Multi-tissue integrated Mendelian randomization method identifies disease risk genes

Abstract

Mendelian randomization (MR) leverages genetic variants as instrumental variables to infer causal relationships between molecular traits and diseases, however, identifying the specific causal genes underlying disease risk remains challenging. Here, we present MULTI (Multi-tissue Unified Likelihood-based Transcriptomic Integration), a Bayesian MR framework that integrates genetic information across tissues to improve the accuracy of causal inference. MULTI yields reliable estimates even when the number of instruments in a single tissue is limited, and further increases statistical power by adaptively integrating information from similar tissues without inflating type I error rates. Extensive simulations confirm its robustness under diverse genetic architectures, and applications to real datasets demonstrate its capacity to reveal tissue-specific causal mechanisms and coordinated cross-tissue regulation. MULTI offers a practical and extensible framework for elucidating the molecular architecture of complex human diseases.

Introduction

Two-sample Mendelian randomization (MR) is an epidemiological approach that leverages genetic variants as instrumental variables (IVs) to infer the causal effects of complex human traits and diseases [1, 2]. In recent years, however, researchers have become increasingly interested in pinpointing the causal genes underlying disease susceptibility, as this can provide deeper insights into the molecular mechanisms driving observed associations [3–5]. An intuitive approach is to extend the conventional two-sample MR framework by using gene expression levels in place of traditional exposures, thereby enabling the evaluation of gene-trait/outcome causal relationships. The growing availability of expression quantitative trait loci (eQTL) data—such as those from the Genotype-Tissue Expression (GTEx) project [6]—has greatly facilitated this integration, allowing the combination of eQTL resources with genome-wide association studies (GWAS) summary statistics to uncover genes whose genetically regulated expression contributes to disease risk.

However, integrating eQTL data with GWAS summary statistics also introduces several methodological challenges that complicate reliable causal inference. First, unlike GWASs that evaluate variants across the entire genome, eQTL analyses are typically confined to cis-regulatory variants near the gene locus [6]. This narrow cis-window substantially reduces the number of eligible IVs, resulting in unstable causal estimates—particularly for genes with low expression heritability [7–9]. Second, linkage disequilibrium (LD) among cis-eQTLs violates the requirement of valid IVs for standard MR framework [10]. Thus, in practice, incomplete or overly aggressive LD pruning can both introduce bias. Although recent studies have shown that explicity modeling LD structure is essential for obtaining unbiased estimates, most exiting MR frameworks neglect this dependency [11, 12]. Third, existing methods confound direct IV-to-exposure effects (DIEs, i.e. IV affects exposure directly) and indirect IV-to-exposure effects (IIEs, i.e. IV affects exposure via unmeasured confounders). Most approaches account for only one of them, failing to reflect the biological reality that some IVs may simultaneously exhibit both effects [13]. Finally, the direct application of conventional MR frameworks often overlooks the biological reality that gene regulatory mechanisms are partially shared across related tissues. Such simplification limits their ability to detect truly causal genes and to delineate the tissue-specific architecture of complex diseases [14, 15].

To address these challenges, we developed MULTI (Multi-tissue Unified Likelihood-based Transcriptomic Integration), a Bayesian MR framework designed for robust and interpretable causal gene discovery. MULTI unifies the modeling of LD, horizontal pleiotropic effects, and cross-tissue information within a single probabilistic framework, enabling coherent estimation of causal effects even with a limited number of valid instruments. To improve power without compromising inference accuracy, MULTI introduces an adaptive tissue-integration strategy that selectively borrows information from genetically similar tissues. Tissue similarity is quantified using the Hellinger distance (HD) between tissue-specific causal effect distributions, and the algorithm dynamically adjust the amount of shared information according to the tissue similarity matrix. Parameters estimation is performed via a variational expectation–maximization (VEM) [16] algorithm, which ensures both computational stability and scalability for genome-wide analyses. Together, these features enable MULTI to overcome key limitations of existing MR methods and to provide a unified approach for elucidating tissue-specific causal mechanisms of complex diseases.

We conducted extensive simulation studies that covered a wide range of genetic epidemiological scenarios to assess the robustness of MULTI. Compared with conventional MR frameworks, MULTI consistently achieved better control of type I error rates, while its statistical power increased progressively as the tissue-similarity threshold was relaxed. Simulation results were further supported by applications to large-scale transcriptomic and disease-association data, where MULTI successfully identified tissue-dependent causal relationships and coherent cross-tissue regulatory patterns. Together, these results highlight MULTI as a reliable and versatile framework for uncovering causal genes and delineating the tissue-specific regulatory mechanisms that underlie complex human diseases.

Results

Overview of MULTI

MULTI is designed to identify tissue specific gene-disease/trait causal relationship. As depicted in Fig. 1, MULTI is implemented using tissue-specific two-sample summary statistics and their corresponding eQTL correlation matrices as input data. For each tissue, MULTI estimates the causal effect of the target gene on the outcome. For notational simplicity, tissue indices are henceforth omitted. Similar as our previous method RARE (MVMR incorporating Rare variants Accounting for multiple Risk factors and shared horizontal plEiotopy) [13], we allow the presence of DIE (), IIE (), correlated horizontal pleiotropy (CHP) (), and uncorrelated horizontal pleiotropy (UHP) (). Based on the above definitions, the IV-to-outcome effect can be written as:

Figure 1.

MULTI workflow. (A) Required data input. MULTI requires tissue-specific summary-level statistics, including reference SNP cluster ID (rsID), , , , and , together with the paired correlation matrices from the reference panel that characterize linkage disequilibrium. (B) Single-tissue inference. MULTI relaxes the core assumptions of MR by allowing DIE (), IIE (), CHP (), and UHP () to coexist within a unified model. Within this framework, MULTI applies a VEM algorithm to estimate the gene-disease/trait causal effect for each tissue, assuming a normal prior for the effect distribution. MULTI then computes the Hellinger distances between the target tissue and all supportive tissues. (C) Adaptive tissue integration and re-estimation. MULTI determines tissue integration strategies based on user-specified similarity thresholds. Summary-level statistics from supportive tissues deemed similar to the target tissue are incorporated into the target-tissue dataset, and a new SNP correlation matrix is constructed for the expanded SNP set. Based on the integrated datasets, MULTI generates the final causal effect estimate for the target tissue.

(1)

where is the causal effect of gene on outcome. In accordance with the previous literatures and to avoid over-parameterization, we define to capture both DIE and IIE. Thus, Equation (1) can be expressed as follows:

(2)

Parameters, including the causal effect , are estimated using VEM algorithm based on a Bayesian hierarchical model. The details are provided in Methods and Supplementary Note 1.

To improve the power for detecting causal genes in the target tissue, MULTI quantifies the similarity between target tissue and each supportive tissue using the Hellinger distance. Tissues with distance below predefined cutoff (e.g. 0.1, 0.3, or 0.5) are regarded as similar and are incorporated into target tissue for analysis. By borrowing information from these selected tissues, MULTI expands the pool of cis-eQTL instruments and subsequently re-estimates the gene-specific causal effect in the target tissue.

MULTI controls the type I error rate and increases statistical power

We conducted extensive simulation studies to evaluate the robustness and efficiency of MULTI under diverse genetic architectures. Four configurations were examined: using the IVs of target tissue only for causal estimation (MULTI_s) and incorporating IVs from supportive tissues at three similarity cutoffs (0.1, 0.3, 0.5), denoted as MULTI_0.1, MULTI_0.3, and MULTI_0.5, respectively. The performance of the four configurations was compared with several well-established MR methods, including Inverse-Variance Weighted (IVW) [17], MR-Egger [18], median-based MR [19], mode-based MR [20], and ConMix [21], across multiple simulated scenarios. Among all competing approaches, the median- and mode-based methods required strong and valid IVs that are unaffected by either UHP or CHP, whereas IVW and MR-Egger allowed for UHP but assumed the absence of CHP. In contrast, both MULTI framework and ConMix accommodated the coexistence of UHP and CHP. Notably, MULTI further handled moderate-to-strong LD structures, enabling robust inference without LD pruning of IVs.

It should be emphasized that, in the simulation study, conventional MR methods were applied without additional LD pruning. This design was based on the aforementioned characteristics of cis-eQTL data. In the context of cis-eQTL data, aggressive LD pruning may further reduce the number of usable instruments and compromise the stability of causal-effect estimation. Accordingly, the simulation comparison was intended to assess method performance in LD-structured cis-eQTL settings, rather than to represent the standard best-case implementation of conventional MR methods under their recommended preprocessing pipelines.

In all main settings, we considered two putative similar tissues (), and the exposures and the outcome were generated based on polygenic architecture as shown in Equation (7). All IVs contribute to the exposure through the heritability of DIE () and the heritability of IIE (), while the contribute to outcome can be decomposed into components of the causal effect , the heritability of UHP (), and the heritability of CHP (), as detailed in the Methods and Supplementary Note 2.

We first assessed the control of type I error rates (Fig. 2A) under varying levels of LD () and correlation between IIE and CHP (). We set and examined multiple combinations of . Across all scenarios, the MULTI framework—including MULTI_s, MULTI_0.1, MULTI_0.3, and MULTI_0.5—maintained well-controlled type I error rates and consistently outperformed competing methods, even in settings with limited IV numbers, the presence of LD, and correlations between IIE and CHP. It could be seen that, although incorporating IVs from similar tissues led to a slight increase in the type I error rate of the framework, it remained at or below the nominal 0.05 level in most simulation scenarios (Fig. 2A). Importantly, this slight inflation in type I error rate control was accompanied by a notable gain in statistical power (Fig. 2B), aligning with the primary motivation underlying the design of MULTI framework. We next examined the robustness of MULTI to different LD structures. As shown in Fig. 2A and Supplementary Note 3.1, MULTI was robust to low-to-moderate LD (  v.s.  ) and even to strong LD (). This robustness eliminates the need for IV selection in most applications, allowing MULTI to retain as many eligible IVs as possible, which in turn contributes to its superior performance when the number of instruments is limited. Nevertheless, when and jointly accounted for a substantial proportion of the outcome heritability (10%) and the correlation between IIE and CHP increased (  v.s.  ), MULTI exhibited an inflation in type I error rates. Broader simulations varying both and yielded consistent results, confirming that MULTI achieved robust type I error rate control across a wide range of genetic architectures (Supplementary Note 3.1–3.2).

Figure 2.

Comparison of MULTI and existing methods in simulation studies. (A) Type I error rate comparison. Type I error rates of MULTI and competing methods across combinational of and , under (top panel) and (bottom panel). (B) Power comparison. Powers of MULTI and other methods as a function of under the settings of and , with (left panel), and (right panel), respectively. (C) Bias comparison. Boxplots of point estimates across 100 replications for each method under different number of tissues (K) and different number of IVs (p). Boxes denote the IQR, with center lines indicating the median. IQR, interquartile range.

We next compared the statistical power of all methods by varying , defined as , with . LD was fixed at and the correlation between IIE and CHP was set at . Two combinations of pleiotropic components were considered as , and , (Fig. 2B). Under a low heritability of , the power of the MULTI framework increased progressively as the similarity threshold was relaxed (from MULTI_s to MULTI_0.5), yielding approximately a 10%~15% gain. When the exposure heritability increased to , MULTI_0.5 still yielded additional gain in power compared with MULTI_s, although its advantage over MULTI_0.3 was minimal. Overall, MULTI_0.5 achieved comparable statistical power compared with that of competing methods, suggesting that a similarity cutoff of 0.5 may serve as a reasonable and empirically effective choice in real-data applications. Supplementary analyses further indicate that, when sufficient number of cis-eQTLs are available, moderate LD pruning can further improve power, although it is not required under the MULTI framework (Supplementary Note 3.3).

To further assess estimation stability, we set the true causal effect in the target tissue and calculated the estimation bias. As shown in Fig. 2C, all MULTI configurations generated estimates centered around the true effect, demonstrating strong stability across simulation settings. Notably, this robustness persisted even under limited-IV scenarios, in which many conventional MR estimators became unstable. Such instability may partly account for the type I error inflation observed for these methods in earlier simulations.

Additional considerations including the heritability of UHP and CHP, the correlation between tissues, and the sample size of the cis-eQTL and GWAS data were also provided in the Supplementary Note 3.4–3.7. Although the main simulations considered only two tissues, we extended the framework to as many as 30 tissues in the Supplementary Note 3.8 and obtained consistent conclusions. Furthermore, to assist researchers in better utilizing MULTI, we provided a reference figure detailing its computational performance under different numbers of IVs and tissues (Supplementary Note 3.9).

In summary, the choice of the tissue-similarity threshold reflects a trade-off between type I error rate control and statistical power. In brief, a lower threshold (e.g. cutoff = 0.1) provides more stringent control of type I error, but at the cost of reduced statistical power of detecting true causal effects. In contrast, a higher threshold (e.g. cutoff = 0.5) improves the ability to identify true causal effects, although it may lead to a slight increase in type I error rate. In real-data applications, the choice of threshold should depend on the analytical objective. When the primary goal is to minimize false positives, we recommend using a lower similarity threshold; conversely, when the goal is to explore a broader set of potential causal effects, a higher similarity threshold may be more appropriate.

Identifying causal genes for depression along the gut-brain axis

To evaluate the performance of MULTI in real-world data, we applied it to identify causal genes for depression along the gut-brain axis (GBA). The eQTL-gene expression associations were derived from the GTEx database [6] (“colon transverse” and “colon sigmoid”), and the Single Nucleotide Polymorphism (SNP)-depression associations were obtained from large-scale GWAS summary statistics.

Using the colon transverse as the target tissue and the colon sigmoid as the supportive tissue, we first selected genes with at least 20 valid cis-eQTLs between both tissues. For each gene, the MULTI framework was used to estimate its causal effect on depression in the target tissue. Following the simulation-derived criterion, a similarity threshold of 0.5 was adopted to determine whether supportive-tissue SNPs should be incorporated. The same procedure was then repeated with the sigmoid colon as the target tissue.

Among all the putative causal genes, DRAM1 emerged as the most promising candidate. DRAM1 encodes a lysosomal membrane protein that modulates autophagy and cellular stress responses, which has been implicated in maintaining intestinal epithelial homeostasis, particularly under oxidative or microbial stress [22, 23]. Notably, recent studies suggest that DRAM1 expression in colonic tissues is responsive to luminal factors such as hydrogen sulfide (H₂S) [24], a gut-derived gasotransmitter known to influence epithelial integrity and local immune tone. Since epithelial barrier dysfunction and gut inflammation are central components of GBA dysregulation, the implication of DRAM1 in depression risk suggests a plausible mechanistic link: genetic variation affecting autophagy regulation may influence gut barrier health, thereby contributing to peripheral inflammatory signaling that ultimately impacts central nervous system mood regulation. In addition to DRAM1, we identified CLEC2D and CLECL1, members of the C-type lectin-like receptor family. Perturbation in CLEC2D/CLECL1 signaling could lead to dysregulated immune responses at the gut interface, potentially priming the neuroinflammatory cascades implicated in depression [25]. Finally, KLKB1, encoding plasma kallikrein B1, was also prioritized. This gene participates in the kallikrein-kinin system and regulates bradykinin production, vascular permeability, and inflammatory cell recruitment [26, 27]. It is plausible that KLKB1 may influence gut barrier integrity and neurovascular interactions. Considering the gut-vascular-brain route is increasingly recognized in the pathophysiology of depression, KLKB1 may serve as a molecular mediator linking gut-derived inflammation to cerebral outcomes.

Collectively, these findings suggest that MULTI can effectively capture biologically meaningful gene-trait relationships along the GBA and highlight potential molecular links between intestinal gene regulation and depression susceptibility. Detailed results, including sensitive analysis using different tissue-similarity cutoff, can be found in Supplementary Results 1 and Supplementary Note 4.

Identifying regional gene-disease causal pattern

Previous integrative genomic analyses combining GWAS, transcriptome-wide association studies, and colocalization approaches have consistently implicated WDR6 as a susceptibility gene for Parkinson’s disease (PD), with supporting evidence pointing to eQTL-mediated regulatory effects in dopaminergic brain regions [28, 29]. However, the spatial patterning and directional consistency of these putative causal effects across anatomically distinct brain regions remain poorly characterized. To address this gap, we applied the MULTI framework to dissect the tissue-specific causal patterns linking WDR6 to PD risk, using its cross-region effects as an illustrative case study.

We applied the MULTI framework using eQTL summary statistics derived from 11 GTEx brain tissues as exposures, and PD GWAS summary statistics from UK Biobank as outcome. As illustrated in Fig. 3B, pairwise HD values across brain regions revealed striking patterns of similarity and divergence. Closely related regions, including the amygdala, cerebellar hemisphere, spinal cord and substantia nigra exhibited high tissue similarity (HD < 0.1), indicating consistent regulatory architecture and enabling effective information borrowing during causal effect estimation. In contrast, regions such as caudate, cerebellum, and cortex demonstrated distinct causal profiles (HD > 0.5), reflecting region-specific regulatory dynamics and the necessity for separate estimation. Notably, increased WDR6 expression in the substantia nigra showed a significant protective effect on PD risk (), a region widely recognized as the primary site of dopaminergic neurodegeneration in PD [30, 31]. A similarly protective association was observed in the putamen (), which serves as a major target of the nigrostriatal pathway [32, 33]. Overall, the spatial clustering of HD values across anatomical hierarchies supports the biological validity of MULTI’s tissue-adaptive integration strategy. By quantifying and visualizing causal effect concordance at the tissue level, MULTI provides a powerful framework to elucidate how gene regulation contributes to disease vulnerability in a regionally specific manner.

Figure 3.

MULTI analysis of two real data. (A) Summary results for causal genes of depression along the GBA. The upper and lower panels correspond to the colon transverse and colon sigmoid, respectively. Each point represents a gene across 22 chromosomes, with red indicating significant causal effects and green indicating non-significance. Triangular markers denote analyses based on tissue-specific SNPs, whereas circular markers represent those incorporating supportive-tissue SNPs. The bar plots on the right show the number of significant causal genes per chromosome. (B) Heatmap of the similarity matrix. Pairwise HD matrix representing the similarity of causal effect distributions of WDR6 on PD across 11 brain regions. Lighter blue indicates higher similarity (lower HD), while red indicates greater divergence. The similarity cutoff was set as 0.5. Symbols (±) along the diagonal represent the estimated causal direction of WDR6 in each region, and the corresponding P-values are reported beneath the symbols. Caudate: caudate (basal ganglia); Cerebellar: cerebellar hemisphere; Frontal: frontal cortex; Nucleus: nucleus accumbens (basal ganglia); Spinal: spinal cord (cervical c-1); Substantia: substantia nigra.

These results demonstrate that MULTI yields interpretable, anatomy-aware causal maps of gene function across complex tissue landscapes. In this case, our findings refine WDR6’s role in PD etiology and highlight the importance of the substantia nigra as a key site of regulatory action. Full quantitative estimates and confidence intervals are provided in Supplementary Results 2.

Discussion

In this study, we proposed MULTI, a novel multi-tissue integrative MR framework that provides a statistically robust and biologically interpretable approach for identifying disease risk genes. MULTI extends the conventional two-sample MR design by explicitly modeling LD, horizontal pleiotropy, and cross-tissue genetic similarity within a unified Bayesian hierarchical model. Unlike previous MR methods that rely on aggressive LD pruning and assume independent IVs [18–20], MULTI incorporates the LD structure directly into the likelihood, thereby preserving more genetic information and improving estimation stability when the number of valid instruments is limited. Moreover, the introduction of the Hellinger distance-based adaptive tissue-integration strategy allows MULTI to dynamically borrow information from supportive tissues that exhibit similar causal architectures, improving power while maintaining well-controlled false-positive rates.

Through extensive simulation analyses covering a broad range of genetic scenarios, MULTI consistently demonstrated superior performance compared with currently widely used MR methods. It should be noted that we were already aware of another multi-tissue MR framework, mintMR [34], during the development of MULTI. However, because mintMR could not be applied to our data input, it was not included as a benchmark in this study. Specifically, mintMR is essentially a joint estimation framework based on multiple exposures, multiple tissues, and multiple modalities, whereas the input data for MULTI include only measurements of a single gene across different tissues. Compared with mintMR, MULTI imposes less stringent input data requirements and is therefore more feasible in practical applications. When integrating similar tissues, MULTI maintained robust type I error control even under complex conditions involving strong LD and correlated pleiotropy, while achieving a 10% to 15% increase in statistical power relative to single-tissue analyses. More importantly, as the tissue-similarity threshold was relaxed, the gain in power provided by this integrative strategy increased gradually, suggesting that moderate integration (i.e. HD ≤ 0.5) can effectively balance robustness and sensitivity. This design makes MULTI particularly advantageous for transcriptome-based MR analyses, in which the limited number of cis-eQTLs and tissue heterogeneity often undermine the reliability of causal inference. In addition, the Bayesian VEM algorithm ensures computational efficiency and stable convergence, enabling the method to scale readily to genome-wide analyses across dozens of tissues.

The application of MULTI to real datasets further highlights its practical value in elucidating biologically meaningful gene-trait relationships. Along the GBA, MULTI identified several plausible causal genes for depression, including DRAM1, CLECL1, CLEC2D, and KLKB1, whose functions are associated with neurotransmission, immune signaling, and metabolic pathways [25–27]. These genes have been implicated in neuroinflammatory responses and intestinal homeostasis, supporting the hypothesis that gut epithelial gene regulation contributes to depression pathogenesis. By integrating supportive-tissue SNPs under the HD-based criterion, MULTI enhanced detection sensitivity without compromising type I error control, illustrating the importance of cross-tissue information sharing for capturing subtle regulatory effects. Similarly, in the analysis of WDR6 across multiple brain regions in PD, MULTI revealed distinct yet coherent regional causal patterns. Significantly protective associations were observed in the substantia nigra and the putamen, which suggesting that WDR6 may exert neuroprotective effects through the regulation of dopaminergic neuron survival, thereby validating the biological relevance of MULTI’s results. Taken together, our findings underscore that MULTI bridges the gap between tissue-specific and cross-tissue inference paradigms. Traditional MR approaches often treat tissues as independent units, neglecting the coordinated regulatory networks underlying complex diseases. MULTI, in contrast, adapts to tissue similarity in a data-driven manner, effectively capturing shared genetic regulation while preserving tissue-specific causal heterogeneity. This framework enables a more nuanced understanding of how genetic perturbations manifest differently across biological contexts, offering new insights into systemic diseases such as depression, which involve inter-organ communication, and neurodegenerative disorders like PD, where region-specific vulnerability shapes disease progression.

Despite these strengths, several limitations warrant consideration. MULTI assumes linearity between gene expression and complex traits, which may overlook non-linear or interaction effects among regulatory elements. The similarity metric based on Hellinger distance reflects statistical proximity of causal estimates but may not fully capture molecular-level interactions such as co-expression or chromatin conformation. Moreover, while the current implementation focuses on cis-eQTLs in European-ancestry populations, extending MULTI to incorporate trans-eQTLs, rare variants, and multi-ancestry data will be essential for improving its generalizability. Future methodological extensions could integrate additional omics layers-such as chromatin accessibility, methylation, or spatial transcriptomics-to construct a more comprehensive view of tissue-specific causal mechanisms.

In conclusion, MULTI represents a flexible and powerful framework for cross-tissue causal inference in integrative genomics. By jointly modeling LD, pleiotropy, and tissue similarity within a Bayesian framework, MULTI enhances the accuracy and interpretability of causal gene discovery. Its adaptive integration strategy provides a principled means of leveraging biological similarity while maintaining statistical rigor. We anticipate that MULTI will serve as a versatile tool for dissecting the molecular mechanisms underlying complex traits, guiding future functional validation studies, and informing precision medicine strategies through tissue- and system-level causal insights.

Methods

Data input for MULTI

Assume tissues, , and let denote the expression level of gene measured in tissue . Let be the target tissue in which we aim to estimate the causal effect of on the outcome , while the remaining tissues are regarded as supportive tissues to be integrated. For each tissue , let denote the number of eligible IVs for , selected as variants associated with at a prespecified significance threshold in eQTL or GWAS analyses.

The estimated IV-to-exposure effect size vector and its corresponding diagonal standard error matrix are defined as

and similarly, the estimated IV-to-outcome associations are denoted as

The pairwise correlation structure among the IVs is represented by the matrix , which is estimated from an independent reference panel data.

Estimating LD matrix from a reference panel

To estimate the LD matrix, we used an independent reference panel derived from European-ancestry samples in the 1000 Genomes Project Phase 3. Individuals with missing genotype rate exceeding 5% were excluded to ensure data quality and consistency across variants. Variant identifiers (rsIDs) were harmonized using dbSNP Build 151, enabling accurate alignment between the 1000 Genomes Project variants and standardized dbSNP annotations.

MULTI used the union of IVs across tissues to construct the final LD matrix. From the perspective of constructing the new LD matrix, duplicate instruments across tissues are not allowed, because their presence would make the reconstructed LD matrix rank-deficient, thereby causing the MULTI algorithm to fail. In real-data applications, to ensure a full-rank LD matrix, MULTI removes variants in the supportive tissues that are duplicated in the target tissue.

Modeling linkage disequilibrium in MULTI

Within the MULTI framework, a unified algorithmic procedure is used for both single-tissue causal-effect estimation and causal-effect estimation after information integration. Accordingly, tissue-specific subscripts are omitted throughout this section for notational simplicity. Within MULTI framework, the IV-to-exposure effect is decomposed into DIE () and IIE () [13], and we define to capture both components, with the prior . Similarly, the CHP () and UHP () follow the prior and , respectively. The probabilistic relationships between the estimated and true effect sizes are modeled as¹²

(3)

Combing Equation (3) with Equation (2), the conditional distribution of [[ineq82]] is

(4)

To facilitate Bayesian inference, we assign a normal prior . The joint probabilistic model can then be expressed as

(5)

Variational EM for MULTI

Define , . Let be a variational distribution approximating the posterior of , the marginal likelihood can be decomposed as

(6)

where

At each iteration, is updated by minimizing in the E-step, and the evidence lower bound (ELBO) is maximized w.r.t.   in M-step, thereby increase the marginal likelihood.

We adopt mean-field variational Bayes (VEM) and assume the following factorization

where collects all the variational parameters. Detailed updating equations for each parameter are provided in the Supplementary Note 1.

Generation of simulated data

For the k-th tissue, genotype matrices , were generated for the exposure and outcome, respectively. An additional reference panel was simulated to compute the covariance matrix .

Individual-level tissue-specific exposure data and the shared outcome data were simulated under a two-stage structural model as

(7)

and represent unmeasured confounders for exposures and outcomes, while and are the corresponding coefficient vectors. , , , and are the DIE, IIE, CHP, and UHP, respectively. , and denote the exposure traits in the two samples and the outcome, while , and represent random noise terms. The coefficient corresponds to the causal effect of the exposure on the outcome in the k-th tissue. Single-variant analyses were performed between and to yield marginal SNP-exposure statistics and . Analogous analyses between and produced and . In the simulation studies, the magnitudes of , , , were controlled through their heritability, defined as ,  ,  ,  , respectively. Further implementation details are provided in the Supplementary Note 2.

Key Points

• This paper proposes a novel multi-tissue Mendelian randomization method named MULTI.

• MULTI jointly models linkage disequilibrium and multiple types of horizontal pleiotropy within a Bayesian framework.

• MULTI adaptively integrates genetically similar tissues using a Hellinger distance-based similarity criterion.

• MULTI achieves well-controlled type I error and improved statistical power even when the number of cis-eQTL IVs is limited.

• MULTI provides interpretable tissue-specific causal effect estimates.

Author contributions

Fangrong Yan: Conceptualization, Funding acquisition, Supervision, Writing – review & editing. Tiantian Liu: Conceptualization, Funding acquisition, Supervision, Writing – review & editing. Liyun Jiang: Conceptualization, Funding acquisition, Supervision. Yu Cheng: Conceptualization, Formal analysis, Software, Writing – original draft. Shuhan Liu: Formal analysis, Writing – original draft. Xinjia Ruan: Formal analysis, Software. Zhonghua Li: Formal analysis. Yu Cheng, Shuhan Liu, and Xinjia Ruan contributed equally to this work.

Conflicts of interest

The authors declared no competing interests.

Funding

This work was supported by the Fundamental Research Funds for the Central Universities of China Pharmaceutical University [2632023FY04, 2632023FY05], the National Natural Science Foundation of China [No. 82273735, No. 82304252, No. 82473737], the National Science Foundation for Young Scientists of China [No. 82404380], the 2024 National Key Research on Multi-Target Natural Drugs [SKLNMZZ2024JS32], and Key Laboratory of Scientific and Engineering Computing (Ministry of Education) and the Shanghai Frontiers Science Center of Modern Analysis.

Data availability

The eQTL summary statistics are available from the Genotype-Tissue Expression (GTEx) Portal (gtexportal.org/home/). GWAS summary statistics can be obtained from fastGWA (yanglab.westlake.edu.cn/data/ukb_fastgwa). The 1000 Genomes Project Phase 3 data and dbSNP Build 151 data are accessible at ftp.1000genomes.ebi.ac.uk/vol1/ftp and ftp.ncbi.nlm.nih.gov/snp, respectively. Processed chromosome data are available from the corresponding author upon reasonable request.

Code availability

All the analysis code is publicly available. MULTI is implemented in an open-source R package available through Github at https://github.com/Hide-in-lab/MULTI/tree/main. The code used to reproduce the simulation can be found at https://github.com/Hide-in-lab/MULTI/blob/SupplementaryResults/simulation.R. The code used to reproduce the data analysis can be found at https://github.com/Hide-in-lab/MULTI/blob/SupplementaryResults/DataAnalysis.R.