A nutritionally informed model for Bayesian variable selection with metabolite response variables

Summary

Understanding the pathways through which diet affects human metabolism is a central task in nutritional epidemiology. This article proposes novel methodology to identify food items associated with blood metabolites in two cohorts of healthcare professionals. We analyze 244 metabolites characterized by statistical complexities that include skewness, left-censoring, and structural missingness. Though existing methods can address such factors in low-dimensional settings, they cannot exploit the nutritional or statistical relationships among the 30 considered food intake variables, and they are unsuitable for performing high-dimensional inference. To address these challenges, we develop a novel Bayesian variable selection framework for metabolite response variables based on a skew-normal censored mixture model, while exploiting substantive information on the considered food items via a Markov random field prior. Applying this methodology to the cohort data identifies multiple metabolite-diet associations that are consistent with previous research as well as several potentially novel associations that were not detected using standard methods. The proposed approach is implemented in the R package multimetab, facilitating its use in high-dimensional metabolomic analyses.

1. Introduction

A central goal of metabolomics is to understand how factors such as age, sex, health, and nutrition affect the human body’s abundance of blood metabolites. Metabolites are small molecules produced or expended by metabolism—an umbrella term for the chemical reactions that keep the human body alive and functioning. Metabolites play essential roles in processes such as signaling, immune modulation, and energy production, among others (Baker and Rutter, 2023). Most metabolites are derived from digested compounds or produced as a chemical byproduct of the gut microbiome, making nutritional intake an important driver of their abundance (Jia and others, 2022; Caspani and Swann, 2019). Quantifying the relationships between nutritional factors and metabolite abundance is essential for elucidating the metabolic pathways that underpin human health.

The objective of this article is to identify dietary predictors of metabolites using data from two cohort studies of healthcare professionals: the Mind Body Study (MBS) and the Men’s Lifestyle Validation Study (MLVS) (Huang and others, 2019; Wang and others, 2021). Recent analyses of MBS and MLVS data have established relationships between specific types of metabolites and diet, cardiometabolic risk, and the gut microbiome (Li and others, 2022b; Hamaya and others, 2024; Malik and others, 2019; Li and others, 2022c). These analyses have mainly been narrow in scope, targeting a subset of the measured metabolites and considering only a few food intake variables, or aggregated versions of them, individually. We expand on this research by performing a comprehensive scan over 244 metabolites of diverse types. Each metabolite is treated as the response variable in a separate regression model that adjusts for all 30 food intake variables jointly, allowing us to pinpoint specific food items that are most informative. We accomplish this by developing a novel Bayesian variable selection framework for metabolite responses that exploits external information on the nutritional similarity of the considered food items.

In common with most metabolomics datasets, the considered data present an array of challenges for statistical analysis. First, mass spectrometry-based metabolomics data exhibit a specific type of missing values termed point mass values (PMVs) (Hrydziuszko and Viant, 2012; Do and others, 2018). “Biological” PMVs arise when the mass spectrometer fails to detect a metabolite because it is completely absent from the sample. “Technical” PMVs occur when a metabolite’s intensity is below a threshold called the “detection limit,” producing a spurious missing value that reflects censoring, not true absence. It is generally unknown whether individual PMVs are technical or biological (Taylor and others, 2013). In the MLVS and MBS data, 12% of metabolites have at least one PMV. The metabolite missing a value the most often has 54% PMVs (fig. 1A).

Fig. 1. MBS and MLVS data challenges.

(A) The point mass value (PMV) percentages of 30 metabolites with at least one PMV in the Mind Body Study (MBS) and Men’s Lifestyle Validation Study (MLVS). (B) Pairwise correlations among the 30 dietary intake predictor variables after combining the two datasets. (C) The skewness index of all 244 metabolites, where more positive values indicate stronger right skewness. We used half-minimum imputation for panels (B) and (C).

Second, the dietary predictors exhibit complex dependence structure through their correlations and nutritional relationships. With Pearson correlations among the 30 food intake variables ranging from −0.24 to 0.54 (fig. 1B), these data require advanced statistical techniques, such as variable selection, that can extract sparse signals from complex feature spaces. Adding to the complexity, some pairs of foods have similar metabolic signatures due to their source or nutritional content, such as types of fruit. Satija and others (2016) exploited such information to identify dietary influences of type II diabetes using cohort data overlapping with MBS and MLVS, grouping foods by their nutritional value to calculate healthy and unhealthy diet scores. An ideal modeling framework should similarly use nutritional relationships to promote powerful, biologically informed inference.

Finally, metabolites tend to be strongly right skewed and have many outliers. Excessive skewness can invalidate statistical methods that assume Gaussian errors, introducing bias and distorting inference (Wilson and others, 2020). The metabolites in MBS and MLVS have a mean skewness index of 2.3, indicating heavy right skewness (fig. 1C).

There are two general frameworks for handling PMVs in statistical analysis. One strategy is to use imputation methods designed specifically for mass spectrometry data, which are essential because PMVs are missing-not-at-random, invalidating standard imputation tools (Wei and others, 2018). The second strategy, adopted in this article, is to use advanced statistical methods that handle PMVs internally (Taylor and others, 2013; Gleiss and others, 2015; Huang and Wang, 2022). The simplest approach of this type is to perform one-part tests based on tobit or accelerated failure time (AFT) models, assuming the PMVs are technical PMVs caused by censoring. In contrast, two-part models compare the PMVs and non-PMVs separately and combine the p-values, treating the PMVs as biological. To flexibly account for PMVs of both types, hybrid approaches—such as tobit or AFT mixture models—have been developed. Recent extensions of this framework have used Bayesian modeling and empirical Bayes shrinkage techniques (Shah and others, 2019; Huang and others, 2020). These methods address outcome modeling—but not predictor selection—in the presence of PMVs.

Although mixture models are well established for accommodating PMVs in metabolomics studies, their existing variants are unsuitable for handling correlated and nutritionally related dietary factors as predictors. Specifically, no existing mixture models offer frameworks for performing variable selection, rendering them inadequate for detecting signals obscured by correlated, high-dimensional sets of covariates. Nor do they offer architecture for using external information on the substantive relationships among predictors, meaning the nutritional patterns of the food intake variables cannot be exploited.

Existing mixture models also rely on ad hoc strategies for handling skewness, such as assuming the log-transformed metabolites have Gaussian errors. The main weakness of transformations is that some metabolites are still highly skewed after being transformed. Though distributional violations are usually inconsequential for standard linear regression models, they can greatly impact the performance of censored data methods such as tobit models (Wilson and others, 2020). A preferable strategy might be to incorporate skewness estimation into the model framework, adapting to the specific skewness of each metabolite. Lastly, though not essential, it is helpful to have the option to analyze untransformed data to preserve the original interpretation of the regression coefficients.

We address these challenges by developing an innovative Bayesian variable selection framework that: (1) internally accounts for technical and biological PMVs; (2) identifies predictive covariates within complex feature spaces; (3) exploits structural information on the substantive relationships among covariates; and (4) directly models the skewness of each metabolite. The proposed framework assumes a skew-normal censored mixture (SNCM) model, flexibly handling missingness and skewness without requiring imputation or transformation. To identify dietary factors associated with each metabolite, we perform Bayesian variable selection via spike-and-slab priors. To exploit the nutritional relationships among dietary factors, we adopt a Markov random field (MRF) hyperprior that promotes the joint selection of nutritionally similar variables (Stingo and others, 2011). Recognizing the computational burden of standard MRF implementations, which often require repeated model fitting for hyperparameter tuning, we propose a computationally efficient, data-independent strategy for hyperparameter specification. We provide user-friendly software in the R package multimetab.

2. Model and Posterior Inference

2.1. Model formulation

For $i\in 1,\dots ,n$, let $Y_i$ denote the observed value of a given metabolite for subject $i$, possibly recorded as missing. Let $Y_i^{\mathrm{*}}$ be the corresponding “true” value that is observed only if $Y_i^{\mathrm{*}}⩾\psi$, where $\psi >0$ is the detection limit that induces censoring. Let ${\mathit{X}}_i={\left(X_{1i},\dots ,X_{pi}\right)}^{\mathbf{\top }}$ denote a possibly high-dimensional vector of $p$ covariates, some associated with $Y_i^{\mathrm{*}}$ and others not. The variables in ${\mathit{X}}_i$ are the main covariates of interest and will undergo variable selection. Let ${\mathit{C}}_i={\left(C_{1i},\dots ,C_{si}\right)}^{\mathrm{\top }}$ denote a low-dimensional vector of $s$ covariates, $s\ll n$, that are excluded from variable selection because they are identified a priori as potential confounding variables. For consistency of language, we will refer to $X_{1i},\dots ,X_{pi}$ as “predictors” and $C_{1i},\dots ,C_{si}$ as “confounders.” Let $U_i\sim \mathrm{B}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{o}\mathrm{u}\mathrm{l}\mathrm{l}\mathrm{i}\left(\rho \right)$ be indicator variables marking whether the metabolite is present in the sample or absent, and let $V_i$ be the value of the metabolite when it is present. We adopt a hierarchical model:

$$V_i={\beta }_0+\sum _{j=1}^p{\beta }_j{X}_{ij}+\sum _{t=1}^s{\alpha }_t{C}_{it}+{ϵ}_i$$

$$Y_i^{\mathrm{*}}=U_i{V}_i$$

$$Y_i=\left\{\begin{array}{ll}{Y}_i^{\mathrm{*}}& Y_i^{\mathrm{*}}⩾\psi \\ \mathrm{P}\mathrm{M}\mathrm{V}& Y_i^{\mathrm{*}}<\psi \end{array}\right.$$

The PMVs, where $Y_i$ is missing, have either $U_i=0,$ $V_i<\psi$, or both. PMVs with $U_i=0$ denote biological PMVs caused by the total absence of the metabolite from the sample, whereas PMVs with $U_i=1$ and $V_i<\psi$ denote technical PMVs caused by censoring. We treat $V_i$ as the response variable while assuming ${ϵ}_i\stackrel{iid}{\sim }\mathrm{S}\mathrm{N}\left(0,{\sigma }^2,\delta \right)$, a skew-normal distribution with variance parameter ${\sigma }^2$ and skewness $\delta$ under the parameterization given by Sahu and others (2003). This is equivalent to assuming the error distribution is $\mathrm{N}\left(0,{\sigma }^2\right)+\delta \left|\mathrm{N}\right(\mathrm{0,1}\left)\right|$, with mean $\delta \sqrt{2/\pi }$ and variance ${\sigma }^2+(1-2/\pi ){\delta }^2$. Our main interest lies in $\mathit{\beta }={\left({\beta }_1,\dots ,{\beta }_p\right)}^{\mathbf{\top }}$, which represents the linear associations between the predictors and the metabolite among samples where the metabolite is present (i.e., those with $U_i=1$). The vector $\mathit{\alpha }={\left({\alpha }_1,\dots ,{\alpha }_s\right)}^{\mathrm{\top }}$ denotes the associations corresponding to the confounding variables.

Let $F_{\mathrm{S}\mathrm{N}}$ denote the cumulative distribution function of the skew-normal distribution and $f_{\mathrm{S}\mathrm{N}}$ the corresponding density. Let $W_i=\mathbb{1}(Y_i^{\mathrm{*}}⩾\psi )$ be an indicator variable for whether $Y_i$ is observed (i.e., not a PMV), and define ${\mu }_i={\beta }_0+{\mathit{X}}_i^{\mathbf{\top }}\mathit{\beta }+{\mathit{C}}_i^{\mathbf{\top }}\mathit{\alpha }$. Then, letting $\mathit{\theta }=\left({\beta }_0,\mathit{\beta },\mathit{\alpha },{\sigma }^2,\delta ,\rho \right)$, the likelihood for observation $i$ is

$${ℒ}_i\left(\mathit{\theta };y_i,{\mathit{x}}_i,{\mathit{c}}_i\right)={\left(\rho f_{\mathrm{S}\mathrm{N}}\left(y_i;{\mu }_i,\sigma ,\delta \right)\right)}^{w_i}{\left(1-\rho +\rho F_{\mathrm{S}\mathrm{N}}\left(\psi ;{\mu }_i,\sigma ,\delta \right)\right)}^{1-w_i}.$$

2.2. Prior specification and incorporation of external information

To identify predictors associated with the abundance of a given metabolite, we place variable selection priors on the entries in $\mathit{\beta }$. Specifically, letting ${\gamma }_j$ be a latent indicator variable for whether ${\beta }_j\ne 0$, $j\in 1,\dots ,p$, we apply the spike-and-slab priors ${\beta }_j^{\mathrm{*}}{\sim }_{iid}\mathrm{N}\left(0,{\nu }^2\right)$, ${\beta }_j={\beta }^{\mathrm{*}}{\gamma }_j$. While it is more common to write such priors in the simpler form ${\beta }_j\sim \mathrm{N}\left(0,{\gamma }_j{\nu }^2\right)$, omitting the latent variable ${\beta }_j^{\mathrm{*}}$, we use the former version because it allows the parameters be updated by using Gibbs sampling (Brown and others, 1998; Kuo and Mallick, 1998).

In Bayesian variable selection methods based on spike-and-slab priors, ${\gamma }_1,\dots ,{\gamma }_p$ are typically assigned independent Bernoulli hyperiors that allot equal inclusion probability to all predictors. However, many of the dietary predictors in the motivating data share nutritional components that may inform their metabolic effects, for example, fermented and non-fermented dairy products. Collapsing such categories risks obscuring any differences between them and may reduce sensitivity to detect their metabolic signatures. To exploit such structure in the variable selection scheme, we adopt a structured multivariate prior, the MRF prior (Li and Zhang, 2010; Zhao and others, 2024), on the vector $\gamma ={\left({\gamma }_1,\dots ,{\gamma }_p\right)}^{\mathrm{\top }}$.

Let $\mathit{R}=[r_{j,j^{\prime }}]\in {\mathbb{R}}^{p\times p}$ denote a symmetric matrix with zero diagonal and non-negative off-diagonal entries, where $r_{j,j^{\prime }}$ quantifies the strength of the relationship between predictors $X_j$ and $X_{j^{\prime }}$. The form of the MRF prior is

$$P(\mathit{\gamma })\propto \mathrm{e}\mathrm{x}\mathrm{p}\left(\omega \sum _{j=1}^p{\gamma }_j+\eta {\mathit{\gamma }}^{\mathrm{\top }}\mathit{R}\mathit{\gamma }\right),$$

where $\omega \in \mathbb{R}$ and $\eta ⩾0$ are hyperparameters controlling the model’s sparsity such that higher values tend to produce models with more variables. $\omega$ controls overall sparsity uniformly across variables, and $\eta$ regulates the influence of the relationship structure given by $\mathit{R}$. When $\eta =0$, the prior reduces to an independent Bernoulli prior with inclusion probability equal to ${\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}}^{-1}(\omega )$. When $\eta >0$, the prior encourages the joint selection of variables that are mutually related, such that greater values of $r_{j,j^{\prime }}$ increase the probability of selecting $X_j$ once $X_{j^{\prime }}$ is selected. Greater values of $\eta$ enhance this effect.

For the remaining parameters, we assume ${\beta }_0\sim \mathrm{N}\left(0,{\nu }_0^2\right),{\sigma }^2\sim \mathrm{I}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e}-\mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}\left(0.5{\xi }_0,0.5{\xi }_0{\sigma }_0^2\right)$, $\delta \sim \mathrm{N}\left(0,{\nu }_{\mathrm{d}}^2\right),{\alpha }_t\sim \mathrm{N}\left(0,{\lambda }_t^2\right),t\in 1,\dots ,s$, and $\rho \sim \mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}\left({\rho }_0,{\rho }_1\right)$. These priors reduce to conjugate priors after the introduction of additional latent variables, discussed next.

2.3. Posterior inference and computation

We perform inference via Markov chain Monte Carlo (MCMC) using Gibbs sampling. The specified prior distributions can be viewed as conjugate priors after introducing latent variables to handle non-standard components of the model. First, we decompose the skew-normal error term into a normal and a half-normal component as suggested by Sahu and others (2003). Specifically, an equivalent definition of $V_i$ is

$$V_i={\beta }_0+\sum _{j=1}^p{\gamma }_j{\beta }_j^{\mathrm{*}}{X}_{ji}+\sum _{t=1}^s{\alpha }_t{C}_{ti}+\delta Z_i+{\zeta }_i,$$ (2.1)

$i\in 1,\dots ,n$, where $Z_i$ is a positive random variable following the folded standard normal distribution, and ${\zeta }_i\stackrel{iid}{\sim }\mathrm{N}\left(0,{\sigma }^2\right)$. Conditional on the latent variables $\mathit{\gamma },\mathit{V}={\left(V_1,\dots ,V_n\right)}^{\mathrm{\top }}$, and $\mathit{Z}={\left(Z_1,\dots ,Z_n\right)}^{\mathrm{\top }}$, the model reduces to a standard linear regression model with Gaussian errors, for which the priors on ${\beta }_0$, ${\sigma }^2$, $\delta$, ${\mathit{\beta }}^{\mathrm{*}}$, and $\mathit{\alpha }$ are conjugate. Likewise, the prior on $\rho$ is conjugate conditional on the latent variables $\mathit{U}={\left(U_1,\dots ,U_n\right)}^{\mathbf{\top }}$.

We use data augmentation to introduce $\mathit{Z}$, $\mathit{V}$, and $\mathit{U}$ to the inferential scheme (Albert and Chib, 1993). Each $Z_i$ is updated based on its full conditional distribution assuming a half-normal conjugate prior. Each $V_i$ equals $Y_i$ if $W_i=1$ (non-PMVs), follows a $\mathrm{N}\left({\mu }_i+\delta Z_i,{\sigma }^2\right)$ distribution if $W_i=0$ and $U_i=0$, and follows a $\mathrm{N}\left({\mu }_i+\delta Z_i,{\sigma }^2\right)$ distribution truncated to $(-\mathrm{\infty },\psi )$ if $W_i=0$ and $U_i=1$. Finally, each $U_i$ equals 1 if $W_i=1$, equals 0 if $W_i=0$ and $V_i>\psi$, and follows a $\mathrm{B}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{o}\mathrm{u}\mathrm{l}\mathrm{l}\mathrm{i}(\rho )$ distribution if $W_i=0$ and $V_i<\psi$. The augmented likelihood in terms of latent variables is

$${ℒ}_i\left(\mathit{\theta };V_i,U_i,Z_i,{\mathit{X}}_i,{\mathit{C}}_i\right)=(1-\rho {)}^{1-U_i}{\rho }^{U_i}\frac{2}{\delta \sigma }\varphi \left(\frac{Z_i}{\delta }\right)\varphi \left(\frac{v_i-{\mu }_i-\delta Z_i}{\sigma }\right),$$

where $\varphi (\cdot )$ is the standard normal density.

The convenient structure of the Kuo and Mallick (1998) spike-and-slab prior allows $\mathit{\beta }\mathrm{*}$ and $\mathit{\alpha }$ to be updated by their joint full conditional distribution. The ${\beta }_j^{\mathrm{*}}$ parameters for which ${\gamma }_j=1$ are updated jointly with $\mathit{\alpha }$ using the standard update for Bayesian linear regression. If ${\gamma }_j=0$, then ${\beta }_j^{\mathrm{*}}$ does not affect the likelihood and can be updated by sampling directly from the prior. Finally, we update each ${\gamma }_j$ by its full conditional, exploiting the fact that ${\gamma }_j\mid {\left\{{\gamma }_{j^{\prime }}\right\}}_{j^{\prime }\ne j}$ has a tractable Bernoulli prior where

$$\mathrm{P}\mathrm{r}\left({\gamma }_j=1\mid {\left\{{\gamma }_{j^{\prime }}\right\}}_{j^{\prime }\ne j}\right)={\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}}^{-1}\left(\omega +2\eta \sum _{j^{\prime }\ne j}{r}_{j,j^{\prime }}{\gamma }_{j^{\prime }}\right).$$

Further details are provided in the supplement (Section S1).

2.3.1. A prior-based method for MRF hyperparameter specification

An important consideration when implementing the MRF prior is specifying $\eta$, which controls the degree to which the relationship matrix, $\mathit{R}$, affects variable selection. Prior work has shown that MRF-based priors can exhibit ‘phase transition’ behavior: as $\eta$ increases beyond a critical threshold, the prior can shift abruptly from favoring sparse models to selecting nearly all variables (Li and Zhang, 2010; Zhao and others, 2024). This behavior may lead to unstable posterior inference and obscure the interpretability of identified associations.

Several approaches have been proposed to address this challenge. One option is to place a hyperior on $\eta$ to infer it directly (Stingo and others, 2011). Another is to conduct a data-driven grid search to find the largest $\eta$ value below the phase transition point (Li and Zhang, 2010; Lee and others, 2017). However, both strategies require exploring large $\eta$ values that can produce highly parameterized models, often creating a substantial computational burden.

To circumvent these issues, we introduce a new approach for choosing $\eta$ that depends only on $\omega$ and $\mathit{R}$. The idea is to curtail the prior probability of a large model by finding an MRF prior with summary statistics similar to those from a reasonable independent prior. The first step is to select $\omega$ based on a target baseline sparsity level. For example, choosing $\omega =\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}(0.01)$ corresponds to a 1% prior inclusion rate under independent Bernoulli hyperpriors $(\eta =0)$. After setting $\omega ={\omega }_0$, we choose the largest $\eta$ such that the prior 95^th percentile model size is below that of the $\mathrm{M}\mathrm{R}\mathrm{F}\left(\omega =\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}\left(2{\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}}^{-1}\left({\omega }_0\right)\right),\eta =0\right)$, an independent Bernoulli “reference” prior with inclusion probability $2{\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}}^{-1}\left({\omega }_0\right)$. We search for such an $\eta$ over a list of candidates by sampling from the corresponding prior distributions, which is substantially more computationally efficient than existing posterior- or data-driven tuning methods.

3. Simulation Studies

We evaluated the performance of the proposed inferential scheme through a series of simulation studies. The goal was to assess its robustness under varying levels of skewness, censoring, residual variance, and high-dimensionality—challenges commonly encountered in mass spectrometry-based metabolomics data.

3.1. Data generation and settings

For a baseline setting, we generated data under the SNCM model with $n=400$ observations, $p=300$ predictors, and no confounding variables $(s=0)$. We set ${\beta }_0=5$ and generated predictors as ${\mathit{X}}_i\stackrel{iid}{\sim }\mathrm{M}\mathrm{V}\mathrm{N}(0,\mathit{I})$. We chose $\sigma$ and $\delta$ so that the residual variance in $V_i$ was fixed at 8, with 75% explained by the half-normal latent variable $Z_i$. To induce an average censoring rate of 36%, we set $\rho =0.8$ and chose $\psi$ such that $P\left(V_i<\psi \right)=0.20$.

To mimic the hierarchical dietary relationships in the MLVS/MBS data (section 4.2.2), we defined $\mathit{R}\in {\mathbb{R}}^{300\times 300}$ as a block-diagonal matrix with 15 blocks (categories) of 20 connected predictors. Within each category, predictors 6–10 and 11–20 formed two subcategories, and predictors 16–20 formed a sub-subcategory (fig. 2). For predictors $j$ and $j^{\prime }$, we let $r_{j,j^{\prime }}=\mathrm{e}\mathrm{x}\mathrm{p}\left(a_{j,j^{\prime }}/c\right)/b$, where $a_{j,j^{\prime }}\in \{1,2,3\}$ is the depth of their deepest shared category/subcategory; $b_{j,j^{\prime }}\in \{1,\dots ,20\}$ is the size of their deepest shared category/subcategory; and $c=3$ is the maximum depth in the hierarchy. For example, since predictors 15 and 16 are in a subcategory $(a=2)$ of size 10, their relationship strength is exp(2/3)/10 ≈ 0.19. Since predictors 16 and 17 are in a sub-subcategory $(a=3)$ of size 5, their relationship is exp(3/3)/5 ≈ 0.54. Predictors in different blocks have a relationship strength of 0.

Fig. 2. Relationships within a block of 20 predictors in the simulation design.

The heatmap depicts one of 15 identical predictor blocks in the dietary relationship matrix $\mathit{R}$. Predictors within the same block have structured relationships based on the hierarchical tree shown on the left, and predictors in different blocks are unrelated.

The construction of $\mathit{\beta }\in {\mathbb{R}}^{300}$ was motivated by two goals: to induce variety in the signal strengths and directions among predictors, and to create correspondence with the relationship structure defined in $\mathit{R}$. Specifically, in each of the first four predictor blocks/categories used to construct $\mathit{R}$, we assigned five predictors to be associated with the response (20 total). We chose predictors 1–5 (weakly connected through $\mathit{R}$) in block 1, 6–10 (highly connected) in block 2, 11–15 (moderately connected) in block 3, and 16–20 (very highly connected) in block 4. The regression coefficients within each block were set to {0.4, −0.6, 0.8, −1.0, 1.2}, varying the signal strength and direction.

To evaluate performance under varied conditions, we considered seven additional data-generating scenarios in each of which one aspect of the baseline setting was perturbed. (i) Higher noise was induced by multiplying both $\sigma$ and $\delta$ by 1.5, which raised the residual variance of $V_i$ to 18; (ii) Heavier censoring was created by setting $\rho =0.7$ and $P\left(V_i<\psi \right)=0.30$, resulting in an overall censoring rate of 51%; (iii) Greater skewness was obtained by adjusting $\sigma$ and $\delta$ so that the half-normal latent variable explained 95% (instead of 75%) of the error variance while keeping total variance unchanged; (iv) Misspecified dependence structure was simulated by randomly permuting the rows and columns of $\mathit{R}$ in each replicate, thereby breaking the link between relationships among predictors and their associations with the response; (v) Non-normal errors were examined by drawing ${ϵ}_i$ from a log-normal distribution, matched in mean and variance to the skew-normal error used in the baseline scenario; (vi) Correlated predictors were introduced by sampling ${\mathit{X}}_i\stackrel{iid}{\sim }\mathrm{M}\mathrm{V}\mathrm{N}(0,\mathbf{\Sigma })$, where $\mathbf{\Sigma }$ was obtained by replacing the diagonal of $\mathit{R}$ with ones; and (vii) Large sample size was considered by raising $n$ to 1,000. Finally, in additional simulation scenarios described in the supplement (Section S2), we explored data-driven scenarios that aligned closely with the MBS/MLVS datasets in terms of the sample size size, number of predictor variables, confounding adjustment, correlations, and relationship matrix.

3.2. Analysis and hyperparameters

We fit the SNCM model to 800 simulated replicates of each setting and summarized performance in terms of the mean overall true positive rate (TPR), variable-specific TPR, and false discovery rate (FDR). Letting ${\tilde{\gamma }}_j$ be an indicator variable for whether predictor $j$ is selected for a given dataset, the variable-specific true positive rate is $\mathbb{E}\left[{\tilde{\gamma }}_j\mid {\gamma }_j=1\right]$, the mean overall TPR is $\mathbb{E}[\sum _j{\gamma }_j{\tilde{\gamma }}_j/\sum _j{\gamma }_j\mid \mathit{\gamma }]$, and the FDR is $\mathbb{E}[\sum _j\left(1-{\gamma }_j\right){\tilde{\gamma }}_j/\sum _j{\tilde{\gamma }}_j\mid \mathit{\gamma }]$. We also calculated the variable-specific bias and root mean squared error (rMSE) for estimating the regression coefficients of the effective predictors. Hyperparameters were fixed at ${\nu }_0^2=5^2$, ${\nu }_d^2=5^2$, ${\nu }^2=2^2$, ${\xi }_0=5$, ${\sigma }_0^2=4$, and $\omega =\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}(0.02)$. We searched for $\eta$ over the range $0.01r^{-1},0.02r^{-1},\dots ,1r^{-1}$, where $r=\mathrm{m}\mathrm{a}\mathrm{x}(\mathit{R})$. We adapted ${\rho }_0$ and ${\rho }_1$ to each data set by setting ${\rho }_0=5\sqrt{\bar{W}}$ and ${\rho }_1=5(1-\sqrt{\bar{W}})$, where $\bar{W}=\sum _{i=1}^n{W}_i/n$; this yields a weakly informative prior centered on $P\left(U_i=0\right)=P\left(V_i<\right.\psi$). Since the $\psi$ is unknown in practice, we estimated it from the data as $\hat{\psi }=\mathrm{m}\mathrm{i}\mathrm{n}\left\{Y_i:W_i=1\right\}$.

For each simulation scenario, we assessed the performance of the SNCM model with either independent Bernoulli priors (setting $\eta =0$) or the MRF prior. Though the use of mixture models is well-established (Taylor and others, 2013; Huang and Wang, 2022), for completeness, we additionally benchmarked the SNCM model against two ad hoc treatments of PMVs for the baseline scenarios: (i) treating all PMVs as technical (i.e., forcing $\rho =1$), and (ii) half-minimum imputation (Wei and others, 2018). For both approaches we used independent Bernoulli priors.

To analyze each dataset, we drew 125,000 posterior samples after a burn-in period of 25,000, retaining only 1 in 25 samples to reduce autocorrelation. We considered variables to be selected by the model if their posterior inclusion probability (PIP), ${\hat{\gamma }}_j=\hat{\mathbb{E}}\left[{\gamma }_j\mid \mathit{X}=\mathit{x},\mathit{Y}=\mathit{y}\right]$, exceeded a threshold chosen to achieve a 5% Bayesian FDR (Newton and others, 2004), where $\mathit{y}$ and $\mathit{x}$ denote the observed realizations of $\mathit{Y}$ and $\mathit{X}$, respectively. We estimated the regression coefficients as ${\hat{\beta }}_j=\hat{\mathbb{E}}\left[{\beta }_j\mid \mathit{X}=\mathit{x},\mathit{Y},{\gamma }_j=1\right]$.

3.3. Results

Under baseline simulation conditions, the SNCM model achieved uniformly higher TPR with the MRF prior than with the independent Bernoulli priors within all four relationship-strength clusters (fig. 3). The improvement was relatively small for weakly connected predictor clusters and became increasingly pronounced in the moderately, highly, and very highly connected clusters. For example, within the very highly connected cluster, the TPR for a predictor whose effect size was 0.4 increased from < 5% to nearly 60%. Variable-specific bias remained negligible and similar across variables and priors, and the rMSE was slightly lower for the MRF prior (Supplementary Table S3).

Fig. 3. Variable-specific true positive rate (TPR) among clusters of substantively connected variables in baseline simulations.

Each panel depicts TPR of five predictors with different effect sizes in a given cluster. Predictors in the same cluster are either weakly (0.07), moderately (0.19), highly (0.39), or very highly (0.54) connected through the dietary relationship matrix $\mathit{R}$. Lines indicate the TPR change when using the Markov random field (MRF) prior to exploit relationship information instead of using independent Bernoulli priors.

The above-described TPR gains were obtained without inflating the the FDR (table 1). Across all the settings, FDR remained within 0.01–0.03 for the MRF prior and 0.01–0.04 for the independent Bernoulli priors, close to the 5% nominal level. Even in more challenging settings—under higher variance, heavier censoring, increased skewness, correlated predictors, misspecified dependence, or misspecified errors—the MRF prior consistently outperformed the independent Bernoulli prior based on TPR while maintaining comparable FDRs. In contrast, treating all PMVs as technical or imputing them with the half-minimum rule led to substantially poorer TPR and FDR (Supplementary Table S2).

Table 1.

Overall true positive rate (TPR) and false discovery rate (FDR).

Data generation	TPR		FDR
	Independent	MRF	Independent	MRF
Baseline	0.53 (0.11)	0.75 (0.09)	0.01 (0.03)	0.01 (0.03)
High Variance	0.22 (0.10)	0.39 (0.15)	0.04 (0.11)	0.03 (0.09)
High Censoring	0.36 (0.12)	0.60 (0.14)	0.02 (0.06)	0.02 (0.05)
High Skewness	0.54 (0.11)	0.77 (0.09)	0.01 (0.03)	0.01 (0.03)
Misspecified R	0.53 (0.11)	0.56 (0.11)	0.01 (0.03)	0.02 (0.04)
Log-normal Errors	0.55 (0.11)	0.77 (0.09)	0.01 (0.03)	0.01 (0.03)
Correlated Predictors	0.38 (0.11)	0.60 (0.14)	0.02 (0.05)	0.02 (0.05)
Large n	0.87 (0.06)	0.95 (0.04)	0.00 (0.01)	0.01 (0.02)

This table displays the average (standard deviation) overall TPR and FDR when using independent Bernoulli priors for variable selection or the Markov random field (MRF) prior.

4. Analysis of the MBS and MLVS

4.1. Data

HPFS is an ongoing prospective cohort study that began in 1986 and includes 51,529 US male health professionals (Li and others, 2022c, 2021). NHSII is an ongoing prospective cohort study that began in 1989 and includes 116,429 female registered nurses (Huang and others, 2019; Ke and others, 2023). In both studies, biennial measurements are collected on lifestyle factors such as diet, exercise, sleep, physical and mental health, and medication use (Rimm and others, 1991; Bao and others, 2016). The present analysis centers on the substudies MLVS and MBS. Briefly, MLVS ran from 2011 to 2013 and included 700 randomly selected participants from HPFS aged 52–81. MBS was a 2013–2014 substudy of NHSII that included 233 randomly selected women aged 49–67 (Huang and others, 2019). Participants of either substudy were free of coronary heart disease, stroke, cancer, and major neurological disease at the time of enrollment. Both substudies involved the collection of two blood samples and two pairs of stool samples at six-month intervals for the purpose of metabolome and microbiome profiling. For consistency with recent analyses on the relationship between metabolites and microbiome composition (Li and others, 2022b,c), we restricted analysis to the 203 participants in MBS and 287 participants in MLVS for whom microbiome and metabolite data are both available. Further details on metabolite measurement are given by Manghi and others (2024).

We quantified diet based on a validated food frequency questionnaire (Willett and others, 1985; Gu and others, 2024; Yuan and others, 2018) that measured how often respondents had eaten specific foods over the previous 12 months, from “never or less than once/month” to “⩾ 6 times/day.” The questionnaire contained > 130 items and was administered every 4 years beginning in 1986 for HPFS and in 1991 for NHSII. To summarize each person’s long-term consumption habits for each survey item, we averaged over the survey responses that preceded blood sample collection. Finally, we aggregated the responses into 30 variables representing the average daily servings of specific foods types, listed in section 4.2.

4.2. Analysis

We analyzed the 244 metabolites with < 80% PMVs in both cohorts. For each metabolite in each cohort, we fit four regression models that assumed normal or skew-normal errors and used either independent Bernoulli priors or the MRF prior. All models adjusted for age as a confounding variable; continuous variables were $z$-standardized. To place metabolites on a comparable scale, we temporarily replaced PMVs with half-minimum values, standardized, and re-introduced the PMVs into the data before analysis. For the metabolite cortisone, which exhibited poor MCMC convergence in the MLVS because it had only one PMV and an extreme outlier, analysis was based on the imputed data.

4.2.1. Hyperparameters and implementation

We chose $\psi$, ${\rho }_0$, and ${\rho }_1$ on a metabolite-specific basis by the approach described in section 3.2. We set ${\nu }^2$ to the empirical variance of the single-predictor regression coefficients, a common practice in Bayesian variable selection (Li and Zhang, 2010). These models were adjusted for the same confounding variables as in the main analyses and used the normalized, imputed metabolites as responses. We used standard weakly informative priors for the intercept, skewness, and confounder effects, setting ${\nu }_0^2={\nu }_d^2={\lambda }_1^2={\lambda }_2^2={\lambda }_3^2={10}^2$ (Wadsworth and others, 2017). We set $\left({\xi }_0,{\sigma }_0^2\right)=(\mathrm{3,1})$ so that ${\sigma }^{-2}$, the precision, would have prior mean 1 and variance 1.5. We let $\omega =\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}(0.05)$ to promote sparsity, and we searched for $\eta$ over the range 0.01, 0.02, . . . , 1.00.

For the MCMC, we drew 4 chains of 50,000 posterior samples following a burn-in period of 20,000 samples. We thinned the samples by 95% to reduce autocorrelation and pooled the results across chains. Convergence was evaluated by visually comparing the trace plots and posterior means among chains and by calculating the $\hat{R}$ and bulk and tail effective sample size (ESS) (Gelman and others, 2020). To account for testing associations in multiple models, we declared signals to be significant if their PIP exceeded a 5% Bayesian FDR threshold over all combinations of metabolites and food intake variables.

4.2.2. Informative variable selection prior

To construct $\mathit{R}$, we used previously defined groupings based on the nutritional and biological relationships among food items. Satija and others (2016) classified the food items in HPFS and NHSII as healthy plant-based, unhealthy plant-based, or animal-based foods, given their established associations with risk of type II diabetes and related health outcomes. Within these broad classes, we defined subcategories for fruits, vegetables, and healthy beverages (within healthy plant-based), unhealthy beverages (within unhealthy plant-based), and dairy and meat/fish/poultry (within animal-based). We derived $\mathit{R}$ from this structure using the approach described in section 3.1 (fig. 4).

Fig. 4. Dietary relationship matrix ($\mathit{R}$).

The figure depicts empirical pairwise relationships among the 30 dietary intake variables considered for variable selection. Variables were grouped hierarchically based on their nutritional properties. $\mathit{R}$ was incorporated into the variable selection scheme via the Markov random field prior.

4.2.3. Model Validation

We evaluated the quality of the model fit in two complementary ways. First, for the three metabolites with the strongest signals, we visually inspected the posterior predictive distribution of $Y_i$ by overlaying it on the empirical distribution. Second, for all metabolites, we compared the normal and skew-normal error models by estimating their expected log posterior predictive density (ELPD), which quantifies the out-of-sample predictive ability (Gelman and others, 2020). Models with higher ELPD values capture the true distribution of the outcome more accurately. The ELPD can be estimated within-sample by using importance sampling (IS) or the Watanabe-Akaike Information Criterion (WAIC), based on the log-likelihood of posterior draws (Gelfand and others, 1992; Watanabe, 2010). For each combination of cohort, error distribution, and prior specification, we summed log-likelihoods across individual metabolite models to compute a single multivariate log-likelihood. We used this procedure to estimate the ELPD using both IS and WAIC, summarizing the quality of fit over all metabolites.

4.3. Results

When implementing the SNCM model with an MRF prior, we identified 35 food-metabolite associations in the MBS and 45 in the MLVS (fig. 5). The signals comprised 60 unique metabolites and 13 unique foods. The most commonly selected foods were processed red meat (29 signals), fish (13), nuts (9), and coffee (8). Only 14 associations (17.5%) were negative—including every association for nut consumption, compared with 66 positive (82.5%) associations. Only 5 of the 80 signals appeared in both cohorts, 3 involving coffee and 2 involving unprocessed red meat. The processed red meat variable, which composed the bulk of signals in the MBS, produced none in the MLVS.

Fig. 5. Metabolite-dietary associations identified by SNCM model.

The figure depicts the associations detected in MBS or MLVS when fitting the skew-normal censored mixture (SNCM) model with a Markov random field prior (MRF). Associations with a black dot were not identified when using independent Bernoulli priors, only the MRF prior. Coefficient estimates are on the scale of 1 standard deviation of the response and predictor.

The MRF-based analysis detected 11 associations—4 in the MBS and 7 in the MLVS—that were not found when using independent Bernoulli priors. In the MBS, the additional signals were between processed red meat and the metabolites lysine, TG(48:2), DG(36:1), and TG(43:2). In the MLVS, they included an array of signals involving nuts, juice, carotenoid vegetables, unprocessed red meat, and non-fermented dairy products. No signals in either cohort were detected only with the independent prior and not the MRF prior.

In terms of MCMC convergence, all parameters achieved satisfactory diagnostics for every model, with $\hat{R}$ below 1.05 and bulk and tail ESS above 100.

4.3.1. Validation

Across all metabolites and both cohorts, models with skew-normal errors offered better fit than their normal-error counterparts as assessed by ELPD (table 2). Using the MRF prior instead of the independent prior resulted in improved model fit, though the gain was less pronounced than the improvement achieved by switching from a normal to a skew-normal error model.

Table 2.

Comparison of model fit by estimated expected log posterior density (ELPD).

Cohort	Selection Prior	${\text{EL}\widehat{\text{P}}\text{D}}_{\text{IS}}$		${\text{EL}\widehat{\text{P}}\text{D}}_{\text{WAIC}}$
		Normal	Skew-normal	Normal	Skew-normal
MBS	Independent	−70345.61	−64296.55	−70717.85	−64486.52
MBS	MRF	−70291.39	−64281.21	−70704.21	−64455.42
MLVS	Independent	−99430.68	−89402.57	−99964.20	−89632.53
MLVS	MRF	−99327.46	−89341.04	−99888.23	−89576.83

Each column displays the estimated ELPD for the normal or skew-normal censored mixture model. Datasets were from the Mind Body Study (MBS) or Men’s Lifestyle Validation Study (MLVS). Models used independent Bernoulli priors or the Markov random field (prior) for variable selection. We estimated ELPD using importance sampling (IS) and the Watanabe-Akaike information criterion (WAIC). Higher values indicate better model fit.

TG(49:2), TG(49:3), and TG(50:2; 16:0_16:0_18:2) were the three metabolites associated with the largest number of food items. Based on the posterior predictive distributions of these metabolites among MRF-based models, the skew-normal model (orange curves) more closely matched the empirical density than the normal model (purple curves) in most cases (fig. 6). While the orange curves tended to be centered properly around the peak of the empirical density, the purple curves overestimated the true mode of the density due to skewness, demonstrating the skew-normal model’s superior fit. These patterns were subtle in some cases but were pronounced for TG(49:2).

Fig. 6. Posterior predictive distributions from the normal and skew-normal models.

The shaded region depicts the empirical density of a metabolite in either the Mind Body Study (MBS) or Men’s Lifestyle Validation Study (MLVS). For models assuming either a normal (purple) or skew-normal (orange) error term, we show 8 densities sampled from the joint posterior distribution of the model parameters, approximating the posterior predictive distribution.

5. Discussion

To characterize the metabolomic signatures of 30 food intake variables in two cohorts of healthcare professionals, we developed a novel Bayesian variable selection framework that combines a SNCM model with a substantively informed MRF prior. A broad spectrum of simulation experiments demonstrated that the SNCM model effectively identifies true predictors of metabolite abundance in the presence of high PMV rates, heavy skewness, and inter-predictor correlations. Using the MRF prior to incorporate external information led to substantial improvements in overall TPRs without inflating FDRs, while greatly improving the individual TPR of predictors with weak signals. In simulations where the external knowledge was uninformative of the predictive variables, the MRF prior had little effect on TPR or FDR, suggesting there is no intrinsic penalty if $\mathit{R}$ is misspecified. In analyses of real data, the MRF-based SNCM model identified more dietary predictors than models using independent Bernoulli selection priors while offering superior model fit. The framework can implemented using the R package multimetab, available on GitHub (https://github.com/dclarkboucher/multimetab). Though the design was motivated by metabolomics data, the proposed method should prove useful for other omics response types, such as proteomics or viral load data, which share many of the same analytical challenges (Schwämmle and others, 2020; Dagne and Huang, 2013).

Several findings align with known metabolic pathways. Two established caffeine-derived metabolites—5-acetylamino-6-amino-3-methyluracil and 1,7-dimethyluric acid (Batista-da Silva and others, 2024)—and trigonelline, a metabolite found in green coffee beans (Nguyen and others, 2024), were positively associated with coffee intake in both cohorts. PC(40:6) and PE(P-40:6)/PE(0–40:7), which were associated with fish consumption in the MLVS, contain substituents such as docosahexaenoic acid (DHA) which is derived from fish oils (Vauzour and others, 2023). The amino acid homoarginine, which was negatively associated with unhealthy fruit juice consumption the MLVS, is positively associated with better cardiovascular health (Tsikas, 2023). In the MBS, the metabolites creatine, C3 carnatine, lysine, and leucine, all of which serve as potential red meat biomarkers (Guan and others, 2021; Li and others, 2022a; Cuparencu and others, 2019), were associated with processed red meat intake.

Eleven signals were not detected when using independent Bernoulli priors; based on the simulation results, it is plausible to speculate that the nutritionally informed model identified some true metabolomic signatures that were missed by the model with indpendent priors. For example, in the MLVS, MRF-based models selected fermented dairy intake for the metabolites C49:2 TG and C49:3; both triacylglycerols contain fatty acyl (pentadecanoic acyl chain, 15:0) chains that serve as potential dairy biomarkers (Abdullah and others, 2015). However, neither association was detected using independent Bernoulli priors.

Comparing the cohorts, we detected 45 metabolite-diet relationships in the MLVS and 35 in the MBS, with limited overlap. This difference may partly reflect cohort-specific demographic features, since the MLVS includes male participants aged 65–82 whereas the MBS includes female participants aged 49–66. It may also reflect lower statistical power in the MBS due to its smaller sample size (203 versus 287). Nevertheless, we observed consistent signals related to coffee intake in both cohorts, suggesting that certain diet–metabolite relationships may be more robust across populations. Future work should examine these associations in larger cohorts to better assess their reproducibility and cohort specificity.

A key strength of the proposed framework is its sensitivity and practical utility in settings with high-dimensional and correlated predictors, as evidenced by the simulation experiments with $p=300$. Such scenarios arise often in omic and multi-omic applications and pose substantial challenges for standard statistical methods. This work demonstrates the value of adopting sophisticated statistical frameworks—such as the MRF prior—that enhance signal detection by exploiting domain knowledge about the relationships under study.

The framework’s strong performance lays the foundation for statistical and applied extensions to more complex types of covariates, such as microbial taxa. Microbiome studies often involve hundreds of taxa simultaneously, with measurements that are noisy, zero-inflated, and compositional (Gloor and others, 2017). Although there are existing methods for microbiome covariates incorporating MRF priors to exploit taxonomic or phylogenetic structure, such approaches are not engineered for omics responses generated by mass-spectrometry workflows (Zhang and others, 2021). We therefore plan to extend the proposed framework to incorporate microbial covariates while explicitly accounting for these added features.

Another contribution is the newly described strategy for specifying the MRF prior’s hyperparameters, which is convenient and effective. Although this strategy lacks a rigorous theoretical basis—in common with most other techniques available—the simulation studies demonstrated strong performance across a range of assumptions and relationship structures, without sacrificing computational efficiency. Certain aspects of the strategy could be generalized to provide greater flexibility. For instance, instead of setting the prior inclusion probability of the reference prior to $2\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}{\mathrm{t}}^{-1}\left({\omega }_0\right)$, one could assign a separate value, ${\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}}^{-1}\left({\omega }_1\right),{\omega }_1>{\omega }_0$, such that $\mathrm{B}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{o}\mathrm{u}\mathrm{l}\mathrm{l}\mathrm{i}{(\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}}^{-1}\left({\omega }_1\right))$ is considered a reasonable independent prior. This approach may be preferable if a $2\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}{\mathrm{t}}^{-1}\left({\omega }_0\right)$ prior inclusion probability is considered too high.

Explicitly modeling the skewness of the metabolites via the skew-normal distribution circumvented the need for ad hoc logarithmic (or other) transformations and yielded uniformly better fit than a normal model. One could generalize this framework by instead adopting a skewed t model, potentially capturing skewed and heavy-tailed metabolite distributions even more effectively. The main drawback of this approach would be the additional sampling of the degrees of freedom parameter (Sahu and others, 2003).

A potential methodological limitation of the proposed framework is that it treats $\rho$, the biological PMV rate, as the same for all individuals. Some analysis methods for metabolite responses make the same simplifying assumption (Shah and others, 2019), whereas others allow $\rho$ to vary across subjects (Huang and others, 2020). An ambitious extension would be to model $\rho$ as a function of ${\mathit{C}}_i$ and ${\mathit{X}}_i$, inferring relationships between the biological PMV rate and the primary predictors. However, this approach could present substantial computational and identifiability challenges due to the complexity of performing variable selection in zero-inflated models—especially if the biological PMV rate is low (Lee and others, 2020). Similarly, another direction would be to replace the shared metabolite-specific skewness parameter with a hierarchical structure allowing $\delta$ and, potentially ${\sigma }^2$ to vary across subjects or subgroups, thereby accommodating heterogeneity in residual skewness and variance. Such extensions could provide better characterization of the outcome distribution and potentially yield higher statistical power.

Another methodological limitation is that the proposed framework analyzes the metabolites univariately, disregarding their potentially rich correlation structure. We are currently developing a multivariate architecture that can estimate and exploit such correlations more fully. Such an approach could provide greater statistical power through its enhanced precision, improved imputation accuracy, and better characterization of the joint outcome distribution.

When implementing variable selection, we adopted the spike-and-slab prior formulation by Kuo and Mallick (1998) instead of the more commonly used variance-mixture form, ${\beta }_j\sim N\left(0,{\gamma }_j{\nu }^2\right)$. Early versions of the proposed method implemented the latter formulation using stochastic search variable selection (SSVS), which requires a Metropolis-Hastings algorithm since Gibbs sampling is not feasible (George and McCulloch, 1997). However, the SSVS struggled in multiple facets of MCMC convergence: it required many more posterior samples to reach the target distribution, exhibited high autocorrelation between successive draws, and produced unstable posterior means of $\gamma$ across chains. While a rigorous comparison of these implementations is beyond the current scope, these observations suggest that for high-dimensional models with complex responses, researchers should carefully assess the stability of SSVS-style sampling schemes and consider alternatives when appropriate.

Identifying dietary predictors of metabolite abundance is essential for understanding the nutritional foundations of human health. Our work sheds light on such mechanisms by presenting a general, flexible framework for performing variable selection with metabolite responses. The framework was greatly enhanced by incorporating external nutritional information via the MRF prior, offering stronger performance in simulations and detecting more food-metabolite associations on the MLVS data compared with analyses using independent priors. The framework’s efficacy in high-dimensional covariate settings paves the way for multi-omics analyses that can contribute to an integrative understanding of biological pathways. As the breadth and complexity of omics-based applications continues to expand, statistical frameworks that can robustly handle complex types of data will be essential for making transformative biological insights.

Supplementary Material

Supplementary material is available online at http://biostatistics.oxfordjournals.org.