Fast autoregressive model for multivariate dependent outcomes with application to lipidomics analysis for Alzheimer’s disease and APOE-ε4

Abstract

Association analysis of multivariate omics outcomes is challenging due to the high dimensionality and inter-correlation among outcome variables. In practice, the classic multi-univariate analysis approaches are commonly employed, utilizing linear regression models for each individual outcome followed by adjustments for multiplicity through control of the false discovery rate (FDR) or family-wise error rate (FWER). While straightforward, these multi-univariate methods overlook dependencies between outcome variables. This oversight leads to less accurate statistical inferences, characterized by lower power and an increased false discovery rate, ultimately resulting in reduced replicability across studies. Recently, advanced frequentist and Bayesian methods have been developed to account for these dependencies. However, these methods often pose significant computational challenges for researchers in the field. To bridge this gap, a computationally efficient autoregressive multivariate regression model is proposed that explicitly accounts for the dependence structure among outcome variables. Through extensive simulations, it is demonstrated that the approach provides more accurate multivariate inferences than traditional methods and remains robust even under model misspecification. Additionally, the proposed method is applied to investigate whether the associations between serum lipidomics outcomes and Alzheimer’s disease differentiate in ε4 allele carriers and non-carriers of the apolipoprotein E (APOE) gene.

1. Introduction

Advancements in data acquisition technologies have made multivariate data increasingly prevalent. In line with this, analyzing multivariate response data is gaining popularity across various fields, including neuroimaging and omics data, as it enables comprehensive analyses that provide deeper insights into research questions compared to using univariate responses separately. One notable characteristic of high-dimensional response variables is that they often exhibit correlation due to their inherent interconnectedness and complex dependencies (Langfelder and Horvath, 2008). For example, in multi-omics research, it is common to encounter sets of co-regulated biological features—such as lipids, metabolites, and gene expression profiles—that exhibit synchronized behaviors governed by shared biological pathways or regulatory mechanisms. This co-regulation leads to correlated response variables (Hawe et al., 2019). Addressing these correlations in the model is crucial for enabling accurate data analysis by simultaneously increasing statistical power and reducing false positive findings (Leek and Storey, 2008). However, the popular practice of statistical analysis often overlooks these correlations to gain computational efficiency and simplicity, which can lead to biased estimates and misconclusions, reducing the reliability of the findings.

For example, the standard procedure for analyzing omics data typically involves i) initially applying statistical models to each response variable (e.g., differential expression analysis) and ii) subsequently performing multiple testing corrections to identify supra-threshold findings (Yamada et al., 2021). Commonly used multiple testing corrections for controlling the false discovery rate (FDR), such as the Benjamini–Hochberg (BH) procedure (Benjamini and Hochberg, 1995), assume that the test statistics or p-values are independent. However, test statistics for omics and imaging outcomes can be highly correlated and applying conventional correction methods by treating them as independent can lead to misleading conclusions, such as an overestimation of statistical significance (increasing false positives) or reduced statistical power (increasing false negatives) (Stevens et al., 2017). Since most omics studies have small to moderate sample sizes (e.g., a hundred or less Guo et al. (2021); Kirpich et al. (2018)), the corrected thresholds often pose the analysis (e.g., assessing moderation effects) underpowered and at the risk of a high false negative rate. Therefore, studying these types of data necessitates the utilization of methods that accommodate the dependencies among response variables to ensure high sensitivity, low false discovery rate, and improved reliability of statistical inferences.

In light of this, one can consider using the Mixed-effects Model for Repeated Measures (MMRM; Mallinckrodt et al., 2001), which can be implemented with various covariance structures for outcome variables, including unstructured covariance, compound symmetry, and Toeplitz. However, MMRM can pose a challenge for high-dimensional omics data as it involves estimating a large number of parameters, which can result in computational intractability and issues with model convergence. Particularly, in an MMRM with an unstructured covariance matrix, the number of parameters to be estimated is $q(q+1)∕2$, where $q$ resents the dimension of the outcome variable.

To address the issue of dependence among high-throughput outcomes, advanced statistical methods have been developed that model the covariance matrix by leveraging the spatial information of the data (Bowman, 2005; Bowman et al., 2008; Derado et al., 2010; Hyun et al., 2014; Zhang et al., 2016; Castruccio et al., 2018; Mejia et al., 2020). To overcome the difficulty of estimating an ultra-high number of the parameters in the covariance matrix, these models often leverage the biological prior knowledge, for example, biological pathways in omics data (Paczkowska et al., 2020) and spatial adjacency to inform the large covariance matrix parametrization. The commonly used assumption is that the outcomes within the (omics) pathway and imaging spatial neighborhood are more correlated. However, these assumptions may not fully capture the correlation patterns because features among pathways and spatial neighborhoods can be highly correlated whereas adjacent features can be less related. The implementation of these methods is computationally intensive, necessitating considerable resources, which can be a constraint in resource-limited scenarios.

In addition to statistical methods that account for dependence, there have been rapid advancements in methodologies for analyzing high-dimensional outcomes. However, though these advanced statistical methods each have their own pros and cons . For example, penalized multivariate regression techniques for multivariate responses (Chen et al., 2013; Wang et al., 2013) are effective for variable selection, but they often assume independence or impose simple correlation structures among outcomes, limiting their utility when complex dependencies are present. Latent factor-augmented regression models (Fan et al., 2024; Fan and and, 2024; Leek and Storey, 2008) can improve the modeling of hidden confounding and enhance inference, though they typically require computationally intensive procedures for estimating latent factors. Conditional Gaussian graphical models (CGGMs) (Yin and Li, 2011; Chen et al., 2016) provide interpretable network structures, but they often rely on complex optimization algorithms that can be computationally prohibitive in high-dimensional settings.

Beyond statistical methods, the past decade has witnessed increasing interest in deep learning algorithms for high-dimensional omics data analysis (e.g., Zou et al. (2019); Lopez et al. (2018); Lin et al. (2018); Tian et al. (2019)). While deep learning models often achieve superior predictive performance, they are generally limited in their ability to conduct population-level statistical inference for individual biomedical outcomes or to identify specific biological signals associated with health conditions under rigorous control of Type I and Type II error rates.

Therefore, despite these advancements, the classic approach (i.e., multi-univariate analysis with multiple testing corrections) remains the most commonly used strategy, even though it may lead to less accurate inference. To bridge the aforementioned methodological gaps, we develop a new fast autoregressive model for multivariate dependent outcomes (FARM). Built on the autoregressive framework, FARM utilizes data-driven approaches to specify the structure of covariance matrix rather than relying on prior knowledge. We further derive closed-form estimation of dependence parameters, resulting in a significant reduction in computational time and complexity compared to the commonly used Fisher’s scoring algorithm in autoregressive models (Derado et al., 2010). This makes FARM highly scalable and straightforwardly applicable to general high-throughput data when the covariance matrix presents a structured pattern. By accounting for the dependence patterns of the multivariate correlated response (Leek and Storey, 2008; Wu, 2021), FARM can yield more accurate statistical inferences than conventional methods (see Section 3). The improved true positive rate and lower false discovery rate by FARM can make the findings more reliable and replicable across labs, which is central for omics and imaging data analysis.

Motivation study.

This research is largely motivated by lipidomics findings from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) study (Barupal et al., 2018), which reported serum lipid levels for 514 unique lipid species measured in 806 ADNI1 participants. The cohort included 188 Alzheimer’s disease (AD) patients (66.1%), 392 individuals with mild cognitive impairment (53.3%), and 226 cognitively normal subjects (26.5%), with the percentages indicating the proportions of APOE-ε4 carriers within each group. Since lipid metabolisms are central to Alzheimer’s disease and are related to the apolipoprotein E (APOE) gene ε4 allele carrier status (Haney et al., 2024), it is both biologically and clinically important to understand the comprehensive lipid profile. We are interested in comparing the lipidomics profiles of patients with Alzheimer’s disease vs. healthy controls, and further testing whether the differential patterns vary across APOE genotypes. The genotype-specific disease effects on lipidomics may suggest precision treatment strategies tailored for individuals based on their genetic profiles

The rest of this paper is organized as follows. In Section 2, we describe FARM in detail and introduce the parameter estimation strategies. In Section 3, we perform extensive simulation analyses for model validation and comparison. In Section 4, we apply FARM to lipidomics data to investigate the differential Alzheimer’s disease impact on lipid levels for APOE-ε4 carriers and non-carriers. We conclude with discussions in Section 5.

2. Method

2.1. Background: a multivariate autoregressive model

Multivariate autoregressive models are commonly used to account for correlations among multiple response variables. Consider $n$ observations, indexed by $i=1,\cdots ,n$. Each observation has a $q$-dimensional response variable ${\mathbf{Y}}_i=({\mathbf{Y}}_{i1},\cdots ,{\mathbf{Y}}_{iq})$, and a $p$-dimensional predictor ${\mathbf{X}}_i=({\mathbf{X}}_{i1},\cdots ,{\mathbf{X}}_{ip})$. Then, for the $j$-th response variable, the autoregressive model is specified as:

$${\mathbf{Y}}_{ij}={\mathbf{X}}_i^{\mathsf{T}}{\beta }_j+\frac{1}{\mid {𝒩}_k\mid -1}\sum _{j^{\prime }\in {𝒩}_k,j^{\prime }\ne j}{\rho }_k({\mathbf{Y}}_{i{j}^{\prime }}-{\mathbf{X}}_i^{\mathsf{T}}{\beta }_{j^{\prime }})+{\epsilon }_{ij},$$ (1)

where ${\beta }_j\in {\mathbb{R}}^p$ is the coefficient vector, and the error ${\epsilon }_{ij}$ is assumed to be normally distribution, i.e., ${\epsilon }_{ij}\sim N(0,{\sigma }_j^2)$. The set ${𝒩}_k$ denotes the $k$-th block of variables, for $k=1,\dots ,K$, with size $\mid {𝒩}_k\mid$, which defines the neighbors of the $j$-th outcome variable. Specifically, the $j$-th outcome variable is included in ${𝒩}_k$ and is influenced by other outcomes, $j^{\prime }$, within the same block. The block-level dependence parameter ${\rho }_k$ captures the strength of the intra-block dependence.

The autoregressive model in (1) encompasses several extreme cases. For instance, when all outcome variables are assigned to a single neighborhood block ($K$ = 1), the model reduces to a global autoregressive structure with a single dependence parameter ${\rho }_1$. Conversely, when each outcome variable forms its own block ($K=q$), the model degenerates to standard multivariate linear regression, with no dependence modeled across outcomes.

While autoregressive models effectively address localized dependencies among response variables, estimating the dependence parameters, ${\rho }_k$ becomes computationally intensive (Derado et al., 2010) when applied to high-dimensional responses. This poses significant challenges for omics data analysis. To overcome this limitation, we propose a novel method that efficiently estimates ${\rho }_k$, thereby enhancing the practicality of autoregressive models in omics data analysis.

2.2. The FARM model

We base the FARM model on the conventional autoregressive model in (1) and develop a novel estimating procedure. To capture specific dependencies of outcome variables and to facilitate the modeling process, the original response variables ${\mathbf{Y}}_i={({\mathbf{Y}}_{i1},\cdots ,{\mathbf{Y}}_{iq})}^{\mathsf{T}}$ can be reordered using a mapping function, i.e., $\pi :\{1,\cdots ,q\}\to \{1,\cdots ,q\}$. This reordering results in the response variable ${\mathbf{Y}}_{ij}^{\pi }={\mathbf{Y}}_{i\pi (j)}$, which explicitly reveals the latent neighborhood structure. Specifically, reflecting the inherent modularity of omics data, the latent neighborhood pattern is assumed to have a block diagonal structure (Yamada et al., 2017; Devijver and Gallopin, 2018). Consequently, the mapping function $\pi$ is determined through a data-driven manner to uncover the neighborhood blocks ${𝒩}_k$, wherein correlations are predominantly concentrated within blocks and remain negligible between distinct blocks. In this paper, we employ the iterative adaptive dense-subnetwork extraction method (Wu et al., 2021) throughout all simulation studies and the real-data application to identify the underlying covariance structure, i.e., to determine the neighborhoods ${𝒩}_k$ and the total number of neighborhood blocks $K$. The $\mathtt{R}$ package for implementing the dense-subnetwork extraction method is available at https://github.com/xavienzo/ICONS. Other commonly used methods such as WSBM (weighted stochastic block model; Faskowitz et al., 2018) and WGCNA (weighted correlation network analysis; Langfelder and Horvath, 2008) generally yield similar network structures. To simplify notation and streamline subsequent model development, we slightly abuse notation by denoting the reordered response variables ${\mathbf{Y}}_{ij}^{\pi }$ as ${\mathbf{Y}}_{ij}$. Therefore, hereafter, we assume that the response variables are reordered according to their respective modules.

The FARM model is expressed as follows:

$$\begin{gathered} {\mathbf{Y}}_i={\mathbf{BX}}_i+\Gamma ({\mathbf{Y}}_i-{\mathbf{BX}}_i)+{\epsilon }_i, \\ {\mathbf{Y}}_i\sim \text{MVN}({\mathbf{BX}}_i,\Sigma ),\text{where}\Sigma =\underset{k=1}{\overset{K}{\oplus }}{\Sigma }_k=\text{diag}({\Sigma }_1,\cdots ,{\Sigma }_K), \end{gathered}$$ (2)

where $\mathbf{B}\in {\mathbb{R}}^{q\times p}$ is the regression coefficient matrix (i.e., ${\mathbf{B}}^{\mathsf{T}}=({\beta }_1,\cdots ,{\beta }_q)$), and $\Gamma \in {\mathbb{R}}^{q\times q}$ adjusts inter-variable dependencies within the block, whose ($j,j^{\prime }$)-th element is given by ${\Gamma }_{j{j}^{\prime }}={\rho }_k∕(\mid {𝒩}_k\mid -1)\mathbb{I}(j\ne j^{\prime })\mathbb{I}(j,j^{\prime }\in {𝒩}_k)$. The operator $\oplus$ denotes the direct sum of matrices, which forms a block diagonal structure in $\Sigma$. Note that the error terms ${\epsilon }_i$ follow a multivariate normal distribution, denoted as ${\epsilon }_i\sim \mathrm{MVN}(0,\mathrm{diag}({\sigma }_1^2,\cdots ,{\sigma }_q^2))$. The schematic illustration of the proposed FARM model is presented in Figure 1.

Figure 1:

Overview of FARM: an autoregressive multivariate regression model for correlated outcomes. The left heatmap demonstrates the correlations among all omics outcomes. The middle heatmap is the re-ordered covariance matrix revealing the modularity. The dependence parameters of {${𝒩}_k$, ${\rho }_k$} in FARM account for the dependence patterns and modularity of the omics outcomes (i.e., via co-expression analysis). The right heatmap shows the covariance matrix of the residuals of the FARM model. Since the second term of the FARM model takes into account most of the inter-correlations among omics variables, the right heatmap represents a covariance matrix which close to a diagonal matrix.

Reparameterizing ${\rho }_k$ by $\Sigma$.

As the link between ${\rho }_k$ and $\Sigma$ is implicit, we reparameterize ${\rho }_k$ by $\Sigma$ to facilitate likelihood-based parameter estimation. To do this, we further assume that each block ${\Sigma }_k$ follows a compound symmetry structure, where all variables within a block have identical variances and every pair of distinct variables within the same block shares a uniform covariance (Waller and Gotway, 2004; Bowman, 2005). Compound symmetry covariance structures are widely used in multivariate analyses, including neuroimaging (Bowman et al., 2008; Derado et al., 2010), omics data (Yang et al., 2024), and repeated measures designs such as mixed models for repeated measures, as they simplify modeling by reducing the number of parameters to estimate. Throughout the rest of this paper, we will refer to the covariance structure of $\Sigma$ as block-diagonal with compound symmetry (BDCS).

The BDCS covariance structure is well-suited for omics data applications, where genes or molecular entities within the same functional module are typically co-regulated, resulting in similar expression patterns, thereby rendering the compound symmetry (CS) assumption plausible. Additionally, assuming uniform covariance across all gene pairs within a block reduces the number of parameters, thereby simplifying the model—a particularly advantageous feature for high-dimensional omics datasets.

Next, we express ${\rho }_k$ using parameters in ${\Sigma }_k$ with a BDCS structure. Specifically, we have the equation of each diagonal block ${\Sigma }_k$ (for the neighborhood ${𝒩}_k$) and ${\rho }_k$ in (1) as

$${\Sigma }_k={({\mathbf{I}}_{\mid {𝒩}_k\mid }-{\rho }_k{\mathbf{W}}_k)}^{-1}{\mathbf{M}}_k{({\mathbf{I}}_{\mid {𝒩}_k\mid }-{\rho }_k{\mathbf{W}}_k)}^{-1},$$ (3)

where ${\mathbf{W}}_k=-({\mathbf{I}}_{\mid {𝒩}_k\mid }-{\mathbf{J}}_{\mid {𝒩}_k\mid })∕(\mid {𝒩}_k\mid -1)$, ${\mathbf{M}}_k=\mathrm{diag}({\sigma }_1^2,\cdots ,{\sigma }_{\mid {𝒩}_k\mid }^2),{\mathbf{I}}_{\mid {𝒩}_k\mid }$ is an identity matrix, and ${\mathbf{J}}_{\mid {𝒩}_k\mid }$ is a matrix of ones. By using the properties of the compound symmetry matrix, ${\Sigma }_k$ can be further simplified to

$${\Sigma }_k={(1-c_k)}^{-2}({\mathbf{I}}_{\mid {𝒩}_k\mid }+d_k{\mathbf{J}}_{\mid {𝒩}_k\mid }){\mathbf{M}}_k,$$ (4)

where

$$\begin{gathered} c_k=-\frac{{\rho }_k}{\mid {𝒩}_k\mid -1} \\ {d}_k=\frac{c_k}{1+(\mid {𝒩}_k\mid -1)c_k}\left(\frac{\mid {𝒩}_k\mid c_k}{1+(\mid {𝒩}_k\mid -1)c_k}-2\right). \end{gathered}$$

The detailed derivation of (4) is provided in Appendix A.

We further assume that errors belonging to the same neighbor have a common variance, i.e., ${\sigma }_1^2=\cdots ={\sigma }_{\mid {𝒩}_k\mid }^2={\xi }_k^2$. That gives ${\Sigma }_k(j,j^{\prime })=(d_k+\mathbb{I}\{j=j^{\prime }\}){\xi }_k^2∕{(1-c_k)}^2$, where $\mathbb{I}\{\cdot \}$ is an indicator function. In addition, since the correlation matrix can be obtained from the covariance matrix, i.e., ${\mathbf{R}}_k=\mathrm{diag}{({\Sigma }_k)}^{-1∕2}{\Sigma }_k\mathrm{diag}{({\Sigma }_k)}^{-1∕2}$, it can be expressed as ${\mathbf{R}}_k=(1-r_k){\mathbf{I}}_{\mid {𝒩}_k\mid }+r_k{\mathbf{J}}_{\mid {𝒩}_k\mid }$, where $r_k=r_{j,j^{\prime }\in {𝒩}_k}=d_k∕(1+d_k)$ is the correlation parameter within the neighborhood. We then have the following proposition, which indicates that the dependence parameter ${\rho }_k$ in the autoregressive model can be reparameterized by the correlation parameter $r_k$ in ${\Sigma }_k$.

Proposition 1. The dependence parameter ${\rho }_k$ can be written as the function of the correlation parameter $r_k$ and the size of ${𝒩}_k$,

$${\rho }_k=\frac{(1-\mid {𝒩}_k\mid )\left(1-\sqrt{1+\mid {𝒩}_k\mid (\frac{r_k}{1-r_k})}\right)}{\mid {𝒩}_k\mid +(1-\mid {𝒩}_k\mid )\left(1-\sqrt{1+\mid {𝒩}_k\mid (\frac{r_k}{1-r_k})}\right)}.$$ (5)

The proof is provided in the Appendix B.

2.3. FARM model estimation

The parameters to be estimated in FARM include $\Theta =(\mathbf{B},\rho ,\xi )$, where $\rho =({\rho }_1,\cdots ,{\rho }_K)$ and $\xi =({\xi }_1,\cdots ,{\xi }_K)$. We use the maximum likelihood (ML) method based on the estimated ${\widehat{𝒩}}_k$. The log-likelihood is given by

$$\begin{gathered} ℒ(\Theta \mid \mathbf{Y})\propto \sum _{i=1}^n\sum _{k=1}^K-\frac{1}{2}\text{ln}\mid (\text{diag}{({\Sigma }_k)}^{1∕2}{\mathbf{R}}_k\text{diag}{({\Sigma }_k)}^{1∕2}\mid \\ -\frac{1}{2}{({\mathbf{Y}}_{i{𝒩}_k}-{\mathbf{B}}_k{\mathbf{X}}_i)}^{\mathsf{T}}{\left(\text{diag}{({\Sigma }_k)}^{1∕2}{\mathbf{R}}_k\text{diag}{({\Sigma }_k)}^{1∕2}\right)}^{-1}({\mathbf{Y}}_{i{𝒩}_k}-{\mathbf{B}}_k{\mathbf{X}}_i), \end{gathered}$$ (6)

where ${\mathbf{Y}}_{i{𝒩}_k}\in {\mathbb{R}}^{\mid {𝒩}_k\mid }$ is the collection of outcome variables in the $k$-th neighborhood and ${\mathbf{B}}_k\in {\mathbb{R}}^{\mid {𝒩}_k\mid \times p}$ is the corresponding regression coefficient matrix. It is well established that the whole regression coefficients matrix $\widehat{\mathbf{B}}\in {\mathbb{R}}^{q\times p}$ can be estimated without bias through ordinary least squares (OLS) and are independent of the covariance parameters. Thus, the estimation of the parametric covariance matrix ${\Sigma }_k$ is performed using the partially maximized likelihood function $ℒ(\rho ,\xi \mid \mathbf{Y},\widehat{\mathbf{B}})$ (Derado et al., 2010).

Conventionally, the dependence parameter ${\rho }_k$ in the autoregressive model is estimated by Fisher’s scoring algorithm, which optimizes the likelihood function in (6), as detailed in (Waller and Gotway, 2004; Bowman, 2005; Derado et al., 2010). However, the estimation procedure entails calculating the inverse of a large covariance matrix, which is computationally intensive. Such computational intensity may hinder subsequent inferential steps, such as bootstrapping for confidence interval estimation of ${\rho }_k$ and conducting permutation tests for multiple testing corrections. We introduce a closed-form estimate for the dependence parameter that accelerates the estimation process, rendering multivariate autoregressive models more practical for analyzing high-dimensional omics data.

Given that the variance and correlation parameters can be estimated separately (Barnard et al., 2000; Fan et al., 2016), and considering the straight-forward estimation of variance parameters, our primary focus lies on estimating the off-diagonal correlation parameters $r_k$ for each neighborhood ${𝒩}_k$. Let ${\widehat{\Sigma }}_k=n^{-1}{({\mathbf{Y}}_{{𝒩}_k}-\mathbf{X}{\widehat{\mathbf{B}}}_k^{\mathsf{T}})}^{\mathsf{T}}({\mathbf{Y}}_{{𝒩}_k}-\mathbf{X}{\widehat{\mathbf{B}}}_k)$, where ${\mathbf{Y}}_{{𝒩}_k}^{\mathsf{T}}=({\mathbf{Y}}_{1{𝒩}_k},\cdots ,{\mathbf{Y}}_{n{𝒩}_k})\in {\mathbb{R}}^{\mid {𝒩}_k\mid \times n}$ and ${\mathbf{X}}^{\mathsf{T}}=({\mathbf{X}}_1,\cdots ,{\mathbf{X}}_n)\in {\mathbb{R}}^{p\times n}$, be the sample covariance matrix for the neighborhood ${𝒩}_k$, and denote the sample correlation matrix by ${\mathbf{S}}_k=\mathrm{diag}{({\widehat{\Sigma }}_k)}^{-1∕2}{\widehat{\Sigma }}_k\mathrm{diag}{({\widehat{\Sigma }}_k)}^{-1∕2}$. Then, for the $k$-th neighborhood, we obtain the part of the likelihood function associated with the correlation matrix as

$$ℒ({\mathbf{R}}_k\mid {\mathbf{S}}_k)=-\log \text{det}({\mathbf{R}}_k)-\text{tr}({\mathbf{S}}_k{\mathbf{R}}_k^{-1}).$$ (7)

By using the properties of the compound symmetric matrix, the determinant and inverse of ${\mathbf{R}}_k$ can be expressed as:

$$\begin{gathered} \text{det}({\mathbf{R}}_k)={(1-r_k)}^{\mid {𝒩}_k\mid -1}(1+(\mid {𝒩}_k\mid -1)r_k) \\ {\mathbf{R}}_k^{-1}=\frac{1}{1-r_k}\left({\mathbf{I}}_{\mid {𝒩}_k\mid }-\frac{r_k}{1+(\mid {𝒩}_k\mid -1)r_k}{\mathbf{J}}_{\mid {𝒩}_k\mid }\right). \end{gathered}$$

As (7) can be expressed in terms of the off-diagonal correlation parameters $r_k$ in ${\mathbf{R}}_k$, the maximum likelihood estimate of $r_k$ can be obtained by solving ${\widehat{r}}_k={\mathrm{argmax}}_{r_k}ℒ(r_k\mid {\mathbf{S}}_k)$, where

$$ℒ(r_k\mid {\mathbf{S}}_k)=\log \left(\frac{{(1-r_k)}^{1-\mid {𝒩}_k\mid }}{1+(\mid {𝒩}_k\mid -1)r_k}\right)-\frac{1}{1-r_k}\left(\text{trace}({\mathbf{S}}_k)-r_k\frac{\text{trace}({\mathbf{S}}_k{\mathbf{J}}_{\mid {𝒩}_k\mid })}{1+(\mid {𝒩}_k\mid -1)r_k}\right).$$ (8)

We begin by computing the first derivative of the likelihood function with respect to $r_k$,

$$\frac{\partial }{\partial r_k}ℒ(r_k\mid {\mathbf{S}}_k)=\frac{1+(\mid {𝒩}_k\mid -1)r_k^2}{{(1-r_k)}^2(1+(\mid {𝒩}_k\mid -1)r_k)}\left(\frac{\text{trace}({\mathbf{S}}_k{\mathbf{J}}_{\mid {𝒩}_k\mid })}{(1+(\mid {𝒩}_k\mid -1)r_k)}-\mid {𝒩}_k\mid \right).$$

By setting the first derivative to zero, i.e., $\partial ∕\partial r_kℒ(r_k\mid {\mathbf{S}}_k)=0$, we obtain the solution,

$${\widehat{r}}_k=\frac{\text{trace}({\mathbf{S}}_k{\mathbf{J}}_{\mid {𝒩}_k\mid })-\mid {𝒩}_k\mid }{\mid {𝒩}_k\mid (\mid {𝒩}_k\mid -1)}.$$ (9)

Thus, for the $k$-th neighborhood, we derive the closed-form estimate for the dependence parameter ${\widehat{\rho }}_k$ by plugging ${\widehat{r}}_k$ into (5) in Proposition 1. Overall, because the FARM procedure for obtaining the dependence parameter ${\rho }_k$ does not require an iterative algorithm, it significantly improves computational efficiency.

2.4. Inference

The dependence among the multivariate outcomes and their test statistics can influence both the false discovery rate and the true positive rate in multiple testing (Leek and Storey, 2008; Efron, 2007; Fan et al., 2012; Fan and Han, 2017). The impact of dependence on multiple testing corrections can be more pronounced when the correlations among the multivariate outcomes are strong. Applying our method, we first obtain individual test statistics across the whole response variable while accounting for the correlation patterns. Specifically, the test statistic for the $j$-th omics outcome and the $l$-th predictor is given by $t_{jl}={\widehat{\beta }}_{jl}∕\mathrm{se}({\widehat{\beta }}_{jl})$, where $\mathrm{se}({\widehat{\beta }}_{jl})=\sqrt{{\widehat{\sigma }}_j^2{({\mathbf{X}}^{\mathsf{T}}\mathbf{X})}_{ll}^{-1}}$, and ${\widehat{\sigma }}_j^2={\sum }_{i=1}^n{\widehat{\epsilon }}_{ij}^2∕(n-p)$. We then adjust the resulting p-values using standard multiple-testing procedures—for example, the Benjamini–Hochberg false-discovery-rate (FDR) method—to control for multiplicity across all outcomes.

Because the revised residuals ${\widehat{\epsilon }}_{ij}$ in the autoregressive model are independent given the estimated ${\rho }_k$ and ${𝒩}_k$, the resulting tests and their corresponding p-values are also independent. This independence is critical for simultaneous multivariate inference, as the commonly used methods (e.g., FDR) typically assume independent test statistics (Benjamini and Hochberg, 1995; Fan and Han, 2017). As demonstrated in the following section, FARM has been shown to be effective in reducing FDR compared to OLS and other multivariate regression models.

3. Simulation

In this section, we conducted simulation studies to evaluate the performance of FARM. We simulated the multivariate correlated response variable ${\mathbf{Y}}_i\in {\mathbb{R}}^q$ for each subject $i$ from a multivariate normal distribution as follows:

$${\mathbf{Y}}_i\sim \text{MVN}({\mathbf{BX}}_i,\Sigma ),i=1,\cdots ,n,$$

where ${\mathbf{X}}_i\in {\mathbb{R}}^p$ denotes the clinical status and demographic variables of the $i$-th subject, and $\mathbf{B}\in {\mathbb{R}}^{q\times q}$ denotes the coefficient matrix. The covariance matrix $\Sigma$ was generated from a BDCS structure (i.e., $\Sigma ={\otimes }_{k=1}^K{\Sigma }_k$, where ${\Sigma }_k=\mathbf{I}(1-r_k)+\mathbf{J}{r}_k)$.

We generated data in different settings by varying the dimension of the response variable $q$ = {100, 200}, the number of subjects $n$ = {100, 200}, and the correlation parameter $r_k$ = {0.4, 0.8}. We simulated the data with two dense blocks, while all other blocks were singletons (i.e., $\mid {𝒩}_k\mid$ = 1 for $k$ > 2). Specifically, for the dimension of the response variable $q$, half were allocated to dense blocks and the rest to singletons. In the dense blocks, the ratio of the first to the second block was 4:1. Consequently, when $q$ = 100, $\mid N_1\mid$ = 40 and $\mid N_2\mid$ = 10; when $q$ = 200, $\mid N_1\mid$ = 80 and $\mid N_2\mid$ = 20. We generated the univariate covariate ${\mathbf{X}}_i$ from a standard normal distribution. We set the effect size of non-zero elements in $\mathbf{B}$ to be {0.1, 0.2}.

We set 95% of the elements within the dense blocks to be non-zero, while 5% of the elements outside the blocks are non-zero. This configuration reflects the biological phenomenon wherein regulatory interactions predominantly occur within tightly connected modules. Conversely, the sparse connectivity outside the blocks suggests that singleton response elements—those not belonging to any module—are minimally associated with covariates such as clinical status and demographics.

We considered SVA, OLS and MMRM with an unstructured covariance structure as competing methods. We assessed the performance of all methods by measuring their FDR, sensitivity, and specificity at a nominal level of $\alpha$ = 0.05, as well as the area under the ROC curve (AUC). While the importance of each measure can vary depending on the research question and specific goals of the analysis, FDR holds particular importance in our study type. In situations where ensuring the validity of the findings and preventing spurious results are essential, FDR allows scientists to establish a threshold for the maximum acceptable rate of false positives. A lower FDR indicates that the method produces fewer false positives, which can be a marker of increased precision. We also considered AUC as a metric to assess the impact of dependence adjustment in multivariate regression on the multiple testing correction. Note that unlike sensitivity, and specificity, AUC is invariant to the choice of the cut-off value of the multiple testing correction method (e.g., $q$ values for FDR). Therefore, if a statistical method provides more accurate inference, the ranking of statistical significance levels (e.g., absolute values of test statistics and $p$ values) will align more closely with the truth, yielding a higher AUC value. Statistical methods that yield multivariate inference with higher AUC values are generally superior because a more accurate ranking of multivariate features can uniformly lead to better performance in large-scale multiple testing.

The results for $n$ = 100 are summarized in Table 1, with the setting $n$ = 200 deferred to Supplementary Material. ROC curves under $n$ = 100 with different effect sizes are also shown in Figure 2. FARM consistently outperforms competing methods across most evaluated metrics (3 out of 4) and all settings. The exception is specificity, where the difference between FARM and other competing methods is only marginal. FARM achieves a balance between sensitivity and specificity, resulting in the best AUC, which enables the drawing of more accurate and reliable scientific conclusions from statistical analyses, thereby enhancing replicability. Conversely, OLS and MMRM demonstrate high specificity, yet they exhibit poor performance in both sensitivity and FDR. This pronounced difference between high specificity and low sensitivity is due to the tests’ low rejection power. Specifically, the test that is prone to failing to reject the null hypothesis leads to an increased incidence of false negatives, thereby diminishing sensitivity, while specificity remains elevated as the rate of true negatives is largely unaffected. These lower-powered methods pose significant challenges in scientific research due to their susceptibility to Type II errors, which can cause true effects or associations to remain undetected. (i.e., missed discoveries). Furthermore, it struggles to replicate findings, thus contributing to the reproducibility crisis across various scientific fields.

Table 1:

Simulation results ($n$ = 100). The accuracy of multivariate inference is measured by FDR, sensitivity, specificity, and AUC. Each criterion is averaged over 100 replications, with standard errors provided in parentheses. The best-ranked method for each criterion is given in bold.

$\beta$	$q$	$r_k$	Method	FDR	Sensitivity	Specificity	AUC
0.1	100	0.4	FARM	0.228 (0.16)	0.258 (0.16)	0.940 (0.04)	0.599 (0.08)
			SVA	0.523 (0.24)	0.059 (0.04)	0.936 (0.04)	0.498 (0.03)
			OLS	0.358 (0.26)	0.141 (0.12)	0.945 (0.03)	0.543 (0.06)
			MMRM	0.357 (0.25)	0.145 (0.12)	0.942 (0.03)	0.544 (0.06)
		0.8	FARM	0.170 (0.16)	0.507 (0.27)	0.929 (0.04)	0.718 (0.14)
			SVA	0.539 (0.23)	0.090 (0.11)	0.919 (0.04)	0.505 (0.06)
			OLS	0.501 (0.35)	0.140 (0.19)	0.945 (0.03)	0.542 (0.09)
			MMRM	0.505 (0.35)	0.143 (0.19)	0.942 (0.03)	0.542 (0.10)
	200	0.4	FARM	0.177 (0.13)	0.318 (0.20)	0.951 (0.02)	0.634 (0.10)
			SVA	0.469 (0.16)	0.056 (0.02)	0.950 (0.02)	0.503 (0.02)
			OLS	0.274 (0.23)	0.199 (0.17)	0.955 (0.02)	0.577 (0.08)
			MMRM	0.273 (0.21)	0.205 (0.17)	0.952 (0.02)	0.578 (0.09)
		0.8	FARM	0.149 (0.16)	0.568 (0.31)	0.938 (0.03)	0.753 (0.16)
			SVA	0.463 (0.19)	0.108 (0.14)	0.934 (0.03)	0.521 (0.07)
			OLS	0.393 (0.32)	0.205 (0.26)	0.953 (0.02)	0.579 (0.13)
			MMRM	0.393 (0.32)	0.211 (0.27)	0.951 (0.02)	0.581 (0.13)
0.2	100	0.4	FARM	0.095 (0.07)	0.641 (0.19)	0.940 (0.04)	0.790 (0.10)
			SVA	0.395 (0.22)	0.161 (0.14)	0.922 (0.04)	0.541 (0.08)
			OLS	0.118 (0.10)	0.493 (0.19)	0.945 (0.03)	0.719 (0.10)
			MMRM	0.123 (0.10)	0.501 (0.19)	0.942 (0.03)	0.721 (0.10)
		0.8	FARM	0.090 (0.07)	0.824 (0.20)	0.929 (0.04)	0.876 (0.11)
			SVA	0.282 (0.24)	0.416 (0.33)	0.918 (0.04)	0.667 (0.17)
			OLS	0.158 (0.16)	0.475 (0.29)	0.945 (0.03)	0.710 (0.15)
			MMRM	0.162 (0.17)	0.484 (0.29)	0.942 (0.03)	0.713 (0.15)
	200	0.4	FARM	0.074 (0.04)	0.677 (0.20)	0.951 (0.02)	0.814 (0.10)
			SVA	0.385 (0.17)	0.138 (0.14)	0.934 (0.03)	0.536 (0.07)
			OLS	0.092 (0.07)	0.534 (0.22)	0.955 (0.02)	0.744 (0.11)
			MMRM	0.094 (0.06)	0.541 (0.22)	0.952 (0.02)	0.747 (0.11)
		0.8	FARM	0.075 (0.05)	0.851 (0.19)	0.938 (0.03)	0.895 (0.10)
			SVA	0.266 (0.23)	0.448 (0.36)	0.925 (0.03)	0.687 (0.19)
			OLS	0.140 (0.16)	0.533 (0.33)	0.953 (0.02)	0.743 (0.17)
			MMRM	0.138 (0.15)	0.540 (0.33)	0.951 (0.02)	0.746 (0.16)

Figure 2:

The figure displays ROC curves that depict the model’s performance under various effect sizes ($\beta$ = {0.1, 0.2, 0.3}) and correlations ($r$ = {0.4, 0.8}) among responses, with $q$ = 100, and $\mid {𝒩}_1\mid$ = 40, $\mid {𝒩}_2\mid$ = 10. The curves were generated by varying the cutoff values for the p-values over a range of values.

Computational Time Comparison.

To assess the computational efficiency of the proposed FARM, we conducted a simulation study comparing its runtime to that of other competing methods. In this simulation study, we first varied the sample size $n$ from 100 to 500 in increments of 50 while keeping the number of outcome variables fixed at $q$ = 100. Second, we varied the dimension of the outcome variables $q$ from 100 to 500 while fixing the sample size at $n$ = 500. For each setting, computation times were averaged over 20 replications. The results are shown in Figure 3, demonstrating that FARM is computationally fast, with average computation time not significantly deviating from that of OLS. In contrast, MMRM exhibits the worst computational performance, and the computation time for SVA increases rapidly with larger sample sizes.

Figure 3:

Comparison of computation time (in seconds), averaged over 20 replications. Left panel: Sample size ($n$) is varied from 100 to 500 with a fixed outcome dimension ($q$ = 100). Right panel: Outcome dimension ($q$) is varied from 100 to 500 with a fixed sample size ($n$ = 500).

Robustness to model misspecification.

In practice, the actual covariance structure may deviate from the assumed BDCS structure, rendering this assumption overly restrictive and potentially leading to model misspecification issues. To evaluate FARM’s performance when this assumption is violated, we conducted additional simulation studies. Specifically, we generated data from three different underlying correlation structures. First, we considered a Toeplitz correlation structure, where off-diagonal entries follow a first-order autoregressive pattern. This setting simulates scenarios commonly observed in spatial and time series data, where outcomes are more strongly correlated with spatially or temporally proximate outcomes and the correlation gradually decreases as spatial or temporal distance increases. Second, we explored an inter-correlated block structure characterized by negative correlations between blocks, a scenario often observed in biological data such as gene expression. Third, we examined an unstructured correlation within the blocks, along with some correlations between elements outside the blocks. This configuration mimics situations where relationships between variables are not uniformly patterned or predictable, exhibiting irregular interdependencies. The heatmaps of the correlation matrices used in this simulation study are displayed in Figure C.6 in Appendix C. The performance of FARM and comparable methods under these three scenarios is summarized in Table 2. Overall, FARM demonstrates robust performance by consistently outperforming other competing methods across three measures out of four, with the only exception being specificity.

Table 2:

The results ($n$ = 100) of the simulation study under model misspecification settings. Each criterion is averaged over 100 replications, with standard errors provided in parentheses. The best-ranked method for each criterion is given in bold. Scenarios 1, 2, and 3 represent the Toeplitz (AR), inter-correlated block, and unstructured block within the correlation matrix, respectively.

Scenario	$r_k$	Method	FDR	Sensitivity	Specificity	AUC
Scenario 1	0.4	FARM	0.100 (0.06)	0.546 (0.15)	0.943 (0.03)	0.744 (0.08)
		SVA	0.216 (0.15)	0.276 (0.15)	0.938 (0.03)	0.607 (0.08)
		OLS	0.108 (0.07)	0.488 (0.15)	0.945 (0.03)	0.716 (0.08)
		MMRM	0.112 (0.07)	0.498 (0.15)	0.942 (0.03)	0.720 (0.08)
	0.8	FARM	0.096 (0.09)	0.728 (0.21)	0.935 (0.04)	0.832 (0.11)
		SVA	0.323 (0.21)	0.283 (0.25)	0.916 (0.04)	0.600 (0.13)
		OLS	0.130 (0.12)	0.480 (0.23)	0.945 (0.03)	0.712 (0.12)
		MMRM	0.134 (0.12)	0.486 (0.23)	0.942 (0.03)	0.714 (0.12)
Scenario 2	0.4	FARM	0.090 (0.06)	0.646 (0.16)	0.940 (0.04)	0.793 (0.08)
		SVA	0.208 (0.12)	0.372 (0.20)	0.916 (0.04)	0.644 (0.11)
		OLS	0.106 (0.07)	0.500 (0.15)	0.944 (0.03)	0.722 (0.08)
		MMRM	0.111 (0.07)	0.508 (0.15)	0.942 (0.03)	0.725 (0.08)
	0.8	FARM	0.090 (0.07)	0.828 (0.19)	0.928 (0.04)	0.878 (0.10)
		SVA	0.202 (0.18)	0.491 (0.30)	0.922 (0.04)	0.706 (0.16)
		OLS	0.133 (0.11)	0.481 (0.26)	0.945 (0.03)	0.713 (0.13)
		MMRM	0.136 (0.11)	0.491 (0.26)	0.942 (0.03)	0.716 (0.13)
Scenario 3	0.4	FARM	0.101 (0.08)	0.613 (0.20)	0.940 (0.04)	0.776 (0.10)
		SVA	0.350 (0.20)	0.192 (0.15)	0.922 (0.04)	0.557 (0.08)
		OLS	0.123 (0.11)	0.479 (0.21)	0.946 (0.03)	0.712 (0.10)
		MMRM	0.125 (0.11)	0.488 (0.21)	0.943 (0.03)	0.715 (0.11)
	0.8	FARM	0.109 (0.10)	0.633 (0.21)	0.934 (0.04)	0.783 (0.11)
		SVA	0.302 (0.20)	0.305 (0.23)	0.906 (0.05)	0.606 (0.12)
		OLS	0.144 (0.16)	0.466 (0.24)	0.944 (0.04)	0.705 (0.12)
		MMRM	0.146 (0.16)	0.477 (0.24)	0.941 (0.04)	0.709 (0.12)

The robustness of the proposed FARM model under model misspecification—such as covariance structure or neighborhood misspecification—can be attributed to two key factors. i) Data-driven neighborhood identification via dense subgraph extraction ensures that variables within each block are strongly and uniformly correlated, thereby forming neighborhoods that closely approximate compound symmetry and reducing the risk of assumption violations. ii) The averaging effect of the autoregressive model applies a weighted smoothing over residuals within each neighborhood block via the dependence parameter ${\rho }_k$, which mitigates the impact of local misspecification in the correlation structure, even when compound symmetry is substantially violated.

We conclude this section by noting a potential limitation of the proposed FARM model. While FARM performs well even when the correlation structure deviates from the BDCS assumption, it may fail when additional correlations arise from unmeasured or unmodeled factors—such as technical artifacts (e.g., batch effects). Due to the lack of an explicit term to account for latent sources of variation that can introduce correlations among omics outcomes, the performance of autoregressive-type models may be suboptimal. Under these conditions, methods like SVA, which model unobserved variation through low-dimensional latent factors, typically perform better. We demonstrate this limitation through an additional simulation study presented in Supplementary Material.

4. Data Application

In this section, we applied the proposed method to lipidomics data for Alzheimer’s disease research. Lipids are a broad group of fatty compounds that play various roles in the body, including energy storage and hormone regulation. Impaired lipid metabolism is central to Alzheimer’s disease pathology, and linked to the genetic predisposition to APOE-ε4, one of the most significant genetic risk factors for Alzheimer’s disease (Chew et al., 2020; Haney et al., 2024). Over the past decades, lipidomics, a field of research that elucidates the multifaceted roles of lipids within biological systems, has progressively gained attention in investigating diseases such as cardiovascular disease and Alzheimer’s disease. Specifically, by examining alterations in blood lipid concentrations linked to these diseases, lipidomics enables researchers to gain deeper insights into disease mechanisms and unearth novel biomarkers for assessing the effectiveness of interventions and preventions.

We investigated lipid profile differences between individuals with Alzheimer’s disease (AD) and a healthy control group, and between carriers and noncarriers of the APOE-ε4 allele. Additionally, we investigated their interaction effects to assess the differential impact of AD on lipid levels. We utilized ADNI lipidomics data, collected by the Alzheimer’s Disease Metabolomics Consortium (ADMC; https://sites.duke.edu/adnimetab/). Specifically, lipid concentrations (serum levels) of 349 known lipids, measured from fasting blood samples of 806 subjects within the ADNI1 cohort, were used as multivariate responses. Details of the preprocessing steps for extracting lipid levels are provided in Barupal et al. (2018). For predictors, we used AD diagnosis, APOE-ε4 carrier status, and their interaction. Of the 806 subjects, 189 (23%) were diagnosed with AD, and 395 (49%) were carriers of the APOE-ε4 allele. AD was clinically diagnosed according to the NINCDS-ADRDA criteria for probable Alzheimer’s disease (McKhann et al., 1984). Within the AD group, 127 subjects (67%) were APOE-ε4 carriers. Our analysis accounted for the effects of sex-469 (58%) males and 337 (42%) females- and age, with a mean of 75.23 years and a standard deviation of 6.88. The data used in this analysis are available from the ADNI database (adni.loni.usc.edu).

We compared our method with competing methods including SVA, OLS and MMRM. Results are summarized in Table 3. In analyzing the main effects of AD diagnosis and APOE-ε4 carrier status on lipid level differences, FARM identified 16 and 17 significant lipids, respectively. In contrast, SVA identified 1 and 6 significant lipids; OLS detected only 2 and 1; while MMRM failed to identify any lipids associated with AD diagnosis and detected only one significant lipid for APOE-ε4 status. For interaction effects, FARM identified 12 significant lipids, SVA identified 1, while none of the classical methods (OLS, and MMRM) detected any interactions, due to their lower statistical power.

Table 3:

Results from the ADNI lipidomics data. The values represent the number of lipids with significant effects of AD diagnosis, APOE-ε4 carrier status, and their interaction, as identified through various methods. The values in parentheses indicate the ratio of significant lipids to the total number of lipids analyzed ($q$ = 349).

	FARM	SVA	OLS	MMRM
AD Diagnosis	16 (4.58%)	1 (0.29%)	2 (0.57%)	0 (0.00%)
APOE-ε4	17 (4.87%)	6 (1.72%)	1 (0.29%)	1 (0.29%)
Interaction	12 (3.43%)	1 (0.29%)	0 (0.00%)	0 (0.00%)

Our findings from FARM align well with established biological knowledge. For example, unlike saturated fatty acids, unsaturated fatty acids are generally known to be healthy because they reduce low-density lipoprotein (LDL) cholesterol (bad cholesterol) levels. The fatty acid lipids identified as significant by FARM are all unsaturated fatty acids. Specifically, two types are identified: monounsaturated fatty acids (MUFAs)— FA (14:1), FA (16:1), FA (17:1), and FA (18:1) —and polyunsaturated fatty acids (PUFAs), FA (18:2), and FA (18:3).

Both MUFAs and PUFAs are essential components of cell membranes, enhancing membrane fluidity which boosts neuronal function and efficient neurotransmission; they also offer neuroprotective effects through their anti-inflammatory properties that may reduce neuroinflammation, a key factor in neurodegenerative diseases such as Alzheimer’s disease. Among MUFAs and PUFAs, the omega 3-6-9 fatty acids are particularly recognized for their roles in cognitive function. In the lipids identified by FARM, FA (18:3) ($\alpha$-linolenic acid) corresponds to omega-3, FA (18:2) (linoleic acid) to omega-6, and FA (18:1) (oleic acid) to omega-9, respectively. Among these, the first two are “essential fatty acids (EFAs)” that must be obtained through the diet, while omega-9 can be synthesized by the human body. These three unsaturated fatty acids are reported to be associated with Alzheimer’s disease, with reduced levels specifically observed in patients with AD compared to healthy controls (Snowden et al., 2017). Inadequate intake of linoleic acid and $\alpha$-linolenic acid is linked to neuropathology, including conditions such as dementia (Kim and Song, 2024).

There is substantial evidence suggesting that EFAs are protective against Alzheimer’s disease and other forms of dementia (Gustafson et al., 2020). However, the benefits might not uniformly apply to everyone, especially those with the APOE-ε4 genotype. APOE-ε4 carriers often exhibit altered fatty acid metabolism (Chouinard-Watkins and Plourde, 2014), which can significantly impact the absorption and utilization of essential fatty acids. In addition, the APOE-ε4 carriers are at an increased risk of Blood-Brain Barrier (BBB) dysfunction (Montagne et al., 2020), which may hinder the transport of EFAs to the brain (Pan et al., 2015). Thus, while a diet enriched with essential fatty acids proves beneficial for non-APOE-ε4 Alzheimer’s subjects in mitigating disease symptoms and progression (Wei et al., 2023), it has been found to be ineffective in APOE-ε4 carriers (Whalley et al., 2008; Plourde et al., 2009) potentially due to unique lipid metabolic characteristics (Barberger-Gateau et al., 2011; Jeong et al., 2019). Our novel findings suggest that alternative nutrition therapy is likely needed for AD patients with APOE-ε4 genotype to mitigate the deficits of (protective) unsaturated fatty acids.

5. Discussion

In this paper, we developed a novel, fast multivariate autoregressive model (FARM) to address the computational challenges associated with analyzing high-dimensional correlated omics data. By accounting for latent dependencies among outcome variables, our model more accurately reflects the real-world modular patterns present in omics data. In addition, by leveraging structural covariance (BDCS), we derived a closed-form estimate for the model parameters, significantly enhancing the efficiency of our analysis.

The major contribution of FARM lies in its capacity to mitigate prevalent concerns regarding the low reproducibility and replicability of findings from high-throughput biomedical data (Ioannidis, 2005; Baggerly and Coombes, 2009; Leek et al., 2010). Importantly, the dependencies among multivariate outcomes are known to significantly affect the outcomes of multiple testing procedures (Leek and Storey, 2008; Fan et al., 2012; Fan and Han, 2017). By effectively accounting for these dependencies, FARM not only produces more precise results but also improves the replicability of findings across various studies. Additionally, due to its closed-form solution, FARM is highly scalable to high-dimensional applications prevalent in omics data, such as those involving hundreds of thousands of outcome variables. This scalability addresses the computational challenges inherent in analyzing large-scale omics datasets. Lastly, in contrast to multivariate autoregressive models developed for neuroimaging research, which rely on spatial adjacency as prior knowledge, FARM selects neighbors based on functional correlations. This approach renders FARM more suitable for analyzing omics data, where spatial information is typically unavailable.

We conclude this paper by outlining directions for future research. In this work, we adopt a compound symmetry structure within each block ${\Sigma }_k$ to ensure model simplicity and computational efficiency. Extending the framework to accommodate more flexible covariance structures—such as unstructured or Toeplitz covariance matrices within each block—could enhance modeling flexibility and represents an interesting avenue for future investigation.

A software package that implements FARM is available online at https://github.com/hwiyoungstat/FARM.

Figure 4:

Correlation heat maps of residuals of the ADNI lipids data at different stages of the modeling process. (a) Correlation heat map of unordered residuals after fitting OLS, exhibiting numerous high correlation values (indicated by scattered red and blue colors across the heatmap), which suggests significant unexplained dependencies among the variables. (b) Reordered correlation heat map of the residuals, revealing modular patterns due to latent neighborhood structures among the lipids. (c) Correlation heat map of residuals after fitting FARM, showing values close to zero (predominantly green), indicating that the model has effectively accounted for the dependencies, resulting in nearly uncorrelated residuals.

Figure 5:

Forest plot illustrating the effects of Alzheimer’s disease (AD) on lipid levels. Effect sizes are depicted as circles, representing the magnitude of AD’s impact on each lipid type among APOE-ε4 carriers (i.e., ${\widehat{\beta }}_{\mathrm{AD}}+{\widehat{\beta }}_{\mathrm{AD}\times \mathrm{APOE}\text{-}\epsilon 4}$, displayed in blue), among APOE-ε4 noncarriers (i.e., ${\widehat{\beta }}_{\mathrm{AD}}$, displayed in black), and interaction effect (i.e., ${\widehat{\beta }}_{\mathrm{AD}\times \mathrm{APOE}\text{-}\epsilon 4}$, displayed in red).

Appendix A. Derivation of ${\Sigma }_{\mathit{k}}$ (the $\mathit{k}$-th block of the covariance matrix)

In this section, we provide a detailed derivation of (4) in Section 2.

$$\begin{gathered} {\Sigma }_k={({\mathbf{I}}_{\mid {𝒩}_k\mid }-{\rho }_k{\mathbf{W}}_k)}^{-1}{\mathbf{M}}_k{({\mathbf{I}}_{\mid {𝒩}_k\mid }-{\rho }_k{\mathbf{W}}_k)}^{-1} \\ ={\left(\left(1+\frac{{\rho }_k}{\mid {𝒩}_k\mid -1}\right){\mathbf{I}}_{\mid {𝒩}_k\mid }-\frac{{\rho }_k}{\mid {𝒩}_k\mid -1}{\mathbf{J}}_{\mid {𝒩}_k\mid }\right)}^{-1}{\mathbf{M}}_k{\left(\left(1+\frac{{\rho }_k}{\mid {𝒩}_k\mid -1}\right){\mathbf{I}}_{\mid {𝒩}_k\mid }-\frac{{\rho }_k}{\mid {𝒩}_k\mid -1}{\mathbf{J}}_{\mid {𝒩}_k\mid }\right)}^{-1} \\ ={\left((1-c_k){\mathbf{I}}_{\mid {𝒩}_k\mid }+c_k{\mathbf{J}}_{\mid {𝒩}_k\mid }\right)}^{-1}{\mathbf{M}}_k{\left((1-c_k){\mathbf{I}}_{\mid {𝒩}_k\mid }+c_k{\mathbf{J}}_{\mid {𝒩}_k\mid }\right)}^{-1} \\ =\frac{1}{{(1-c_k)}^2}{\left({\mathbf{I}}_{\mid {𝒩}_k\mid }-\frac{c_k}{(1+(\mid {𝒩}_k\mid -1)c_k)}{\mathbf{J}}_{\mid {𝒩}_k\mid }\right)}^{-2}{\mathbf{M}}_k(\ast ) \\ =\frac{1}{{(1-c_k)}^2}\left({\mathbf{I}}_{\mid {𝒩}_k\mid }-\frac{c_k}{1+(\mid {𝒩}_k\mid -1)c_k)}\left(\frac{\mid {𝒩}_k\mid c_k}{1+(\mid {𝒩}_k\mid -1)c_k)}-2\right){\mathbf{J}}_{\mid {𝒩}_k\mid }\right){\mathbf{M}}_k(\ast \ast ) \\ ={(1-c_k)}^{-2}({\mathbf{I}}_{\mid {𝒩}_k\mid }+d_k{\mathbf{J}}_{\mid {𝒩}_k\mid }){\mathbf{M}}_k. \end{gathered}$$

In $(\ast )$, we use the inverse of the compound symmetry matrix, i.e., ${(a{\mathbf{I}}_n+{b_J}_n)}^{-1}=({\mathbf{I}}_n-b∕(a+nb){\mathbf{J}}_n)∕a$, and in $(\ast \ast )$, we use ${\mathbf{J}}_n{\mathbf{J}}_n=n{\mathbf{J}}_n$, which leads to ${({\mathbf{I}}_n-\alpha {\mathbf{J}}_n)}^2={\mathbf{I}}_n+\alpha (\alpha n-2){\mathbf{J}}_{\mid {𝒩}_k\mid }$.

Appendix B. Proof of Proposition 1

Proof. We first rewrite $d_k$ in terms of ${\rho }_k$,

$$\begin{gathered} d_k=\frac{c_k}{1+(\mid {𝒩}_k\mid -1)c_k}\left(\frac{\mid {𝒩}_k\mid c_k}{1+(\mid {𝒩}_k\mid -1)c_k}-2\right) \\ =\mid {𝒩}_k\mid {\left(\frac{{\rho }_k}{(1-{\rho }_k)(1-\mid {𝒩}_k\mid )}\right)}^2-2\frac{{\rho }_k}{(1-{\rho }_k)(1-\mid {𝒩}_k\mid )}, \end{gathered}$$

and set $f({\rho }_k)={\rho }_k∕((1-{\rho }_k)(1-\mid {𝒩}_k\mid ))$. Since $d_k=r_k∕(1-r_k)$, solution of the quadratic equation $\mid {𝒩}_k\mid f{({\rho }_k)}^2-2f({\rho }_k)-r_k∕(1-r_k)$ = 0 with the two constraints ${\rho }_k$ > 0, and $\mid {𝒩}_k\mid$ > 1, which restricts $f({\rho }_k)$ ≤ 0, is given by

$$f({\rho }_k)=\frac{1-\sqrt{1+\mid {𝒩}_k\mid (\frac{r_k}{1-r_k})}}{\mid {𝒩}_k\mid }.$$

We then rewrite the above equation in terms of ${\rho }_k$, which completes the proof.

Appendix C. Illustrations of the sample correlation matrices used in the model misspecification setting.

Figure C.6:

Heatmaps of the correlation matrices used in the simulation study are displayed. In the left panel (BDCS), correlation matrices with varying $r_k$ values are shown (i.e., $r_k$ = {0.4, 0.8}), while in the right panel (non-BDCS), correlation matrices under conditions of model misspecification are presented. Scenario 1 features a Toeplitz correlation structure, with first-order autoregressive relations within the block, where the diagonal area shows strong correlations (shown in red) that gradually decrease (shown in yellow). Scenario 2 exhibits an inter-correlated block structure with a distinct negative correlation pattern (shown in blue) between the two blocks. Scenario 3 involves an unstructured correlation pattern within the block, including correlations among some elements outside the block. For each scenario, the heatmap in the top panel represents the ground truth from which the simulated data were generated, while the bottom panel displays sample correlation matrices obtained from the generated data.

Alzheimer’s Disease Neuroimaging Initiative:

Data used in preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu). As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this report. A complete listing of ADNI investigators can be found at: https://adni.loni.usc.edu/wp-content/uploads/how_to_apply/ADNI_Acknowledgement_List.pdf.

Alzheimer’s Disease Metabolomics Consortium:

Data used in preparation of this article were generated by the Alzheimer’s Disease Metabolomics Consortium (ADMC). As such, the investigators within the ADMC provided data but did not participate in analysis or writing of this report. A complete listing of ADMC investigators can be found at: https://sites.duke.edu/adnimetab/team/.