Our framework starts with a set of structural equations that jointly specify the generative model on the disease Y that relies on K observed risk factors X = (X₁, ⋯, X_K) of interest, and the vector Z = (Z₁, Z₂, ⋯) containing all genetic information of an individual (Fig 1a). Y = X T β + f ( U , Z , E Y ) ( if Y is a continuous trait ) logit [ P ( Y = 1 ) ] = X T β + f ( U , Z , E Y ) ( if Y is a binary trait ) X k = g k ( U , Z , E X k ) , k = 1 , ⋯ , K (1) Here U represents unknown non-heritable confounding factors and E X k and E_Y are random noise acting on X_k and Y respectively. The parameter of interest, β, quantifies the causal effect of the vector of risk factors X on Y. Mendel’s laws of inheritance suggest that the genotypes Z are randomized during conception and are generally independent of the environmental factors (U , E Y , E X k). The function f(U, Z, E_Y) represents the causal effect of unmeasured risk factors on Y, which can be heritable (contributed by Z) or non-heritable (contributed by U). The non-parametric functions f(⋅) and g_k(⋅) allow interactions among SNPs in Z and variables (U , E Y , E X k) in their causal effects on X and Y. Under this model, there is horizontal pleiotropy for a SNP j if Z_j has nonzero association with f(U, Z, E_Y). This is the case, for example, when Z_j acts on Y through a pathway affecting unmeasured risk factors, or when Z_j is in linkage disequilibrium (LD) with such a locus.

Now consider the case where only GWAS summary statistics, i.e. the estimated marginal associations between each SNP j and the risk factors/disease traits, are available and there are in total p SNPs selected. Let Γ_j be the true association between SNP j and Y, and γ_j be the vector of true marginal associations between SNP j and X. Later, we will denote their estimated values from GWAS summary statistics as Γ ^ j and γ ^ j. Then, as shown in Materials and Methods, the model (1) results in the linear relationship Γ j = γ j T β + α j (2) where for binary Y, the parameter β in model (2) is a conservatively biased version of β in model (1). This relationship holds even when the functions f(⋅) and g_k(⋅) in (1) are not linear. Here, α_j is the marginal association between Z_j and f(U, Z, E_Y), representing the unknown horizontal pleiotropy of SNP j.

One can immediately see that identifying β is impossible without further assumptions regarding α_j. Early MR methods such as IVW [5] made the assumption that all instruments are valid satisfying α_j = 0. Other methods such as Weighted Median [7] or MR-PRSSO [9] assume that α_j is sparsely nonzero. However, the no or sparse pleiotropy assumption follows from statistical convenience rather than biological insights. As discussed in Introduction, horizontal pleiotropy is pervasive for most complex traits. One assumption that allows pervasive pleiotropy is to assume the InSIDE assumption [6] where α j⫫ γ j, or alternatively, the random effect model [10, 16] where α j ∼ N ( 0 , τ 2 ) for most genetic instruments. Unfortunately, the InSIDE assumption can be easily violated if the pleiotropic effects of selected genetic variants are driven by shared pleiotropic mechanisms.

Some more recent MR methods such as LCV [26], MRMix [12], Contamination mixture [13] and CAUSE [15] have noticed this limitation of the InSIDE assumption and allow a subset of the genetic instruments to be associated with a common hidden pleiotropic pathway. For instance, using the above notation, both CAUSE and MRMix assumed that when for the SNPs that violate the inSIDE assumption, their pleiotropic effects satisfy α j = a γ j + α ˜ j (when K = 1) where aγ_j represents the correlated pleiotropic effects due to a confounding pathway and α ˜ j ⫫ γ j. This is a more realistic assumption than InSIDE, though there would then be an issue in distinguishing the true causal effect β from the pleiotropic direction β + a. Allowing for only one pleiotropic pathway also makes the model restrictive for real datasets.

The key idea underlying GRAPPLE is that multiple pleiotropic pathways can be detected by using the shape of the profile likelihood function under no pleiotropy assumption. This allows us to probe the underlying causal mechanism, without explicit assumptions on pleiotropic patterns (Fig 1b). When K = 1, the GWAS summary statistics reduce to the scalar γ ^ j and Γ ^ j, with their standard errors σ_1j and σ_2j. From the central limit theorem, the joint distribution of ( γ ^ j , Γ ^ j ) approximately follows a multivariate normal distribution ( γ ^ j Γ ^ j ) ∼ N ( ( γ j Γ j ) , ( σ 1 j 2 σ 1 j σ 2 j θ σ 1 j σ 2 j θ σ 2 j 2 ) ) (3) where θ is a shared sample correlation that can be estimated as θ ^ (see Materials and methods).

When there is no horizontal pleiotropy in the p selected independent genetic instruments (α_j = 0 for j = 1, 2, ⋯, p), the robust profile likelihood [10] is given by, l ( b ) = - ∑ j = 1 p ρ ( Γ ^ j - b γ ^ j σ 1 j 2 + b 2 σ 2 j 2 + 2 b θ ^ σ 1 j σ 2 j ) (4) where ρ(⋅) is the Tukey’s Biweight loss, or any other robust loss functions. As described with more details in Materials and Methods, the profile likelihood is obtained by profiling out nuisance parameters γ₁, ⋯, γ_p in the full likelihood from (3), which is further robustified by replacing the L₂ loss with Tukey’s Biweight loss to increase the sensitivity of mode detection. Under no pleiotropy or InSIDE assumption, this function l(b) should have only one mode near the true causal effect b = β.

Now consider the case where a second genetic pathway (Pathway 2) also contributes substantially to the disease, and some instrument loci are also associated with Pathway 2 (Fig 1b). In this scenario, SNPs that are associated with X only through Pathway 2 can contribute to a second mode in the profile likelihood at location β + κ/δ, where κ and δ quantifies the causal effect of Pathway 2 on Y and its marginal association with X, respectively (Materials and Methods). Similarly, multiple pleiotropic pathways generally result in multiple modes of l(b). Thus, we can use multiple modes in a plot of l(b) to diagnose the presence of horizontal pleiotropic effects that are grouped by different pleiotropic pathways.

The existence of pleiotropic pathways complicates MR and makes the causal effects of the risk factors potentially unidentifiable. Specifically, when Pathway 2 exists, the GWAS summary statistics alone cannot provide information to distinguish β from β + κ/δ. Instead of making further untestable assumptions such as one pathway “dominates” the other, when multiple modes are detected, we suggest that whenever multiple modes are detected, the investigator should try to find biomarkers for each mode and collect more GWAS data to adjust for confounding risk factors. Specifically, GRAPPLE facilitates this by identifying marker SNPs of each mode, as well as the mapped genes and GWAS traits of each marker SNP (see Materials and methods). This allows researchers to use their expert knowledge to infer possible confounding risk factors that contribute to each mode. With GWAS summary statistics of these confounding traits, GRAPPLE can perform a multivariable MR analysis assuming the InSIDE assumption applies for the remaining horizontal pleiotropic effects (Materials and Methods).

The detection of multiple modes can be also used to determine the causal direction (Fig 1c). If the wrong causal direction is specified in model (1) and Y is a cause of X, the genetic variants associated with X can be classified in two groups: those associated with X through Y, and those associated with X through another pathway unrelated to Y. In the former case, γ_j = βΓ_j where β is the causal effect of Y on X, and these SNPs should contribute to a mode around 1/β. In the latter case, a SNP j satisfies γ_j ≠ 0 but Γ_j = 0, and would contribute to a mode of l(b) at 0. Thus, there will be two modes in the robust profile likelihood with one mode being around 0. This idea can be viewed as an extension of the bidirectional MR [29, 30]. Bidirectional MR is based on the assumptions that when X is a cause of Y, most of the genetic instruments for Y should be unassociated with X, because they affect Y through a different pathway, thus the reserve MR would indicate a zero effect of Y on X. GRAPPLE makes this inference more robust by making use of the fundamentally different shape of the robust profile likelihood plots in different directions. In the correct causal direction, the plot should only show one mode around the true causal effect β. In the incorrect reverse direction - when the true outcome is treated as the risk factor and the true risk factor is treated as the outcome - the plot of the robust profile likelihood will have two modes, one around 0, representing the variants directly related to the true outcome, and one around 1/β, representing the variants indirectly related to the true outcome through the true risk factor.

Besides the assumption of no-horizontal-pleiotropy, for a SNP to be a valid genetic instrument, it needs to have a non-zero association with the risk factor of interest. In most MR pipelines, SNPs are selected as instruments only when their p-values are below 10⁻⁸, which is required to guarantee a low family-wise error rate (FWER) for GWAS data. Using such a stringent threshold also helps to avoid weak instrument bias [31], where measurement errors in γ ^ j k are not ignorable and lead to bias in β ^. However, such a stringent selection threshold may result in very few, or even no instruments being selected with under-powered GWAS, and may still not be adequate to avoid weak instrument bias. Further, when our goal is to jointly model the effects of multiple risk factors (the setting where X as a vector), it is unrealistic to assume that all selected SNPs have strong effects on every risk factor. In addition, the high polygenecity of complex traits indicates that the weak instruments far outnumbers strong instruments, and collectively, they may substantially improve the estimation accuracy.

In GRAPPLE, we use a flexible p-value threshold, which can be either as stringent as 10⁻⁸ or as relaxed as 10⁻², for instrument selection. Based on the profile likelihood framework of MR-RAPS [10], GRAPPLE can provide valid inference for β ^ that avoids weak instrument bias for multiple risk factors even when the p-value threshold is as large as 10⁻². This flexible p-value threshold is beneficial for several reasons. First, including moderate and weak instruments may increase power, especially for under-powered GWAS. Second, for MR with multiple risk factors where it is inevitable to include SNPs that have weak associations with some of the risk factors, we can obtain much more accurate causal effect estimations than methods that can only deal with jointly strong SNPs. More importantly, we can investigate the stability of the estimates across a series of p-value thresholds and get a more complete picture of the underlying horizontal pleiotropy. In practice, we suggest researchers to vary the selection p-value thresholds from a stringent one (say 10⁻⁸) to a relaxed one (say 10⁻²), both in the detection of multiple modes and in estimating causal effects.

The current two-sample design of MR uses one GWAS data for the risk factor and one for the disease. The selection of genetic instruments is performed with p-values reported in the GWAS data for the risk factor. However, selecting instruments from GWAS summary statistics can introduce bias, which is commonly referred to as the “winner’s curse”. Conditional on being selected, the magnitude of γ ^ j k is generally larger than γ_jk and introduces bias to the estimation of β. When K = 1 where there is only one risk factor, the estimate will be biased towards 0, but there is no guarantee on the direction of the bias when K > 1. Among practitioners, a common belief is that the selection bias is negligible when only the strongly associated SNPs are selected as instruments.

However, this rule of thumb may not hold even when we only use that are genome-wide significant (p-value ≤10⁻⁸) (S1(a) Fig). Thus, we strongly advocate using a three-sample GWAS summary statistics design (Fig 1d). To avoid the selection bias, selection of genetic instruments is done on another GWAS dataset for the risk factor, whose cohort has no overlapping individuals with both the risk factor and disease cohorts. In addition, to simplify calculation and avoid bias due to different LD structure in heterogeneous populations, we use LD clumping [32] to select independent SNPs in GRAPPLE (see Materials and methods). The three-sample design will also avoid possible selection bias introduced during clumping.

Summarizing the above points, a complete diagram of the GRAPPLE workflow is shown in Fig 1d. A researcher may start with a single target risk factor of interest. The shape of the robust profile likelihood provides information on possible pleiotropic pathways. If only a single mode is detected, one can use GRAPPLE for the target risk factor. This is equivalent to using the original MR-RAPS. If multiple modes are detected, the researcher needs to seriously consider how to adjust for pleiotropic pathways. Researchers can use the marker SNP/gene/trait information that GRAPPLE provides to investigate each mode, decide on which confounding risk factors to adjust for, and collect extra GWAS data for them. GRAPPLE can then be used to jointly estimate the causal effects of the original and the additional risk factors.

We first examine whether GRAPPLE provides reliable statistical inference using instruments with different strength under an artificial setting with real GWAS summary statistics. In this setting, we make the “artificial risk factor” X and the “artificial disease” Y be the same trait from two non-overlapping cohorts, thus γ_j = Γ_j while γ ^ j ≠ Γ ^ j for any SNP j. Though the structural equation describing the causal effect of X on Y is not well defined., the linear relationship model (2) from which we estimate β still holds with β = 1 and α_j = 0. In other words, we are not estimating a meaningful “casual” effect, but are in a special case where the true β is known. This setup can be used to verify the validity of MR methods under no pleiotropy.

Specifically, we consider three traits: Body mass index (BMI), Type II diabetes (T2D) and height from the GIANT and DIAGRAM consortia where sex-specific GWAS data are available [33–35]. The female cohort is used to get γ ^ j and the male cohort is used to get Γ ^ j. As a three-sample design, the UK Biobank data for corresponding traits are used for SNP selection. If we assume that all selected instruments have no gender-specific association with the traits, the true β would equal 1. For benchmarking, we compare the performance of GRAPPLE with CAUSE [15] and three other widely adopted MR methods: inverse-variance weighted (IVW) [5], MR-Egger [6] and weighted median [7] with the same three-sample design.

We compare different p-value thresholds for instrument selection, ranging from a stringent threshold of 10⁻⁸ to a relaxed threshold of 10⁻² (Fig 2a). GRAPPLE provides roughly unbiased estimates of β no matter which threshold is used, showing that it does not suffer from weak instrument bias. Surprisingly, the other MR methods are biased even with a stringent p-value threshold.

Notice that for T2D, the confidence intervals of GRAPPLE do get narrower with increasing p-value thresholds (Fig 2a), showing the potential power gain of including weak instruments in less powerful GWAS studies. In addition, we simulate synthetic GWAS summary statistics of the risk factor and disease (see S1 Text for details) and confirm that the estimated β indeed gets more accurate with the inclusion of weakly associated SNPs (S2(a) Fig). In the simulations, we also use GRAPPLE to adjust for measured confounding risk factors and compare the performance with MVMR [36], a commonly used multivariable MR method. As discussed earlier, in multivariable MR, the inclusion of SNPs that are weakly associated with at least one risk factors is inevitable. As GRAPPLE does not suffer from weak instrument bias, we see that it provides accurate estimates of the causal effects as well as reliable confidence intervals with both stringent and mild p-value thresholds (S3–S5 Figs).

Finally, we demonstrate that to avoid bias, the three-sample design is necessary no matter which MR method is used. As shown in S1(a) Fig, the two-sample design where we use the same cohort of the risk factor for selection can result in biased casual effects estimation, and the bias occurs with most MR methods even when we only select the strongly associated SNPs.

Next, we examine whether or not the weak instruments are more vulnerable to pleiotropy, which can be a concern for including the weak SNPs. We compare four risk factor and disease pairs that cover eight different complex traits, including the effect of BMI on T2D, low-density cholesterol concentrations (LDL-C) on coronary artery disease (CAD), height on smoking, and systolic blood pressure (SBP) on stroke (Fig 2b). The GWAS summary data are collected from the original study repositories [37–42].

We test whether independent sets of strongly and weakly associated SNPs can provide consistent estimates of the causal effects of the risk factors. SNPs passing the p-value threshold 10⁻² in the cohort for selection are divided into three non-overlapping groups after LD clumping: “strong” (p_j ≤ 10⁻⁸), “moderate” (10⁻⁸ < p_j ≤ 10⁻⁵), and “weak” (10⁻⁵ < p_j ≤ 10⁻²). The SNPs across groups are used separately to obtain group-specific estimates of the causal effect β. We observe that for all the four pairs, the estimates β ^ are stable across groups (Fig 2b). Though the “weaker” SNPs provide estimates with more uncertainty due to limited power, the estimates are consistent with those from the “strong” group. Other MR methods also show some level of consistency in estimating β across different sets of instruments, but perform less well due to weak instrument bias (S1(b) Fig). To conclude, in the analysis of these four pairs of traits, we do not see any evidence that weakly associated SNPs provide more biased estimates than strong instruments due to horizontal pleiotropy. In contrast, as with the strong instruments, the weakly associated SNPs may also provide useful information to infer the causal effects of the risk factors.

We also examine the performance of GRAPPLE in identifying the causal direction with the shape of the profile likelihood. For the causal direction, we focus on the two pairs of traits with known causal relationship: BMI on T2D, and LDL-C on CAD. We switch the roles of the risk factor and disease to see if the correct direction can be revealed. Specifically, we treat T2D and CAD as the “risk factor”, and BMI and LDL-C as the corresponding “disease” (Fig 2c). For T2D, the cohort for the other gender is used for SNP selection and for CAD, the risk factor cohort used is from [43] and the selection p-values are from [44]. As expected, we see that when the roles of the risk factor and disease are reversed, the robust profile likelihood shows a main mode at 0, and a weaker mode around 1 / β ^.

Finally, we test the ability of GRAPPLE to identify multiple pleiotropic pathways with the analysis of the C-reactive protein (CRP) effect on CAD. C-reactive protein has been found to be strongly associated with the risk of heart disease while many SNPs that are associated with the C-reactive protein also seem to have pleiotropic effect on lipid traits [45]. Previous MR analyses only included SNPs that are near the CRP gene to guarantee a free-of-pleiotropy analysis [46, 47] and found that CRP has no causal effect on CAD. Now, instead of only using SNPs near the CRP gene, by using associated SNPs across the whole genome that are known to involve pleiotropy pathways, can GRAPPLE identify the existence of these pathways and still obtain the correct estimate of the C-reactive protein effect?

CRP GWAS data from [48] are used for selection and the data from [49] using a larger cohort is used for getting γ ^ j. Similar to a multi-modality pattern already reported in [11], our robust profile likelihood shows a pattern of three modes, indicating the existence of at least three different pathways (Fig 2d). One mode is negative, one is positive and the third is around zero. The negative mode involves a few marker genes including HNF1A and PVRL2, with a marker trait LDL-C. The positive mode has marker traits pulmonary function and the C-reactive protein, and the few markers genes (IL6R, ARHGAP10, BCL7B, PABPC4) are also involved in immune response and lung cancer progression [50, 51]. The mode at 0 has marker genes CRP and LEPR, and only one marker trait, C-reactive protein.

We compare across 3 p-value thresholds (10⁻⁸, 10⁻⁵, 10⁻³) and check how the existence of multiple pathways affects causal estimates of the effect of C-reactive protein in MR methods using SNPs across the genome. Including C-reactive protein as the only risk factor, all bench-marking methods give a negative estimate of the CRP effect, which is possibly driven by the bias from an LDL-C induced pleiotropic pathway (Fig 2e). MR-RAPS is the estimation method used in GRAPPLE if we only use one risk factor, and the three other bench-marking methods give incorrect inference of the CRP effect with a p-value of β below 0.01 for at least one SNP selection threshold (notice that the weak instrument bias is towards 0 as shown in Fig 2a, thus the significance at p-value threshold 10⁻³ for MR-Egger and IVW cannot be explained by weak instrument bias). In contrast, after using two risk factors: C-reactive protein and LDL-C, where LDL-C is an identified confounding risk factor from the marker SNPs in Fig 2d, the estimates of CRP effect are much closer to 0 compared with that without including LDL-C. This analysis illustrates how GRAPPLE can detect pleiotropic pathways, provide information to identify the confounding risk factors to adjust for, and obtain correct inference after adjusting for these risk factors.

As a complement to the analysis on CRP, we also use simulations (S1 Text) and generate synthetic disease traits to evaluate the precision and recall in the detection of multiple modes and marker SNPs when there are pleiotropic pathways. We consider scenarios with one or two pleiotropic pathways caused by hidden confounding risk factors, and vary the genetic correlations between these hidden factors and the target risk factor. A higher genetic correlation corresponds to a larger proportion of SNPs that have a correlated pleiotropic effect. We observe that the detection of multiple modes is most powerful when the genetic correlation is neither too large nor too small (S2(b) Fig). If the genetic correlation is too high, then there are not enough SNPs to contribute to the mode of the true causal effect, while if the genetic correlation is too low there will be too few SNPs to contribute to the pleiotropic modes. Including weaker SNPs will decrease the sensitivity in mode detection but can increase the recall of true marker SNPs. In our simulations, we also observe that all univariable MR methods can perform poorly in estimating the true causal effect in the presence of pleiotropic pathways (S3–S5 Figs).