study

3.2.1. Single-Mediator Model

To evaluate the performance of the GPQ method and compare it with the other available methods, we performed a simulation study where we simulated data using the two regression models from (1) and (2). We used 16 combinations of values for $\alpha$ and $\beta$ as performed before in [3,8] ($\alpha$ and $\beta$ take the values: 0, 0.14, 0.39, and 0.59). The study involved 5000 simulated data sets; for each simulated data set, the bootstrap methods used 5000 resamples, and the GPQ and Monte Carlo methods used 5000 simulations each. The sizes of the simulated data sets were 50, 100, and 200, respectively. The independent variable Z was simulated as a standard normal. The intercepts ${\delta }_1$ and ${\delta }_2$ from (1) and (2) were set as 0 and the coefficient $\tau$ from (2) was set as 0.5. We considered a total of 16 scenarios and compared the performance of the GPQ, Sobel, PRODCLIN, Monte Carlo, Goodman, percentile bootstrap, and BCa bootstrap methods. The results of the simulations, specifically the simulated coverage probability and average interval length for 95% confidence intervals, are presented in Appendix A,

Table A1. Simulated results using 5000 simulations for the 95% confidence intervals for the mediated effect for the single-mediator model (n = 50).

		Coverage Probability							Average Interval Length
$\mathit{\alpha }$	$\mathit{\beta }$	GPQ	Sobel	Prod	MC	Goodman	Percentile	BCa	GPQ	Sobel	Prod	MC	Goodman	Percentile	BCa
0.00	0.00	0.9978	1.0000	0.9978	0.9974	1.0000	0.9978	0.9916	0.1492	0.1029	0.1457	0.1456	0.1364	0.1476	0.1595
0.00	0.14	0.9966	0.9998	0.9952	0.9952	0.9998	0.9926	0.9780	0.1679	0.1265	0.1635	0.1634	0.1549	0.1655	0.1776
0.00	0.39	0.9736	0.9904	0.9662	0.9670	0.9914	0.9594	0.9260	0.2656	0.2366	0.2597	0.2595	0.2530	0.2613	0.2708
0.00	0.59	0.9550	0.9762	0.9482	0.9492	0.9782	0.9418	0.9088	0.3678	0.3423	0.3652	0.3651	0.3591	0.3677	0.3733
0.14	0.00	0.9956	0.9988	0.9948	0.9946	0.9992	0.9922	0.9766	0.1692	0.1278	0.1642	0.1642	0.1558	0.1661	0.1781
0.14	0.14	0.9928	0.9546	0.9892	0.9902	0.9988	0.9856	0.9404	0.1854	0.1484	0.1793	0.1792	0.1721	0.1808	0.1942
0.14	0.39	0.9530	0.9272	0.9560	0.9550	0.9752	0.9484	0.9162	0.2759	0.2499	0.2700	0.2699	0.2656	0.2717	0.2835
0.14	0.59	0.9534	0.9540	0.9448	0.9452	0.9582	0.9380	0.9202	0.3784	0.3550	0.3704	0.3702	0.3661	0.3740	0.3819
0.39	0.00	0.9654	0.9876	0.9670	0.9674	0.9916	0.9640	0.9270	0.2669	0.2379	0.2611	0.2610	0.2542	0.2633	0.2728
0.39	0.14	0.9558	0.9460	0.9484	0.9484	0.9708	0.9414	0.9160	0.2788	0.2528	0.2714	0.2713	0.2669	0.2736	0.2858
0.39	0.39	0.9508	0.9192	0.9364	0.9374	0.9226	0.9312	0.9398	0.3444	0.3266	0.3372	0.3371	0.3383	0.3408	0.3568
0.39	0.59	0.9528	0.9332	0.9516	0.9506	0.9418	0.9420	0.9500	0.4281	0.4110	0.4212	0.4210	0.4223	0.4243	0.4379
0.59	0.00	0.9488	0.9686	0.9480	0.9484	0.9736	0.9414	0.9122	0.3730	0.3473	0.3671	0.3669	0.3608	0.3702	0.3759
0.59	0.14	0.9560	0.9536	0.9448	0.9438	0.9600	0.9368	0.9158	0.3790	0.3555	0.3727	0.3726	0.3683	0.3757	0.3831
0.59	0.39	0.9508	0.9348	0.9436	0.9442	0.9388	0.9394	0.9428	0.4337	0.4161	0.4220	0.4218	0.4230	0.4257	0.4388
0.59	0.59	0.9512	0.9314	0.9426	0.9432	0.9386	0.9332	0.9388	0.5020	0.4854	0.4919	0.4917	0.4940	0.4954	0.5095

,

Table A2. Simulated results using 5000 simulations for the 95% confidence intervals for the mediated effect for the single-mediator model (n = 100).

		Coverage Probability							Average Interval Length
$\mathit{\alpha }$	$\mathit{\beta }$	GPQ	Sobel	Prod	MC	Goodman	Percentile	BCa	GPQ	Sobel	Prod	MC	Goodman	Percentile	BCa
0.00	0.00	0.9988	1.0000	0.9986	0.9988	1.0000	0.9982	0.9926	0.0717	0.0502	0.0708	0.0708	0.0663	0.0714	0.0773
0.00	0.14	0.9904	0.9982	0.9898	0.9902	0.9984	0.9874	0.9704	0.0892	0.0720	0.0881	0.0880	0.0842	0.0890	0.0948
0.00	0.39	0.9508	0.9756	0.9490	0.9490	0.9788	0.9474	0.9174	0.1702	0.1599	0.1683	0.1682	0.1652	0.1692	0.1717
0.00	0.59	0.9428	0.9542	0.9410	0.9408	0.9580	0.9390	0.9216	0.2453	0.2365	0.2426	0.2425	0.2400	0.2439	0.2452
0.14	0.00	0.9914	0.9986	0.9908	0.9904	0.9990	0.9894	0.9696	0.0900	0.0727	0.0889	0.0888	0.0850	0.0897	0.0955
0.14	0.14	0.9756	0.9034	0.9726	0.9732	0.9906	0.9622	0.9082	0.1034	0.0898	0.1022	0.1021	0.0997	0.1032	0.1101
0.14	0.39	0.9476	0.9402	0.9444	0.9440	0.9504	0.9404	0.9292	0.1775	0.1691	0.1755	0.1755	0.1741	0.1763	0.1807
0.14	0.59	0.9548	0.9570	0.9526	0.9526	0.9602	0.9482	0.9374	0.2523	0.2444	0.2494	0.2493	0.2478	0.2507	0.2532
0.39	0.00	0.9550	0.9786	0.9532	0.9534	0.9810	0.9502	0.9208	0.1707	0.1605	0.1688	0.1687	0.1657	0.1699	0.1725
0.39	0.14	0.9514	0.9442	0.9492	0.9492	0.9528	0.9448	0.9352	0.1784	0.1700	0.1764	0.1764	0.1750	0.1777	0.1822
0.39	0.39	0.9506	0.9330	0.9488	0.9488	0.9382	0.9426	0.9496	0.2290	0.2240	0.2265	0.2264	0.2276	0.2273	0.2341
0.39	0.59	0.9496	0.9390	0.9482	0.9480	0.9410	0.9448	0.9462	0.2899	0.2846	0.2868	0.2867	0.2874	0.2879	0.2930
0.59	0.00	0.9500	0.9590	0.9458	0.9454	0.9612	0.9418	0.9274	0.2480	0.2391	0.2451	0.2449	0.2425	0.2460	0.2477
0.59	0.14	0.9484	0.9504	0.9456	0.9454	0.9540	0.9398	0.9296	0.2521	0.2441	0.2492	0.2491	0.2475	0.2508	0.2533
0.59	0.39	0.9500	0.9384	0.9470	0.9478	0.9412	0.9452	0.9462	0.2904	0.2850	0.2873	0.2871	0.2879	0.2886	0.2939
0.59	0.59	0.9530	0.9416	0.9500	0.9504	0.9438	0.9468	0.9480	0.3405	0.3352	0.3367	0.3365	0.3376	0.3377	0.3431

and

Table A3. Simulated results using 5000 simulations for the 95% confidence intervals for the mediated effect for the single-mediator model (n = 200).

		Coverage Probability							Average Interval Length
$\mathit{\alpha }$	$\mathit{\beta }$	GPQ	Sobel	Prod	MC	Goodman	Percentile	BCa	GPQ	Sobel	Prod	MC	Goodman	Percentile	BCa
0.00	0.00	0.9988	0.9996	0.9988	0.9988	0.9996	0.9986	0.9952	0.0354	0.0250	0.0352	0.0352	0.0330	0.0356	0.0385
0.00	0.14	0.9802	0.9968	0.9798	0.9798	0.9976	0.9788	0.9536	0.0516	0.0446	0.0513	0.0513	0.0496	0.0519	0.0545
0.00	0.39	0.9506	0.9630	0.9496	0.9492	0.9658	0.9480	0.9282	0.1142	0.1104	0.1136	0.1136	0.1122	0.1145	0.1152
0.00	0.59	0.9514	0.9570	0.9510	0.9506	0.9594	0.9468	0.9414	0.1683	0.1653	0.1674	0.1674	0.1665	0.1688	0.1692
0.14	0.00	0.9808	0.9968	0.9806	0.9804	0.9974	0.9786	0.9550	0.0518	0.0449	0.0515	0.0515	0.0497	0.0520	0.0545
0.14	0.14	0.9434	0.9064	0.9432	0.9444	0.9318	0.9424	0.9422	0.0630	0.0587	0.0627	0.0627	0.0624	0.0633	0.0669
0.14	0.39	0.9488	0.9448	0.9482	0.9466	0.9498	0.9424	0.9408	0.1196	0.1169	0.1191	0.1190	0.1186	0.1197	0.1215
0.14	0.59	0.9502	0.9546	0.9502	0.9500	0.9558	0.9492	0.9418	0.1729	0.1703	0.1721	0.1720	0.1715	0.1731	0.1740
0.39	0.00	0.9436	0.9600	0.9424	0.9404	0.9628	0.9438	0.9258	0.1144	0.1107	0.1138	0.1138	0.1125	0.1146	0.1153
0.39	0.14	0.9488	0.9450	0.9460	0.9452	0.9488	0.9396	0.9374	0.1201	0.1173	0.1195	0.1195	0.1191	0.1205	0.1221
0.39	0.39	0.9542	0.9464	0.9532	0.9536	0.9476	0.9536	0.9558	0.1574	0.1559	0.1567	0.1566	0.1572	0.1575	0.1603
0.39	0.59	0.9534	0.9480	0.9526	0.9530	0.9484	0.9516	0.9524	0.2003	0.1985	0.1993	0.1992	0.1995	0.2005	0.2026
0.59	0.00	0.9504	0.9566	0.9488	0.9488	0.9578	0.9496	0.9404	0.1692	0.1661	0.1683	0.1682	0.1673	0.1695	0.1700
0.59	0.14	0.9448	0.9456	0.9440	0.9434	0.9468	0.9422	0.9360	0.1727	0.1701	0.1718	0.1717	0.1712	0.1731	0.1740
0.59	0.39	0.9538	0.9486	0.9532	0.9520	0.9496	0.9488	0.9514	0.2001	0.1984	0.1991	0.1990	0.1994	0.2006	0.2026
0.59	0.59	0.9490	0.9456	0.9494	0.9500	0.9468	0.9446	0.9458	0.2358	0.2341	0.2346	0.2345	0.2350	0.2360	0.2382

. We used the R package RMediation (see [13]) for the Goodman and PRODCLIN methods.

We have considered an empirical coverage of 94.5% or more to be acceptable from a practical point of view.

Table 1. The percentage of simulated coverage probabilities for the 16 scenarios which are above the 94.5% level. The seven methods used were: generalized pivotal quantity (GPQ), Sobel, PRODCLIN (Prod), Monte Carlo (MC), Goodman, percentile bootstrap, and bias-corrected and accelerated (BCa) bootstrap.

	GPQ	Sobel	Prod	MC	Goodman	Percentile	BCa
$n=50$	100.00%	68.75%	68.75%	75.00%	75.00%	43.75%	25.00%
$n=100$	93.75%	56.25%	87.50%	87.50%	75.00%	56.25%	43.75%
$n=200$	81.25%	87.50%	81.25%	81.25%	93.75%	62.50%	43.75%

provides the percentage of simulated coverage probabilities that are above the 94.5% level across the 16 scenarios. Our simulation results are similar to those from [8] in the sense that the GPQ method had more instances than the other methods when the simulated coverage was above the 94.5% level, especially for the small sample sizes. For the larger sample size, $n=200$, the Sobel and Goodman methods had larger percentages than the GPQ method. However, the lowest coverage probabilities for the Sobel and Goodman methods were 90.64% and 93.18%, respectively, for the situation when both parameters were 0.14, while the lowest coverage probability for the GPQ method was 94.34%. This may indicate that the GPQ method is more stable than the other methods. As shown before, and also as seen in our simulation study, the GPQ method tended to have the largest average interval length.

3.2.2. Multiple Mediators Model

To evaluate the performance of the GPQ method for the multiple-mediator model and compare it with other methods, we performed a simulation study where we simulated data using the three regression models from (8) to (10). We used combinations of values for ${\alpha }_1$, ${\alpha }_2$, ${\beta }_1$, and ${\beta }_2$ as performed in the previous section. Therefore, there are 256 scenarios in total. The study involved 5000 simulated data sets, the bootstrap methods used 5000 resamples, and the GPQ method involved 5000 simulations. The sample sizes of the simulated data sets were 50, 100, and 200, respectively. The independent variable was simulated as a standard normal. The intercepts ${\delta }_1$, ${\delta }_2$, and ${\delta }_3$ from (8) to (10) were set as 0 and the coefficient $\tau$ from (2) was set as 0.5. We compared the performance of the GPQ, Sobel, Monte Carlo, percentile bootstrap, and BCa bootstrap methods. The results for 16 scenarios, selected out of the total of 256 scenarios, including the simulated coverage probability and average interval length for 95% confidence intervals, are presented in Appendix A,

Table A4. Simulated results using 5000 simulations for the 95% confidence intervals for the mediated effect for the two-mediator model (n = 50).

				Coverage Probability					Average Interval Length
${\mathit{\alpha }}_{\mathbf{1}}$	${\mathit{\beta }}_{\mathbf{1}}$	${\mathit{\alpha }}_{\mathbf{2}}$	${\mathit{\beta }}_{\mathbf{2}}$	GPQ	Sobel	MC	Percentile	BCa	GPQ	Sobel	MC	Percentile	BCa
0.00	0.00	0.00	0.00	0.9984	0.9996	0.9992	0.9962	0.9920	0.2136	0.1560	0.2090	0.2123	0.2178
0.00	0.00	0.39	0.39	0.9616	0.9346	0.9582	0.9522	0.9568	0.3812	0.3505	0.3717	0.3741	0.3852
0.00	0.14	0.00	0.59	0.9684	0.9764	0.9648	0.9586	0.9412	0.4086	0.3736	0.3987	0.4033	0.4069
0.00	0.39	0.59	0.59	0.9578	0.9468	0.9482	0.9494	0.9478	0.5769	0.5502	0.5638	0.5641	0.5728
0.14	0.00	0.59	0.00	0.9670	0.9740	0.9488	0.9564	0.9340	0.4160	0.3807	0.4058	0.4090	0.4132
0.14	0.39	0.00	0.00	0.9818	0.9690	0.9784	0.9692	0.9492	0.3192	0.2800	0.3101	0.3148	0.3217
0.14	0.39	0.59	0.39	0.9576	0.9396	0.9496	0.9452	0.9456	0.5211	0.4944	0.5077	0.5145	0.5236
0.14	0.59	0.14	0.59	0.9618	0.9572	0.9488	0.9482	0.9404	0.5371	0.5071	0.5246	0.5277	0.5329
0.39	0.00	0.00	0.00	0.9792	0.9868	0.9786	0.9742	0.9538	0.3146	0.2732	0.3048	0.3099	0.3155
0.39	0.00	0.59	0.39	0.9624	0.9492	0.9500	0.9450	0.9426	0.5185	0.4899	0.5080	0.5122	0.5197
0.39	0.39	0.59	0.59	0.9486	0.9364	0.9510	0.9466	0.9506	0.6176	0.5931	0.6045	0.6074	0.6183
0.39	0.59	0.14	0.14	0.9598	0.9430	0.9530	0.9488	0.9512	0.4740	0.4471	0.4619	0.4648	0.4748
0.59	0.00	0.59	0.14	0.9478	0.9504	0.9488	0.9464	0.9304	0.5434	0.5121	0.5272	0.5346	0.5383
0.59	0.14	0.39	0.00	0.9600	0.9626	0.9526	0.9504	0.9360	0.4762	0.4436	0.4626	0.4682	0.4721
0.59	0.39	0.00	0.14	0.9630	0.9470	0.9532	0.9512	0.9514	0.4697	0.4416	0.4601	0.4625	0.4720
0.59	0.59	0.59	0.59	0.9520	0.9412	0.9488	0.9446	0.9460	0.7188	0.6942	0.7043	0.7106	0.7216

,

Table A5. Simulated results using 5000 simulations for the 95% confidence intervals for the mediated effect for the two-mediator model (n = 100).

				Coverage Probability					Average Interval Length
${\mathit{\alpha }}_{\mathbf{1}}$	${\mathit{\beta }}_{\mathbf{1}}$	${\mathit{\alpha }}_{\mathbf{2}}$	${\mathit{\beta }}_{\mathbf{2}}$	GPQ	Sobel	MC	Percentile	BCa	GPQ	Sobel	MC	Percentile	BCa
0.00	0.00	0.00	0.00	0.9992	0.9998	0.9990	0.9988	0.9954	0.1018	0.0754	0.1011	0.1015	0.1042
0.00	0.00	0.39	0.39	0.9630	0.9434	0.9530	0.9536	0.9598	0.2402	0.2309	0.2376	0.2384	0.2439
0.00	0.14	0.00	0.59	0.9530	0.9610	0.9530	0.9534	0.9420	0.2637	0.2517	0.2590	0.2613	0.2627
0.00	0.39	0.59	0.59	0.9568	0.9520	0.9512	0.9484	0.9478	0.3836	0.3752	0.3777	0.3803	0.3838
0.14	0.00	0.59	0.00	0.9598	0.9664	0.9478	0.9562	0.9440	0.2636	0.2515	0.2614	0.2637	0.2650
0.14	0.39	0.00	0.00	0.9632	0.9536	0.9658	0.9544	0.9436	0.1925	0.1793	0.1910	0.1916	0.1949
0.14	0.39	0.59	0.39	0.9570	0.9470	0.9516	0.9472	0.9442	0.3439	0.3355	0.3409	0.3414	0.3453
0.14	0.59	0.14	0.59	0.9556	0.9558	0.9502	0.9420	0.9394	0.3572	0.3475	0.3527	0.3540	0.3560
0.39	0.00	0.00	0.00	0.9656	0.9766	0.9610	0.9592	0.9412	0.1868	0.1718	0.1845	0.1851	0.1870
0.39	0.00	0.59	0.39	0.9560	0.9510	0.9498	0.9448	0.9428	0.3406	0.3315	0.3367	0.3391	0.3424
0.39	0.39	0.59	0.59	0.9524	0.9450	0.9556	0.9496	0.9494	0.4135	0.4058	0.4092	0.4113	0.4159
0.39	0.59	0.14	0.14	0.9510	0.9400	0.9530	0.9440	0.9470	0.3095	0.3012	0.3053	0.3081	0.3128
0.59	0.00	0.59	0.14	0.9548	0.9570	0.9444	0.9476	0.9398	0.3563	0.3459	0.3530	0.3547	0.3559
0.59	0.14	0.39	0.00	0.9550	0.9558	0.9500	0.9504	0.9442	0.3072	0.2965	0.3043	0.3064	0.3080
0.59	0.39	0.00	0.14	0.9504	0.9392	0.9476	0.9506	0.9506	0.3045	0.2959	0.3014	0.3059	0.3104
0.59	0.59	0.59	0.59	0.9556	0.9484	0.9532	0.9492	0.9498	0.4849	0.4772	0.4779	0.4822	0.4867

and

Table A6. Simulated results using 5000 simulations for the 95% confidence intervals for the mediated effect for the two-mediator model (n = 200).

				Coverage Probability					Average Interval Length
${\mathit{\alpha }}_{\mathbf{1}}$	${\mathit{\beta }}_{\mathbf{1}}$	${\mathit{\alpha }}_{\mathbf{2}}$	${\mathit{\beta }}_{\mathbf{2}}$	GPQ	Sobel	MC	Percentile	BCa	GPQ	Sobel	MC	Percentile	BCa
0.00	0.00	0.00	0.00	0.9986	0.9996	0.9982	0.9996	0.9958	0.0503	0.0377	0.0498	0.0502	0.0515
0.00	0.00	0.39	0.39	0.9538	0.9406	0.9510	0.9528	0.9566	0.1614	0.1585	0.1601	0.1618	0.1643
0.00	0.14	0.00	0.59	0.9514	0.9558	0.9546	0.9482	0.9382	0.1767	0.1725	0.1758	0.1766	0.1772
0.00	0.39	0.59	0.59	0.9560	0.9544	0.9492	0.9502	0.9506	0.2624	0.2597	0.2610	0.2624	0.2638
0.14	0.00	0.59	0.00	0.9502	0.9542	0.9526	0.9462	0.9372	0.1776	0.1735	0.1768	0.1783	0.1786
0.14	0.39	0.00	0.00	0.9550	0.9514	0.9534	0.9500	0.9446	0.1255	0.1210	0.1252	0.1258	0.1273
0.14	0.39	0.59	0.39	0.9492	0.9448	0.9524	0.9506	0.9498	0.2343	0.2316	0.2333	0.2351	0.2367
0.14	0.59	0.14	0.59	0.9558	0.9548	0.9500	0.9482	0.9452	0.2442	0.2410	0.2434	0.2446	0.2455
0.39	0.00	0.00	0.00	0.9556	0.9632	0.9526	0.9570	0.9466	0.1205	0.1151	0.1199	0.1204	0.1211
0.39	0.00	0.59	0.39	0.9506	0.9492	0.9538	0.9490	0.9450	0.2321	0.2291	0.2301	0.2324	0.2336
0.39	0.39	0.59	0.59	0.9538	0.9472	0.9572	0.9536	0.9518	0.2840	0.2816	0.2829	0.2842	0.2860
0.39	0.59	0.14	0.14	0.9514	0.9442	0.9480	0.9452	0.9480	0.2108	0.2081	0.2099	0.2104	0.2123
0.59	0.00	0.59	0.14	0.9546	0.9552	0.9488	0.9488	0.9436	0.2434	0.2400	0.2422	0.2437	0.2442
0.59	0.14	0.39	0.00	0.9484	0.9508	0.9534	0.9484	0.9432	0.2084	0.2048	0.2077	0.2097	0.2103
0.59	0.39	0.00	0.14	0.9588	0.9506	0.9492	0.9480	0.9470	0.2076	0.2048	0.2068	0.2084	0.2104
0.59	0.59	0.59	0.59	0.9534	0.9500	0.9540	0.9494	0.9508	0.3341	0.3318	0.3328	0.3346	0.3362

.

Table 2. The percentage of simulated coverage probabilities for the 256 scenarios which are above the 94.5% level.

	GPQ	Sobel	MC	Percentile	BCa
$n=50$	100.00%	74.61%	99.61%	88.28%	51.17%
$n=100$	100.00%	84.77%	98.83%	89.45%	46.09%
$n=200$	98.83%	89.84%	96.88%	91.80%	63.28%

provides a summary of the results for the 256 scenarios regarding the percentage of simulated coverage probabilities that are above the 94.5% level. The GPQ method performed better than the other methods for the case of two mediators. We can see that, regardless of the sample size, the GPQ method had the largest percentages of scenarios where the simulated coverage probability was above $94.5\%$, while the other methods had much smaller percentages.

Table 3. The number of scenarios (out of 256 scenarios for each sample size) where the method had the largest average interval length of the 95% confidence intervals.

	GPQ	Sobel	MC	Percentile	BCa
$n=50$	147	0	0	0	109
$n=100$	53	0	0	0	203
$n=200$	5	0	0	0	251

provides a summary of the results across the 256 scenarios, separately for each sample size, regarding the number of scenarios when the specific method had the largest average interval length of the 95% confidence intervals. When the sample size is 50, the number of scenarios for the GPQ method is higher than that for the BCa bootstrap method, while when the sample sizes are 100 and 200, the BCa bootstrap method has the highest number of scenarios. As shown in Table 2, the GPQ method provides good coverage probabilities relative to the nominal level, being competitive with the other approaches for all simulation setups. The cost of that performance is a small increase in average interval length, as seen from the results from Appendix A, Table A4, Table A5 and Table A6.

4. Discussion

We have described GPQ methods to construct confidence intervals for mediation effects for the case of a single-mediator model, and also for the more complex case of two mediators. While the GPQ method for the simple case is not novel, being proposed before in [8] albeit using a different mathematical presentation, the extension to the multiple-mediator model is new. Our simulation study has provided results consistent with those from [8], showing that for the single-mediator model, the GPQ-based confidence intervals have better coverage performance than the alternative methods, although at the price of being somewhat conservative. Based on the results of our simulation studies, we recommend the use of the GPQ method for both the one-mediator and the two-mediator models because it is simpler to use and offers improved coverage.

The proposed new GPQ method for the two-mediator model is an important step in extending the use of the GPQ methods for mediation analysis. Our current work is somewhat limited by focusing solely on a simple two-mediator model. This is a direct consequence of our goal to provide a comprehensive evaluation, involving 256 simulation scenarios, of a simple (but practically important) model involving two mediators (which are allowed to be correlated) as a proof-of-concept study.

Future work will include the extension of the GPQ approach to causal mediation analysis for more complex models with multiple mediators involving sequential mediation, interactions between exposure and mediators, interactions between mediators, a multilevel structure, and the evaluation of interventional mediation effects (see [14,15]), and to meta-analysis of mediated effects.