A statistical method for synthesizing mediation analyses using the product of coefficient approach across multiple trials

Abstract

Mediation analysis often requires larger sample sizes than main effect analysis to achieve the same statistical power. Combining results across similar trials may be the only practical option for increasing statistical power for mediation analysis in some situations. In this paper, we propose a method to estimate: (1) marginal means for mediation path a, the relation of the independent variable to the mediator; (2) marginal means for path b, the relation of the mediator to the outcome, across multiple trials; and (3) the between-trial level variance–covariance matrix based on a bivariate normal distribution. We present the statistical theory and an R computer program to combine regression coefficients from multiple trials to estimate a combined mediated effect and confidence interval under a random effects model. Values of coefficients a and b, along with their standard errors from each trial are the input for the method. This marginal likelihood based approach with Monte Carlo confidence intervals provides more accurate inference than the standard meta-analytic approach. We discuss computational issues, apply the method to two real-data examples and make recommendations for the use of the method in different settings.

Appendix: Restricted maximum likelihood estimates for \(\Sigma \)

The differential $d (\mathbf{V}_i^{-1})$ of this symmetric matrix can be expressed in terms of the the differential of the matrix itself by noting that by the chain rule,

$$\begin{aligned} d ( \mathbf{V}_i^{-1} \mathbf{V}_i ) = d ( \mathbf{V}_i^{-1} ) \mathbf{V}_i + \mathbf{V}_i^{-1} d ( \mathbf{V}_i ) \end{aligned}$$

(21)

and using $d ( \mathbf{V}_i^{-1} \mathbf{V}_i ) = d ( I_K ) = 0$, we have

$$\begin{aligned} d ( \mathbf{V}_i^{-1}) = - ( \mathbf{V}_i^{-1} ) d \mathbf{V}_i \mathbf{V}_i^{-1} \end{aligned}$$

(22)

To find the differential $d log | \mathbf{V}_i |$ we note that a determinant of any positive definite symmetric matrix $\varPsi$ can be related to an integral for a multivariate normal distribution.

$$\begin{aligned} 1 = \int e^{ - { 1 \over 2 }\mathbf{u}' \varPsi { ^{-1}}\mathbf{u}} / (2 \pi )^{K/2} |\varPsi |^{- { 1 \over 2 }} \mathbf{d}\mathbf{u}\end{aligned}$$

(23)

Thus

$$\begin{aligned} |\varPsi | = J^2(\varPsi ) \end{aligned}$$

(24)

where the definite multivariate integral

$$\begin{aligned} J(\varPsi ) = \int e^{ - { 1 \over 2 }\mathbf{u}' \varPsi { ^{-1}}\mathbf{u}} / (2 \pi )^{K/2} \mathbf{d}\mathbf{u} \end{aligned}$$

(25)

Differentiating the logarithm on each side of Eq. 24,

$$\begin{aligned} d \log |\varPsi |= & {} | \varPsi |{ ^{-1}}d |\varPsi | \\= & {} | \varPsi |{ ^{-1}}d ( J^2 ( \varPsi ) ) \nonumber \\= & {} 2 | \varPsi |{ ^{-1}}J( \varPsi ) d ( J( \varPsi )) \nonumber \\= & {} 2 | \varPsi |^{- { 1 \over 2 }} d ( J( | \varPsi | ) \nonumber \end{aligned}$$

(26)

Differentiating the integral can be done by bringing the differential inside the integrand of Equation and using the chain rule, so

$$\begin{aligned} d \log |\varPsi |= & {} 2 | \varPsi |^{- { 1 \over 2 }} d ( J( | \varPsi | ) \nonumber \\= & {} 2 | \varPsi |^{- { 1 \over 2 }} \int d ( e^{ - { 1 \over 2 }\mathbf{u}' \varPsi { ^{-1}}\mathbf{u}} )/ (2 \pi )^{K/2} \mathbf{d}\mathbf{u}\nonumber \\= & {} 2 | \varPsi |^{- { 1 \over 2 }} \int d \left( - { 1 \over 2 }\mathbf{u}' \varPsi { ^{-1}}\mathbf{u}\right) e^{ - { 1 \over 2 }\mathbf{u}' \varPsi { ^{-1}}\mathbf{u}} / (2 \pi )^{K/2} \mathbf{d}\mathbf{u}\nonumber \\= & {} - | \varPsi |^{- { 1 \over 2 }} \int tr [ \mathbf{u}\mathbf{u}' d ( \varPsi { ^{-1}}) ] e^{ - { 1 \over 2 }\mathbf{u}' \varPsi { ^{-1}}\mathbf{u}} / (2 \pi )^{K/2} \mathbf{d}\mathbf{u}\nonumber \\= & {} - | \varPsi |^{- { 1 \over 2 }} tr [ E ( \mathbf{U}\mathbf{U}' ) d ( \varPsi { ^{-1}}) ] \quad | \varPsi |^{{ 1 \over 2 }} \end{aligned}$$

(27)

$$\begin{aligned}= & {} - tr [ \varPsi d ( \varPsi { ^{-1}}) ] \end{aligned}$$

(28)

$$\begin{aligned} d \log |\varPsi |= & {} tr [ d ( \varPsi ) \varPsi { ^{-1}}] \end{aligned}$$

(29)

Equation 27 comes from the definition of the expectation of the random matrix $\mathbf{U}\mathbf{U}'$, and this expectation evaluates to the variance-covariance matrix for a multivariate normal distribution as in Eq. 28. The final equation in this series comes from Eq. 22. Substituting Eqs. 22 and 29 into Eq. 9 we finally obtain,

$$\begin{aligned} d R\log L= & {} { \sum _{i = 1}^K}{ 1 \over 2 }tr [ ( \mathbf{Y}_i - {\varvec{\mu }})( \mathbf{Y}_i - {\varvec{\mu }})' \mathbf{V}_i{ ^{-1}}d(\mathbf{V}_i) \mathbf{V}_i{ ^{-1}}] - \nonumber \\&{ {K - 1 } \over {2K} } tr [ d ( \mathbf{V}_i) \mathbf{V}_i { ^{-1}}] \nonumber \\= & {} { \sum _{i = 1}^K}{ 1 \over 2 }tr \left[ \left\{ ( \mathbf{Y}_i - {\varvec{\mu }})( \mathbf{Y}_i - {\varvec{\mu }})' \mathbf{V}_i{ ^{-1}}- { { K - 1 } \over K } I_2 \right\} d(\mathbf{V}_i) \mathbf{V}_i{ ^{-1}}\right] \qquad \end{aligned}$$

(30)

Since $\mathbf{V}_i = diag ( \varPsi _i , \varPhi _i ) + \Sigma , i = 1 , \ldots , K$, we have

$$\begin{aligned} d\mathbf{V}_i= & {} d \Sigma \nonumber \\= & {} \begin{pmatrix} d \sigma _{11} &{} 0 \\ 0 &{} 0 \end{pmatrix} + \begin{pmatrix} 0&{} d\sigma _{12} \\ d\sigma _{12} &{} 0 \end{pmatrix} + \begin{pmatrix} 0 &{} 0 \\ 0 &{} d \sigma _{22} \end{pmatrix} \end{aligned}$$

(31)

Substituting this expression into the previous differential results in three scalar equations, one for each component of the variance-covariance matrix.