MAJA: multivariate Bayesian model for discovery of shared epigenetic pathways across human phenotypes

Abstract

Genomic measurements of DNA methylation, gene expression or protein levels are becoming more prevalent and are increasingly used to study health outcomes. However, most proposed association testing methods consider only marginal effects of each feature on a single outcome variable and are not set up to handle highly correlated, continuous data. Here, we introduce MAJA, a method to learn shared and outcome-specific effects for multiple traits in multi-omics data. MAJA determines the unique contribution of individual loci, genes, or molecular pathways to variation in one or more traits, conditional on all other measured “omics” data genome-wide. Simulations show MAJA accurately finds shared and distinct associations between omics-data and multiple traits and estimates omics-specific (co)variances, allowing for sparsity and correlations within the data. Applying MAJA to 12 outcome traits in Generation Scotland methylation data (n = 18 264), we find novel shared epigenetic probes among cholesterol metabolism, osteoarthritis, blood pressure and asthma. In contrast to marginal testing, we find only 10 CpG probes with significant effects above the genome-wide background. This highlights the need for joint association testing in highly correlated methylation data from whole blood and for studies of increased sample size in order to refine epigenomic associations in observational data.

1 Introduction

Epigenetic mechanisms influence gene expression, cell differentiation, tissue development, and disease susceptibility (Jones 2012, Ahsan et al. 2017, Bell et al. 2019). Measuring and tracking epigenetic changes through disease progression can provide insight into disease pathogenesis (Hillary et al. 2023), elucidate environmental and lifestyle factors influencing health, and provide biomarkers for disease diagnosis and risk stratification (Trejo Banos et al. 2020). Epigenetic data for many individuals are increasingly being collected (for example Generation Scotland (Smith et al. 2013), UK Biobank), but to date, most epigenetic studies have focused on modeling traits individually. As human phenotypes are highly correlated, with shared risk factors and underlying pathways, estimating the degree to which epigenetic effects are shared across human traits has the potential to reveal shared disease etiology, improve biomarker discovery and maximize outcome prediction.

In genomics, existing methods for the analysis of multiple correlated traits lack flexibility as they: (i) handle only single nucleotide polymorphism (SNP) data (Cheng et al. 2018) (ii) only model at most two phenotypes with multiple variance components (Gianola and Fernando 2020, Hatton et al. 2023); (iii) are targeted only for prediction (Maier et al. 2018); (iv) perform multi-trait fine-mapping on genomic regions associated independently for each trait where the independent regions have to be identified beforehand (Zou et al. 2026); and/or (v) conduct association testing one variable at a time (Zhou and Stephens 2014). Thus, we lack general efficient methods suitable for a range of large scale “omics” data that analyze multiple outcomes jointly, allowing for the inclusion of different data modalities (i.e. methylation, expression, sequence variation, etc.).

A predominant focus has been on the estimation of genome-wide correlations (Hatton et al. 2023), which estimate the degree of similarity in the effects underlying these traits, but do not provide direct insights into specific underlying shared processes. Ideally, we wish to identify individual loci, genes, or molecular pathways that are both shared and unique between traits, and estimate their effects conditional on all other loci, genes, or pathways genome-wide, determining their unique contribution to phenotype. This joint modelling of effects between traits would improve our ability to use multi-modal “omics” data for risk prediction and patient stratification.

Here, we present MAJA, a multivariate multiple linear regression Bayesian joint sparse model. Our Bayesian approach jointly estimates shared effect sizes for multiple traits, potentially across different omics-data, while correcting for correlations within the data and allowing for sparsity. It thus simultaneously finds shared and distinct associations between omics-data and multiple traits, and estimates group-specific (co)variances. It is scalable, flexible, and suitable for all existing high-dimensional genomics data. We demonstrate our approach using the Generation Scotland data (Smith et al. 2013), a cohort of 18 264 individuals with blood-based methylation measures, where we find both trait-specific and shared probe effects and improve out-of-sample prediction as compared to single-trait models.

2 Methods

We developed a multivariate Bayesian multiple regression model (MAJA—MultivAriate Joint bAyesian model) that jointly estimates omics effects and their corrections on multiple traits and performs variable selection, all while taking into account correlations within the data. Our model is suitable for the case where a number of q phenotypes for n individuals is measured within the matrix $\mathit{Y}$. The phenotype matrix is modelled to be linearly related to the matrix containing p genomics measures $\mathit{X}$ (e.g. SNPs, epigenetic probes, gene expression) as,

$$\mathit{Y}=\mathit{X}\mathit{\beta }+\mathit{ϵ},$$ (1)

where the matrix $\mathit{\beta }$ denotes the effect sizes for q traits and $\mathit{ϵ}$ represents the residual error matrix. Each column of the $\mathit{Y}$ and $\mathit{X}$ matrices is standardized. When binary traits are included in $\mathit{Y}$, the continuous output of the binary variable can be interpreted directly as a probability: $P(Y=1|X)=X\beta$, making the effects highly interpretable and the model easy to fit. Gaussian linear models have been shown to be applicable to binary traits in GWAS before in a long history (see for example, Pirinen et al. 2013; Wray and Visscher 2015; Cook et al. 2017). The associations for binary traits are expected to be unaffected by the choice of linear versus logistic regression as a direct mapping exists as shown in Lloyd-Jones et al. (2018). The parameters of (1) are estimated using a Gibbs sampler, an iterative Markov Chain Monte Carlo method. All details on MAJA can be found in the Appendix section.

The effects of each genomic location j on the multiple traits, ${\mathit{\beta }}_{\mathit{j}}$, are assumed to have a multivariate spike-and-slab prior distribution to accommodate zero effects sizes

$${\mathit{\beta }}_{\mathit{j}}\sim (1-\pi )\mathit{MVN}({\mathbf{0}}_q,\mathit{V})+\pi {\delta }_0,$$ (2)

where $\pi$ is the probe exclusion probability common to all traits, $\mathit{MVN}({\mathbf{0}}_q,\mathit{V})$ is a multivariate normal distribution with mean 0 and (co)variance $\mathit{V}$ and ${\delta }_0$ the Dirac delta distribution. A probe is not included in the model if the effects of all traits are estimated to be 0. Through sampling the effects of each probe conditional on the other probes, correlations between the probes are automatically taken into account in our model.

Moreover, MAJA is able to handle multiple $\mathit{X}$ matrices. For example: (i) (epi)genetic information split into groups where the (epi)genetic covariances are estimated within each group; and/or (ii) multi-modal data, where different data sets are combined, like CpG sites and single nucleotide polymorphisms (SNPs). Effect sizes are determined jointly, thus the effects of each column of $\mathit{X}$ are estimated conditional on all others, taking into account correlations across groups and omics layers.

We demonstrate that MAJA accurately infers (co)variances in one or multiple groups using simulations as described in the Appendix and shown in Figs S1–S3, available at supplementary material  Bioinformatics Advances online. We also show that the estimated effects can be used to predict into a test data set, to achieve out-of-sample prediction accuracy that conforms to theoretical expectations and improves over single-trait models, as can be seen in Figs S4–S6, available at supplementary material  Bioinformatics Advances online. Finally, we demonstrate the ability of MAJA to localise effects accurately to the single-variable level, conditional on all other variables, by calculating the true positive (TPR) and false discovery (FDR) rate across all simulation scenarios. MAJA is designed such that a probe or locus affects all the traits or none of them, but the estimated effect size is allowed to differ for each trait, e.g. it can be zero for some traits. Thus, to detect significant associations we use the posterior inclusion probability (PIP $>95\%$) in combination with the posterior distribution of effect sizes to determine which of the estimated effects do not include 0 within either one or two standard deviations. Using both the inclusion probability and the strength of the effect size provides a robust test statistic for fine-mapping CpG effects, as shown in Figs S7–S9, available at supplementary material  Bioinformatics Advances online.

3 Application to multi-trait epigenetics data in Generation Scotland

We apply MAJA to 18 264 individuals in Generation Scotland for whom DNA methylation measures from whole blood were available at 831 349 CpG sites for twelve outcome traits, split into six cognitive, two metabolic and four disease traits. For disease outcomes that were commonly self-reported at the time of blood sampling, the phenotypic variance attributable to the methylation probes ranged between 24% for both depression and asthma, to 69% for hypertension. For clinically measured variables, we find that the phenotypic variance attributable to the methylation probes was 81% for body mass index (BMI) and 73% for ratio of high density lipoprotein over total cholesterol. In addition, we extend our analysis to a series of cognitive evaluations and educational attainment metrics, finding that between 32% and 73% of the phenotypic variation can be attributed to the CpG probes. The estimated variances, covariances between the traits and correlations are shown in Figs 1 and 2. The values along with the 95% credible intervals are listed in Supplementary Tables S1–S5, available at supplementary material  Bioinformatics Advances online. Heat maps of the correlations can be found in Fig. S10, available at supplementary material  Bioinformatics Advances online.

Figure 1.

Estimated epigenetic variances (top), covariances (middle) and correlations (bottom) for body mass index (BMI), ratio of high density lipoprotein over total cholesterol (CHL), high blood pressure (BP), depression (DEP), osteoarthritis (OA) and asthma (AT) in the Generation Scotland methylation data using 18 264 individuals and 831 349 probes. The error bars in the top and middle panels represent the 95% credible interval. The correlations are calculated as covariances scaled by the corresponding variances. The uncertainties are calculated using the posterior means $\pm$ 95% credible interval.

Figure 2.

Estimated epigenetic variances (top), covariances (middle) and correlations (bottom) for years in education (EY), highest qualification in education (EQ), digit symbol (DS), logical memory (LM), verbal fluency (VT) and vocabulary (VO) tests in the Generation Scotland methylation data using 18 264 individuals and 831 349 probes. The error bars in the top and middle panels represent the 95% credible interval. The correlations are calculated as covariances scaled by the corresponding variances. The uncertainties are calculated using the posterior means $\pm$ 95% credible interval.

We find a strong negative correlation of epigenetic effects between BMI and ratio of high density lipoprotein over total cholesterol and that CpG probe effects were positively correlated for BMI and all other traits. CpG effects for both ratio of high density lipoprotein over total cholesterol and self-reported hypertension are positively correlated with those for self-reported osteoarthritis. Interestingly, CpG effects for self-reported asthma are negatively correlated with those for hypertension, implying asthma-associated epigenetic probes have an inverse association for hypertension. These findings, except the negative correlation between asthma and hypertension, were reproduced by the phenotypic (in case of dichotomous traits tetrachoric) correlations, shown in Table S4, available at supplementary material  Bioinformatics Advances online.

We find weak correlations of epigenetic effects among digit symbol and vocabulary cognitive tests, but generally strong correlations among all other tests, which are reproduced by the phenotypic correlations given in Table S6, available at supplementary material  Bioinformatics Advances online. Cognitive tests share underlying methylation probe effects with both years of education and educational attainment. Note here that the highest educational attainment is scored as a “1” (see Appendix) and thus the negative correlation reflects methylation effects acting in the same direction for longer years in education and higher educational attainment.

There are no strong residual correlations, as can be seen from Figs S11 and S12, available at supplementary material  Bioinformatics Advances online, which implies that methylation probe variation captures the vast majority of the signal of trait correlations. Residual covariances that are non-zero are often in contrast to the methylation covariance, implying relationships among risk factors not captured by methylation patterns in whole blood differ to those reflected in the covariance of methylation probes effects.

We find nine unique probes whose effects, conditional on those of all other probes, have PIP above 95%. A list of the probes, their related genes and their associated traits is given in Table 1.

Table 1.

Associated probes with inclusion probability $\ge 95\%$.

Probe	Gene	Traits	Group
cg17075888	PDK4, AC002451.3	(BMI), CHL	Meta-
cg00574958	CPT1A	BMI, CHL, (BP)	Bolic/
cg05325763	CPT1A	BMI, (CHL)	Disease
cg06307915	CETP	(CHL)
cg11024682	SREBF1	BMI, (CHL)
cg17739917	RARA	(CHL)
cg27243685	ABCG1	BMI, CHL, (BP)
cg06500161	ABCG1	BMI, CHL, BP
cg07741821	RP5-1007F24.1, KIAA0087	(EY), EQ, (DS), (LM), (VT), (VO)	Cognitive
cg17739917	RARA	EY, EQ, DS, LM, (VT), (VO)

Traits listed are those where the effect size of the probe does not include 0 within their standard deviations (SD). Traits tested with 1SD are in brackets, 2SD without brackets.

None of the probes with $\mathit{PIP}\ge 0.95$ within the metabolic and disease traits group are associated with depression, osteoarthritis or asthma. Of the seven we discover, almost all are shared between BMI, ratio of high density lipoprotein over total cholesterol, and hypertension. These probes are located near ABCG1 which controls lipoprotein lipase (LPL) activity and promotes lipid accumulation in human macrophages in the presence of triglyceride-rich lipoproteins; CPT1A which is the gatekeeper enzyme for mitochondrial fatty acid oxidation; and PDK4, a regulator of pyruvate dehydrogenase (PDH), which influences acetyl-CoA from beta-oxidation into the citric acid (TCA) cycle, thereby leading to enhanced fatty acid (FA) oxidation and slowing of glycolysis or glycolytic intermediates to alternative metabolic pathways. We then additionally find three genes linked to ratio of high density lipoprotein over total cholesterol: CETP which is a hydrophobic plasma glycoprotein that mediates the transfer and exchange of cholesteryl ester and triglyceride between plasma lipoproteins, playing an important role in high density lipoprotein metabolism; SREBF1 which regulates the uptake and synthesis of cholesterol; and RARA a key regulator of lipid/glucose metabolism. All of these associations have been reported in the epigenome wide association study (EWAS) catalogue for these, or related, traits. However, here we are able to explicitly determine for which traits their effects are shared and for which they act in a trait-dependent manner and to show that the association holds conditional on all other methylation loci.

Interestingly, the methylation effects of probe cg17739917 near RARA which is associated with ratio of HDL over total cholesterol is also linked to all cognitive tests. Also cg07741821 near genes RP5-1007F24.1 and KIAA0087 is associated with variation in all cognitive traits. These two associations have not been reported before. Taken together, our results show that key CpG probes are identified by our model whose effects are determined conditional on the data structure and all other probe effects.

Finally, we wished to demonstrate that our approach facilitates improved out-of-sample prediction as compared to single-trait approaches. Taking the CpG effects estimated in Generation Scotland, we predict traits that were measured in the Lothian Birth Cohort (LBC) 1936 (Taylor et al. 2018). We find that multi-trait predictors generally outperform the comparable single-trait predictors calculated using the BayesR model (Trejo Banos et al. 2020), as shown in Table 2. Of particular note is the predictor of general cognitive function, which explained up to 8.6% of the variance. This is more than double the performance of a previous predictor, derived from a subset of the Generation Scotland dataset (McCartney et al. 2022).

Table 2.

Out-of-sample prediction accuracy of episcores (incremental test $R^2$) created from MAJA as compared to estimates made using the single-trait BayesR model.

Trait	Episcore	${\mathit{R}}^{\mathit{2}}$  [%]
		BayesR	MAJA
		Univariate	Multivariate
BMI	BMI	14.94 (2.90)	16.86 (2.84)
BMI (log)	BMI	14.84 (2.91)	16.8 (2.84)
CHL	CHL	7.02 (3.17)	5.06 (3.24)
Digit symbol	Digit symbol	3.17 (3.31)	3.35 (3.30)
Logical memory	Logical memory	0.47 (3.40)	1.06 (3.38)
Verbal fluency	Verbal fluency	0.88 (3.38)	0.74 (3.39)
Years in education	Years in education	4.10 (3.27)	5.32 (3.23)
WTAR	Vocabulary	3.72 (3.29)	4.03 (3.28)
NART	Vocabulary	4.62 (3.26)	4.68 (3.25)
General cognitive function	Logical memory	3.2 (3.3)	5.9 (3.21)
General cognitive function	Vocabulary	5.2 (3.24)	5.5 (3.23)
General cognitive function	Verbal total	2.2 (3.34)	4.8 (3.25)
General cognitive function	Digit symbol	5.2 (3.24)	6.7 (3.19)
General cognitive function	Education years	6.3 (3.20)	7.6 (3.15)
General cognitive function	All scores		8.6 (3.12)

Episcores are created for a given trait (“Episcore”), using CpG probe estimates from either single trait (“Univariate”), or multi-trait (“Multivariate”) models. The episcores are then used to predict a series of outcome traits (“Traits”) in the Lothian Birth Cohort (LBC) 1936 study (n = 861). The $R^2$ values give the incremental test $R^2$ of including the episcore in a linear model to predict each outcome, adjusting for age and sex. The standard errors given in brackets are approximated using Ref (Bonett 2008). WTAR refers to the Wechsler Test of Adult Reading; NART to the National Adult Reading Test; BMI to body mass index; CHL to ratio of high density lipoprotein over total cholesterol in whole blood. All scores refers to the variance explained by including all predictors together within the model.

4 Discussion

We presented MAJA, a Bayesian method that jointly estimates the effect sizes of (epi)genomic variants, as well as correlations of the effects for multiple traits, while correcting for correlations among variables and allowing for sparsity. We extend previous studies both in terms of methodology and in the phenotypes studied. The variance estimates obtained for BMI agree with previous estimates (Trejo Banos et al. 2020, Hatton et al. 2023), as does our finding of a strong negative correlation of epigenetic effects between BMI and ratio of high density lipoprotein over total cholesterol (Hillary et al. 2023). We highlight novel CpG covariances among hypertension and osteoarthritis, ratio of high density lipoprotein over total cholesterol and osteoarthritis, BMI and hypertension, and BMI and asthma. Our results imply that methylation patterns of cholesterol metabolism related genes in whole blood are associated with osteoarthritis pathogenesis and that there is potentially a complex, yet to be fully explored, relationship between hypertension and asthma.

In this work, we focused on developing a statistical model and associated software, and to give a demonstration of how multi-trait Bayesian models can improve the discovery of shared loci and external trait predictions. We refrain from comparisons to other multivariate methods as they are not set up to handle DNA methylation data that are highly correlated across all chromosomes (Cheng et al. 2018, Zou et al. 2026) or only model the variances of at most two phenotypes (Gianola and Fernando 2020, Hatton et al. 2023). There are several limitations to our study, mainly the sample size of Generation Scotland, which whilst representing one of the largest single cohorts with methylation data available, still has very limited power to detect associations at $\ge 95\%$ confidence and to produce high out-of-sample accuracy, relative to the estimated total variance attributable to all CpGs on the array. Our model will likely return fewer associations than standard one-probe-at-a-time significance testing. However, single probe analyses do not control for correlations across probes and effects are not estimated conditional on all other probes effects. Thus, single-probe testing likely gives estimates that are inflated by correlations and by general data structure and confounding. In contrast, effect sizes and significance are determined jointly within MAJA which we expect (and show in simulation) to provide an accurate determination and localisation of specific probe effects.

Note that, within our model, a probe will be included for all traits when it has an effect on at least one of the traits. This setup has the advantage of helping discover shared effects, in particular when a probe has a strong effect on one trait and a small effect on another that would have been missed when modelling traits independently. The estimates for each trait are freely sampled from a multivariate normal with zero mean and thus there is no reason to expect a directional bias in the effect size estimates for traits for which the probe is not associated. However, estimation error may increase if many small effects are sampled for traits with strong covariance (BMI-ratio of high density lipoprotein over total cholesterol, for example) and this is likely the reason for the loss of out-of-sample accuracy, which we see for the multi-trait predictor of ratio of high density lipoprotein over total cholesterol in the LBC1936. This is a modelling choice to facilitate improved association testing and can be overcome by simply setting the effects of these probes to zero if the posterior estimate includes zero within the standard deviation.

We assume linearity between omics measures and Gaussian-distributed traits, thus not testing for non-linear effects.

Additionally, we highlight that CpG measures in whole blood do not necessarily represent the correct tissue for understanding mechanistic pathways among outcomes. A full characterisation of methylation across multiple tissues is needed to fully capture these relationships. With increasing cross-tissue data, we expect that the ability of MAJA to fit multiple groups could be useful, where multiple cross-tissue methylation measures could be fit within the model to determine patterns of shared effects across tissues. Further limitations are that while MAJA is able to handle larger biobank scale data sets through the use of message-passing interface (MPI) coding, at present it remains computationally expensive and is set up in such a way that all data needs to fit into RAM.

Current multi-trait models for genetic association studies are limited to analysing segments of the DNA (Zou et al. 2026), which is unsuitable for omics data with long-range correlations. MAJA is unique as it finds genome-wide trait-specific and trait-shared associations, it has the capability of fitting multiple omics data jointly through a grouped prior, and has run-times that makes these analyses feasible on current omics data. Having demonstrated the effectiveness of this framework, our future work will now focus on alternative algorithms for joint inference from this model, including more flexible effects priors, within a genomics setting.

In summary, our approach provides a method to learn shared and trait-specific epigenetic effects and to improve prediction of outcomes from omics data. Our approach can be used in future to understand the multi-stage transition from a “pre-disease” to “disease” state, with the overall goal of improving primary prevention, patient stratification, and subsequent clinical management.

Appendix. Materials and methods

Statistical model

Consider n individuals with q observed phenotypes and p recorded (epi)genetic markers. The relationship between the phenotype matrix, $\mathit{Y}$, of dimensions (n × q) and the design matrix, $\mathit{X}$, of dimensions (n x p) is modelled as

$$\mathit{Y}=\mathit{X}\mathit{\beta }+\mathit{ϵ},$$ (3)

where the (p x q) matrix $\mathit{\beta }$ represents the effect sizes while $\mathit{ϵ}$ is the residual error matrix with dimensions (n x q). The design matrix can be split into various groups according to biological annotations. The design and phenotype matrix are standardized for each column.

We assume that $\mathit{Y}$ is distributed like a matrix normal with mean $\mathit{X}\mathit{\beta }$, among-row variance ${\mathbb{I}}_{\mathit{n}}$ (where ${\mathbb{I}}_{\mathit{n}}$ is the identity matrix with dimension (n x n)) and among-column variance $\Sigma$ with dimension (q x q):

$$\mathit{Y}\sim MN(\mathit{X}\mathit{\beta },{\mathbb{I}}_{\mathit{n}},\Sigma )$$ (4)

The matrix normal distribution is related to the multivariate normal as

$$\mathrm{vec}(\mathit{Y})\sim \mathit{MVN}(\mathrm{vec}(\mathit{X}\mathit{\beta }),\Sigma \otimes {\mathbb{I}}_{\mathit{n}})$$ (5)

where $\mathrm{vec}(\mathit{Y})$ represents the vectorization of $\mathit{Y}$ and $\otimes$ the Kronecker product. The prior distribution for the residual error matrix $\mathit{ϵ}$ is also assumed to be a matrix normal $MN({\mathbf{0}}_{(n\mathrm{x}q)},{\mathbb{I}}_{\mathit{n}},\Sigma )$.

The effects of each marker j on the multiple traits, ${\mathit{\beta }}_{\mathit{j}}$, is modelled as a multivariate normal with mean $\mathbf{0}$ and variance ${\mathit{V}}_{\mathit{g}}$ of dimensions (q × q):

$${\mathit{\beta }}_{\mathit{j}}\sim \mathit{MVN}({\mathbf{0}}_q,{\mathit{V}}_{\mathit{g}}),$$ (6)

where g refers to the group the marker is attributed to. The group variance ${\mathit{V}}_{\mathit{g}}$ is specific to each group. To be able to model sparsity in the effects, the Dirac delta ${\delta }_0$ is included in the prior distribution of ${\mathit{\beta }}_{\mathit{j}}$ with the prior group-specific exclusion probability ${\pi }_g$:

$${\mathit{\beta }}_{\mathit{j}}\sim (1-{\pi }_g)\mathit{MVN}({\mathbf{0}}_q,{\mathit{V}}_{\mathit{g}})+{\pi }_g{\delta }_0,$$ (7)

where ${\pi }_g$ is modelled by the Dirichlet distribution.

Covariances ${\mathit{V}}_{\mathit{g}}$ as well as $\Sigma$ (jointly denoted as $\mathrm{cov}$) are modelled as outlined in Section 2 of Chan and Jeliazkov (2009), using a modified Cholesky decomposition:

$$\mathbf{cov}={\mathit{L}}^{-1}\mathit{D}{({\mathit{L}}^{-1})}^T,$$ (8)

where

$$\begin{array}{c}\mathit{D}=\left(\begin{array}{cccc}{d}_1& 0& \cdots & 0\\ 0& d_2& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& \cdots & \cdots & d_q\end{array}\right),\hfill \end{array}$$ (9)

$$\begin{array}{c}\mathit{L}=\left(\begin{array}{cccc}1& 0& \cdots & 0\\ l_{21}& 1& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ l_{q1}& l_{q2}& \cdots & 1\end{array}\right).\hfill \end{array}$$ (10)

This parameterisation of the variance matrices is advantageous compared to the inverse Wishart distribution (which is the conjugate of the multivariate normal distribution) as the elements in L are unrestricted. They are modelled with a multivariate normal distribution with prior mean 0 and variance $s_0=0.0001$. The prior distribution of the diagonal elements of $\mathit{D}$, which have to be positive, are set to an inverse Gamma distribution $G^{-1}(a/2,ab/2)$, where a and b are the prior shape and scale parameter of the inverse Gamma distribution (here $a=2$ and $b=0.1$).

Gibbs sampler

To estimate the unknown parameters in (3), a Gibbs sampler which is a Markov Chain Monte Carlo (MCMC) method, is set up. The Gibbs sampler runs the following steps for a chosen number of iterations:

1. Sample intercept from a normal distribution.

2. Randomly pick a marker j and sample ${\mathit{\beta }}_{\mathit{j}}$ from its conditional posterior distribution

   $${\mathit{\beta }}_{\mathit{j}}\sim (1-{\tau }_j)\mathit{MVN}({\mathit{\mu }}_{\mathit{j}}^{\mathit{*}},{\Omega }_{\mathit{j}}^{\mathit{*}})+{\tau }_j{\delta }_0$$ (11)

   with posterior covariance

   $${\Omega }_{\mathit{j}}^{\mathit{*}}={((n-1){\Sigma }^{-1}+{\mathit{V}}_{\mathit{g}}{ }^{-1})}^{-1},$$ (12)
   Posterior mean

   $${\mathit{\mu }}_{\mathit{j}}^{\mathit{*}}=\mathrm{vec}(({\mathit{X}}_{\mathit{j}}{ }^T\mathit{ϵ}+(n-1){\mathit{\beta }}_{\mathit{j}}^{\mathbf{prev}}){\Omega }_{\mathit{j}}^{\mathit{*}}{\Sigma }^{-1}),$$ (13)

   with ${\mathit{\beta }}_{\mathit{j}}^{\mathbf{prev}}$ referring to the effects of marker j in the previous iteration, and posterior exclusion probability

   $${\tau }_j=\frac{r}{r+\left|{\mathit{V}}_{\mathit{g}}{|}^{-1/2}\right|{\Omega }_{\mathit{j}}^{\mathit{*}}{|}^{1/2}{e}^{\frac{1}{2}\hspace{0.17em}{\mathit{\mu }}_{\mathit{j}}^{\mathit{*T}}{\Omega }_{\mathit{j}}^{\mathit{*-1}}{\mathit{\mu }}_{\mathit{j}}^{\mathit{*}}}},$$ (14)

   where $r=\frac{{\pi }_g}{(1-{\pi }_g)}$.

3. Repeat step (2) until all markers are sampled.

4. Sample exclusion probabilities ${\pi }_g$ for each group from $\mathrm{Dirichlet}(p_g-Z_g,Z_g)$, where $p_g$ is the total number of markers and $Z_g$ is the number of non-zero markers in each group.

5. Calculate ${\mathit{V}}_{\mathit{g}}={\mathit{L}}_{\mathit{g}}{ }^{-1}{\mathit{D}}_{\mathit{g}}{({\mathit{L}}_{\mathit{g}}{ }^{-1})}^T$ for each group by:

   1. Sampling the diagonal elements of ${\mathit{D}}_{\mathit{g}}$ from

      $$d_i\sim {\Gamma }^{-1}(\frac{a}{2}+Z_g,\frac{ab}{2}+w_{ii}),$$ (15)

      where $w_{ii}$ is element (i, i) of $\mathit{w}=Z_g{\mathit{L}}_{\mathit{g}}{\mathit{\beta }}_{\mathit{g}}{ }^T{\mathit{\beta }}_{\mathit{g}}{\mathit{L}}_{\mathit{g}}{ }^T$. When sampling with more than one group, $Z_g$ and $\mathit{w}$ are group-specific.

   2. Sampling the elements of the lower triangular matrix ${\mathit{L}}_{\mathit{g}}$ from a multivariate normal distribution with mean

      $$m_i=-\frac{s_i}{d_i}\left(Z_g{\mathit{\beta }}_{\mathit{g}}{\mathit{\beta }}_{\mathit{g}}{ }^T\right)[1:i,i]$$ (16)
      and variance

      $$s_i=\frac{1}{d_i}\left(Z_g{\mathit{\beta }}_{\mathit{g}}{\mathit{\beta }}_{\mathit{g}}{ }^T\right)[1:i,1:i]+s_0{\mathbb{I}}_{\mathit{i}}$$ (17)

      for each row i of ${\mathit{L}}_{\mathit{g}}$ where $[1:i,1:i]$ denotes the submatrix of $(Z_g{\mathit{\beta }}_{\mathit{g}}{\mathit{\beta }}_{\mathit{g}}{ }^T)$ between rows 1 to i and columns 1 to i and $s_0$ is the initial variance of the multivariate normal.

6. Calculate the covariance $\Sigma$ in the same way as ${\mathit{V}}_{\mathit{g}}$.

The effects of the markers are sampled conditional on all the other markers, thus automatically taking into account correlations between markers or linkage disequilibrium (LD). The means and variances of $\mathit{\beta }$ , ${\mathit{V}}_{\mathit{g}}$ and $\Sigma$ averaged across iterations (excluding the results from the burn-in period) are stored.

The Gibbs sampler is run for in total 5000 iterations, whereof 1000 are discarded as burn-in. The burn-in period of 1000 is chosen to be well away from the point where the sampler first reaches convergence to make sure that the values for posterior means are only taken when the estimates are stable. Multiple chains are run after the burn-in period for the estimation of the posterior means in data. Examples of trace plots of the variances and residual variances are shown in Fig. S13, available at supplementary material  Bioinformatics Advances online, showing the convergence of the chains.

The time complexity of MAJA is of order O(npq). The actual runtime depends on the number of parallel processes, which might be limited by the inherent correlation structure of the data, the speed of the available processors and the MPI setup of the computing cluster. The number of groups does not increase the timing considerably, as this affects only the variance estimation which is based on the estimated effects, which is the most time-consuming part.

The code requires as input a phenotype matrix without any missing values and a standardized design matrix which has to fit into RAM. Missing phenotypic data has to be inferred beforehand. Missing omics data should either be excluded (if the probe is missing for more than 5% of the individuals) or should be set to the mean value when preparing the data for analysis. For further details on the implementation of the sampler using a Bulk synchronous parallel Gibbs sampling scheme with message passing interface (Orliac et al. 2022), see links in Code availability.

Generation Scotland methylation data

Generation Scotland is a large population-based, family-structured cohort of over 24,000 individuals aged 18–99 years (Smith et al. 2013). The study baseline took place between 2006 and 2011 and included detailed cognitive, physical, and health questionnaires, along with sample donation for genetic and biomarker data.

The Generation Scotland methylation (GSM) data includes cytosine-phosphate-guanine dinucleotides (CpG sites) for 18,413 individuals. DNA methylation data were processed and quality-controlled in four batches, following broadly similar procedures. Probe and sample quality was assessed using the meffil package in R (Min et al. 2018). Probes were excluded based on low detection P-value ($\ge 0.5\%$ of samples with detection P  $\ge 0.05$ [batch 1]; $\ge 1\%$ of samples with detection P  $\ge 0.01$ [batches 2–4]) and low bead count ($<3$ in $>5\%$ of samples). Samples were removed based on (1) a high proportion of probes with high detection-P-values ($\ge 1\%$ of CpGs with detection P-value $\ge 0.05$ [batch 1]; $\ge 0.5\%$ of CpGs with detection P-value $\ge 0.01$ [batches 2–4]), (2) where recorded sex did not match predicted sex based on information from sex chromosomes, and (3) outlier values based on log median intensites of methylated vs unmethylated signals. The four quality-controlled batches were normalised as a single dataset using the dasen method in wateRmelon (Pidsley et al. 2013). The final set of 831,349 methylation probes were then adjusted for age, sex, smoking and batch and standardized to mean zero and variance one.

We jointly analyze the following phenotypes: body-mass-index (BMI $\mathrm{k}\mathrm{g}/{\mathrm{m}}^2$); ratio of high density lipoprotein over total cholesterol (CHL, both measured in mmol/L); self-reported high blood pressure (BP, 2472 cases); self-reported depression (DEP, 1807 cases); self-reported osteoarthitis (OA, 1355); self-reported asthma (AT, 2097 cases); logical memory (verbal declarative memory), calculated from the Wechsler Logical Memory test by taking the sum of immediate and delayed recall of one oral story (Smith et al. 2013); digit symbol, ascertained from the Wechsler Digit Symbol Substitution test in which participants recoded digits to symbols over a 120 second period (Smith et al. 2013); verbal fluency phenotype, measuring executive functioning, was derived from the phonemic verbal fluency test, using the letters C, F and L, each for 1 min (Smith et al. 2013); vocabulary, measured using the Mill Hill Vocabulary Scale, junior and senior synonyms combined (Smith et al. 2013); year spent in education; and finally highest educational qualification achieved. Years spent in education was self-reported as the total years attended school/study full-time, with coding 0: 0, 1: 1–4,2: 5–9, 3: 10–11, 4: 12–13, 5: 14–15, 6: 16–17, 7: 18–19, 8: 20–21, 9: 22–23, 10: more than 24 years. For highest educational qualification participants were asked what the highest educational qualification they have obtained, with data then coded as: 1—College or University degree, 2—Other professional or technical qualification, 3—NVQ or HND or HNC or equivalent, 4—Higher Grade, A levels, AS levels or equivalent, 5—Standard Grade, O levels, GCSEs or equivalent, 6—CSEs or equivalent, 7—School leavers certificate, 8—Other, 9—No Qualification. All phenotypic data are standardized to mean zero and variance one.

Simulation study

To demonstrate that our model is capable of accurately inferring phenotypic variations and correlations between multiple traits, we simulate epigenetic effects for two traits for the methylation data of chromosome 1 ($p=80,545$ probes), using three different scenarios for the epigenetic (co)variance matrix,

$$\begin{array}{c}\mathit{V}=\left(\begin{array}{cc}{\beta }_1^2& {\beta }_1{\beta }_2\\ {\beta }_1{\beta }_2& {\beta }_2^2\end{array}\right).\hfill \end{array}$$

1. Scenario 1 represents a covariance matrix where there is no correlation between the two traits:

   $$\begin{array}{c}{\mathit{V}}_{\mathit{1}}=\left(\begin{array}{cc}{\beta }_1^2& 0.0\\ 0.0& {\beta }_2^2\end{array}\right).\hfill \end{array}$$
2. The second scenario introduces negative correlation between the two traits:

   $$\begin{array}{c}{\mathit{V}}_{\mathit{2}}=\left(\begin{array}{cc}{\beta }_1^2& -0.5·\sqrt{{\beta }_1^2·{\beta }_2^2}\\ -0.5·\sqrt{{\beta }_1^2·{\beta }_2^2}& {\beta }_2^2\end{array}\right).\hfill \end{array}$$
3. Scenario 3 assumes positive correlations between the two traits:

   $$\begin{array}{c}{\mathit{V}}_{\mathit{3}}=\left(\begin{array}{cc}{\beta }_1^2& +0.5·\sqrt{{\beta }_1^2·{\beta }_2^2}\\ +0.5·\sqrt{{\beta }_1^2·{\beta }_2^2}& {\beta }_2^2\end{array}\right),\hfill \end{array}$$

   where diag(V) $=({\beta }_1^2,{\beta }_2^2)=$ (0.5, 0.2); (0.3, 0.5); (0.5, 0.4) or (0.5, 0.8).

These matrices are scaled by the number of causal markers $p_0=1000$ to sample the epigenetic effects from a multivariate normal distribution. When multiplying the simulated effects with their respective standardized columns of the $\mathbf{X}$ matrix, we obtain an epigenetic value, $\mathit{g}$, for each individual and trait. In each scenario, a vector of residuals is sampled from a normal distribution with variance $({\mathbb{I}}_q-\mathrm{var}(\mathit{g}))$ with the covariance elements set to 0, and added to $\mathit{g}$ to obtain a matrix of phenotypes, $\mathbf{Y}$. We repeat the data generation ten times for each of the three scenarios, where the causal effects are selected randomly. The datasets are then split into training data (n = 17 264) and data (n = 1000) for replication. The training data are anlaysed with MAJA, running the model for 2000 iterations. The posterior mean estimates of the effect sizes and their (co)variances are calculated using the last 1000 iterations. The posterior means of the effects covariances reproduce the true values very well for all three scenarios, as can be seen in Figs S1 and S2, available at supplementary material  Bioinformatics Advances online.

The estimated effect sizes, $\widehat{\mathit{\beta }}$, are then used to create predictors, $Y_{\mathrm{pred},i}={\mathbf{X}}_i\widehat{\mathit{\beta }}$, for each individual i in the test data to obtain the coefficient of determination

$$R^2=1-\frac{{\sum }_i{(Y_i-Y_{\mathrm{pred},i})}^2}{{\sum }_i{(Y_i-\overline{Y})}^2},$$ (18)

where $\overline{Y}$ is the mean of generated phenotypes. Figures S4 and S5, available at supplementary material  Bioinformatics Advances online show that the estimated $R^2$ agrees well with the expected $R^2$ when the traits are correlated. The expected value is calculated according to Equation (34) in Maier et al. (2018) assuming $M_{\mathrm{eff}}=30,000$ independent markers, a number estimated from the training data. The expected $R^2$ is dependent on the assumed number of independent markers which is likely different for the case where the two traits are uncorrelated, which explains the large difference for estimated and expected $R^2$ for $V_1$.

We determine the true positive (TPR) and false discovery (FDR) rates of MAJA across simulation scenarios. True positives are identified as probes for which a causal effects is simulated, where the posterior inclusion probability (PIP) is $\ge 0.95$ and for which the posterior mean effect estimate is $\ge \pm 2$  SD from zero. TPR is calculated as the number of true positives divided by the number of simulated causal variants. False positives are identified as probes that are not simulated to be causal variants, where the posterior inclusion probability is $\ge 0.95$ and for which the posterior mean effect estimate is $\ge \pm 2$  SD from zero. FDR is calculated as the number of false positives divided by the total number of discoveries. Figures S7 and S8, available at supplementary material  Bioinformatics Advances online show the TPR and FDR for the two simulated traits across scenarios.

To demonstrate the necessity of using PIP in combination with the strength of posterior effects to find significant associations, we simulate completely uncorrelated traits by assigning causal effects to 1000 different probes per trait for diag(V) = (0.5, 0.5). Thus, the traits do not share any underlying probes, which is highly unrealistic. There are about 40% more probes included in the model for this simulation scenario compared to the uncorrelated scenario where all probes are shared. However, about 5% less probes have PIP above 0.95. Therefore, TPR is also lower, as shown in Fig. S9, available at supplementary material  Bioinformatics Advances online. TPR stays the same when selecting associated probes just using PIP or using PIP with the strength of posterior effects (mean $\pm$ 1 or 2 SD). FDR is high when just using PIP, but in combination with the strength of posterior effects FDR is controlled.

Finally, to demonstrate that MAJA is also able to handle multiple groups and accurately estimate the covariances of each group, epigenetic effects and phenotypic information for two traits for the methylation data of chromosome 1 ($p=80 545$ probes) and chromosome 2 ($p=60 707$ probes) are generated. The effect sizes in the two chromosomes are generated according to three scenarios, where the second number in the subscript refers to the group (in this case chromosome):

1. $$\begin{array}{c}{\mathit{V}}_{\mathbf{1,1}}=\left(\begin{array}{cc}0.3& 0.0\\ 0.0& 0.5\end{array}\right),{\mathit{V}}_{\mathbf{1,2}}=\left(\begin{array}{cc}0.5& 0.0\\ 0.0& 0.3\end{array}\right)\hfill \end{array}$$
2. $$\begin{array}{c}{\mathit{V}}_{\mathbf{2,1}}=\left(\begin{array}{cc}0.3& -0.5·\sqrt{0.3·0.5}\\ -0.5·\sqrt{0.3·0.5}& 0.5\end{array}\right),\hfill \end{array}$$

    

   $$\begin{array}{c}{\mathit{V}}_{\mathbf{2,2}}=\left(\begin{array}{cc}0.5& -0.5·\sqrt{0.3·0.5}\\ -0.5·\sqrt{0.3·0.5}& 0.3\end{array}\right)\hfill \end{array}$$
3. $$\begin{array}{c}{\mathit{V}}_{\mathbf{3,1}}=\left(\begin{array}{cc}0.3& +0.5·\sqrt{0.3·0.5}\\ +0.5·\sqrt{0.3·0.5}& 0.5\end{array}\right),\hfill \end{array}$$

    

   $$\begin{array}{c}{\mathit{V}}_{\mathbf{3,2}}=\left(\begin{array}{cc}0.5& +0.5·\sqrt{0.3·0.5}\\ +0.5·\sqrt{0.3·0.5}& 0.3\end{array}\right)\hfill \end{array}$$

Each group is generated to have 500 causal markers. In each of the groups, the posterior means of the effects covariances reproduce the true values very well for all three scenarios, as displayed in Fig. S3, available at supplementary material  Bioinformatics Advances online. Figure S6, available at supplementary material  Bioinformatics Advances online shows that the estimated and expected $R^2$ when the effects are estimated for two different groups with different covariances. The $R^2$ values agree well when the traits are correlated.

Prediction into the Lothian Birth Cohort

The Lothian Birth Cohort of 1936 (LBC1936) represents a longitudinal study of aging (Taylor et al. 2018). The 1091 cohort members were all born in 1936 and have been assessed for a wide variety of health and lifestyle outcomes. DNA has been collected at each clinical visit. In the present study, we considered DNA methylation data (Illumina 450k array) from whole blood, taken at mean age 70, for analysis. Details of the collection and processing of the data have been reported previously (McCartney et al. 2018). In brief, after quality control to remove poorly performing methylation sites, samples, and individuals with mismatching genotypes or predicted sex, a sample of 861 individuals was available for prediction analysis. The methylation and phenotypic data were processed in the same manner as GS. Additional phenotypes in LBC1936 were the Wechsler Test of Adult Reading; the National Adult Reading Test; and a general measure of cognitive function.

In LBC1936, BMI is calculated as weight in kilograms divided by height in meters. Weight and height were assessed at the wave 1 (baseline) clinic appointment. HDL cholesterol (mmol/L) and total cholesterol (mmol/L) are blood-based measurements from samples given in clinic at the baseline appointment. The cholesterol ratio is calculated as HDL cholesterol divided by total cholesterol. Scores for thirteen cognitive tests were available across five waves of data collection. Testing was performed triennially from age 70 to 82. Visuospatial ability was measured using the Block Design, Matrix Reasoning (WAIS-IIIUK) and Spatial Span (WMS-IIIUK) tests. Verbal ability was measured using the National Adult Reading Test, Wechsler Adult Reading Test and Verbal Fluency Test (using letters C, F and L). Memory was assessed via the Verbal Paired Associates, Logical Memory—a combination of immediate and delayed memory (WMS-IIIUK) and Digit Span Backwards (WAIS-IIIUK) tests. Processing speed was evaluated via the Digit Symbol Substitution Test, Symbol Search (WAIS-IIIUK), Choice Reaction Time and Inspection Time tests.

A latent measure of general cognitive function was obtained by using confirmatory factor analysis in a structural equation modelling (SEM) framework using the R package Lavaan (version 0.6–12) (Rosseel 2012). A first-order hierarchical cognitive model was specified. Specifically, levels and change in general cognitive functioning were modelled with latent growth curve model (LGCM) using a Factor of Curves specification (McArdle 1988). Intercepts and slopes of each cognitive test were used to indicate a latent intercept and slopes of general cognitive function and change. The growth curve slopes were weighted by mean lag time between each wave and baseline. Marker method was used to scale according to the first variable, and all models used full information maximum likelihood to include all data available. Negative residual variances were fixed to zero. Residual covariance between tests in the same cognitive domain were specified (Tucker-Drob et al. 2014).

Episcores were projected into LBC1936 wave 1 methylation data (n = 861). Linear regression was used to model each episcore (as a predictor) in relation to the outcome variables. Incremental $R^2$ estimates are reported as the differences between models adjusting for age and sex compared to those that additionally include the episcore. For the variance explained in general cognitive function level, linear regression models were performed within Lavaan, with the G intercept from the latent growth curve models used as the outcome (see McCartney et al. 2022). Model fit and test loadings can be found in Table S7, available at supplementary material  Bioinformatics Advances online.

Comparison to other studies and methods

We want to place our methods in the context of other studies and multi-trait methods. First, we want to point out that marginal EWAS have been shown to significantly enlarge the number of associations (Orliac et al. 2022), but also hugely increase the number of false discoveries due to the inherent correlation structure of the data. For example, Smith et al. (2025) find 57 307 significant associations for BMI in a marginal, single-trait EWAS in GS, but only 27 with a joint Bayesian approach. They also find only a single CpG site, cg06500161 (mapped to ABCG1), associated with BMI and cholesterol with PIP $>0.95$, compared to eight probes including cg06500161 that we find with MAJA for metabolic and disease traits.

Other existing multi-trait estimation methods that can handle omics data are not set up or tested for larger datasets ($n>1000,p>5000$) (Gianola and Fernando 2020) or do not provide ready-to-use code (Hatton et al. 2023). mvSuSiE (Zou et al. 2026), a variational inference approach which performs multi-trait fine-mapping on genomic regions associated independently for each trait, requires independent regions to be identified beforehand. Even with pre-selecting small subsets of probes using simulated data, running mvSuSiE is not feasible. We therefore are unable to compare to any existing multi-trait methods.

There are of course other single-trait methods available for modelling shared structure across omics layers. Recent examples are collaborative regression (Gross and Tibshirani 2015), guided network estimation method (Bartzis et al. 2024), and collaborative graphical lasso (Albanese et al. 2026).

Author contributions

IK and MRR conceived and designed the study. EB, MM, DLM and REM contributed to data preparation and design of the analyses. IK wrote the software and conducted the analyses, with assistance from HS for the prediction. REM, MRR, JC, AC, SEH and SRC provided study oversight. IK and MRR wrote the paper.

Supplementary material

Supplementary material is available at Bioinformatics Advances online.

Conflicts of interest

REM is a scientific advisor to the Epigenetic Clock Development Foundation and Optima Partners—this is unrelated to the work presented here. The remaining authors declare no competing interests.

Funding

This work was funded by an SNSF Eccellenza Grant to MRR [PCEGP3-181181], and by core funding from the Institute of Science and Technology Austria. We would like to acknowledge the participants and investigators of the Generation Scotland and Lothian Birth Cohort studies. Generation Scotland received core support from the Chief Scientist Office of the Scottish Government Health Directorates [CZD/16/6] and the Scottish Funding Council [HR03006]. Genotyping and methylation typing of the GS: SFHS samples was carried out by the Genetics Core Laboratory at the Wellcome Trust Clinical Research Facility, Edinburgh, Scotland and was funded by the Medical Research Council UK and the Wellcome Trust (Wellcome Trust Strategic Award “STratifying Resilience and Depression Longitudinally” (STRADL) Reference 104036/Z/14/Z]. DNA methylation data for Generation Scotland was also funded by a 2018 NARSAD Young Investigator Grant from the Brain and Behavior Research Foundation [Ref: 27404; awardee: Dr David M Howard] and by a John, Margaret, Alfred and Stewart Sim Fellowship from the Royal College of Physicians of Edinburgh (Awardee: Dr Heather C Whalley). The LBC1936 is supported by the Biotechnology and Biological Sciences Research Council, and the Economic and Social Research Council [BB/W008793/1] (which supports SEH, and JC), Age UK (Disconnected Mind project), the Milton Damerel Trust, the Medical Research Council [G0701120, G1001245, MR/M013111/1, MR/R024065/1] and the University of Edinburgh. Methylation typing of LBC1936 was supported by the Centre for Cognitive Ageing and Cognitive Epidemiology (Pilot Fund award), Age UK, The Wellcome Trust Institutional Strategic Support Fund, The University of Edinburgh, and The University of Queensland. HS is supported by funding from the Wellcome Trust 4-year PhD in Translational Neuroscience [218493/Z/19/Z]. SRC is also supported by a Sir Henry Dale Fellowship jointly funded by Wellcome and the Royal Society [221890/Z/20/Z]. High-performance computing was supported by the Scientific Service Units (SSU) of IST Austria through resources provided by Scientific Computing (SciComp).

Data availability

Access to the data is available with appropriate permission from the Generation Scotland Access Committee. Applications should be made to access@generationscotland.org. https://www.ed.ac.uk/lothian-birth-cohorts/data-access-collaboration gives information on the Lothian Birth Cohort data, which are available on request from the Lothian Birth Cohort Study, University of Edinburgh. Data from both cohorts are not publicly available as they contain information that could compromise participant consent and confidentiality.

Source code is available at https://github.com/medical-genomics-group/MAJA.