The method of path coefficients

Causal effect identification consists of expressing the effect of an intervention of one variable on the marginal distribution of one or more other variables using knowledge of the underlying causal diagram and joint distribution of all variables under consideration [23]. A causal diagram is a directed acyclic graph (DAG) of directed edges between variables where a causal relationship is known or hypothesized, augmented with bidirected edges between variables known or hypothesized to be affected by unmodelled confounding factors (Fig 1); bidirected edges can be represented equivalently by two directed edges originating from a common latent factor.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 1. Causal diagrams for identification (A-C) and estimation (D-F) of causal effects at GWAS loci with regulatory pleiotropy.

(A) The standard instrumental variable graph is only suitable for modelling the causal effect of gene X on outcome trait Y if X is the only cis-eGene in the locus of the genome-wide significant variant E for Y. (B) A causal graph for MVMR with regulatory pleiotropy, where two or more cis-eGenes X_i are found in a genome-wide significant locus for Y; the causal effects of each X_i on Y can be identified if there exist at least the same number of causal variants as the number of cis-eGenes in the GWAS locus, even when they are in LD with each other, because the variants form an instrumental set. (C) If the number of causal variants is smaller than the number of cis-eGenes and other variants in the locus are merely associated by LD, the causal effects of X_i on Y are non-identifiable. (D,E) When estimating causal effects from finite samples, we use an overdetermined set of genetic variants as instruments, without knowing the number or identity of the causal variants. (F) If the (unknown) number of causal variants in a locus is smaller than the number of cis-eGenes, causal effects can not be estimated, a condition that can be tested by analyzing the rank or determinant of the covariance matrix between the variants and cis-eGenes. Solid arrows represent direct causal effects and dashed (bi)directed arrows represent non-zero covariances due to unobserved (U) confounders.

https://doi.org/10.1371/journal.pgen.1011473.g001

A special class of causal models are those where the causal diagram is assumed to define a linear structural equation model (SEM) [27], that is, a system of linear equations among a set of variables {X₁, …, X_p} of the form

The U_i are error or disturbance terms representing unobserved background factors which are assumed to have mean zero and a variance/covariance matrix [Ψ_ij] = Cov(U_i, U_j). It is often assumed that the error terms are normally distributed, but, as already pointed out by Wright [22], this is not necessary—the results below hold for any error distribution. Every non-zero coefficient c_ij corresponds to a directed edge X_j → X_i in the causal diagram, that is, c_ij is the direct causal effect of X_j on X_i, and every non-zero element Ψ_ij corresponds to a bidirected edge X_i ↔ X_j. The SEM structure implies that the variables {X₁, …, X_p} have mean zero and covariance matrix where C is the matrix of coefficients c_ij, C^T is its transpose, and I is the identity matrix. Causal identification in linear SEMs consists of solving for the non-zero c_ij (and Ψ_ij) in terms of the covariance matrix Σ.

The method of path coefficients is a result derived by Sewall Wright [21, 22, 27], which expresses the covariance between any pair of variables in a linear SEM as a polynomial in the parameters (causal coefficients and error covariances) of the model: (1) where the summation ranges over unblocked paths p connecting X and Y, and T(p) is the product of the parameters (c_ij or Ψ_ij) of the edges along p. A path in a graph is a sequence of edges such that each pair of consecutive edges share a common node and each node appears only once along the path. A path is unblocked if it does not contain a collider, that is, a pair of consecutive edges pointing at the common node. Two nodes are called d-separated if there are no unblocked paths between them. Eq (1) is valid for standardized variables that are normalized to have zero mean and unit standard deviation; if this is not the case, a modified rule can be applied where each T(p) is multiplied by the variance of the variable that acts as the “root” for path p [27].

Instrumental sets

Fix a variable Y in a linear SEM, and assume we have the equation in our SEM. Brito and Pearl [24] derived a sufficient graphical condition for identifying the parameters c₁, …, c_K by application of Wright’s rule (1), of which we present a simplified version. The set of variables {E₁, …, E_K} is said to be an “instrumental set” relative to {X₁, …, X_K} if there exist pairs (E₁, p₁), …, (E_K, p_K) such that for each i the following three conditions hold.

1. E_i is a non-descendant of Y and p_i is an unblocked path between E_i and Y including edge X_i → Y.
2. E_i is d-separated from Y in , the graph obtained after removing the edges X₁ → Y, …, X_k → Y from the original graph .
3. For j > i, the variable E_j does not appear in path p_i, and if paths p_i and p_j have a common variable V, then both p_i[V ∼ Y] and p_j[E_j ∼ V] point to V, where p_i[V ∼ Y] and p_j[E_j ∼ V] denote truncations of p_i and p_j from V and until V, respectively.

If {E₁, …, E_K} is an instrumental set relative to {X₁, …, X_K}, then the parameters of the edges X₁ → Y, …, X_K → Y are identifiable and can be computed by a set of linear equations [24]. For concrete examples verifying the instrumental set conditions, see S1 File.

Causal effect identification in MVMR with correlated instruments

We assume genes X₁, …, X_K are colocated in the same genomic locus and have a potential causal effect on a phenotypic trait Y associated to the locus. We assume there exist K causal variants in the locus such that each gene has at least one causal variant (causal variants may be shared between genes). The variants are correlated by LD, the exposures X_i can be connected by causal relations or unmodelled latent factors, and likewise there can be latent factors affecting the exposures X_i and outcome Y. Randomization of genotypes ensures that there are no unmodelled correlations between the E_j and X_i. An example causal diagram for K = 2 is shown in Fig 1B.

We now show that {E₁, …, E_K} is an instrumental set relative to {X₁, …, X_K}. By relabelling, we may assume that there exists an edge E_i → X_i in the causal diagram. Define the paths p_i = {E_i → X_i, X_i → Y}. Since E_i is d-separated from Y in the diagram where all edges X_j → Y are removed, it follows that the set {(E₁, p₁), …, (E_K, p_K)} satisfies the three instrumental set conditions (see above) and the causal parameters c_i are identifiable. We apply Wright’s rule (Eq (1)), following the general proof of Brito and Pearl [24]. By Eq (1), (2) where the sum is over unblocked paths p. It is clear that any path that starts in E_i, ends in Y, and includes a bidirected edge X_j ↔ Y must be blocked by a collider V → X_j ↔ Y, where V can be either a variant or another gene. Hence no such paths enter the sum in Eq (2). In other words, all unblocked paths must end with an edge X_j → Y contributing a factor c_j to T(p). Collecting all paths by their final edge, the sum can be decomposed as where the inner sum now extends over the subset of paths from E_i to X_j obtained by truncating the unblocked paths from E_i to Y which have X_j as their penultimate node. Since the truncation of an unblocked path is also necessarily unblocked, each p′ is an unblocked path from E_i to X_j. Vice versa, every unblocked path from E_i to X_j can be extended to an unblocked paths from E_i to Y by adding the edge X_j → Y. Hence, the sum over p′ is the sum over all unblocked paths from E_i to X_j, By another application of Wright’s rule (1), and hence

In matrix-vector notation, we obtain: where c = (c₁, …, c_K)^T is the vector of causal effects, and Σ_EX and Σ_EY are the matrix and vector of covariances and , respectively.

Since Σ_EX is not a symmetric matrix, we cannot assume that an inverse exists, but we can always write the solution to the linear set of equations using a generalized left inverse: (3) that is, find the least squares solution. These equations are valid for standardized variables.

The generalized method of moments for causal effect estimation in MVMR with correlated instruments

To estimate the causal effects from observational data, we consider multiple finite-sample approximations to the above equations. For finite-sample estimation, the number of instruments can be greater than the number of exposures, that is, we assume we have N observations of the random variables {E₁, …, E_L}, {X₁, …, X_K}, and Y, with L ≥ K. We use lower-case letters to denote sample values, e.g. e_il denotes the value of E_l in observation i. We use the notation e_i⋅, e_.l, and e_⋅⋅ to represent the vector of sample values for all L instruments in observation i, the vector of all N observations of instrument l, and the N × L matrix of all observations for all instruments, respectively, and similarly for x_i⋅, x_⋅k, and x_⋅⋅. We assume the observations for all variables are standardized to mean zero and unit standard deviation.

The generalized method of moments (GMM) [25] is based on moment functions that depend on observable random variables and unknown parameters, and that have zero expectation in the population when evaluated at the true parameters: (4) where g denotes the moment function, w_i is a vector of samples of random variables in observation i, and c is a vector of unknown parameters. The moment function g can be linear or nonlinear.

Assuming a linear MVMR SEM as before, the natural moment functions are of the form (5) that is, we consider one moment equation per instrument. From the structural equation for Y in the assumed SEM (see above and Fig 1), it follows that Y − X^TC = U_Y, the residual error on Y, and since all instruments E_l are assumed independent of U_Y (no bidirected arrows between E_l and Y), that is, , it follows immediately that the moment conditions are satisfied:

Replacing the expectation with the empirical mean over the observations results in L equations in K unknowns (c₁, …, c_K) which can be written in matrix-vector notation as (6)

If the number of instruments equals the number of unknowns, L = K, this system can be solved exactly, and by the standardization of all variables, the solution reduces to the least-squares estimator , (7) which is also obtained by replacing the true covariances in the instrumental variable formula (3) by their empirical estimates.

If the model is overdetermined, L > K, then in general there is no unique solution to (6), because not all L sample moments will hold exactly. While the least-squares estimator provides one approximate solution to the overdetermined system, Hansen [25] proposed an alternative solution, based on bringing the sample moments as close to zero as possible by minimizing the quadratic form (8) with respect to the parameters c, where Δ is a positive definite L × L weighting matrix, and the quantity between the large square brackets is understood to be the L-vector of moment functions. It can be shown that this yields a consistent estimator of c, under certain regularity conditions. Again using the standardization of all variables, the cost function can be written as with solution (9)

When Δ = I, the identity matrix, we again obtain the least-squares estimator, .

While all choices of Δ lead to a consistent estimator, in finite samples different Δ-matrices will lead to different point estimates. Hansen [28] showed that the GMM estimator with the smallest asymptotic variance is obtained by taking Δ equal to the inverse of the variance-covariance matrix of the moment functions (5). If the errors are homoskedastic, , we have and obtain the estimator (10)

We will refer to this estimator as the GMM estimator. Note that this estimator is equivalent to the two-stage least squares estimator.

Simulations with randomly generated instrument correlations

We simulated data from idealized models corresponding to the causal diagrams in Fig 1 with binomially distributed instruments and normally distributed exposure and outcome variables.

For the diagrams with a single exposure (Fig 1A and 1D), we set the true causal effect c_X = 0.3. In the case of one instrument (Fig 1A) we varied the instrument strength a between 0.01 (weak instrument), 0.3 (mediocre instrument), and 0.8 (strong instrument). In the case of multiple instruments (Fig 1D), we simulated ten instruments with randomly sampled instrument strengths, either all strong (effect sizes between 0.1 − 0.3) or all weak (effect sizes between 0.001 − 0.03).

For the diagrams with multiple exposures and correlated instruments (Fig 1B, 1C, 1E and 1F), we set the true causal effects and in all simulations.

To test the effect of instrument correlation we simulated data with two instruments (Fig 1B) where the pairwise correlation of the instruments varied between 0.01, 0.3, 0.7, 0.9 and 0.95. The matrix A = [a_ij] of instrument effect sizes, where a_ij is the direct causal effect of E_i on X_j, was generated randomly such that its determinant was greater than 0.05 and there were no weak instruments (a_ij between 0.1 − 0.3).

To test the effect of weak instruments, we simulated data with two instruments (Fig 1B) with randomly generated non-zero pairwise correlation of the instruments and randomly generated matrix of instrument effect sizes A with determinant either less than 0.001 and effect size of each instrument less than 0.001 or determinant greater than 0.05 and effect size of each instrument between 0.1 − 0.3.

To test robustness of the causal effect estimates, we simulated data with two instruments and two exposures (Fig 1B), with randomly generated non-zero pairwise correlation between the instruments, randomly generated matrix of instrument effect sizes A with determinant greater than 0.05, and true causal effects of the exposures on the outcome equal to and , but estimated causal effects for each exposure separately under the (false) hypothesis that the data was generated from Fig 1A.

Simulations with instrument correlations derived from real LD matrices

For more realistic simulations, we simulated data with binomially distributed genotypes and normally distributed exposure and outcome variables in such a way that the summary statistics from the simulated data match summary statistics at real GWAS loci for coronary artery disease (CAD). We used eQTL summary statistics from GTEx [12] and CAD GWAS summary summary statistics from the CARDIoGRAMplusC4D meta-analysis [29].

For a given locus of interest with cis-eQTLs E₁, …, E_L for genes X₁, …, X_K (L ≥ K), with LD matrix Σ_E (size L × L), matrix of eQTL summary statistics A (size L × K), and vector of CAD GWAS summary statistics Σ_EY (size L × 1), we simulated data as follows.

To sample binomially distributed genetic instruments according to a predefined LD matrix, we made the simplifying assumption that the correlation structure between the simulated instruments can be described by a Markov chain. We sampled the first SNP’s genotype from a binomial distribution with probability of success equal to its real minor allele frequency (MAF). For every other SNP, we sampled a genotype from a binomial distribution with probability of success conditional on the genotype of the previous SNP in the sequence, such that both the MAF and pairwise LD between successive SNPs match the real data from the simulated locus. Further details are given in S1 File.

We assumed that a randomly selected subset of at least K eQTLs were causal and all others were associated with the exposures only through LD. For the causal eQTLs, we sampled the vector a of instrument effect sizes randomly between 0.1 and 0.3, with σ² = 1.

Finally, we set the causal effect c of X on the outcome Y equal to the value obtained from eq. (3) with the real Σ_EY, and sampled outcome data from with σ² = 1.

We performed simulations to analyze the influence of LD matrix accuracy, weak instrument bias, standard error calculation method, and horizontal pleiotropy.

For the simulations on the accuracy of the LD matrix, we generated datasets for the SLC22A3-LPA-PLG locus (centered on chr 6: 161089307) in liver as exposures. Causal effects of the genes on the outcome variable Y were calculated to be c_SLC22A3 = 0.15, c_LPA = −0.05 and c_PLG = −0.27 when seven instruments were used, and the same values were also used in the other simulations. We compared the situations where L = 3 (LD threshold of 0.25), L = 4 (LD threshold of 0.3), L = 5 (LD threshold of 0.4), L = 6 (LD threshold of 0.5), and L = 7 instruments (LD threshold of 0.8) were used. We performed one-sample simulations where the sample size varied between 500 − 30000. For the instrument effect sizes, we assumed that three eQTLs were causal for all genes (their effect sizes sampled from a uniform distribution within the range 0.1 − 0.3), while additional eQTLs were assumed to be associated with the exposures only through LD.

For the simulations on weak instrument bias, we generated datasets within the same locus and setup as the LD accuracy simulations except here we kept L = 8 eQTLs (LD threshold of 0.96) and varied the sample size between 500 − 10000. For the instrument effect sizes, we again assumed that only three eQTLs were causal for all genes, their effect sizes sampled from a uniform distribution within the range 0.1 − 0.3 for strong instruments and within the range 0.001 − 0.01 for weak instruments. We confirmed that these instruments were indeed weak/strong by employing the conditional F-statistics computed using the Mendelian Randomization package [30] as well as the MVMR package [31] (S3 Fig).

For the simulations on comparing the standard errors computed approximately using summary-level data and exactly using individual-level data, we generated datasets within the same locus and setup as the previous simulations. We kept L = 7 eQTLs (LD threshold of 0.95) and varied the sample size between 500 − 10000. For the instrument effect sizes, we again assumed that only three eQTLs were causal for all genes. Details about the standard error calculation using summary and individual-level data are given in detail in S1 File.

To simulate the effect of horizontal pleiotropy, we generated datasets for the SLC22A3-LPA-PLG locus with causal effects of the genes on the outcome variable Y set to c_LPA = −0.05 and c_PLG = −0.27 and c_SLC22A3 ∈ [0.0, 0.4]. We used eight instruments where only three eQTLs were causal for all genes, their effect sizes sampled from a uniform distribution within the range 0.1 − 0.3. We estimated the effects in a misspecified estimation model where we performed MVMR only using LPA-PLG genes, assuming horizontal pleiotropy through one unknown/unmeasured gene SLC22A3 in the same locus.

To simulate the effect of potential bias introduced by two-sample MR, we generated independently sampled eQTL and GWAS datasets for the MRAS-ESYT3 locus in the Aorta tissue of STARNET, which share L = 5 eQTLs in their locus (centred on chr 3:138121920), with causal effects of the genes on the outcome variable Y set to c_MRAS = 0.208 and c_ESYT3 = −0.294. We simulated eQTL sample sizes between 300 − 4000 and GWAS sample sizes between 10000 − 140000.

Comparison to other MVMR methods

We compared the least squares (LS) (Eq (7)) and GMM (Eq (10)) estimators to seven other multi-variable MR methods: Transcriptome-Wide MR (TWMR) [19], the mv_multiple estimator from TwoSampleMR package [32], and the mv_mvivw estimator from the Mendelian Randomization package [30], estimators MVMR-Robust, MVMR-Median, and MVMR-Lasso [33] and estimator MVMR-cML [34].

The least squares and GMM estimators can also be applied in the univariate MR setting. In this setting we performed a comparison to all available methods in the MR-Base package, namely, Inverse Variance weighted (IVW), Weigted Median (WM), Maximum Likelihood (MaxLik), MR-Egger (Egger). We also included TWMR in the comparison.

In the course of our analysis we identified the presence of a “regularization” term in the TWMR source code that was hard-coded to a specific value, presumably related to the sample size of the use-case from their paper [10]. More precisely, for the weight matrix used in the GMM estimator, H = ((X^TE) ⋅ (E^TE)⁻¹ ⋅ (E^TX))⁻¹, they used shrinkage of the form Here α is where N = 3781 and I is the identity matrix. Results reported here used the published version of TWMR including this hard-coded term.

Prediction of causal genes and tissues at GWAS loci for coronary artery disease

We used eQTL summary statistics from the Stockholm-Tartu Atherosclerosis Reverse Network Engineering Task (STARNET) study, a genetics-of-gene expression study of tissue samples from blood (n = 481), atherosclerotic-lesion-free internal mammary artery (MAM, n = 552), atherosclerotic aortic root (AOR, n = 539), subcutaneous fat (SF, n = 573), visceral abdominal fat (VAF, n = 531), skeletal muscle (SKLM, n = 534), and liver (LIV, n = 576) obtained during open thorax surgery of 600 coronary artery disease (CAD) patients [26]. eQTL summary statistics from matching tissues in GTEx [12] were used for replication analyses. Significant cis-eQTL associations were defined as in the original STARNET study (FDR <5% across all tested SNP-gene combinations where the SNP is within 1Mb up or downstream of the center of the gene).

We used coronary artery disease (CAD) as the outcome trait and used summary statistics from the CARDIoGRAMplusC4D genome-wide association meta-analysis (GWAMA) study [29]. Summary statistics specific to the European population were extracted using the TwoSampleMR package [32] (study IDs ebi-a-GCST003116, n = 141, 217).

To define locus boundaries and candidate genes of interest for MVMR, we considered genome-wide significant SNPs, all their linked eQTLs within a 0.5Mb radius, and all the genes they are associated with in one or more tissues (“eGenes”). Specifically, we first created tissue-specific outcome data with GWAS summary statistics for all SNPs with non-zero (FDR <5%) eQTL effect in the STARNET summary statistics for the tissue of interest. We filtered these eQTLs by their GWAS p-values, retaining only those with p-values below 5 × 10⁻⁸ in the GWAS study. We then iterated through these eQTLs, sorted by GWAS p-value, such that for the first iteration, the first eQTL would be the first lead SNP. We collected all eQTLs within a 0.5Mb radius and with non-zero LD with the lead SNP, and then removed the lead SNP as well as its collection of linked eQTLs from the list, repeating the procedure with the next lead SNP. For each lead SNP and its collection of linked eQTLs, we first removed SNPs in perfect LD with the lead SNP and then treated the remaining eQTLs as instrumental variables in one causal diagram containing all genes associated with these eQTLs either in a specific tissue, or across all tissues, depending on the application, to complete the diagram. This procedure ensures that for each locus of interest, we obtain a a closed causal diagram, where “closed” means that no more genes (exposures) can be added to the diagram that have shared cis-eQTL associations with any eQTLs (instruments) either already in the diagram or linked to eQTLs in the diagram.

We computed conditional F-statistics using the Mendelian Randomization [30] and MVMR [31] packages to verify instrument strength.

In replication analyses using GTEx, the same procedure was implemented independently for the same GWAS loci in matching tissues, using all cis-eQTL associations in GTEx (which may differ from the cis-eQTL associations in STARNET in the same locus). Hence, replications in GTEx represent the best possible causal effect estimation in GTEx and are not biased by the exposure gene sets used in STARNET.

Since in the GWAS study, the log-odds ratio was used as the unit, that is, logistic regression was performed of the dichotomous outcome trait on SNP genotype, the β-value for a SNP is the increase in log-odds of the outcome. Hence in our models, the predicted causal effect of a gene X on the outcome Y (CAD) is to be interpreted as the effect of X on the log-odds ratio of Y.

We computed summary-based standard errors using the sample size from the GWAS study (see S1 File for details) and obtained p-values from its asymptotic normal distribution. We confirmed using simulations that summary-based and individual-level based standard errors were comparable (S6 Fig). We further confirmed standard errors using the “mr_mvivw” method of the “MendelianRandomization” package (S6 Fig). We note that to the best of our knowledge, the only standard error estimator that takes into account variability due to the different sample sizes in the two-sample setting [35] requires individual-level data and can thus not be computed in the two-sample summary-data setting.

Causal effect identification and estimation in MVMR with correlated instruments

To analyze the problem of MVMR with correlated instruments, we consider causal diagrams where one or more genes X_i are colocated in the same genomic locus and have a potential causal effect on a phenotypic trait Y associated to the same locus. We assume that there exist at least as many causal variants in the locus as there are potential causal genes, such that each gene has at least one causal variant (causal variants may be shared between genes).

If there is only one gene in the locus, we have the classical instrumental variable graph that underpins standard (univariate) MR (Fig 1A). Application of the method of path coefficients (see Methods) shows immediately that the causal effect c of the exposure gene X on the outcome Y is given by the usual instrumental variable formula c = σ_EY/σ_EX, where E is a causal variant (or a variant in LD with a causal one) for X and σ_EY and σ_EX are the covariances between E and X, and E and Y, respectively.

In the case of regulatory pleiotropy, there will be multiple putative causal genes in the same locus, and we obtain an MVMR causal graph with correlated instruments (Fig 1B). Using analytical results derived by Brito and Pearl [24], it can be shown that the variants {E₁, …, E_K} in this graph form an instrumental set relative to the exposures (genes) {X₁, …, X_K} (see Methods), which permits identification of the direct causal effects c = (c₁, …, c_K)^T associated to the edges X₁ → Y, …, X_K → Y in the underlying linear structural equation model (SEM). Application of the method of path coefficients shows that the vector c of causal effects satisfies the system of equations (see Methods for details) (11) with solution (12) where Σ_EX and Σ_EY are the matrix and vector of covariances and , respectively. We note the similarity between Eqs (11) and (12) and the standard instrumental variable solution for univariate MR.

Eqs (11) and (12) describe the causal effects as the coefficients in a linear relation between univariate covariances, and this relation is exact in the infinite sample limit, regardless of LD levels between the instruments. Furthermore, Eqs (11) and (12) remain valid if the number of genetic variants is greater than the number of candidate genes, regardless whether the additional variants are causal or not (Fig 1D and 1E)—the linear system is then merely overdetermined. However, if the number of true causal variants is less than the number of exposures (Fig 1C and 1F), adding genetic instruments that are only in LD with the causal variants does not help. In this case, the variants violate the third instrumental set condition (see Methods), and Eq (11) is easily seen to be underdetermined.

The covariances in Eqs (11) and (12) are the true covariances of the random variables E_i, X_j, and Y. To estimate the causal effects from observational data, we need finite-sample approximations to these equations. The most straightforward estimates are obtained by replacing the true covariances by their empirical estimates. We refer to this solution as , the least squares (LS) estimator, that is,

Since the empirical covariances can be obtained from univariate regressions, the least-squares estimator is also called the “regression of regression coefficients” method.

The least-squares estimator belongs to a more general family of estimators obtained by the generalized method of moments (GMM; see Methods for details), which can be written as where Δ is any square invertible matrix with dimensions equal to the number of instrumental variables. If Δ is the identity matrix (or if is invertible), we recover the least-squares estimator. The estimator is consistent for all choices of Δ. With homoskedastic errors, the estimator with theoretically minimal variance is obtained by taking , the inverse of the empirical covariance matrix of the instruments, that is, the inverse of the LD matrix of the variants (see Methods for details). We will refer to this estimator as the GMM estimator :

We observe that the GMM estimator is identical to the two-stage least squares estimator of McDaid et al. [20] and Porcu et al. [10].

Therefore, both the least squares, regression of regression coefficients estimator and the two-stage least squares estimator are consistent estimators for the MVMR instrumental variable formula (12), and both belong to a more general family of GMM estimators.

Causal effect estimation from simulated data

To test the accuracy of the least-squares (LS) and GMM estimators we performed simulations under a wide range of scenarios. In a first set of experiments, we considered exactly determined systems with normally distributed variables, where the LS and GMM estimators are identical (see Methods). As predicted by theory, the LS/GMM estimator is consistent with variance decreasing with increasing sample size in all simulations (Fig 2). In the literature, no simulations of MVMR with correlated instruments have been reported. We observed, perhaps surprisingly, very limited effect of increased instrument correlation on the variance of the causal effect estimates for pairwise correlations between 0.1 and 0.7, and even at correlation 0.9, the variance is only twice as large as at correlation 0.1 (Fig 2B and 2C).

Download:

• PNG
  larger image
• TIFF
  original image

Fig 2. Causal effect estimation from data simulating instrument correlations, instrument strength, and horizontal pleiotropy.

(A) Causal diagram for the simulation of two exposures X₁ and X₂ for an outcome Y, with two shared instruments E₁ and E₂ with pairwise covariance α and matrix of instrument effect sizes A = [a_ij]. (B) Difference () between estimated and true causal effect of X₁ (true effect size c = 0.2) on Y across 1,000 independently simulated datasets for a range of sample sizes. (C,D) Distribution of the difference between estimated and true causal effects for X₁ (left; true effect size c = 0.2) and X₂ (right; true effect size c = 0.6) across 1,000 independently simulated datasets and a range of sample sizes comparing strong (0.1 − 0.3) vs weak (0.001 − 0.01) instruments. (E) False causal diagram used for inference where one causal exposure is missing. (F-G) Difference between estimated and true causal effect of LPA (left, true effect size c_LPA = −0.05) and PLG (right, true effect size c_PLG = −0.27) on Y across 1,000 independently simulated datasets with sample size 2000 across a range of effect sizes of the hidden gene SLC22A3 mediating horizontal pleiotropic effects of the instruments on Y, showing both the estimates with the correct model (green) and the estimates with the hidden exposure model (orange). See Methods for simulation details.

https://doi.org/10.1371/journal.pgen.1011473.g002

Principal component analysis (PCA) has been suggested as a method to prune correlated SNPs in cis-MR studies by using a reduced set of uncorrelated linear combinations of SNPs as instruments instead [36]. We used the multivariable principal component generalized method of moments (MVMR PCA, function mr_mvpcgmm from the MendelianRandomization package) [37] in our simple simulation and found that at an instrument of correlation value of 0.5 or more, the MVMR PCA method reported too few (that is, only one) effective degrees of freedom for causal effect estimation, despite the causal effects still being estimable.

We further tested the effect of weak instruments, that is, instruments with weak effects on the exposures (see Methods for details). As expected, weak instruments increase the variance in the causal effect estimates, more strongly at small sample sizes (n = 500 − 1,000) (Fig 2D and 2E). In general, the increase in variance due to weak instruments is not greater in the MVMR setting than in standard, univariate MR (S1B and S1D Fig).

In our next set of experiments, we used more realistic model parameters taken from real LD correlation matrices and eQTL and GWAS summary statistics from real loci in the human genome (see Methods for details). First we tested the effect of horizontal pleiotropy. Using real summary statistics from the LPA-PLG-SLC22A3 locus, we simulated data from a model where all three genes (exposures) share causal instruments and are causal for the outcome Y, but we assumed only data for LPA and PLG was available (Fig 2F, see Methods for details). In other words, we assumed that the instruments have horizontal pleiotropic effects through the unmeasured gene SLC22A3. As expected, horizontal pleiotropy introduces bias in the estimated causal effects, although the bias remains modest (within the standard deviation of the estimates obtained from the complete model) upto a simulated effect size of around 0.15 of SLC22A3 on the outcome (Fig 2G and 2H).

Next, we tested the effect of LD matrix bias, where the LD matrices in the eQTL and GWAS population differ from the reference LD matrix (e.g. from 1000 Genomes Project). For this purpose, we compared GMM estimates from data generated from the reference LD matrix and data generated from a biased LD matrix sampled from a Wishart distribution centred around the reference LD matrix (see Methods for details), using the reference LD matrix Σ_EE in both GMM estimates (cf. Eq (10)). With a modest degree of LD matrix bias (Wishart distribution with 50 degrees of freedom), the bias in the causal estimates remained small (Fig 3B and 3C).

Download:

• PNG
  larger image
• TIFF
  original image

Fig 3. Causal effect estimation from data simulating LD matrix sampling bias.

The first and second row show a causal diagram for the simulation of overdetermined systems where the number of causal variants was equal to the number of exposures (first row, three exposures; second row, two exposures) and other variants were only correlated with the exposures and outcome by LD. (A, B, C) Comparison between the GMM estimator using an unbiased (B) and biased (C) LD matrix for the causal effect of simulated SLC22A3, for different LD pruning thresholds (corresponding to models with 3, 4, 5, 6 and 7 instruments). (D, E, F) Comparison between the least-squares, GMM, TWMR, MV multiple (TwoSampleMR), and MVIVW (Mendelian Randomization package) estimators using independently sampled datasets of different sizes for exposures and outcomes to emulate a two-sample MVMR setting. (G, H, I) Difference between estimated and true causal effect of SLC22A3 (left, true effect size c_SLC22A3 = 0.15), LPA (center, true effect size c_LPA = −0.05) and PLG (right, true effect size c_PLG = −0.27) on Y showing comparison between the least-squares, GMM, MVMR Robust, MVMR Median, MVMR Lasso and MVMR cML estimators using simulated datasets from causal diagram (A). All distributions plots show the difference between estimated and true causal effects across 1,000 independently sampled datasets. See Methods for simulation details.

https://doi.org/10.1371/journal.pgen.1011473.g003

Next, we compared the LS and GMM causal effect estimators in over-determined models (more instruments than exposures). While both estimators are guaranteed to be consistent, the GMM estimator is the estimator with theoretically lowest variance (see Methods). However, across thousands of simulations with multiple sample sizes and for models of varying degree of over-determination, we did not observe any differences in variance between the LS and GMM estimators (Fig 3B and 3C).

We also simulated a two-sample setting where eQTL and GWAS LD matrices were sampled independently from two different Wishart distributions centred around the reference LD matrix, with different variance parameters to reflect the difference in sample size between eQTL and GWAS studies, and generated eQTL and GWAS summary statistics independently, with sample sizes representative of real eQTL and GWAS studies (see Methods for details). In the two-sample setting, even the effects estimated using the true LD matrix from the GWAS population are biased due to the LD matrix sampling variation between the eQTL and GWAS studies (Fig 3E). Unsurprisingly, the bias increased further when the reference LD matrix was used instead of the GWAS study LD matrix in the estimation (Fig 3F). In this setting, all estimators were biased until an eQTL and GWAS sample size combination of 4,000/140,000 was reached, which corresponds to the upper end of currently available sample sizes (Fig 3F); TWMR remained biased at all sample sizes, due to a hard-coded regularization factor (see Methods for details). We note that the causal effect bias in two-sample settings is not specific to MVMR or the use of correlated instruments, but can already be observed in univariate MR with a single instrument (S5 Fig).

Prediction of causal genes at GWAS loci for coronary artery disease

To test our method on real-world data, we predicted causal genes at 36 genome-wide significant loci for coronary artery disease (CAD) from the CARDIOGRAMC4D meta-analysis study [38] using eQTL data from seven vascular and metabolic tissues from the STARNET study [26] (see Methods for details). Results are presented here for the least squares estimator; the GMM estimates are predominantly consistent and are provided in the Supporting Data.

First, to get an understanding for the range of causal effect sizes one can expect for true causal genes, we identified 17 genome-wide significant CAD loci which had an effect on expression of a single candidate gene in a single tissue, that is, where no cis-regulatory pleiotropy is present and we can plausibly hypothesize that the true causal gene is known (Fig 4A). These genes included well characterized genes such as PCSK9, a risk gene for low-density lipoprotein cholesterol levels and CAD [39], whose visceral adipose (VAF)-specific association to CAD risk SNPs in STARNET was previously confirmed in independent data [26], and GUCY1A3, whose expression correlates with risk of atherosclerosis [40]. The absolute predicted causal effects on the standardized scale for these 17 genes ranged from 0.099 (PCSK9 in VAF) to 0.34 (GSTM3 in mammary artery (MAM)). We therefore considered an absolute causal effect size greater than or equal to 0.1 to be an appropriate threshold to define causal genes.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 4. Predicted causal genes and their tissue-specific effects on CAD at genome-wide significant CAD loci.

(A) Causal effects from univariate MR of genes at 17 CAD GWAS loci with effects on a single gene in a single tissue. (B) Causal effects from tissue-specific, univariate MR of genes at seven CAD GWAS loci with cis-regulatory effects on a single gene in multiple tissues. (C) Tissue color legend for all panels. (D) Causal effects from tissue-specific MVMR with correlated instrumental variable sets of genes at 12 CAD GWAS loci with cis-regulatory effects on multiple genes in multiple tissues. (E) Scatter plot of tissue-specific causal effect estimates from MVMR vs. univariate MR. (F) Scatter plot of tissue-specific causal effect estimates from MVMR vs. CAD regulary trait concordance (RTC) scores. (G) Scatter plot of tissue-specific causal effect estimates from MVMR in STARNET vs. GTEx.

https://doi.org/10.1371/journal.pgen.1011473.g004

We confirmed the validity of this threshold using seven genome-wide significant CAD loci which had an effect on expression of a single candidate gene in at least two tissues in the STARNET data, that is, loci where we can plausibly hypothesize that the true causal gene, but not the tissue, is known. We used standard, univariate MR on a tissue-by-tissue basis and found that in all seven cases, the candidate gene had a predicted absolute causal effect size greater than 0.1 in at least one tissue (Fig 4B). These genes included CDKN2B, located in one of the most robust genetic markers for type 2 diabetes, CAD, and myocardial infarction [41]. GWAS SNPs in the CDKN2B locus were associated with expression of CDKN2B in aorta (AOR), subcutaneous fat (SF), and blood, with predicted causal effects c = −0.75, −0.55, and 0.5, respectively.

We further computed p-values based on the standard errors of the estimated causal effects. As these p-values are based on one-sample estimates for the standard error and may not reflect variability in two-sample summary-data settings (see Methods for details), we conservatively set a Bonferroni threshold of p < 0.05/150 = 3 ⋅ 10⁻⁴, which exclude only five gene-tissue combinations with estimated causal effect .

Having thus found a reasonable threshold to define causal genes for CAD, we analyzed 12 genome-wide significant CAD GWAS loci which had an effect on expression of multiple nearby genes in one or more tissues. Similar to Porcu et al. [10], we applied MVMR on a tissue-by-tissue basis using all genes with cis-eQTL associations in a locus in a given tissue as potential exposures in a causal diagram (cf. Fig 1B and 1E). We note that at only two of the 12 loci, it was possible to reduce the SNPs to a set of independent instruments using the LD threshold (r² ≤ 0.1) used by Porcu et al. [10], while still retaining at least as many instruments as exposures. Hence an approach using correlated instrument sets is essential.

In total, 35 candidate causal genes with cis-eQTL associations were located in the 12 CAD GWAS loci with pleiotropic gene regulatory effects, of which 24 had a predicted absolute causal effect size greater than 0.1 in at least one tissue (Fig 4D). To analyze these causal genes more systematically, we compared the predicted causal effects from MVMR against the predicted causal effects from standard MR (Fig 4E) and regulatory trait concordance (RTC) scores [42] computed by Franzén et al. [26] (Fig 4F), two univariate methods that ignore the pleiotropic gene regulatory effects. We also conducted a replication analysis for loci with cis-associations in matching tissues in both STARNET and GTEx (Fig 4G, see Methods for details).

We observed a general trend where predicted causal genes using MVMR (predicted causal effect |c| ≥ 0.1) were also predicted to be causal (using the same threshold) using univariate MR, had high RTC values, and had concordant effect sizes between STARNET and GTEx (Fig 4E–4H). However several genes with high univariate MR effects and high RTC values were not causal according to MVMR, suggesting that MVMR can indeed correct for likely false predictions from MR or colocalization analyses due to regulatory pleiotropy.

A clear example of a CAD GWAS locus where MVMR and MR give contrasting results is a locus centred around chr 6:12,901,441. In STARNET, the locus has cis-associations with the expression of PHACTR1, GFOD1, TBC1D7, RP1–257A7.4, and RP1–257A7.5 in MAM (Fig 5A and 5B). In univariate MR analyses these genes have predicted causal effects ranging from 0.15 for PHACTR1 to 0.31 for GFOD1, and a previous integrative genomics analysis lists all of them as candidate causal genes for this locus [43]. After pruning nearly identical SNPs (LD r² ≥ 0.95), 12 instruments (with mutual LD ranging from r² = 0.17 to r² = 0.86) were available for conducting MVMR, resulting in a predicted causal effect of 0.19 for PHACTR1, with all other effects below the threshold of 0.1 in absolute value, suggesting that PHACTR1 is the only causal gene in arterial tissue at this locus.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 5. Example CAD GWAS loci with pleitropic gene regulatory effects and predicted causal genes.

Each row shows the predicted causal effects from tissue-specific MVMR of genes with cis-eQTL associations in a CAD GWAS locus of interest across seven vascular and metabolic tissues from the STARNET study (left) and the genetic architecture and cis-regulatory pleiotropy of the locus. The heatmaps show the causal effects on the liability on a standardized scale. Only tissues with non-zero cis-eQTL effects in the STARNET study are included. Values identically zero indicate gene-tissue combinations not included as exposures in the model (no cis-eQTL association in that tissue). The LocusZoom plots show for each SNP in the locus the GWAS −log₁₀(p − value) (y-axis) and associated cis-eQTL genes (symbol, see legends) in a selected tissue. (A, B) Locus centred on chr 6:12,901,44 with LocusZoom plot for MAM. (C, D) Locus centred on chr 15:79,141,784 with LocusZoom plot for AOR. (E, F) Locus centred on chr 2:85,809,989 with LocusZoom plot for AOR.

https://doi.org/10.1371/journal.pgen.1011473.g005

Another example is the locus centred around chr 15:79,141,784, where functional studies support a causal and proatherogenic role for ADAMTS7 [44]. In STARNET, the locus has cis-associations with the expression of ADAMTS7 in AOR, MAM, and VAF(Fig 5C and 5D). In all three tissues, the locus also has cis-associations with the expression of CTSH. After pruning nearly identical SNPs (LD r² ≥ 0.86), 9, 25, and 10 instruments (with mutual LD ranging from r² = 0.03 to r² = 0.94) were available for conducting MVMR in AOR, MAM, and VAF respectively. In the arterial tissues, MVMR confirms that ADAMTS7 is the most likely causal gene (predicted causal effects 0.18 and 0.14 vs. 0.025 and −0.058 for CTSH in AOR and MAM, respectively; Fig 5C). Replication in GTEx AOR tissue confirmed these results (predicted causal effects 0.18 and 0.009 for ADAMTS7 and CTSH, respectively). By contrast, in VAF, both genes are predicted causal with effects in the opposite direction (predicted causal effects −0.18 and 0.25 for ADAMTS7 and CTSH, respectively), the latter due to opposite eQTL effects of the same variants in VAF vs. AOR and MAM. No eQTL associations with either gene were available in GTEx in adipose tissue to confirm these results. The overall picture is further complicated by the fact that the locus also has cis-associations with the expression of CTSH in blood, SKLM, and SF, and with PSM4 in SKLM. The absence of associations with ADAMTS7 in these tissues implies that the association between the locus and CAD is automatically attributed to the other genes (Fig 5C). Given the overwhelming functional evidence for ADAMTS7 [44], these results probably reaffirm the importance of having all true exposures included in the model (cf. Fig 2F–2H).

As a final example we consider the locus centred around chr 2:85,809,989. Previous analyses found a candidate causal SNP in this locus located in the promoter region of MAT2A, but since the SNP was associated with expression of multiple genes in this locus, a causal gene could not be predicted [45]. In STARNET, the locus has cis-associations with the expression of MAT2A in AOR and MAM. In both tissues, the locus also has cis-associations with the expression of GGCX, and both genes have previously been listed as candidate causal genes in arterial tissues [43]. After pruning nearly identical SNPs (LD r² ≥ 0.95), three instruments (with mutual LD ranging from r² = 0.69 to r² = 0.89) were available for conducting MVMR. In both AOR and MAM, MVMR suggests MAT2A, and not GGCX is the causal gene (predicted causal effects −0.23 and −0.33 vs. 0.02 and 0.16 for GGCX in AOR and MAM, respectively; Fig 5E and 5F). The overall picture is again less clear, since the locus also has cis-associations with the expression of GGCX and VAMP8 in liver, and with GGCX alone in SF and VAF. Interestingly, in liver, neither gene is predicted causal (predicted causal effects −0.08 and −0.06; Fig 5E), while in the adipose tissues, the entire association between the locus and CAD is automatically attributed to GGCX (predicted causal effects −0.14 in both tissues; Fig 5E).

The preceding examples suggest that in fact two distinct forms of regulatory pleiotropy exist. A GWAS locus can be associated (in cis) to expression levels of multiple genes in the same locus in a tissue of interest, but can also be associated to one or more genes in multiple tissues. We attempted to conduct MVMR using all potential exposures (across genes and tissues) in a single model, in order to simultaneously predict the causal gene(s) and tissue(s) at loci of interest. However, conclusive results were often elusive, as both the determinant of the LD-matrix and the instrument strength matrix were very small, indicating that the number of causal variants in most loci is less than the often large number of exposures that must be accounted for in such a multi-tissue analysis.

An example of how multi-tissue MVMR can potentially differ from tissue-specific MVMR is given by the locus centred on chr 19:11,061,315. In STARNET, the locus has cis-associations with the expression of CARMI in SF and SKLM, and with RGL3 and SMARCA4 in liver. In tissue-specific analyses, CARMI was causal in SF (c = 0.19) and SKLM (c = 0.18) (univariate MR), while RGL3 (c = −0.19) and SMARCA4 (c = −0.2) were both predicted to be causal in liver (MVMR). Notably, the lead SNP rs35140030 was shared between CARMI in SF and SKLM, and another SNP in the locus, rs113718993 was shared between CARMI in SKLM and SMARCA4 in liver. Since these SNPs were shared between tissues, and two additional instruments were available after pruning nearly identical SNPs (LD r² ≥ 0.95), we conducted an additional MVMR analysis where exposures were gene-tissue pairs. This analysis predicted that (CARM1, SKLM) (c = 0.18) and (RGL3, liver) (c = −0.19) were the causal gene-tissue pairs, while (CARM1, SF) (c = −0.02) and (SMARCA4, liver) (c = −0.01) were not causal.