A Multivariate Generalized Linear Model Approach to Mediation Analysis and Application of Confidence Ellipses

Abstract

Mediation analysis evaluates the significance of an intermediate variable on the causal pathway between an exposure and an outcome. One commonly utilized test for mediation involves evaluation of counterfactual effects, estimated from separate regression models, corresponding to a composite null hypothesis. However, the “compositeness” of this null hypothesis is not commonly acknowledged and accounted for in mediation analyses. We describe a generalized multivariate approach in which these separate regression models are fit simultaneously in a single parsimonious model. This multivariate modeling approach can reproduce standard mediation analysis and has notable advantages over separate regression models, including the ability to combine distributions in the exponential family with any link functions and perform likelihood-based tests of some relevant hypotheses using existing software. We propose the use of a novel visual representation of confidence intervals of the two estimates for the indirect path with the use of a confidence ellipse. The calculation of the confidence ellipse is facilitated by the multivariate approach, can test the components of the composite null hypothesis under a single experiment-wise type I error rate, and does not require estimation of the standard error of the product of coefficients from two separate regressions. This method is illustrated using three examples. The first compares results between the multivariate method and separate regression models. The second example illustrates the proposed methods in the presence of an exposure–mediator interaction, missing data and confounding, and the third example utilizes these proposed methods for an outcome and mediator with negative binomial distributions.

Appendices

Appendix 1: Derivation of Counterfactual effects for the Generalized Linear Model

Note that many of these effects depend on the chosen values ${A}={a}*$, or ${M}={m}*$, or both.

Recall that

$$\begin{aligned}&{h}_{M} \left\{ {{E}\left[ {{M|a,c}} \right] } \right\} =\beta _0 +\beta _1 a+{\beta }'_2 c\,\mathrm{and}\\&{h}_{Y} \left\{ {{E}\left[ {{Y|a,m,c}} \right] } \right\} =\theta _0 +\theta _1 a+\theta _2 m+\theta _3 am+{\theta }'_4 c \\&\quad =\theta _0 +\theta _1 a+\left( {\theta _2 +\theta _3 a} \right) m+{\theta }'_4 c \\&\quad =\theta _0 +\theta _1 a+\varphi _a m+{\theta }'_4 c, \end{aligned}$$

where $\varphi _a =\theta _2 +\theta _3 a$ denotes the effect of M when $A=a$, c is a vector of covariates, and ${\beta }'_2$ and ${\theta }'_4$ are vectors of regression coefficients.

Note: If the two c’s from M and Y are not the same, we would have to condition on their union.

Controlled Direct Effect defined on the scale of the outcome (inverse link)

$$\begin{aligned} \mathrm{CDE}= & {} h_Y^{-1} \left[ {h_Y \left\{ {E\left[ {Y\,|\,a,m,c} \right] } \right\} -h_Y \left\{ {E\left[ {Y\,|\,a^{*},m,c} \right] } \right\} } \right] \\= & {} h_Y^{-1} \left[ {\left( {\theta _0 +\theta _1 a+\theta _2 m+\theta _3 am+{\theta }'_4 c} \right) -\left( {\theta _0 +\theta _1 a^{*}+\theta _2 m+\theta _3 a^{*}m+{\theta }'_4 c} \right) } \right] \\= & {} h_Y^{-1} \left[ {\left( {\theta _1 +\theta _3 m} \right) \left( {a-a^{*}} \right) } \right] , \end{aligned}$$

where $h_Y^{-1} \left\{ \right\}$ denotes the inverse function of $h_Y \left\{ \right\}$ and m is set to a specified value.

Natural direct effect evaluated at $M=m_\mathrm{a}^{*}$

$$\begin{aligned} \mathrm{NDE}= & {} h_Y^{-1} \left[ {h_Y \left\{ {E\left[ {Y\,|\,a,m_{a^{*}} ,c} \right] } \right\} -h_Y \left\{ {E\left[ {Y\,|\,a^{*},m_{a^{*}} ,c} \right] } \right\} } \right] \\= & {} h_Y^{-1} \left[ \left( {\theta _0 +\theta _1 a+\theta _2 m_{a^{*}} +\theta _3 am_{a^{*}} +\theta _4^{\prime } c} \right) \right. \\&\left. -\left( {\theta _0 +\theta _1 a^{*}+\theta _2 m_{a^{*}} +\theta _3 a^{*}m_{a^{*}} +\theta _4^{\prime } c} \right) \right] \\= & {} h_y^{-1} \left[ {\theta _1 \left( {a-a^{*}} \right) +\theta _3 m_{a^{*}} \left( {a-a^{*}} \right) } \right] \\= & {} h_y^{-1} \left[ {\left( {\theta _1 +\theta _3 m_{a^{*}}}\right) \left( {a-a^{*}} \right) } \right] \\= & {} h_y^{-1} \left[ {\left[ {\theta _1 +\theta _3 h_M^{-1} \left\{ {\beta _0 +\beta _1 a^{*}+\beta ^{\prime }_2 c} \right\} } \right] \left( {a-a^{*}} \right) } \right] , \end{aligned}$$

where $m_{a^{*}} =\mathrm{E}\left[ {\mathrm{M\,|\,a}^{\mathrm{*}}\mathrm{,c}} \right] =h_M^{-1} \left\{ {\beta _0 +\beta _1 a^{*}+{\beta }'_2 c} \right\}$.

Natural indirect effect

$$\begin{aligned} \mathrm{NIE}= & {} h_y^{-1} \left[ {h_Y \left\{ {E\left[ {Y\,|\,a,m_a ,c} \right] } \right\} -h_Y \left\{ {E\left[ {Y\,|\,a,m_{a^{*}} ,c} \right] } \right\} } \right] \\= & {} h_y^{-1} \left[ \left( {\theta _0 +\theta _1 a+\theta _2 m_a +\theta _3 am_a +\theta ^{\prime }_4 c} \right) \right. \\&\left. -\left( {\theta _0 +\theta _1 a+\theta _2 m_{a^{*}} +\theta _3 am_{a^{*}} +\theta ^{\prime }_4 c} \right) \right] \\= & {} h_y^{-1} \left[ {\left( {\theta _2 +\theta _3 a} \right) \left( {m_a -m_{a^{*}}}\right) } \right] \\= & {} h_y^{-1} \left[ {\left( {\theta _2 +\theta _3 a} \right) \left( {h_M^{-1} \left\{ {\beta _0 +\beta _1 a+{\beta }'_2 c} \right\} -h_M^{-1} \left\{ {\beta _0 +\beta _1 a^{*}+{\beta }'_2 c} \right\} } \right) } \right] \\= & {} h_y^{-1} \left[ {\varphi _a \left( {h_M^{-1} \left\{ {\beta _0 +\beta _1 a+{\beta }'_2 c} \right\} -h_M^{-1} \left\{ {\beta _0 +\beta _1 a^{*}+{\beta }'_2 c} \right\} } \right) } \right] , \end{aligned}$$

where $m_a =h_M^{-1} \left\{ {\beta _0 +\beta _1 a+{\beta }'_2 c} \right\}$,$m_{a^{*}} =h_M^{-1} \left\{ {\beta _0 +\beta _1 a^{*}+{\beta }'_2 c} \right\}$ and a is set to a specified value when $\theta _3 \ne 0$.

Total effect

$$\begin{aligned} \mathrm{TE}=\mathrm{NIE}+\mathrm{NDE} \end{aligned}$$

We propose that NIE be used to evaluate whether mediation is present. If there is an interaction, a in $\varphi _a =\theta _2 +\theta _3 a$ must be specified. If A is dichotomous (say $A=1$ for males and $A=0$ for females), then $\mathrm{NIE}\left( 1 \right)$ could estimate mediation for males and $\mathrm{NIE}\left( 0 \right)$ for females. If A is continuous, a might be chosen as the mean value of A.

If there is no interaction between the mediator and the exposure (i.e., $\theta _3 =0)$ and $\varphi _a =\theta _2$, then the counterfactuals simplify as follows

$$\begin{aligned}&\mathrm{CDE}=\mathrm{NDE}=h_Y^{-1} \left\{ {\theta _1 \left( {a-a^{*}} \right) } \right\} \,\,\mathrm{and}\\&\mathrm{NIE}=h_y^{-1} \left\{ {\theta _2 \left[ {h_M^{-1} \left\{ {\beta _0 +\beta _1 a+\beta ^{\prime }_2 c} \right\} -h_M^{-1} \left\{ {\beta _0 +\beta _1 a^{*}+\beta ^{\prime }_2 c} \right\} } \right] } \right\} . \end{aligned}$$

If the outcome and the mediator have identity links, such that ${E}\left[ {{Y|a,m,c}} \right] =\theta _0 +\theta _1 a+\theta _2 m+\theta _3 am+{\theta }'_4 c$ and ${E}\left[ {{M|a,c}} \right] =\beta _0 +\beta _1 a+{\beta }'_2 c$, then

$$\begin{aligned} \mathrm{CDE}&=\left( {\theta _1 +\theta _3 m} \right) \left( {a-a^{*}} \right) \\ \mathrm{NDE}&=\left( {\theta _1 +\theta _3}\right) \left( {\beta _0 +\beta _1 a^{*}+\beta ^{\prime }_2 c} \right) \left( {a-a^{*}} \right) \\ \mathrm{NIE}&=\left( {\theta _2 +\theta _3 a} \right) \beta _1 (a-a^{*})\\&=\varphi _a \beta _1 (a-a^{*}) \end{aligned}$$

as reported by Valeri and VanderWeele [31]. In this case, we propose that in the absence of an interaction, $\beta _1 \theta _2$, and in the presence of an interaction, $\beta _1 \varphi _a$, be used to evaluate whether mediation is present when $A=a$.

For the case where the outcome is binary and fit using a logistic regression, Valeri and VanderWeele [31] calculate the direct and indirect effect odds ratios. These can be derived from the estimates provided above as follows:

$$\begin{aligned}&\mathrm{OR}^\mathrm{CDE}=\exp \left\{ {\left( {\theta _1 +\theta _3 m} \right) \left( {a-a^{*}} \right) } \right\} ,\\&\mathrm{OR}^\mathrm{NDE}=\exp \left\{ {\left[ {\left( {\theta _1 +\theta _3}\right) h_M^{-1} \left\{ {\beta _0 +\beta _1 a^{*}+\beta ^ {\prime }_2 c} \right\} } \right] \left( {a-a^{*}} \right) } \right\} \,\,\mathrm{and}\\&\mathrm{OR}^\mathrm{NIE}=\exp \left\{ {\left( {\theta _2 +\theta _3 a} \right) \left[ {h_M^{-1} \left\{ {\beta _0 +\beta _1 a+\beta ^ {\prime }_2 c} \right\} -h_M^{-1} \left\{ {\beta _0 +\beta _1 a^{*}+\beta ^{\prime }_2 c} \right\} } \right] } \right\} . \end{aligned}$$

Appendix 2: Comparison of the Standard Errors Computed from the Multivariate Approach Implemented in NLMIXED and the Classical Separate Univariate Regression Method Used in the SAS Macro Provided by Valeri and VanderWeele [31]

Most regression programs, including REG and GENMOD used in SAS Macro provided by Valeri and VanderWeele [31], compute restricted maximum likelihood (REML) estimates of the residual variances of the regressions of M on A and Y on A and M, MSE$_{1}$ and MSE$_{2}$, respectively. Instead, NLMIXED computes maximum likelihood (ML) estimates $S_{11 }$ and $S_{22 }$ such that MSE$_{1}=n S_{11}/(n-2)$, and MSE$_{2}=n S_{22}/(n-3)$, where n is the number of subjects. The same proportionality will hold for variances of the regression coefficients. For example, $\mathrm{SE}_\mathrm{REML} \left( {\hat{{\theta }}_2}\right) =\mathrm{SE}_\mathrm{ML} \left( {\hat{{\theta }}_2 } \right) \sqrt{\frac{\mathrm{n}}{\mathrm{n}-3}}$, and $\mathrm{SE}_\mathrm{REML} \left( {\hat{{\beta }}_1}\right) =\mathrm{SE}_\mathrm{ML} \left( {\hat{{\beta }}_1}\right) \sqrt{\frac{{n}}{n-2}}$.