Robust multivariable Mendelian randomization based on constrained maximum likelihood

Summary

Mendelian randomization (MR) is a powerful tool for causal inference with observational genome-wide association study (GWAS) summary data. Compared to the more commonly used univariable MR (UVMR), multivariable MR (MVMR) not only is more robust to the notorious problem of genetic (horizontal) pleiotropy but also estimates the direct effect of each exposure on the outcome after accounting for possible mediating effects of other exposures. Despite promising applications, there is a lack of studies on MVMR’s theoretical properties and robustness in applications. In this work, we propose an efficient and robust MVMR method based on constrained maximum likelihood (cML), called MVMR-cML, with strong theoretical support. Extensive simulations demonstrate that MVMR-cML performs better than other existing MVMR methods while possessing the above two advantages over its univariable counterpart. An application to several large-scale GWAS summary datasets to infer causal relationships between eight cardiometabolic risk factors and coronary artery disease (CAD) highlights the usefulness and some advantages of the proposed method. For example, after accounting for possible pleiotropic and mediating effects, triglyceride (TG), low-density lipoprotein cholesterol (LDL), and systolic blood pressure (SBP) had direct effects on CAD; in contrast, the effects of high-density lipoprotein cholesterol (HDL), diastolic blood pressure (DBP), and body height diminished after accounting for other risk factors.

A robust and efficient multivariable Mendelian randomization method is proposed to estimate the direct effect of each of multiple exposures on an outcome after accounting for possible mediating effects through other exposures. An application to infer causal relationships between eight cardiometabolic risk factors and coronary artery disease highlights its usefulness.

Introduction

Mendelian randomization (MR), including its default version, univariable MR (UVMR), is an instrumental variable (IV) method that utilizes genetic variants as IVs to infer the causal relationship between an exposure and an outcome.^1,2,3 With numerous publicly available large-scale genome-wide association study (GWAS) summary data, MR has recently become popular and powerful to infer causal relationships even in the presence of unmeasured confounding and reverse causation.⁴ In UVMR, a valid IV is required to satisfy the following three assumptions:

UV-A1: the IV is (marginally) associated with the exposure;

UV-A2: the IV is independent of the unmeasured confounder;

UV-A3: the IV is independent of the outcome conditional on the exposure and confounder.

Despite promising and wide applications of UVMR to observational data for causal inference, in reality, these assumptions may not always hold. In particular, the widespread pleiotropy is a major concern, violating UV-A2 or UV-A3; that is, a genetic variant is associated with the outcome other than through the exposure of interest.⁵ Screening procedures can be applied to avoid the use of pleiotropic variants, but there may be only few or even no variants solely associated with the exposure, leading to loss of power, in addition to likely biases. A number of UVMR methods robust to pleiotropy have been proposed, but under different untestable assumptions.^{5,6,7,8,9,10,11} Alternatively, one can alleviate the problem by including other associated risk factors as multiple exposures in the model, motivating the use of multivariable MR (MVMR).¹² MVMR includes multiple exposures in the model and allows the genetic variants to be associated with one or more of them without violating the IV assumption. This is useful especially when a set of related risk factors for the outcome of interest share many commonly associated genetic variants, e.g., various lipids.¹³ Another scenario where MVMR would be useful is in mediation analysis.¹⁴ In MVMR, the direct effect of one exposure on the outcome not mediated through the rest of the exposures is estimated, while in UVMR, only the total effect of the exposure on the outcome, including that mediated through other exposures, is estimated. Therefore, MVMR would be useful when the aim is to study the causal mechanism of a set of risk factors on the outcome after accounting for potential causal pathways among the risk factors.

Despite these exciting advantages of MVMR over UVMR, some assumptions are still required to have valid IVs for MVMR. Paralleling that in UVMR, in MVMR a valid IV must satisfy the following:

MV-A1: the IV is associated with at least one exposure conditional on the other exposures included in the model;

MV-A2: the IV is independent of any confounder of each exposure-outcome pair;

MV-A3: the IV is independent of the outcome conditional on all exposures included in the model and the confounders.

Since a marginal association does not imply a conditional association, while the reverse is also true but less likely, in that sense assumption MV-A1 is stronger than UV-A1 while MV-A3 is weaker than UV-A3. We give an example on each of the two cases in the simulation results for mediation analysis. We note that the above assumptions are for one IV instead of a set of IVs as considered by others.¹⁵ The main reason is that, differing from the previous MVMR approaches, we would like to consider robust MVMR in the presence of some invalid IVs violating one or more of the above three assumptions. While there are many robust UVMR methods in the literature, however, only few MVMR methods exist, which may not be sufficiently robust and efficient (as to be shown).^16,17,18 In this work, we propose a robust and efficient MVMR method by extending the univariable MR-cML⁷ (referred to as UVMR-cML throughout the paper) to the multivariable setting, called MVMR-cML. Based on constrained maximum likelihood (cML), UVMR-cML is robust to the violation of all three IV assumptions (in the univariable case): it allows the presence of invalid IVs violating any or all of the three IV assumptions as long as the plurality condition and several other mild conditions hold. Under some mild conditions, it can consistently identify invalid IV(s) with either or both of correlated and uncorrelated pleiotropy, yielding a consistent and asymptotic normal estimator of the causal effect. It has been shown to have robust performance under various simulation setups and real data analyses.⁷ Under a similar constrained maximum likelihood framework, MVMR-cML is expected to enjoy the same good properties as its univariable version. In particular, as its UV counterpart, it is robust to the presence of invalid IVs violating some or all of the three IV assumptions, including those with correlated or uncorrelated pleiotropy as to be confirmed in simulations. The flexible likelihood framework also allows us to account for possible correlations among any sets of GWAS summary statistics (e.g., due to overlapping samples), and thus the method can be widely applied in different scenarios, including one-sample, two-sample, or even mixed-sample designs. We develop an efficient R package with Rcpp to integrate different versions of MR-cML for various uses.

The rest of the paper is organized as follows. First, we extend the plurality condition for model identification from the univariable case¹⁹ to the current multivariable context. Then we introduce MVMR-cML and its two variants based on model selection and data perturbation, respectively. We show the superior performance of the proposed method over other existing MVMR methods through extensive simulations. Lastly, we consider an application to study the direct effects of eight cardio-metabolic risk factors on coronary artery disease (CAD [MIM: 608320]).

Material and methods

Causal model and its interpretation

Suppose we have $m$ independent SNPs, $L$ exposures $X_l$’s with $l=1,\dots ,L$, an unmeasured confounder (ensemble) $U$, and an outcome $Y$. We consider the following true causal model (Figure 1A), which can be viewed as a multivariable version of the true causal model presented in Xue et al.⁷:

$$b_{Yi}={\theta }_1{b}_{X_{1}i}+\dots +{\theta }_L{b}_{X_{L}i}+{\alpha }_i+{\beta }_{YU}{\phi }_i:={\theta }_1{b}_{X_{1}i}+\dots +{\theta }_L{b}_{X_{L}i}+r_i\text{,}$$ (Equation 1)

where $b_{X_{l}i}$ and $b_{Yi}$ are the (true) marginal associations between SNP/IV $G_i$ and the $l$-th exposure and the outcome, respectively; ${\theta }_l$ is the direct causal effect of $X_l$ on $Y$; and $r_i={\alpha }_i+{\beta }_{YU}{\phi }_i$ represents the total pleiotropic effect of $G_i$ on $Y$ (not through $X_l$’s). For example, in the absence of any causal relationship among the exposures in Figure 1A, we have $b_{X_{l}i}={\gamma }_{X_{l}i}+{\beta }_{X_{l}U}{\phi }_i$, for $l=1,\dots ,L$; in Figure 1B with a causal relation from $X_1$ to $X_2$, we have $b_{X_{1}i}={\gamma }_{X_{1}i}+{\beta }_{X_{1}U}{\phi }_i$ and $b_{X_{2}i}={\gamma }_{X_{2}i}+{\beta }_{X_{2}U}{\phi }_i+\xi b_{X_{1}i}$.

Figure 1.

True causal models for multivariable MR

(A and B) A general (A) and a specific (B) causal graph showing the relationships among one IV $\left(G_i\right)$, multiple exposures $\left(X_1,\dots ,X_L\right)$, an unmeasured confounder $\left(U\right)$, and the outcome $\left(Y\right)$.

The three valid IV assumptions for a valid IV $G_i$ require, respectively, ${\gamma }_{X_{l}i}\ne 0$ (for some $1\le l\le L$), ${\phi }_i=0$, and ${\alpha }_i=0$, the last two of which imply no pleiotropic effects with $r_i=0$. On the other hand, ${\alpha }_i\ne 0$ and ${\phi }_i\ne 0$ lead to so-called uncorrelated and correlated pleiotropy, respectively. It is a major goal here to consider robust MVMR analysis in the possible presence of some invalid IVs with $r_i\ne 0$.

In the true causal model depicted in Figure 1A, for MVMR we do not need to specify causal relationships among the $L$ exposures, the presence of which is likely and, importantly, implies that the direct causal effect ${\theta }_l$ is in general different from the total causal effect of $X_l$ on $Y$. For illustration, we consider a simple example with $L=2$ exposures, one of which mediates the effect of the other on the outcome (Figure 1B); more examples can be found elsewhere.^15,20 The corresponding linear regression model for MVMR is

$$Y=X_1{\theta }_1+X_2{\theta }_2+U{\beta }_{YU}+G_i{\alpha }_i+ϵ\text{,}$$

where $ϵ$ is a random error. Accordingly, we can interpret the direct (causal) effect ${\theta }_1$ of $X_1$ on $Y$ as that of changing $X_1$ while holding $X_2$ fixed. In contrast, by replacing $X_2=\xi X_1+G_i{\gamma }_{X_{2}i}+U{\beta }_{X_{2}U}+e_2$ (with $e_2$’s being a random error), we obtain the corresponding linear regression model for UVMR for $X_1$ as

$$Y=X_1\left({\theta }_1+\xi {\theta }_2\right)+U\left({\beta }_{YU}+{\beta }_{X_{2}U}{\theta }_2\right)+G_i\left({\alpha }_i+{\gamma }_{X_{2}i}{\theta }_2\right)+\left(ϵ+e_2{\theta }_2\right)\text{,}$$

giving the total effect of $X_1$ on $Y$ as ${\theta }_1+\xi {\theta }_2$, including both the direct effect ${\theta }_1$ and the indirect effect $\xi {\theta }_2$ mediated through $X_2$. In addition, this example also illustrates an advantage of MVMR over UVMR: if ${\alpha }_i=0$ and ${\phi }_i=0$, $G_i$ is likely a valid IV with no pleiotropic effect (i.e., $r_i=0$) in MVMR; however, in UVMR for $X_1$, $G_i$ is an invalid IV with pleiotropic effect $r_i={\gamma }_{X_{2}i}{\theta }_2\ne 0$.

Model identification

Suppose the ground truth to Equation 1 is given by $b_{Yi}={\theta }_1^{\ast }{b}_{X_{1}i}+\dots +{\theta }_L^{\ast }{b}_{X_{L}i}+r_i^{\ast }$, and the set of valid IVs is ${\mathcal{V}}^{\ast }=\left\{i:r_i^{\ast }=0\right\}$ with $m_0=\left|{\mathcal{V}}^{\ast }\right|$. Here, we use the asterisk to denote the true value of a parameter and only consider relevant IVs with ${\mathit{b}}_{Xi}={\left(b_{X_{1}i},\dots ,b_{X_{L}i}\right)}^T\ne \mathbf{0}$. We say that the true parameters ${\mathit{r}}^{\ast }={\left(r_1^{\ast },\dots ,r_m^{\ast }\right)}^T$ and ${\mathit{\theta }}^{\ast }={\left({\theta }_1^{\ast },\dots ,{\theta }_L^{\ast }\right)}^T$ are identifiable if, given the (true) marginal associations $b_{Yi}$ and ${\mathit{b}}_{Xi}\ne \mathbf{0}$ for $i=1,\dots ,m$, there is a unique solution $\left({\mathit{r}}^{\ast },{\mathit{\theta }}^{\ast }\right)$ (as the ground truth) to

$$b_{Yi}=r_i+{\theta }_1{b}_{X_{1}i}+\dots +{\theta }_L{b}_{X_{L}i},i=1,\dots ,m\text{,}$$ (Equation 2)

under the constraint that any solution is obtained from the largest set(s) of IVs with their corresponding $r_i=0$ (as a part of the solution). Note that the constraint is needed because otherwise we would have many solutions with $m+L$ unknown parameters ($r_i$’s and ${\theta }_j$’s) in $m$ linear equations.

There are two aspects to consider for model identification. First, unique to MVMR (i.e., different from UVMR), even in the ideal case of using only $m_0$ valid IVs (with $r_i=0$), in order to have a unique solution to the system of linear equations in Equation 2, we require that the marginal association matrix ${\mathit{B}}_{{\mathcal{V}}^{\ast }}={\left({\mathit{b}}_{X1},{\mathit{b}}_{X2},\dots ,{\mathit{b}}_{X{m}_0}\right)}^T$ to be of full column rank, which implies at least one valid IV for each exposure and thus $m_0\ge L$. This condition ensures the marginal associations between the IVs and exposures are not multicollinear, as pointed out by others; a conditional F-test has been proposed to test for possible violation of this condition.¹⁷ The other aspect, similar to that in UVMR, is a plurality condition in the presence of invalid IVs with $r_i\ne 0$: the (true) valid IVs form the largest group to give the same causal parameter estimate. The two aspects are combined into the following assumption.

Assumption 1: Suppose the matrix of ${\mathit{b}}_{Xi}$’s for $i\in {\mathcal{V}}^{\ast }$, ${\mathit{B}}_{{\mathcal{V}}^{\ast }}={\left[{\mathit{b}}_{Xi}\right]}_{i\in {\mathcal{V}}^{\ast }}\in {\mathbb{R}}^{m_0\times L}$, has full column rank $L$. Moreover, the following multivariable plurality condition holds:

$$\left|{\mathcal{V}}^{\ast }\right|>{\max }_{\left\{\mathit{c}\ne \mathbf{0},\mathit{c}\in {\mathbb{R}}^L\right\}}\left|\left\{i:r_i^{\ast }={\mathit{b}}_{Xi}^T\mathit{c}\right\}\right|\text{.}$$ (Equation 3)

Theorem 1: Given $b_{Yi}$ and ${\mathit{b}}_{Xi}$ for $i=1,\dots ,m$, the true parameters ${\mathit{\theta }}^{\ast }$ and ${\mathit{r}}^{\ast }$ in Equation 2 are identifiable if and only if Assumption 1 holds.

The proof of Theorem 1 is given in section S1.1 in the supplemental information. We first give some intuitive explanation on why the plurality condition is needed. If it is violated, it means that some invalid IVs form the largest group with a solution $\mathit{c}+{\mathit{\theta }}^{\ast }\ne {\mathit{\theta }}^{\ast }$ to Equation 2. Next, we note that the plurality condition in Equation 3 is a generalization of that for the univariable case with $L=1$. Theorem 1 in Guo et al.¹⁹ states that, given $b_{Yi}$ and $b_{X_{1}i}$, the model parameters ${\theta }_1$ and $r_1,\dots ,r_m$ are identifiable if and only if the following plurality rule condition holds:

$$\left|{\mathcal{V}}^{\ast }\right|>{\max }_{c\ne 0}\left|\left\{i:r_i^{\ast }/b_{X_{1}i}=c\right\}\right|\text{,}$$ (Equation 4)

where ${\mathcal{V}}^{\ast }$ is the set of valid IVs with $r_i=0$ and $b_{X_{1}i}\ne 0$. It is clear that Equation 3 reduces to Equation 4 with $L=1$.

New method: Multivariable MRcML (MVMR-cML)

We extend the UVMR-cML proposed in Xue et al.⁷ to the multivariable case. The goal of MVMR-cML is to estimate the direct causal effect of each exposure on the outcome (while relaxing the no-pleiotropy condition for valid IVs). Denote $\{{\hat{\beta }}_{X_{1}i},\dots ,{\hat{\beta }}_{X_{L}i},{\hat{\beta }}_{Yi}$, ${{\hat{\sigma }}_{X_{1}i}^2,\dots ,{\hat{\sigma }}_{X_{L}i}^2,{\hat{\sigma }}_{Yi}^2\}}_{i=1}^m$ as the GWAS summary statistics of $m$ (nearly) independent SNPs/IVs, each of which is selected on the basis of its significant marginal association with at least one of the exposures (e.g., at the usual genome-wide significance level of p value $<$ 5e−8). We have

$$\begin{array}{l}{\hat{\beta }}_{X_{l}i}=b_{X_{l}i}+{ϵ}_{X_{l}i},l=1,\dots ,L\text{,}\\ {\hat{\beta }}_{Yi}=b_{Yi}+{ϵ}_{Yi}\text{,}\end{array}$$

with $\text{var}\left({ϵ}_{X_{l}i}\right)={\sigma }_{X_{l}i}^2$ and $\text{var}\left({ϵ}_{Yi}\right)={\sigma }_{Yi}^2$. We assume ${\sigma }_{X_{l}i}^2$ and ${\sigma }_{Yi}^2$ are known or well estimated as ${\hat{\sigma }}_{X_{l}i}^2$ and ${\hat{\sigma }}_{Yi}^2$, respectively. Then we have the model for SNP $i$:

$${\hat{\mathit{\beta }}}_i={\left({\hat{\beta }}_{X_{1}i},\dots ,{\hat{\beta }}_{X_{L}i},{\hat{\beta }}_{Yi}\right)}^T\sim \mathcal{N}\left({\mathit{b}}_i={\left(b_{X_{1}i},\dots ,b_{X_{L}i},\sum _{l=1}^L{\theta }_l{b}_{X_{l}i}+r_i\right)}^T,{\mathbf{\Sigma }}_i\right)\text{,}$$ (Equation 5)

where ${\mathbf{\Sigma }}_i=\left(\begin{array}{ccccc}{\sigma }_{X_{1}i}^2& {\rho }_{12}{\sigma }_{X_{1}i}{\sigma }_{X_{2}i}& \dots & {\rho }_{1L}{\sigma }_{X_{1}i}{\sigma }_{X_{L}i}& {\rho }_{1Y}{\sigma }_{X_{1}i}{\sigma }_{Yi}\\ & {\sigma }_{X_{2}i}^2& \dots & {\rho }_{2L}{\sigma }_{X_{2}i}{\sigma }_{X_{L}i}& {\rho }_{2Y}{\sigma }_{X_{2}i}{\sigma }_{Yi}\\ ⋮& & \ddots & & ⋮\\ & \ddots & & {\sigma }_{X_{L}i}^2& {\rho }_{LY}{\sigma }_{X_{L}i}{\sigma }_{Yi}\\ & & \dots & & {\sigma }_{Yi}^2\end{array}\right)$, ${\rho }_{l{l}^{\prime }}$ ($l,l^{\prime }=1,\dots ,L$ and $l\ne l^{\prime }$) is the correlation between the two GWAS summary estimates for exposures $X_l$ and $X_{l^{\prime }}$, and ${\rho }_{lY}$ ($l=1,\dots ,L$) is the correlation between the two GWAS summary datasets for exposure $X_l$ and outcome $Y$. When the $L+1$ GWAS summary datasets are calculated from $L+1$ sets of non-overlapping samples, respectively, ${\rho }_{l{l}^{\prime }}={\rho }_{lY}=0$ and ${\mathbf{\Sigma }}_i$ is diagonal. Otherwise, we can use bivariate linkage disequilibrium score regression (LDSC)²¹ or the sample correlation between two sets of GWAS Z scores across null SNPs²² to estimate all the $\rho$’s.

Since the $m$ IVs are independent, the log likelihood of the observed GWAS data (up to some constants) is

$$l\left(\mathit{\theta },\left\{{\mathit{b}}_{Xi}\right\},\left\{r_i\right\};\left\{{\hat{\mathit{\beta }}}_i,{\mathbf{\Sigma }}_i\right\}\right)=\sum _{i=1}^m{l}_i\left(\mathit{\theta },{\mathit{b}}_{Xi},r_i;{\hat{\mathit{\beta }}}_i,{\mathbf{\Sigma }}_i\right)=-\frac{1}{2}\sum _{i=1}^m{\left({\hat{\mathit{\beta }}}_i-{\mathit{b}}_i\right)}^T{\mathbf{\Sigma }}_i^{-1}\left({\hat{\mathit{\beta }}}_i-{\mathit{b}}_i\right)\text{,}$$ (Equation 6)

where $\mathit{\theta }={\left({\theta }_1,\dots ,{\theta }_L\right)}^T$, and we use $\left\{{\mathit{b}}_{Xi}\right\}=\left\{{\left(b_{X_{1}i},\dots ,b_{X_{L}i}\right)}^T,i=1,\dots ,m\right\}$ to represent a set of the parameters and similarly for $\left\{r_i\right\}$ and $\left\{{\hat{\mathit{\beta }}}_i,{\mathbf{\Sigma }}_i\right\}$.

Under the constraint that the number of invalid IVs is $K$, we estimate the unknown parameters by solving the following constrained maximum likelihood:

$$\begin{array}{l}\left(\hat{\mathit{\theta }},\left\{{\hat{\mathbf{b}}}_{Xi}\right\},\left\{{\hat{r}}_i\right\}\right)=\underset{\mathit{\theta },\left\{{\mathbf{b}}_{Xi}\right\},\left\{r_i\right\}}{\arg \max }l\left(\mathit{\theta },\left\{{\mathbf{b}}_{Xi}\right\},\left\{r_i\right\};\left\{{\hat{\mathit{\beta }}}_i,{\mathbf{\Sigma }}_i\right\}\right)\\ \text{subject}\text{to}\sum _{i=1}^{m}I\left(r_i\ne 0\right)=K\text{.}\end{array}$$

For a given number of invalid IVs, $K$, a coordinate descent-like algorithm is implemented as follows: at the $\left(t+1\right)$-th iteration,

Step 1: calculate ${\hat{r}}_i^{\left(t+1\right)}$ by solving ${\frac{\partial l_i}{\partial r_i}|}_{{\mathit{\theta }}^{\left(t+1\right)},{\mathit{b}}_{Xi}^{\left(t\right)}}=0$, order $d_i^{\left(t+1\right)}=l_i\left({\mathit{\theta }}^{\left(t\right)},{\mathit{b}}_{Xi}^{\left(t\right)},{\hat{r}}_i^{\left(t+1\right)};{\hat{\mathit{\beta }}}_i,{\mathbf{\Sigma }}_i\right)-l_i\left({\mathit{\theta }}^{\left(t\right)},{\mathit{b}}_{Xi}^{\left(t\right)},0;{\hat{\mathit{\beta }}}_i,{\mathbf{\Sigma }}_i\right)$ decreasingly, then for $i=1,\dots ,K$, update $r_{\left(i\right)}^{\left(t+1\right)}={\hat{r}}_{\left(i\right)}^{\left(t+1\right)}$; for $i=K+1,\dots ,m$, update $r_{\left(i\right)}^{\left(t+1\right)}=0$;

Step 2: update ${\mathit{b}}_{Xi}^{\left(t+1\right)}$ by solving ${\frac{\partial l_i}{\partial {\mathit{b}}_{Xi}}|}_{{\mathit{\theta }}^{\left(t\right)},r_i^{\left(t+1\right)}}=0$ for $i=1,\dots ,m$;

Step 3: update ${\mathit{\theta }}^{\left(t+1\right)}$ by solving ${\frac{\partial l}{\partial \mathit{\theta }}|}_{\left\{{\mathit{b}}_{Xi}^{\left(t+1\right)},r_i^{\left(t+1\right)}\right\}}=0$.

We repeat the above three steps until convergence, obtaining the final estimates $\hat{\mathit{\theta }}\left(K\right)$ and ${\left\{{\hat{\mathit{b}}}_{Xi}\left(K\right),{\hat{r}}_i\left(K\right)\right\}}_{i=1}^m$. As in UVMR-cML,⁷ it is notable that at the convergence the (estimated) invalid IVs (with ${\hat{r}}_i\ne 0$) do not contribute to estimating $\mathit{\theta }$, and the resulting cMLE of $\mathit{\theta }$ is the same as the maximum (profile) likelihood estimator being applied to all (selected) valid IVs.

We select the number of invalid IVs, $K$, from a candidate set $\mathcal{K}$ based on the following Bayesian information criterion (BIC):

$$\text{BIC}\left(K\right)=-2l\left(\hat{\mathit{\theta }}\left(K\right),\left\{{\hat{\mathit{b}}}_{Xi}\left(K\right),{\hat{r}}_i\left(K\right)\right\};\left\{{\hat{\mathit{\beta }}}_i,{\mathbf{\Sigma }}_i\right\}\right)+\mathit{log}\left(N\right)·K\text{,}$$ (Equation 7)

where $N$ is the minimum sample size of all GWAS datasets used in the model. We select $\hat{K}=\arg {\min }_{K\in \mathcal{K}}\text{BIC}\left(K\right)$ and $\hat{\mathcal{V}}=\left\{i|{\hat{r}}_i\left(\hat{K}\right)=0,i=1,\dots ,m\right\}$. Then the final cMLE of $\mathit{\theta }$ is $\hat{\mathit{\theta }}=\hat{\mathit{\theta }}\left(\hat{K}\right)$. The standard errors are calculated on the basis of the observed Fisher information matrix from the (profile) likelihood with all selected valid IVs (in $\hat{\mathcal{V}}$).^7,23 With $\hat{\mathit{\theta }}={\left({\hat{\theta }}_1,\dots ,{\hat{\theta }}_L\right)}^T$ and corresponding standard errors, we draw inference on the basis of the asymptotic normal distribution. We call this method MVMR-cML-BIC. We note that our proposed cMLE $\hat{\mathit{\theta }}$ also enjoys the nice statistical properties of estimation and selection consistency as its univariable counterpart UVMR-cML.⁷ Here, we state the main conclusions with the proofs relegated to the supplemental information.

Assumption 2: For every SNP $i=1,\dots ,m$,

$${\hat{\mathit{\beta }}}_i={\left({\hat{\beta }}_{X_{1}i},\dots ,{\hat{\beta }}_{X_{L}i},{\hat{\beta }}_{Yi}\right)}^T\sim \mathcal{N}\left({\mathit{b}}_i={\left(b_{X_{1}i},\dots ,b_{X_{L}i},\sum _{l=1}^L{\theta }_l{b}_{X_{l}i}+r_i\right)}^T,{\mathit{\Sigma }}_i\right)\text{,}$$

with known covariance matrix ${\mathbf{\Sigma }}_i$. Furthermore, the $m$ vectors ${\left\{{\hat{\mathit{\beta }}}_i\right\}}_{i=1}^m$ are mutually independent.

Assumption 3: Let $N=\min \left(N_{X_1},\dots ,N_{X_L},N_Y\right)$. There exist positive constants $c_1$ and $c_2$ such that we have $c_1/N\le {\sigma }_{X_{l}i}^2\le c_2/N$, and $c_1/N\le {\sigma }_{Yi}^2\le c_2/N$, for $l=1,\dots ,L$, $i=1,\dots ,m$.

Theorem 2: With Assumptions 1 to 3 satisfied, our proposed BIC consistently selects valid IVs, i.e., $P\left(\hat{K}=m-m_0\right)\to 1$ and $P\left(\hat{\mathcal{V}}={\mathcal{V}}^{\ast }\right)\to 1$ as $N\to \infty$. Furthermore, the proposed constrained maximum likelihood estimator $\hat{\mathit{\theta }}$, combined with the use of the BIC, is consistent for the true causal parameter ${\mathit{\theta }}^{\ast }$, and asymptotically normal with

$${\mathit{V}}^{\frac{1}{2}}\left(\hat{\mathit{\theta }}-{\mathit{\theta }}^{\ast }\right)\stackrel{d}{\to }\mathcal{N}\left(\mathbf{0},\mathit{I}\right)\text{,}$$

where $\mathit{V}=\mathbb{E}\left[-{\partial }^2\tilde{l}\left(\mathit{\theta }\right)/\partial \mathit{\theta }\partial {\mathit{\theta }}^{\prime }\right]$ is the expected Fisher information matrix for the profile log likelihood that can be consistently estimated by its sample version.

For typical GWAS summary data with large sample sizes, Assumptions 2 and 3 are reasonable. Based on the plurality condition, the range of $K$ can be varied from 0, i.e., no invalid IV, up to $m-\left(L+1\right)$. However in practice, we suggest first try a smaller range of $K$, for example from 0 to $m/2$, or based on other methods such as MVMR-Lasso,¹⁶ and one can keep expanding the range if the best $K$ selected is on or close to the upper bound. One reason is that the proportion of invalid IVs is relatively low in many real data examples using UVMR,^6,7 which is expected to be lower in MVMR when we explicitly include other exposures in the model. Moreover, this can speed up the computation of MVMR-cML dramatically, especially for the data perturbation version to be described next.

To better account for the uncertainty in model selection described above, we adopt the data perturbation approach⁷; in a UVMR context, the data perturbation on GWAS summary data is shown to be equivalent to bootstrapping GWAS individual-level data.⁶ For the $b$-th perturbation, $b=1,\dots ,B$, we generate the perturbed GWAS summary data:

$${\hat{\mathit{\beta }}}_i^{\left(b\right)}={\left({\hat{\beta }}_{X_{1}i}^{\left(b\right)},\dots ,{\hat{\beta }}_{X_{L}i}^{\left(b\right)},{\hat{\beta }}_{Yi}^{\left(b\right)}\right)}^T\sim \mathcal{N}\left({\hat{\mathit{\beta }}}_i={\left({\hat{\beta }}_{X_{1}i},\dots ,{\hat{\beta }}_{X_{L}i},{\hat{\beta }}_{Yi}\right)}^T,{\mathit{\Sigma }}_i\right)\text{,}$$

for $i=1,\dots ,m$ independently. Then we apply MVMR-cML-BIC on the $b$-th perturbed sample and obtain ${\hat{\mathit{\theta }}}^{\left(b\right)}$. We use the (element-wise) sample mean and sample covariance of ${\hat{\mathit{\theta }}}^{\left(1\right)},\dots ,{\hat{\mathit{\theta }}}^{\left(B\right)}$ as the final estimate of $\mathit{\theta }$ and its covariance matrix, respectively. The number of perturbations $B$ is suggested to be at least 100. We call this method MVMR-cML-DP.

Simulations

Comparison of MVMR-cML and other MVMR methods in the presence of pleiotropy

We first compare the performance of our proposed method, MVMR-cML, with other existing MVMR methods in the presence of pleiotropy. Following Grant and Burgess,¹⁶ we simulated data as follows:

$$\begin{array}{l}\mathbf{U}=\mathbf{G}\mathit{\phi }+{\mathbf{e}}_U\text{,}\\ {\mathbf{X}}_l=\mathbf{G}{\mathbf{\gamma }}_{X_l}+0.25\mathbf{U}+{\mathbf{e}}_{X_l},l=1,\dots ,4\text{,}\\ \mathbf{Y}=\sum _{l=1}^4{\theta }_l·{\mathbf{X}}_l+\mathbf{G}\mathbf{\alpha }+\mathbf{U}+{\mathbf{e}}_Y\text{,}\end{array}$$

where each component of ${\mathit{e}}_U,{\mathit{e}}_{X_l},{\mathit{e}}_Y$ was independently and identically distributed (iid) as the standard normal $\mathcal{N}\left(\mathrm{0,1}\right)$, $\mathbf{G}$ was the genotype matrix with 20 IVs, each generated independently from a binomial distribution with minor allele frequency (MAF) 0.3 and ${\gamma }_{X_{l}i}\sim \text{Uniform}\left(\mathrm{0,0.22}\right)$ iid. Two sets of values for the causal effects were considered: (1) $\left({\theta }_1,{\theta }_2,{\theta }_3,{\theta }_4\right)=\left(\mathrm{0.2,0.1,0.3,0.4}\right)$ and (2) $\left({\theta }_1,{\theta }_2,{\theta }_3,{\theta }_4\right)=\left(0,-\mathrm{0.1,0.1,0.2}\right)$. We considered three scenarios with different patterns of pleiotropy, and considered 30% and 50% invalid IVs for each scenario.

S1: Balanced and uncorrelated pleiotropy (with InSIDE satisfied): $\phi$ was set to 0 and the ${\alpha }_i$’s corresponding to invalid IVs were generated from $\mathcal{N}\left(\mathrm{0,0}.2^2\right)$.

S2: Directional and uncorrelated pleiotropy (with InSIDE satisfied): $\phi$ was set to 0 and the ${\alpha }_i$’s corresponding to invalid IVs were generated from $\mathcal{N}\left(\mathrm{0.1,0}.2^2\right)$.

S3: Directional and correlated pleiotropy (with InSIDE violated): for invalid IVs, ${\phi }_i$’s were generated from Uniform(0,0.1) and the ${\alpha }_i$’s were generated from $\mathcal{N}\left(\mathrm{0,0}.2^2\right)$.

We also considered a scenario (S4) where some IVs violated assumption MV-A1 and some had uncorrelated pleiotropy $\left(\phi =0\right)$. Specifically, we randomly selected 30% SNPs and set ${\gamma }_{X_{1}i}={\gamma }_{X_{2}i}={\gamma }_{X_{3}i}={\gamma }_{X_{4}i}=0$. We also randomly selected 30% SNPs that had pleiotropic effect with ${\alpha }_i\sim \mathcal{N}\left(\mathrm{0.1,0}.1^2\right)$. Note that these two sets of (30%) invalid IVs could have overlap.

The GWAS sample sizes for all traits were set as $N=\mathrm{50,000}$. The GWAS summary statistics for the four risk factors were calculated with the same 50,000 individuals, and the outcome GWAS summary statistics were calculated with the other non-overlapping 50,000 individuals. We calculated ${\rho }_{l{l}^{\prime }}$ as the sample correlation between ${\mathbf{X}}_l$ and ${\mathbf{X}}_{l^{\prime }}$, and ${\rho }_{lY}$ was set to zero in MVMR-cML.

For each simulation set-up, we ran 500 replications and compared MVMR-cML-BIC, MVMR-cML-DP, and some existing MVMR methods including MVMR-IVW,¹² MVMR-Egger,¹⁸ MVMR-median, MVMR-robust, and MVMR-Lasso.¹⁶

Comparison of MVMR-cML and other MVMR methods in the presence of weak IVs

In this section, we simulated an MVMR model with two exposures and 45 IVs. We considered two different ways to generate the strengths of SNP-exposure associations. In the first scenario, some weak IVs were simulated with relatively small ${\gamma }_{X_{1}i}$ and ${\gamma }_{X_{2}i}$, i.e., the exposures were marginally weakly associated with the IVs. In the second scenario, some marginally strong but conditionally weak IVs were generated by introducing a strong correlation between ${\gamma }_{X_{1}i}$ and ${\gamma }_{X_{2}i}$; that is, each exposure was strongly associated with the IVs marginally but only weakly associated conditional on the other exposure. In both scenarios, we maintained the two-sample conditional F-statistic proposed in Sanderson et al.¹⁷ smaller than 10, the conventional cut-off for detecting weak IVs in MVMR. Details of simulation set-ups are given in the supplemental information. For each set-up, we ran 500 replications and compared the proposed MVMR-cML with other existing MVMR methods.

Mediation analysis: MVMR versus UVMR

In this simulation, we illustrate two advantages of using MVMR over UVMR. First, when there are causal relationships among the risk factors, MVMR can distinguish a direct effect from a total effect of one risk factor on the outcome, while UVMR estimates the total effect on the outcome. Second, ignoring causal pathways via other risk factors may lead to the violation of the IV assumptions for some IVs and thus of the plurality condition in UVMR.

We considered $L=2$ risk factors $X_1$ and $X_2$ in Figure 1B, with a causal effect $X_1\to X_2$, and simulated the GWAS summary statistics according to the three scenarios in Figure 2. More details are given in the supplemental information.

Figure 2.

True causal models in simulations

(A–C) Three scenarios of simulated genetic instruments for $i=1,\dots ,K$ (A); $i=K+1,\dots ,K+m_1$ (B); and $i=K+m_1+1,\dots ,20$ (C). ${\gamma }_{X_{1}i},{\gamma }_{X_{2}i}\stackrel{iid.}{\sim }\mathcal{U}\left(\mathrm{0,0.22}\right)$, ${\alpha }_i\sim \mathcal{N}\left(\mathrm{0.1,0}.2^2\right)$, and $\left({\theta }_1,{\theta }_2\right)=\left(\mathrm{0.1,0.2}\right)$.

We applied MVMR-cML and other MVMR methods with both $X_1$ and $X_2$ as exposures and applied UVMR-cML and UVMR-IVW with only $X_1$ as the exposure. In particular, the SNPs in Figure 2A had direct/pleiotropic effects on $Y$, not going through either $X_1$ or $X_2$, and thus they were invalid IVs in both UVMR and MVMR analyses; the SNPs in Figure 2B did not have direct effects on $Y$ conditional on $X_1$, so they were valid IVs in both UVMR (for $X_1$ only) and MVMR analyses; the SNPs in Figure 2C were directly associated with $X_2$ in addition to a mediating route via $X_1$, and thus they were invalid IVs in UVMR but not in MVMR. We considered different combinations of $\left(K,m_1\right)$ from $\left\{\left(\mathrm{0,1}\right),\left(\mathrm{0,14}\right),\left(\mathrm{0,18}\right),\left(\mathrm{3,11}\right),\left(\mathrm{3,15}\right),\left(\mathrm{10,4}\right)\right\}$. We note that when $K=0$ (i.e., no SNPs generated from scenario Figure 2A), all IVs were valid in MVMR analysis; when $\left(K,m_1\right)=\left(\mathrm{0,1}\right)$, there was only one valid IV for UVMR and thus the plurality condition for UMVR⁷ was violated.

GWAS data

In the real data application, we focused on assessing the causal effects of eight cardio-metabolic risk factors on CAD.²⁴ The eight risk factors were triglyceride (TG), low-density lipoprotein cholesterol (LDL), high-density lipoprotein cholesterol (HDL),²⁵ body mass index (BMI),²⁶ height,²⁷ fasting glucose (FG),²⁸ systolic blood pressure (SBP), and diastolic blood pressure (DBP).²⁹ The sample size of the nine GWAS datasets ranges from around 130,000 to around 750,000. We extracted the SNPs as IVs by using the mv_extract_exposures function in the R package TwoSampleMR.³⁰ After harmonizing data, we retained 201 IVs; for each of the eight exposures, the number of significantly associated IVs ranged from 14 to 79.

Results

Simulations: Better performance of MVMR-cML over other MVMR methods

Robustness to pleiotropy

Here, we show some representative results for estimation and inference of the direct causal effect of exposure $X_1$ on the outcome, ${\theta }_1$, while the results for ${\theta }_2,{\theta }_3,{\theta }_4$ are provided in the supplemental information with the same conclusion. Tables 1 and 2 show the results of different MVMR methods under scenarios S1–S3 when ${\theta }_1=0.2$ and ${\theta }_1=0$, respectively. Table 3 shows the results under scenario S4.

Table 1.

Mean and standard deviation (SD) of estimates, mean standard error (SE), coverage rate (Cov), power, and mean squared error (MSE) when $m=20$ and ${\theta }_1=0.2$

Method	30% invalid						50% invalid
	Mean	SD	SE	Cov	Power	MSE	Mean	SD	SE	Cov	Power	MSE
Scenario 1: Balanced pleiotropy, InSIDE met
MVMR-cML-BIC	0.200	0.070	0.054	0.872	0.912	0.005	0.195	0.227	0.064	0.678	0.794	0.052
MVMR-cML-DP	0.199	0.073	0.084	0.974	0.706	0.005	0.193	0.182	0.192	0.968	0.322	0.033
MVMR-Egger	0.206	0.440	0.428	0.950	0.092	0.194	0.156	0.599	0.559	0.940	0.070	0.360
MVMR-IVW	0.207	0.372	0.361	0.944	0.114	0.138	0.163	0.502	0.475	0.938	0.080	0.253
MVMR-Lasso	0.194	0.095	0.058	0.894	0.870	0.009	0.205	0.255	0.084	0.686	0.714	0.065
MVMR-median	0.198	0.114	0.083	0.922	0.730	0.013	0.209	0.279	0.121	0.740	0.580	0.078
MVMR-robust	0.200	0.085	0.096	0.921	0.661	0.007	0.191	0.381	0.413	0.927	0.156	0.145
Scenario 2: Directional pleiotropy, InSIDE met
MVMR-cML-BIC	0.200	0.076	0.055	0.860	0.896	0.006	0.203	0.223	0.067	0.694	0.796	0.050
MVMR-cML-DP	0.199	0.075	0.085	0.958	0.712	0.006	0.204	0.190	0.173	0.930	0.370	0.036
MVMR-Egger	0.204	0.509	0.484	0.946	0.086	0.259	0.207	0.649	0.601	0.926	0.098	0.422
MVMR-IVW	0.267	0.432	0.409	0.926	0.130	0.191	0.297	0.555	0.510	0.914	0.128	0.317
MVMR-Lasso	0.203	0.113	0.063	0.840	0.844	0.013	0.244	0.389	0.093	0.580	0.714	0.153
MVMR-median	0.210	0.134	0.090	0.904	0.686	0.018	0.249	0.402	0.142	0.658	0.568	0.164
MVMR-robust	0.204	0.109	0.102	0.899	0.662	0.012	0.267	0.429	0.474	0.919	0.136	0.188
Scenario 3: Directional pleiotropy, InSIDE violated
MVMR-cML-BIC	0.198	0.069	0.055	0.872	0.908	0.005	0.203	0.201	0.065	0.698	0.780	0.041
MVMR-cML-DP	0.200	0.069	0.083	0.974	0.714	0.005	0.202	0.159	0.180	0.950	0.354	0.025
MVMR-Egger	0.221	0.507	0.452	0.922	0.138	0.258	0.265	0.609	0.569	0.912	0.098	0.376
MVMR-IVW	0.211	0.433	0.384	0.916	0.128	0.187	0.268	0.508	0.484	0.926	0.108	0.263
MVMR-Lasso	0.202	0.141	0.059	0.878	0.902	0.020	0.221	0.334	0.089	0.664	0.718	0.112
MVMR-median	0.206	0.163	0.087	0.924	0.688	0.027	0.227	0.337	0.133	0.762	0.576	0.114
MVMR-robust	0.202	0.089	0.100	0.907	0.659	0.008	0.243	0.404	0.405	0.929	0.190	0.165

Table 2.

Mean and standard deviation (SD) of estimates, mean standard error (SE), coverage rate (Cov), type-I error, and mean squared error (MSE) when $m=20$ and ${\theta }_1=0$

Method	30% invalid						50% invalid
	Mean	SD	SE	Cov	Type-I	MSE	Mean	SD	SE	Cov	Type-I	MSE
Scenario 1: Balanced pleiotropy, InSIDE met
MVMR-cML-BIC	0.001	0.056	0.044	0.882	0.118	0.003	−0.004	0.149	0.054	0.700	0.300	0.022
MVMR-cML-DP	0.001	0.056	0.069	0.974	0.026	0.003	−0.003	0.127	0.143	0.952	0.048	0.016
MVMR-Egger	0.008	0.441	0.427	0.946	0.054	0.194	−0.043	0.601	0.559	0.930	0.070	0.363
MVMR-IVW	0.008	0.371	0.361	0.944	0.056	0.138	−0.037	0.504	0.475	0.936	0.064	0.256
MVMR-Lasso	−0.003	0.085	0.049	0.906	0.094	0.007	0.007	0.243	0.074	0.684	0.316	0.059
MVMR-median	0.000	0.096	0.070	0.920	0.080	0.009	0.005	0.276	0.108	0.736	0.264	0.076
MVMR-robust	0.003	0.077	0.079	0.907	0.093	0.006	−0.006	0.381	0.417	0.921	0.079	0.145
Scenario 2: Directional pleiotropy, InSIDE met
MVMR-cML-BIC	−0.002	0.055	0.045	0.880	0.120	0.003	0.009	0.156	0.055	0.746	0.254	0.024
MVMR-cML-DP	−0.001	0.057	0.069	0.970	0.030	0.003	0.006	0.144	0.144	0.954	0.046	0.021
MVMR-Egger	0.006	0.508	0.483	0.942	0.058	0.258	0.009	0.648	0.601	0.930	0.070	0.420
MVMR-IVW	0.067	0.431	0.408	0.918	0.082	0.190	0.096	0.553	0.510	0.910	0.090	0.315
MVMR-Lasso	0.002	0.104	0.052	0.868	0.132	0.011	0.042	0.380	0.084	0.590	0.410	0.146
MVMR-median	0.008	0.122	0.075	0.916	0.084	0.015	0.046	0.392	0.128	0.640	0.360	0.156
MVMR-robust	0.000	0.078	0.076	0.906	0.094	0.006	0.060	0.422	0.478	0.921	0.079	0.182
Scenario 3: Directional pleiotropy, InSIDE violated
MVMR-cML-BIC	0.002	0.062	0.045	0.876	0.124	0.004	0.007	0.197	0.054	0.706	0.294	0.039
MVMR-cML-DP	0.001	0.056	0.067	0.966	0.034	0.003	0.007	0.136	0.145	0.938	0.062	0.019
MVMR-Egger	0.024	0.506	0.451	0.918	0.082	0.257	0.066	0.610	0.569	0.908	0.092	0.377
MVMR-IVW	0.012	0.433	0.383	0.914	0.086	0.187	0.068	0.508	0.483	0.926	0.074	0.263
MVMR-Lasso	0.004	0.116	0.051	0.874	0.126	0.014	0.029	0.326	0.079	0.674	0.326	0.107
MVMR-median	0.007	0.154	0.074	0.922	0.078	0.024	0.024	0.329	0.119	0.740	0.260	0.109
MVMR-robust	0.003	0.070	0.078	0.887	0.113	0.005	0.049	0.396	0.413	0.925	0.075	0.159

Table 3.

Results for scenario S4

Method	${\mathit{\theta }}_{\mathbf{1}}=\mathbf{0.2}$						${\mathit{\theta }}_{\mathbf{1}}=\mathbf{0}$
	Mean	SD	SE	Cov	Power	MSE	Mean	SD	SE	Cov	Type-I	MSE
MVMR-cML-BIC	0.205	0.142	0.067	0.802	0.814	0.020	0.013	0.216	0.058	0.806	0.194	0.047
MVMR-cML-DP	0.207	0.129	0.137	0.960	0.496	0.017	0.014	0.137	0.140	0.970	0.030	0.019
MVMR-Egger	0.259	0.313	0.325	0.948	0.152	0.101	0.054	0.324	0.330	0.936	0.064	0.108
MVMR-IVW	0.266	0.294	0.309	0.946	0.176	0.091	0.062	0.297	0.315	0.936	0.064	0.092
MVMR-Lasso	0.213	0.179	0.078	0.762	0.756	0.032	0.019	0.208	0.069	0.786	0.214	0.044
MVMR-median	0.226	0.181	0.112	0.836	0.618	0.033	0.022	0.226	0.099	0.812	0.188	0.052
MVMR-robust	0.204	0.169	0.118	0.851	0.632	0.029	0.023	0.166	0.095	0.814	0.186	0.028

Mean and standard deviation (SD) of estimates, mean standard error (SE), coverage rate (Cov), power/type-I error, and mean squared error (MSE).

First, across all considered scenarios, our proposed method MVMR-cML had the smallest bias and mean squared error (MSE) among all the methods, followed by MVMR-Lasso and MVMR-robust. The advantage of our methods over other methods was more pronounced when there was a higher proportion of invalid IVs at 50%. MVMR-Egger and MVMR-IVW had much less precise estimates than other methods, leading to much lower power and larger MSEs in general. Second, MVMR-cML-DP was the only method that could control the type-I error below the nominal level 5% (Table 2), followed by MVMR-IVW, MVMR-Egger, and MVMR-robust, while at the same time MVMR-cML-DP had much higher power than the other three methods (Table 1). On the other hand, MVMR-Lasso had the largest inflated type-I error rate. Lastly, although MVMR-cML-BIC had good performance in estimation in terms of a small bias and MSE, its inference might not be satisfactory. In the presence of invalid IVs, due to the uncertainty in model selection, its mean standard error was lower than the sample standard deviation of the estimates, leading to an anti-conservative coverage rate and an inflated type-I error rate (Table 2), And the issue was more severe when 50% IVs were invalid. Nevertheless, the data perturbation approach was able to alleviate this problem, giving a satisfactory coverage rate close to the nominal level 95% with a well-controlled type-I error rate. In addition, MVMR-cML-DP had a smaller MSE than MVMR-cML-BIC in the presence of 50% invalid IVs. Hence, overall, MVMR-cML-DP performed best and would be recommended.

Robustness to weak IVs

In this section, we considered a scenario with marginally (and conditionally) weak IVs (Table 4) and a scenario with marginally strong but conditionally weak IVs (Table 5), while all IVs were valid with no pleiotropic effects. The average conditional F-statistics across 500 simulation replicates were 6.73 and 6.70 for the two exposures, respectively, in the first scenario, and 9.35 and 9.38 in the second scenario. We can see that, only MVMR-cML yielded (almost) unbiased estimates (with the smallest MSE) in both scenarios, suggesting its robustness to weak instrument bias. We note that data perturbation might produce a slightly conservative confidence interval as observed in the previous section. In the null causal effect case (Tables S7 and S8), all methods gave unbiased results.

Table 4.

Simulation results for weak IVs

Method	${\mathit{\theta }}_{\mathbf{1}}=\mathbf{0.5}$						${\mathit{\theta }}_{\mathbf{2}}=-\mathbf{0.3}$
	Mean	SD	SE	Cov	Power	MSE	Mean	SD	SE	Cov	Power	MSE
MVMR-cML-BIC	0.501	0.081	0.073	0.924	1.000	0.007	−0.301	0.082	0.073	0.922	0.972	0.007
MVMR-cML-DP	0.500	0.081	0.102	0.984	1.000	0.007	−0.301	0.081	0.103	0.984	0.878	0.007
MVMR-Egger	0.449	0.097	0.094	0.904	0.992	0.012	−0.225	0.078	0.077	0.824	0.826	0.012
MVMR-IVW	0.433	0.068	0.066	0.824	1.000	0.009	−0.233	0.068	0.067	0.820	0.920	0.009
MVMR-Lasso	0.430	0.080	0.158	0.868	0.764	0.011	−0.233	0.080	0.165	0.872	0.502	0.011
MVMR-median	0.430	0.083	0.090	0.906	1.000	0.012	−0.233	0.083	0.091	0.894	0.748	0.011
MVMR-robust	0.432	0.069	0.069	0.828	1.000	0.009	−0.233	0.069	0.069	0.818	0.900	0.009

The average conditional F-statistics across 500 replicates for $X_1$ and $X_2$ are 6.73 and 6.70 (with SD 1.44 and 1.41), respectively. Mean and standard deviation (SD) of estimates, mean standard error (SE), coverage rate (Cov), power, and mean squared error (MSE).

Table 5.

Simulation results for conditionally weak IVs

Method	${\mathit{\theta }}_{\mathbf{1}}=\mathbf{0.5}$						${\mathit{\theta }}_{\mathbf{2}}=-\mathbf{0.3}$
	Mean	SD	SE	Cov	Power	MSE	Mean	SD	SE	Cov	Power	MSE
MVMR-cML-BIC	0.501	0.064	0.059	0.940	1.000	0.004	−0.301	0.045	0.042	0.932	1.000	0.002
MVMR-cML-DP	0.503	0.065	0.081	0.984	1.000	0.004	−0.302	0.046	0.057	0.982	1.000	0.002
MVMR-Egger	0.450	0.057	0.057	0.854	1.000	0.006	−0.265	0.040	0.040	0.846	1.000	0.003
MVMR-IVW	0.450	0.055	0.056	0.846	1.000	0.006	−0.265	0.039	0.039	0.850	1.000	0.003
MVMR-Lasso	0.449	0.066	0.134	0.869	0.860	0.007	−0.265	0.047	0.094	0.869	0.801	0.003
MVMR-median	0.449	0.071	0.073	0.882	1.000	0.008	−0.265	0.050	0.051	0.882	1.000	0.004
MVMR-robust	0.450	0.058	0.058	0.822	1.000	0.006	−0.265	0.041	0.041	0.826	1.000	0.003

The average conditional F-statistics across 500 replicates for $X_1$ and $X_2$ are 9.35 and 9.38 (with SD 1.95 and 1.96), respectively. Mean and standard deviation (SD) of estimates, mean standard error (SE), coverage rate (Cov), power, and mean squared error (MSE).

As shown in the supplemental information, we also considered other scenarios with both weak and pleiotropic IVs. Under these more challenging situations, all MVMR methods yielded biased estimates, but MVMR-cML-DP was least biased and controlled type-I error best.

Simulations: Advantages of MVMR-cML for mediation analysis

In this simulation, we considered a scenario for mediation analysis with two exposures, one of which $\left(X_2\right)$ mediated the effect of the other $\left(X_1\right)$ on the outcome. We compared MVMR-cML with other MVMR methods and two representative UVMR methods. We show some representative results here while more results are given in the supplemental information. First, Table 6 shows the results with the number of invalid IVs $K=10$ and the number of valid IVs (for both MVMR and UVMR) $m_1=4$ (the remaining six IVs are valid for MVMR but not for UVMR). With 50% invalid IVs for MVMR, all other methods yielded severely biased estimates for the direct causal effect of each exposure on the outcome, while only MVMR-cML gave (almost) unbiased estimates. As shown in the supplemental information, when the number of invalid IVs $K$ decreased, the performance of other robust MVMR methods improved and was more similar to that of MVMR-cML. In the UVMR analysis, since most of the IVs were invalid, UVMR-IVW yielded much more biased estimates for the total causal effect of $X_1$ than those of UVMR-cML. As shown in the supplemental information (Tables S14–S17), with more valid IVs, UVMR-cML yielded almost unbiased estimates for the total causal effect.

Table 6.

Mean and standard deviation (SD) of estimates, mean standard error (SE), and power when $K=10$, $m_1=4$

Method	${\mathit{\theta }}_{\mathbf{1}}=\mathbf{0.1}$$\left({\mathit{\theta }}_{\mathbf{1}\mathit{T}}=\mathbf{0.2}\right)$				${\mathit{\theta }}_{\mathbf{2}}=\mathbf{0.2}$
	Mean	SD	SE	Power	Mean	SD	SE	Power
MVMR-cML-DP	0.102	0.029	0.032	0.866	0.199	0.025	0.027	0.992
MVMR-Egger	0.389	0.580	0.638	0.062	−0.261	0.344	0.589	0.004
MVMR-IVW	0.699	0.438	0.495	0.270	−0.142	0.288	0.571	0.002
MVMR-Lasso	0.201	0.290	0.035	0.920	0.144	0.181	0.029	0.946
MVMR-median	0.227	0.290	0.055	0.900	0.130	0.180	0.042	0.892
MVMR-robust	0.269	0.339	0.491	0.123	0.105	0.204	0.287	0.319
UVMR-cML-DP	0.209	0.064	0.025	0.928	N/A	N/A	N/A	N/A
UVMR-IVW	0.591	0.249	0.266	0.628	N/A	N/A	N/A	N/A

The total causal effect of $X_1$ is ${\theta }_{1T}=0.2$.

Second, Table 7 shows the results when $K=0$ and $m_1=1$, in which case only one IV was valid in UVMR analysis, thus the plurality condition required by UVMR-cML was violated. MVMR-cML had similar performance to MVMR-IVW, which can be considered as the oracle estimator as all Ivs were valid in MVMR analysis. All MVMR methods yielded (almost) unbiased estimates for the direct causal effect. On the other hand, as the plurality condition was violated in UVMR, both UVMR-cML and UVMR-IVW performed poorly in estimating the total effect of $X_1$.

Table 7.

Mean and standard deviation (SD) of estimates, mean standard error (SE), and power when $K=0$, $m_1=1$

Method	${\mathit{\theta }}_{\mathbf{1}}=\mathbf{0.1}$$\left({\mathit{\theta }}_{\mathbf{1}\mathit{T}}=\mathbf{0.2}\right)$				${\mathit{\theta }}_{\mathbf{2}}=\mathbf{0.2}$
	Mean	SD	SE	Power	Mean	SD	SE	Power
MVMR-cML-DP	0.101	0.020	0.021	0.992	0.199	0.014	0.015	1.000
MVMR-Egger	0.100	0.021	0.020	0.988	0.199	0.017	0.017	1.000
MVMR-IVW	0.100	0.019	0.018	0.994	0.199	0.014	0.013	1.000
MVMR-Lasso	0.100	0.023	0.045	0.729	0.199	0.017	0.031	0.980
MVMR-median	0.101	0.023	0.025	0.980	0.199	0.017	0.018	1.000
MVMR-robust	0.100	0.020	0.019	0.994	0.199	0.014	0.014	1.000
UVMR-cML-DP	0.298	0.045	0.019	1.000	NA	NA	NA	NA
UVMR-IVW	0.344	0.029	0.031	1.000	NA	NA	NA	NA

The total causal effect of $X_1$ is ${\theta }_{1T}=0.2$.

Note that we did not apply UVMR with exposure $X_2$ because all of the Ivs were invalid for UVMR, although the total effect of $X_2$ on the outcome was the same as its direct effect. This example illustrates an advantage of MVMR over UVMR: the valid IV assumption MV-A3 is weaker than the corresponding UV-A3. On the other hand, in the case of $K=0$, $m_1=20$ and ${\theta }_1=0$ with a true causal graph of $G_i\to X1\to X2\to Y$ for all $i$, there is no valid IV for $X_2$ in MVMR while all the IVs are valid in UVMR for either $X_1$ or $X_2$, illustrating that the valid IV assumption MV-A1 is stronger than UV-A1.

Real data application: The causal effects of cardio-metabolic risk factors on CAD

In this section, we studied the causal effects of the eight cardio-metabolic risk factors on CAD. The correlation matrix for the SNP-trait association estimates was estimated with bivariate LDSC as discussed in the section “new method: multivariable MRcML (MVMR-cML)” (with details provided in supplemental information, section S4). We first applied MVMR-cML (and MVMR-IVW as the standard method) by using the eight risk factors as exposures and CAD as the outcome. Then we applied UVMR-cML on each risk factor-CAD pair by using the set of IVs significantly associated (p value < 5e−8) with the corresponding risk factor (Figure 3). We calculated the conditional F-statistics¹⁷ for each of the eight exposures. As shown in Figure 3, all of the conditional F-statistics were larger than 10 except for BMI (8.15). This suggests that the weak instrument bias should not be severe in this analysis, and as shown in the previous simulations, our proposed method was expected to be robust. Furthermore, as discussed later, estimating the direct effect of BMI was potentially problematic and would not be a focus here.

Figure 3.

The estimated effects (and 95% confidence intervals) of each of the eight risk factors on CAD by various UVMR and MVMR methods

The conditional F-statistic is given in the parentheses following each exposure name.

Direct causal effects estimated by MVMR

The results are shown in Figure 3, and we summarize a few main findings here. First, MVMR methods suggested a null effect of HDL on CAD after adjustment for other seven risk factors, while UVMR-cML suggested a protective effect. Second, the positive effect of DBP on CAD diminished in MVMR-cML but stayed nearly significant in MVMR-IVW. There are several possible reasons for a risk factor to show an effect in UVMR analysis but not in MVMR. The first is that some genetic variants for that risk factor might have pleiotropic effects, leading to biased inference in UVMR. The second is that the effect of that risk factor on CAD is mediated through other risk factors in the model. The third is possible loss of power in MVMR. We will take a closer look at these possibilities in the following sections. Moreover, we note that the GWAS summary statistics for SBP and DBP were obtained after adjusting for BMI. While using covariate-adjusted SNP-trait associations may lead to bias in MR analyses,³¹ as discussed in Gilbody et al.,³² including the covariate, BMI in this case, as an additional exposure in an MVMR analysis can recover the direct causal effects of the exposures of interest, DBP and SBP in this case. However, the estimated direct effect of the covariate (BMI) on the outcome (CAD) was potentially biased and should be interpreted with caution.

MVMR-cML-BIC identified ten invalid IVs out of the 201 IVs used in the MVMR analysis. We calculated the conditional F-statistics on the basis of the selected set of valid IVs. The results were similar to those when using the whole set of 201 IVs: only BMI had a conditional F-statistic smaller than 10 (Table S19), suggesting no severe issue of multicollinearity. Lastly, to detect possible outlying or influential IVs, Cook’s distance for each IV in the MVMR-IVW model was calculated. None of them exceeded the recommended threshold of the median of the corresponding F-distribution,^33,34 suggesting that there was no influential point in the analysis (Figure S2). We also applied a leave-one-out analysis with MVMR-cML-DP, reaching the same conclusion (Figures S3 and S4).

Causal effects of lipids on CAD

The causal effect of various lipid fractions, including LDL, HDL, and TG, on CAD is an important issue that has been studied widely. While increased LDL doubtlessly has a deleterious causal effect on CAD, the roles of TG and HDL are still under debate. Numerous MR analyses have been conducted to investigate this issue. In particular, in standard UVMR analyses using genome-wide significant variants for HDL, a protective role of HDL on CAD has been suggested by several UVMR methods.⁷ However, this could be due to the fact that many variants associated with HDL are also associated with other risk factors like LDL, TG, and BMI, suggesting possible pleiotropic effects. For example, Holmes et al.³⁵ showed that, when an unrestricted allele score was used, a protective effect of HDL was identified, while when a restricted allele score (by removing any SNPs associated with either of the other two lipid traits) was used, no significant effect was found.

In our example here, we reached a similar conclusion. First, in the univariable analysis, UVMR-cML-BIC suggested that HDL had a protective effect on CAD while the result of UVMR-cML-DP was not significant. As noted in Xue et al.,⁷ model selection based on BIC might miss some invalid IVs due to their small effect sizes, leading to inflated type-I errors, especially when pleiotropic effects were weak, while data perturbation could help. We found that out of the 26 IVs associated with HDL, 17 were also associated with at least one of the three other likely causal risk factors, including LDL, TG, and BMI (at the significance level of 5e−8). However, UVMR-cML-BIC only identified eight of them, leading to possibly biased inference. On the other hand, for UVMR-cML-DP, 12 of them were identified as invalid at least 10 times out of the 100 perturbed datasets. Although UVMR-cML-DP performed more robustly than UVMR-cML-BIC in the presence of many invalid IVs, it also yielded a much wider confidence interval, even than that of MVMR-cML. We also applied UVMR-cML with only nine IVs for HDL by removing the 17 potentially invalid IVs. In this case, both UVMR-cML-BIC and UVMR-cML-DP suggested a null effect of HDL on CAD (though it could be due to the reduce power of using less SNPs). On the other hand, both MVMR-cML-BIC and MVMR-cML-DP suggested that both TG and LDL, but not HDL, had a causal effect on CAD after adjusting for other risk factors included in the model; this could be partly due to the relaxed non-pleiotropy assumption in MVMR.

Diminished causal effects of DBP on CAD after accounting for SBP

As shown in Figure 3, DBP had a significant positive effect on CAD in UVMR-cML, but the effect completely diminished in MVMR-cML. This was concordant with some previous studies finding that the effect of DBP on CAD disappeared after adjusting for SBP.^36,37 First, there were 79 SNPs associated with DBP, and 63 of them were also associated with at least one of the other seven risk factors (at the significance level of 5e−8). We removed these 63 variants and performed UVMR-cML on DBP to CAD with the remaining set of 16 SNPs. Unlike what we observed for HDL, the UVMR-cML result still showed a significant positive effect, suggesting that the diminished effect of DBP in MVMR (but significant in UVMR) might be due to other reasons besides the presence of pleiotropic variants.

Next, we applied MVMR-cML-DP with a subset of exposures to investigate some potential mediating effects. We separately used each of the other seven risk factors along with DBP as exposures and CAD as the outcome. As shown in the left panel in Figure 4, the effect of DBP on CAD after adjusting for SBP changed from that in UVMR-cML-DP, while adjusting for any one of the other six risk factors (height, BMI, FG, HDL, LDL, and TG) did not lead to much change. Alternatively, we started from the "full" model as shown in Figure 3 with all eight risk factors and deleted each of the seven risk factors (other than DBP). The right panel in Figure 4 told a similar story: after SBP was removed from the full model, DBP still had an effect on CAD, although slightly smaller than that in UVMR, after adjusting for the other six remaining risk factors. This suggested that a mediating effect of DBP on CAD was likely via SBP. However, we should be cautious in interpreting any mediating effect, as it relies on the correct inference by both UVMR and MVMR. It is also possible that the total effect of DBP on CAD shown in UVMR was due to the fact that many of the IVs for DBP had pleiotropic effects on SBP (or a closely related trait), some of which may be too weak to be detectable, so the significant result with the restricted set of 16 IVs might still be biased, and the different results between UVMR-cML and MVMR-cML could be due to their differing adjustments for pleiotropy via SBP.

Figure 4.

The estimated effects (and 95% confidence intervals) of DBP on CAD with various sets of exposures by MVMR-cML-DP

Left corresponds to the sets of two exposures (DBP plus one of the other seven risk factors). Right corresponds to the sets of the six exposures after excluding one of seven risk factors marked out in the left. Results from UVMR-cML-DP and MVMR-cML-DP in Figure 3 are also added at bottom for comparison.

Lastly, we applied other robust MVMR methods. All methods gave results in line with that of MVMR-cML, suggesting a null direct effect of DBP on CAD, and their point estimates were also in the same direction (see Figure S1).

Discussion

We have proposed a robust and efficient multivariable Mendelian randomization method based on constrained maximum likelihood called MVMR-cML. It is an important and useful extension of the UVMR-cML.⁷ We have shown in both simulations and a real data application that, compared to its univariable counterpart (and other UVMR methods), MVMR-cML has two main advantages (while maintaining the major advantages of UVMR-cML, such as its estimation efficiency and robustness). First, MVMR estimates the direct effect of an exposure on the outcome after accounting for the other exposures included in the model, while UVMR only estimates the total effect. When there are causal pathways among the exposures, the direct effect of an exposure is in general different from its total effect on an outcome. Second, MVMR can account for some known pleiotropic effects through other exposures included in the MVMR model, making it more robust to pleiotropy than its univariable counterparts. This is important especially when (putative causal) risk factors, e.g., various lipids such as TG, LDL, and HDL, share some genetic associations. Although UVMR-cML and some other UVMR methods allow for invalid IVs under the plurality condition, it is still possible that some invalid IVs with weak pleiotropic effects cannot be selected out, leading to incorrect inference. On the other hand, an application of MVMR requires some assumptions beyond those of UVMR; in particular, MVMR requires at least one valid IV for each exposure; that is, the matrix of the marginal IV-exposure associations has a full column rank. We have given an example in mediation analysis in which UVMR, but not MVMR, would work. More generally, as in multiple regression, if the marginal association matrix is nearly singular, multicollinearity may lead to unstable estimates and thus inflated type-I errors and loss of power in MVMR; a conditional F-test has been proposed to detect such a case.¹⁷ In addition, all the current MVMR methods require the use of (nearly) independent IVs, leading to the use of fewer IVs (than in UVMR) and thus possibly exacerbating the issue of multicollinearity. An extreme example is when there is only one valid IV for one exposure in MVMR; removing this IV would lead to the non-identifiable MVMR model and thus unreliable estimates. These issues can be regarded as a price we pay for using MVMR.

There are a few advantages of our proposed method over other existing MVMR methods. First, MVMR-cML has nice statistical properties such as selection consistency, estimation consistency, and asymptotic normality with strong theoretical support. Second, as highlighted in our simulation studies, MVMR-cML was shown to have the smallest bias and MSE under various pleiotropic and/or weak IV settings among all MVMR methods being compared. In particular, MVMR-cML-DP consistently performed best, especially in controlling type-I errors. Compared to MVMR-cML-BIC, MVMR-cML-DP accounts for model selection uncertainties ignored by the former, often performs better for finite samples, and thus is recommended. Third, most of the MVMR methods are based on the two-sample MR setup, which means that all the exposure and outcome GWAS were performed in non-overlapping (and unrelated) samples. Our likelihood framework can account for overlapping samples by taking into account of the correlations among genetic associations with the exposures and outcome. This avoids sample splitting into non-overlapping subsets for analysis with reduced power (e.g., in Davies et al.³⁸; Sanderson et al.²⁰). Fourth, although in this paper we focused on testing for each direct effect separately, it is straightforward to test for a subset of multiple causal effects jointly in MVMR-cML. Furthermore, unlike some other MVMR methods, we do not need the assumption of no measurement errors of SNP-exposure associations (NOME) in our method by directly accounting for the variation of the estimated associations $\left({\hat{\beta }}_{X_{l}i}\right)$ in the likelihood framework. This contributes to its robustness to weak IV biases.^14,17,39

There are some limitations with our method. First, as in other MVMR methods, we assume a linear and homogeneous effect of each exposure on the outcome.^14,16 When a linear effect is absent, it does not necessarily imply no causal effects. Some exposures might have a non-linear effect on the outcome; for example, a U-shaped or J-shaped effect of DBP on CAD has been reported in previous studies.^40,41 Second and importantly, as in multiple regression, if MVMR fails to detect a direct effect, it could be due to its low power with multicollinearity if many related exposures are included. Third, despite the two advantages of MVMR over UVMR mentioned earlier, when an estimated direct effect in MVMR differs from an estimated total effect in UVMR, it may be difficult to distinguish whether such a difference is due to mediating effects of other exposures or to pleiotropic effects of genetic variants on the outcome or both. Finally, though emerging as a powerful tool for causal inference with observational data, various (MV)MR methods depend on their own assumptions as well as on the quality of the genetic variants as IVs and, more generally, on the GWAS data being used. MR cannot completely replace traditional experimental studies, while triangulation through more applications of various MR methods to real data would be worthwhile and warranted.

Published: March 21, 2023

Web resources

Conditional F-test, https://github.com/WSpiller/MVMR

IEU open GWAS project, https://gwas.mrcieu.ac.uk

MVMR-IVW, MVMR-Egger, https://cran.r-project.org/web/packages/MendelianRandomization

MVMR-Median, MVMR-Robust, MVMR-Lasso, https://github.com/aj-grant/robust-mvmr

TwoSampleMR, https://mrcieu.github.io/TwoSampleMR/

UVMR-cML, https://github.com/xue-hr/MRcML

Data and code availability

The GWAS summary datasets used in the real data analysis are all publicly available. Computer code for simulation studies and real data analysis is available at https://github.com/ZhaotongL/MVMR-paper. The software for MVMR-cML is publicly available on GitHub at https://github.com/ZhaotongL/MVMR-cML.