Identifying genetic variants for brain connectivity using Ball Covariance Ranking and Aggregation

Abstract

Understanding the genetic architecture of brain functions is essential to clarify the biological etiologies of behavioral and psychiatric disorders. Functional connectivity, representing pairwise correlations of neural activities between brain regions, is moderately heritable. Current methods to identify single nucleotide polymorphisms (SNPs) linked to functional connectivity either neglect the complex structure of functional connectivity or fail to control false discoveries. Therefore, we propose a SNP-set hypothesis test, Ball Covariance Ranking and Aggregation (BCRA), to select and test the significance of SNP sets related to functional connectivity, incorporating matrix structure and controlling false discovery rate. Additionally, we present subsample-BCRA, a faster version for large-scale datasets. Simulation studies show both methods effectively detect SNPs with interactive structures, with subsample-BCRA shortens the running time by 700 folds. Applying our method to UK Biobank data from 34,129 individuals, we identify 10 SNP-sets with 29 SNPs significantly impacting functional connectivity. Gene-based analyses reveal three SNPs as eQTLs of gene NBPF15, known to change functional connectivity. We also detect nine novel genes associated with behavioral and psychiatric disorders, whose connections to brain functions remain unexplored. Our findings improve our understanding of the genetic basis for brain connectivity and showcase our method’s utility for broader applications.

1. Introduction

The human brain is a complex system where functional organization and communication between brain regions are essential for behavior and cognition. Neuroimaging techniques, such as functional magnetic resonance imaging (fMRI), have widely been used in both scientific research and clinical applications to understand the human brain. fMRI measures brain activity by detecting changes in blood oxygen level-dependent (BOLD) signals at resting states or performing explicit tasks (Fox & Raichle 2007). Specifically, resting-state fMRI (rsfMRI) captures spontaneous intrinsic brain activities and evaluates regional interactions at rest. (Fox & Raichle 2007). Resting-state functional connectivity matrices, representing pairwise relations between inter-regional ensembles of neuronal elements, are crucial for studying rsfMRI and exploring associations with complex diseases, genetics, and clinical information.

Single nucleotide polymorphisms (SNPs) are germline substitutions of a single nucleotide in the genome and are the most common genetic variation among populations, often linked to susceptibility to neuropsychiatric disorders. For example, a single-base mutation in the APOE (apolipoprotein E) gene is associated with a lower risk for Alzheimer’s disease (Wolf et al. 2013). Thus, the joint analysis of resting-state functional connectivity and SNPs data may ultimately lead to discovering new genetic variations and giving us pathways for many neuropsychiatric and neurological disorders that have not previously been implicated in directly studying those disorders.

Twin and family studies have reported low to moderate heritability for resting-state functional connectivity (Foo et al. 2020). Studies have found that functional connectivity within some networks such as default model network (DMN) is moderately heritable $(h^2:30\%-42\%)$ in both family and twin studies (Ge et al. 2017). Using genome-wide genotyping data (n = 44,190), a recent study showed that the SNP heritability of whole-brain functional connectivity traits can be higher than 30 % (Zhao et al. 2022). Although studies suggest a low to moderate genetic contribution for resting-state functional connectivity phenotypes, currently the identified genetic variants only account for little heritability, which is the motivation for this research to identify more genetic variants to explain the missing heritability (Zhao et al. 2022).

Some statistical methods aiming to find SNPs associated with resting-state functional connectivity were proposed. The common approach, univariate regression like genome-wide association study (GWAS), treats each brain region pair connectivity as an independent response. For example, Zhao et al. (2022) carried out a GWAS analysis on 1,777 intrinsic brain activity traits with 40 million SNPs for 44,190 individuals in the UK Biobank. However, this method has a strict significance level to adjust for multiple comparisons, potentially missing important SNPs with weak effects on brain function. SNP-set methods have demonstrated higher power for discovering SNPs by aggregating weak SNPs (Hu et al. 2020, 2021, Li et al. 2021, Song & Zhang 2014). Additionally, univariate methods also neglect functional connectivity matrix structure and other complex structures like spatial structure, impeding SNP detection. To account for matrix structure, alternative methods incorporating matrix information are proposed. For instance,Kong et al. (2019) propose a low-rank regression model and a rank-one screening procedure based on the spectral norm of the estimated coefficient matrix. Li et al. (2021) introduce a ranking and aggregation approach based on the nuclear norm of the estimated coefficient matrix to screen a set of weak genetic signals.

Existing methods that incorporate matrix structure information do not control the false discovery rate since they are not designed for hypothesis testing (Kong et al. 2019, Li et al. 2021). There is currently no statistical hypothesis testing procedure for identifying high-dimensional SNPs with functional connectivity that controls false discovery rate. Additionally, there are challenges with weak individual genetic signals, complex matrix structures, and potential non-linear relationships between SNPs and resting-state functional connectivity.

To address these challenges, we develop a SNP-set hypothesis test procedure, BCRA, to identify a set of SNPs with resting-state functional connectivity that will: (1) identify and test the significance of a set of SNPs associated with functional connectivity; (2) incorporate the complex matrix structure of functional connectivity; (3) consider nonlinear relationships between SNPs and functional connectivity; (4) achieve high detection power while controlling false discovery rate. This research will advance our understanding of the genetic basis of brain functions and may reveal underlying biological mechanisms for neuropsychiatric and neurological disorders. Specifically, BCRA employs the concept of super-variant to aggregate weak signals in individual SNPs and consider potential SNP interactions. The Ball covariance statistics (Bcov) (Pan et al. 2020) is used to quantify and test the association between SNPs and resting-state functional connectivity matrix, considering matrix structure and nonlinearity. For large sample sizes, subsample-BCRA is proposed as a faster solution by using a subset of samples in Bcov calculation.

This rest of this article is organized as follows: Section 2 introduces BCRA and subsample-BCRA for hypothesis testing; Section 3 presents simulations comparing the proposed methods to existing approaches; Section 4 illustrates the application of the proposed methods using UK Biobank data from 34,129 participants; and Section 5 concludes with discussions.

2. Methods

2.1. BCRA

The proposed method BCRA aims to find associations between matrix response $\mathbf{Y}$, e.g., functional connectivity matrix and ultra-high dimensional covariates $\mathbf{X}$, e.g., SNPs. We start with introducing some notations. Let ${\mathbf{Y}}_i\in {\mathbb{R}}^{q\times q}$ be a matrix response for $i$-th subject and ${\mathbf{X}}_i\in {\mathbb{R}}^p$ be a $p$-dimensional vector of $p$ SNPs for $i$-th subject $(i=1,\dots ,n)$. Observations for different individuals are assumed to be independent and identically distributed. Suppose that $p$ SNPs are divided into $G$ SNP-sets with $p_g$ SNPs in the $g$-th SNP-set, and ${\mathbf{X}}^{jg}$ is the value for $j$-th SNP in the $g$-th SNP-set, our approach aims to associate $g$-th SNP-set ${\mathbf{X}}^g$ with the response $\mathbf{Y}$ as below

$$\mathbf{Y}=\sum _{g=1}^G{f}_g\left({\mathbf{X}}_g\right)+\mathbf{E},f_g:{\mathbb{R}}^{p_g}\to {\mathbb{R}}^{q\times q}.$$ (1)

As depicted in the model, rather than focusing on a single SNP, we adopt the concept of analyzing a combination of SNPs. It has been demonstrated that such approach can deal with weak signals of SNPs and is easy to detect for more stable associations with the disease than that from a single SNP (Hu et al. 2020, 2021, Li et al. 2021, Song & Zhang 2014, Dai et al. 2022). The BCRA aims to test the hypothesis empirically

$$H_0:f_g=\mathbf{0}.H_1:f_g\ne \mathbf{0}.$$

To achieve this, BCRA follows four steps: segmentation, marginal ranking, best cut-off selection, and statistical inference (Fig. 1a–d). The dataset is split into two sub-datasets to prevent Type I error inflation (Dai et al. 2022), denoted as $\left({\mathbf{Y}}^{\left(1\right)},{\mathbf{X}}^{\left(1\right)}\right)$ for the first sub-dataset with sample size $n_1=n/2$ and $\left({\mathbf{Y}}^{\left(2\right)},{\mathbf{X}}^{\left(2\right)}\right)$ for the second sub-dataset with $n_2=n/2$. The first three steps are applied to the first sub-dataset, while the inference step is applied to the second. We selected $n_1=n_2=n/2$ based on prior analyses showing that balanced splitting ratios yield robust results(Wang et al. 2024).

Figure 1:

Overview of BCRA. (a): SNPs on the chromosome are divided into sets by physical locations on the genome (1MB as a set in this work). (b): Within each set, a ranking method based on Ball covariance statistic (Bcov) is utilized to obtain the importance measure of each SNP in terms of their marginal contribution to the functional connectivity and form a ranking of SNPs. (c): Empirically determine the number of top SNPs to form a super-variant. The best cut-off threshold is the one that gives the largest Bcov among all possible cut-offs. (d): Top SNPs within each set are then aggregated into a super-variant to test its association with the functional connectivity with a p-value. To control false discoveries across multiple sets, p-values are adjusted by Bonferroni correction.

2.1.1. Segmentation

Based on our assumption, $p$ SNPs are divided into $G$ SNP-sets with $p_g$ SNPs in the $g$-th SNP-set. We choose to divide them into non-overlapping local sets by the physical position so that each set consists of SNPs within a segment of physical length 1 Mb. There could be other ways to incorporate biological information to form biologically meaningful sets, such as forming SNPs that are in linkage disequilibrium (LD).

2.1.2. Marginal ranking

In line with the work of Li et al. (2021), SNPs are ordered according to their marginal effect sizes. The effect sizes are estimated by Bcov (Pan et al. 2020) since Bcov incorporates complex matrix structure and can capture the nonlinear relationships between SNPs and resting-state functional connectivity matrix. The idea of Bcov is very similar to the concept of covariance. Thus, larger Bcov indicates stronger association between a SNP and resting-state functional connectivity matrix. The details of Bcov between SNPs and resting state functional connectivity are described below.

Let $(𝒳,\rho )$ and $(𝒴,\xi )$ be two sets, where the norms $\rho$ and $\xi$ represent their induced distances. For ${\mathbf{X}}_1^{\left(1\right)}$ and ${\mathbf{X}}_2^{\left(1\right)}\in 𝒳$, denote the closed ball with the center ${\mathbf{X}}_1^{\left(1\right)}$ and the radius $\rho \left({\mathbf{X}}_1^{\left(1\right)},{\mathbf{X}}_2^{\left(1\right)}\right)$ in $𝒳$ as $\bar{B}\left({\mathbf{X}}_1^{\left(1\right)},\rho \left({\mathbf{X}}_1^{\left(1\right)},{\mathbf{X}}_2^{\left(1\right)}\right)\right)$ or ${\bar{B}}_{\rho }\left({\mathbf{X}}_1^{\left(1\right)},{\mathbf{X}}_2^{\left(1\right)}\right)$. Likewise, for ${\mathbf{Y}}_1^{\left(1\right)}$ and ${\mathbf{Y}}_2^{\left(1\right)}\in 𝒴$, denote the closed ball with the center ${\mathbf{Y}}_1^{\left(1\right)}$ and the radius $\xi \left({\mathbf{Y}}_1^{\left(1\right)},{\mathbf{Y}}_2^{\left(1\right)}\right)$ in $𝒴$ as $\bar{B}\left({\mathbf{Y}}_1^{\left(1\right)},\xi \left({\mathbf{Y}}_1^{\left(1\right)},{\mathbf{Y}}_2^{\left(1\right)}\right)\right)$ or ${\bar{B}}_{\xi }\left({\mathbf{Y}}_1^{\left(1\right)},{\mathbf{Y}}_2^{\left(1\right)}\right)$. For the $j$-th SNP in the $g$-th SNP-set, let ${\delta }_{il,k}^{\mathbf{X}jg,\left(1\right)}=\mathbb{I}\left({\mathbf{X}}_k^{jg,\left(1\right)}\in {\bar{B}}_{\rho }\left({\mathbf{X}}_i^{jg,\left(1\right)},{\mathbf{X}}_l^{jg,\left(1\right)}\right)\right)$, which indicates whether ${\mathbf{X}}_k^{jg,\left(1\right)}$ is located in the closed ball $\left.{\bar{B}}_{\rho }\left({\mathbf{X}}_i^{jg,\left(1\right)},{\mathbf{X}}_l^{jg,\left(1\right)}\right)\right)$. Similarly, we can define ${\delta }_{il,k}^{\mathbf{Y},\left(1\right)}=\mathbb{I}\left({\mathbf{Y}}_k^{\left(1\right)}\in {\bar{B}}_{\xi }\left({\mathbf{Y}}_i^{\left(1\right)},{\mathbf{Y}}_l^{\left(1\right)}\right)\right)$ to denote whether ${\mathbf{Y}}_k^{\left(1\right)}$ is located in the closed ball ${\bar{B}}_{\xi }\left({\mathbf{Y}}_i^{\left(1\right)},{\mathbf{Y}}_l^{\left(1\right)}\right)$. The empirical Bcov between the $j$-th SNP in the $g$-th SNP-set and resting-state functional connectivity is defined as

$${\text{Bcov}}_{jg}=\frac{1}{n_1^2}\sum _{i,l=1}^{n_1}{\left({\Delta }_{il,n_1}^{{\mathbf{X}}^{jg,\left(1\right)}{\mathbf{Y}}^{\left(1\right)}}-{\Delta }_{il,n_1}^{{\mathbf{X}}^{jg,\left(1\right)}}{\Delta }_{il,n_1}^{{\mathbf{Y}}^{\left(1\right)}}\right)}^2{\omega }_{1,n_1}\left({\mathbf{X}}_i^{jg,\left(1\right)},{\mathbf{X}}_l^{jg,\left(1\right)}\right){\omega }_{2,n_1}\left({\mathbf{Y}}_i^{\left(1\right)},{\mathbf{Y}}_l^{\left(1\right)}\right),$$ (2)

where ${\Delta }_{il,n_1}^{{\mathbf{X}}^{jg,\left(1\right)}{\mathbf{Y}}^{\left(1\right)}}=\frac{1}{n_1}\sum _{k=1}^{n_1}{\delta }_{il,k}^{{\mathbf{X}}^{jg,\left(1\right)}}{\delta }_{il,k}^{{\mathbf{Y}}^{\left(1\right)}},{\Delta }_{il,n_1}^{{\mathbf{X}}^{jg},\left(1\right)}=\frac{1}{n_1}\sum _{k=1}^{n_1}{\delta }_{il,k}^{{\mathbf{X}}^{jg,\left(1\right)}},{\Delta }_{il,n_1}^{{\mathbf{Y}}^{\left(1\right)}}=\frac{1}{n_1}\sum _{k=1}^{n_1}{\delta }_{il,k}^{{\mathbf{Y}}^{\left(1\right)}},{\omega }_{1,n_1}\left({\mathbf{X}}_i^{jg,\left(1\right)},{\mathbf{X}}_l^{jg,\left(1\right)}\right)$, and ${\omega }_{2,n_1}\left({\mathbf{Y}}_i^{\left(1\right)},{\mathbf{Y}}_l^{\left(1\right)}\right)$ are weight functions. We adopt three variations of weights as BCov might have different performances with various weights as reported in Zhu et al. (2021).

• constant: ${\omega }_{1,n_1}\left({\mathbf{X}}_i^{jg,\left(1\right)},{\mathbf{X}}_l^{jg,\left(1\right)}\right)={\omega }_{2,n_1}\left({\mathbf{Y}}_i^{\left(1\right)},{\mathbf{Y}}_l^{\left(1\right)}\right)=1$;

• probability: ${\omega }_{1,n_1}\left({\mathbf{X}}_i^{jg,\left(1\right)},{\mathbf{X}}_l^{jg,\left(1\right)}\right)={\left({\Delta }_{il,n_1}^{{\text{X}}^{jg,\left(1\right)}}\right)}^{-1},{\omega }_{2,n_1}\left({\mathbf{Y}}_i^{\left(1\right)},{\mathbf{Y}}_l^{\left(1\right)}\right)={\left({\Delta }_{il,n_1}^{{\text{Y}}^{\left(1\right)}}\right)}^{-1}$;

• chi-squared: ${\omega }_{1,n_1}\left({\mathbf{X}}_i^{jg,\left(1\right)},{\mathbf{X}}_l^{jg,\left(1\right)}\right)=\left({\Delta }_{il,n_1}^{{\mathbf{j}}^{jg,\left(1\right)}}{\left(1-{\Delta }_{il,n_1}^{{\mathbf{X}}^{jg,\left(1\right)}}\right)}^{-1},{\omega }_{2,n_1}\left({\mathbf{Y}}_i^{\left(1\right)},{\mathbf{Y}}_l^{\left(1\right)}\right)=\left({\Delta }_{il,n_1}^{{\mathbf{Y}}^{\left(1\right)}}{\left(1-{\Delta }_{il,n_1}^{{\mathbf{Y}}^{\left(1\right)}}\right)}^{-1}\right.\right.$.

Notice that ${\Delta }_{il,n_1}^{{\text{X}}^{jg,\left(1\right)}}$ represents the proportion of subjects with ${\mathbf{X}}^{jg,\left(1\right)}$ falling in ${\bar{B}}_{\rho }\left({\mathbf{X}}_i^{jg,\left(1\right)},{\mathbf{X}}_l^{jg,\left(1\right)}\right)$, which describes the local empirical density of ${\mathbf{X}}_l^{jg,\left(1\right)}$ at ${\mathbf{X}}_i^{jg,\left(1\right)}$ with resolution $\rho \left({\mathbf{X}}_i^{jg,\left(1\right)},{\mathbf{X}}_l^{jg,\left(1\right)}\right)$. The meanings of ${\Delta }_{il,n_1}^{{\mathbf{X}}^{jg,\left(1\right)}{\mathbf{Y}}^{\left(1\right)}}$ and ${\Delta }_{il,n_1}^{{\mathbf{Y}}^{\left(1\right)}}$ follow similarly. Thus, $\Delta$ s describes the empirical density defined on the gradient defined by each pair of subjects $(i,l)$, and the BCov is the difference between the joint density and the product of marginal densities, which equals to zero when independence holds between two variables. Generally speaking, ${\Delta .}_{,n_1}$ represents the local empirical densities at all data points $k=1,\cdots ,n_1$ across all possible resolutions $(i,l),l=1,\cdots ,n_1$, and the expression of ${\text{Bcov}}_{jg}$ is an average of all local empirical density differences (squared) across all locations and resolutions defined by data points in the sample.

After calculating Bcov values, let ${\text{Bcov}}_{jg}$ be the marginal effect size for the $j$-th SNP in the $g$-th SNP-set with $p_g$ SNPs, i.e., $1\le j\le p_g$, and the estimated marginal effects are placed in descending ordered. Once the ordering of the variants is obtained for the $g$-th set, let $d_{jg}$ be the index of the SNP with the $j$-th largest Bcov. Define

$${\mathbf{R}}_i^{g,\left(1\right)}=\left\{\begin{array}{ll}\underset{1\le j\le p_g}{\text{argmin}}\left\{j:{\mathbf{X}}_i^{d_{jg}g,\left(1\right)}>0\right\}& \text{if}\exists {\mathbf{X}}_i^{d_{jg}g,\left(1\right)}>0,\\ p_g+1& \text{otherwise.}\end{array}\right.$$ (3)

In other words, for each subject, the transformation returns the rank of the first SNP with minor allele within a list of Bcov-ordered SNPs, i.e., ${\mathbf{R}}^{g,\left(1\right)}=\left(R_1^{g,\left(1\right)},\cdots ,R_{n_1}^{g,\left(1\right)}\right)$.

2.1.3. Best cut-off selection

To decide the best mapping $f_g$, we first transform ${\mathbf{X}}_g^{\left(1\right)}\to {\mathbf{S}}_g^{\left(1\right)}$. The ${\mathbf{S}}_g^{\left(1\right)}$ is determined based on the ranking vector ${\mathbf{R}}_g^{\left(1\right)}$ and a best cut-off value $c_{\text{best}}^g$ for ${\mathbf{S}}_g^{\left(1\right)}$, i.e.,

$${\mathbf{S}}_g^{\left(1\right)}=\mathbb{I}\left({\mathbf{R}}_g^{\left(1\right)}\le c_{\text{best}}^g\right).$$

The $c_{\text{best}}^g$ is determined empirically where we inspect all possible cut-off values of variable ${\mathbf{R}}_g^{\left(1\right)}$ with observations ${\mathbf{r}}_g^{\left(1\right)}=\left\{r_{1g}^{\left(1\right)},\cdots ,r_{n_{1}g}^{\left(1\right)}\right\}\in \left\{1,\dots ,p_g+1\right\}$. For each cut-off value $c_g\in \left\{r_{1g}^{\left(1\right)},\cdots ,r_{n_{1}g}^{\left(1\right)}\right\}$, a subset of values in $\left\{1,\dots ,p_g+1\right\}$, we have a candidate variable ${\mathbf{T}}_g^{c,\left(1\right)}=\mathbb{I}\left({\mathbf{R}}_g^{\left(1\right)}\le c_g\right)$. The Bcov is used to investigate the dependence between all candidate variables ${\left\{{\mathbf{T}}_g^{c_g,\left(1\right)}\right\}}_{c_g{\in \mathbf{r}}_g^{\left(1\right)}}$ and functional connectivity, and the best cut-off is the one that gives the largest Bcov among all possible cut-offs, i.e.,

$$c_{\text{best}}^g=\underset{c_g}{\text{argmax}}\text{Bcov}\left({\mathbf{T}}_g^{c_g,\left(1\right)},{\mathbf{Y}}^{\left(1\right)}\right).$$

With the value of $c_{\text{best}}^g$, we can have the best $f_g$ where it takes aggregation top $c_{\text{best}}^g$ SNPs with largest BCov values into ${\mathbf{S}}_g^{\left(2\right)}$, then map to ${\mathbb{R}}^{q\times q}$.

2.1.4. Statistical inference

To test if the selected SNPs which forms into a SNP-set ${\mathbf{S}}_g^{\left(2\right)}$ is associated with ${\mathbf{Y}}^{\left(2\right)}$, we test the $H_0:f_g=0$ through permutation test. Specifically, we first calculate the observed Bcov between ${\mathbf{S}}_g^{\left(2\right)}$ and ${\mathbf{Y}}^{\left(2\right)}$ as ${\text{BCov}}^{\text{obs}}\left({\mathbf{S}}_g^{\left(2\right)},{\mathbf{Y}}^{\left(2\right)}\right)$. Then for $h\in \{1,\cdots ,H\}$, we randomly shuffle the order of $n$ samples $\left\{1^h,\dots ,n^h\right\}$ to create a permuted dataset: $\left({\mathbf{Y}}^h=\left\{{\mathbf{Y}}_{1^h},\dots ,{\mathbf{Y}}_{1^{n2}}^{\left(2\right)}\right\},\left\{s_{1g}^{\left(2\right)},\cdots ,s_{n_{2}g}^{\left(2\right)}\right\}\right)$. We recalculate the Bcov between ${\mathbf{S}}_g^{\left(2\right)}$ and ${\mathbf{Y}}^{h,\left(2\right)}$ with the shuffled samples as ${\text{BCov}}^{\text{h}}\left({\mathbf{S}}_g^{\left(2\right)},{\mathbf{Y}}^{h,\left(2\right)}\right)$. The final p-value is given as $p_{\text{permutation}}=\frac{\sum _{h=1}^H\mathbb{I}\left({\text{BCov}}^{\text{h}}\ge {\text{BCov}}^{\text{obs}}\right)}{H}$, which indicates the significance between top SNPs for $g$-th SNP-set that are used to form ${\mathbf{S}}_g^{\left(2\right)}$, and functional connectivity. For testing all $G$ SNP-sets, the significance level for inference will be adjusted to be $\alpha /G$ to address the multiple comparison problem, where $\alpha$ is normally chosen to be 0.05.

2.2. subsample-BCRA

Since the computational complexity for Bcov is $O\left(n^{2}H\right)$ where $H$ is the number of permutations and $n$ is the sample size, the computation will become burdensome when a dataset includes thousands of samples. We here propose subsample-BCRA to improve the computational efficiency. Given the complexity of $O\left(n^{2}H\right)$, the most efficient way to reduce computation complexity is to reduce $n$. The idea of subsample-BCRA is to use a subset of samples when calculating Bcov. Specifically, for equation (2), instead of summing over $n_1$ subjects, the summation will only range from 1 to $n_{\text{subset}}$, where $n_{\text{subset}}$ is a subset of $n_1$ samples. Therefore, the calculation of Bcov will become

$${\text{Bcov}}_{jg}^{\text{sub}}=\frac{1}{n_1^2}\sum _{i,l=1}^{n_{\text{subset}}}{\left({\Delta }_{il,n_{\text{subset}}}^{{\mathbf{\text{X}}}^{jg}\mathbf{Y}}-{\Delta }_{il,n_{\text{subset}}}^{{\mathbf{X}}^{jg}}{\Delta }_{il,n_{\text{subset}}}^{\mathbf{Y}}\right)}^2{\omega }_{1,n_{\text{subset}}}\left({\mathbf{X}}_i^{jg},{\mathbf{X}}_l^{jg}\right){\omega }_{2,n_{\text{subset}}}\left({\mathbf{Y}}_i,{\mathbf{Y}}_l\right).$$ (4)

The choice of $n_{\text{subset}}$ resides in the nature of SNP data. Let $A$ and $a$ be the major and minor allele, respectively, of a SNP, and $m$ be the minor allele frequency (MAF). Under Hardy-Weinberg equilibrium, the frequencies of two minor alleles $\left(aa\right)$, a minor allel $\left(Aa\right)$ and two major alleles $\left(AA\right)$ in an individual is expected to be $m^2,2m(1-m)$ and $(1-m{)}^2$, respectively. Since most SNPs have low MAFs, i.e, small $m$, there will be large difference in the sample sizes between the three genotype groups. We hypothesize that the power of a statistical test will not suffer much if we reduce the sample size in $AA$ group (or also in the $Aa$ group). Our subsample-BCRA approach uses a weighted sampling method, first ranking individuals based on SNP call rates for data completeness. The top percentile of individuals is then chosen, with their selection fine-tuned according to the allele frequencies of SNPs, ensuring our sample mirrors the genetic diversity of the wider population. Simulations were employed to decide the optimal percentage for this top tier of individuals.

With this subsample strategy, the computational burden can be largely reduced. We will demonstrate in the simulation studies that subsample-BCRA achieves comparable performance to BCRA but significantly reduces the computational time.

3. Simulations

3.1. Data generation

In this section, we perform simulation studies to evaluate the performance for the developed methods. The simulation studies are conducted from two perspectives: single SNP-set and multiple SNP-sets. The single SNP-set simulation aims to evaluate how BCRA and subsample-BCRA perform under various sparsity levels of true signals. Whereas, the multiple SNP-set simulation mimics the real applications where more SNP-sets are put together to identify true signals.

For both simulations, we use SNPs data on Chromosome 16 from UKB. We randomly sample $n=\mathrm{4,000}$ subjects to avoid heavy computational burden in simulation studies. The selection of chromosome 16 and the random sampling of n = 4,000 individuals were methodological choices made to demonstrate the applicability of our analytical approach in a controlled simulation setting, without any implications for real-world genetic associations. For SNPs data, we first perform following quality controls to ensure data quality. Specifically, we exclude: 1) subjects with more than 10% missing genotypes; 2) variants with missing genotype rate larger than 10%; and 3) variants that failed the Hardy-Weinberg test at 10⁻⁶ level.

We generated the synthetic functional connectivity matrix based on the model 1 where we set $f_g\left({\mathbf{X}}_g\right)$ to have the following format:

$$f_g\left({\mathbf{X}}_g\right)=\mathbb{I}\left(\sum _{j=1}^{p_g}{\mathbf{X}}_i^{jg}{\beta }^{jg}{\gamma }^{jg}>0\right)\cdot \mathbf{B},$$ (5)

where ${\mathbf{E}}_i~N\left(0,{\sigma }^2\mathbf{I}\right),{\mathbf{Y}}_i,{\mathbf{E}}_i,\mathbf{B},\mathbf{I}\in {\mathbb{R}}^{64\times 64}\cdot {\gamma }^{jg}\in \left\{\mathrm{0,1}\right\}$ indicates if the $j$-th SNP in the $g$-th SNP-set has effects on $\mathbf{Y}$, and the corresponding effect size is quantified by ${\beta }^{\text{ig}}$. Throughout all simulations, $p_g=\mathrm{500,1}\le g\le G$ and $\sigma =1$.

The simulation studies focus on evaluating both Type-I error and power. For evaluating Type-I error, ${\beta }^{jg}=\mathrm{0,1}\le j\le p_g,1\le g\le G$ and $\mathbf{B}=0$. For evaluating the power, parameters are set for the single SNP-set and the multiple SNP-set experiments separately.

• Single SNP-set: $G=1,\mathbf{B}$ has the cross structure as depicted in Fig.2(a), ${\beta }^{jg}~U\left(\mathrm{0.4,0.8}\right)$ and ${\gamma }^{jg}~\text{Bernoulli}\left(\pi \right)$ with $\pi \in \left\{\mathrm{0.01,0.05,0.1,0.2,0.3,0.5}\right\}$.

• Multiple SNP-set: $G=30,\mathbf{B}$ has four structures: cross, butterfly, smile and wink as displayed in Fig.2. Specifically, we define

Figure 2:

The four different structures of the true coefficient matrix for the SNP-set simulations, i.e., $\mathbf{B}$ in equation (5): (a) the cross structure; (b) the butterfly structure; (c) the smile structure; (d) the wink structure. The value at each pixel is either 0 (white) or 1 (blue).

$$\begin{array}{l}\sum _{g=1}^G{f}_g\left({\mathbf{X}}_g\right)=\sum _{g=1}^G\mathbb{I}\left(\sum _{j=1}^{p_g}{\mathbf{X}}_i^{jg}{\beta }^{jg}{\gamma }^{jg}>0\right)\cdot \mathbf{B}\\ =\mathbb{I}\left(0.1{\mathbf{X}}_i^{\mathrm{1,1}}+0.3{\mathbf{X}}_i^{\mathrm{11,1}}+0.5{\mathbf{X}}_i^{\mathrm{21,1}}+0.5{\mathbf{X}}_i^{\mathrm{31,1}}+0.3{\mathbf{X}}_i^{\mathrm{41,1}}+0.1{\mathbf{X}}_i^{\mathrm{51,1}}>0\right)+\mathbb{I}\left(\sum _{j\in \left(\mathrm{51,52}\right)}0.1{\mathbf{X}}_i^{j,2}>0\right)+\mathbb{I}\left(\sum _{j\in \left(\mathrm{61,62}\right)}0.3{\mathbf{X}}_i^{j,2}>0\right)+\mathbb{I}\left(\sum _{j\in \left(\mathrm{71,72}\right)}0.5{\mathbf{X}}_i^{j,2}>0\right)+\mathbb{I}\left(\sum _{j\in \left(\mathrm{11,12,13}\right)}0.1{\mathbf{X}}_i^{j,3}>0\right)+\mathbb{I}\left(\sum _{j\in \left(\mathrm{21,22,23}\right)}0.3{\mathbf{X}}_i^{j,3}>0\right)+\mathbb{I}\left(\sum _{j\in \left(\mathrm{31,31,33}\right)}0.5{\mathbf{X}}_i^{j,3}>0\right).\end{array}$$

Therefore, for the single SNP-set, we vary sparsity level, $\pi$, to explore how sparse and dense signals affect final results. The significance level is set to be 0.05. For the multiple SNP-set, we create the sparse signals where with only 21 causal SNPs from three different SNP sets with three different structures, individual signals, two and three interactive signals. In total, the first three SNP-sets have true signals while the rest 27 sets do not. The significance level is set to be $0.05/30=1.67\times {10}^{-3}$.

3.2. Evaluation Criteria

The Type-I error is defined as the frequency of detecting at least one non-contributing SNP or SNP-set across iterations. The evaluation of power is through five metrics: 1) the detection rate for truth, defined as the frequency of detecting true SNP or true SNP-sets across iterations; 2) sensitivity (TP/(TP+ FN)); 3) specificity (TN/(TN+ FP)); 4) precision (TP/(TP+ FP)) and 5) negative prediction value (TN/(TN+FN)). The evaluations are based on 500 iterations at SNP or SNP-set level depending on the methods. The simple GWAS method and the method of Kong et al. (2019), which we refer to as l2rm, are evaluated at the SNP level, whereas our approaches are evaluated at the SNP-set level. Fig.S4 provides a simplified illustration of how metrics are calculated for different methods, explaining the computational process during simulation evaluations. It does not represent actual results; instead, it simplifies scenarios such as zero type-I error rates for GWAS and BCBA for clarity. Actual simulation outcomes are detailed in Section 3.3.

For competing approaches, we compare the performance of BCRA and subsample-BCRA to the traditional single-SNP (or GWAS) approach (Zhao et al. 2022) and the l2rm method through both Type-I error and five metrics of power. Additionally, we also compare the p-values and running time of subsample-BCRA to BCRA as a demonstration that subsample-BCRA can achieve a comparable performance as BCRA but shorten the running time.

3.3. Simulation results

3.3.1. Single SNP-set

The Type-I errors were around 0.05 for both BCRA and subsample-BCRA under both sparse and dense signal cases. In terms of detection rate, the BCRA performed equally well under both sparse and dense signals while for subsample-BCRA, the average detection rate decreased to around 70% when signals became denser (Fig.3). This indicated that the subsample-BCRA method might work as desirable as BCRA when signals were sparse across the chromosome, which is expected to be common in applications. For the choice of weights in calculating the statistic Bcov, there was not much difference across three weight options for BCRA while using chi-squared weight in the subsample-BCRA yielded slightly a higher detection rate.

Figure 3:

Single SNP-set results. (a) Type I Error rate varied across different proportions of true signals $(\pi \in \left\{\mathrm{0.01,0.05,0.1,0.2,0.3,0.5}\right\})$. The Type-I errors were around 0.05 for both BCRA and subsample-BCRA under both sparse and dense signal cases. (b) Detection rate, defined as the frequency of detecting the SNP-set over iterations, varied across different proportions of true signals $(\pi \in \left\{\mathrm{0.01,0.05,0.1,0.2,0.3,0.5}\right\})$. The BCRA performs equally well under both sparse and dense signals as GWAS. While for subsample-BCRA, with more true signals included, the average detection rate decreases to around 70%, indicating that the subsample-BCRA method works well in the situation where genetic signals are sparse across the chromosome. There is not much difference in results in terms of the choice of weights. Different colors represented different approaches (orange: BCRA; green: subsample-BCRA; blue: GWAS). Different shapes represented different weight choices (circle: constant weight; diamond: probability weight; triangle: chi-squared weight; square: GWAS).

3.3.2. Multiple SNP-set

The family-wise error rate (FWER) were well controlled for single-SNP method (GWAS: 0.0299), BRCA (0.04 for constant weights; 0.049 for density weights; and 0.045 for chi-squared), subsample-BCRA (0.05 for constant weights; 0.05 for density weights; and 0.052 for chi-squared) for the multiple SNP-sets cases.

The FWER was 0.444 for the 12rm method. It failed to control FWER since it was not designed for hypothesis testing. Thus, we did not compare 12rm with the proposed method in power simulations.

We first conducted a p-value comparison between subsample-BCRA and BCRA to ensure that subsample-BCRA generated similar results as BCRA. We plotted the p-values of 30 SNP-sets across iterations for BCRA and subsample-BCRA obtained under the cross scenario in Fig.4. Across different weight choices, the distribution of the points was close to the $y=x$ line, implying that the subsample-BCRA can yield similar p-values to BCRA.

Figure 4:

P-value comparisons for BCRA and subsample-BCRA under three weight choices defined in (2). (a) constant, (b) probability and (c) chi-squared. P-values of 30 blocks across 500 iterations under BCRA (x-axis) and subsample-BCRA (y-axis) are plotted as grey points. The red dashed line is $y=x$.

The power performance of the proposed method was first evaluated by detection rate (Table 1(a) and Tables S3–S5). The three true SNP sets were selected almost every time under the cases of cross structure and butterfly structure. For the other two cases, both the first and second SNP sets were identified at each iteration. However, the third SNP set, where it included three interactive signals, was less frequently selected. For subsample-BCRA, the detection rates were comparable to BCRA except for the third SNP-set, but the detection rate of the third SNP-set for subsample-BCRA still remained over 0.8, indicating that the proposed method including both BCRA and subsample-BCRA can select true SNP sets with a good level of confidence. In short, both approaches are advantageous over GWAS.

Table 1:

The detection rate of true signals for the first three SNP sets containing true signals under the cross coefficient structure (a) and the average running time for BCRA and subsample-BCRA in seconds (b). The detection rate for BCRA or subsample-BCRA is defined at the SNP-set level as the number of iterations detecting a SNP-set divided by the total number of iterations. For GWAS approach, the detection rate is defined at the SNP level as the frequency of detecting an individual SNP over iterations. The average detection rate at the SNP-level of GWAS is the average values across all true SNPs. Different columns represent various choices of weights in calculating the statistic BCov in equation (2).

		(a) Average detection Rate
Method	Set	constant	probability	chis-squared
BCRA	Set1	1	0.998	1
	Set2	1	1	1
	Set3	0.987	0.975	0.995
subsample-BCRA	Set1	0.944	0.944	0.944
	Set2	0.992	0.992	0.992
	Set3	0.806	0.814	0.812
GWAS	Set1	0.494	NA	NA
	Set2	0.56	NA	NA
	Set3	0	NA	NA
		(b) Average Running Time (seconds)
Case	Method	constant	probability	chi-squared
Cross	BCRA	1924266	1932197	1889767
	subsample-BCRA	2682	2660	2652
Butterfly	BCRA	1054166	1060450	1019933
	subsample-BCRA	2764	2728	2712
Smile	BCRA	2216298	2226784	2180670
	subsample-BCRA	2872	2847	2862
Wink	BCRA	2150531	2160101	2107494
	subsample-BCRA	2857	2847	2858

Apart from the detection rate, we also looked into the trade-offs between true and false positives through sensitivity, specificity, positive predictive value and accuracy. At each time, all four measures were calculated, and we reported the mean (SD) of those measures over 500 times under various cases (Table. 2 and Table S6–S8). The sensitivities were close to 1 for BCRA under four scenarios. Compared to BCRA, subsample-BCRA lost a bit sensitivity which was consistent with results of detection rate but still remained over 80% sensitivity. Across all scenarios, both BCRA and subsample-BCRA achieved high specificity, precision and negative predictive value, indicating that both procedures could identify true signals while controlling for the false discovery rates.Performances of BCRA and subsample-BCRA were consistent across different scenarios, which implied that they were powerful for identifying SNP sets when true signals have various complex structures. Compared to GWAS, both approaches achieved much better performance, demonstrating the advantage of inference from the SNP-set level over the single SNP level.

Table 2:

SNP-set (top) and SNP (bottom) level power results: average sensitivities, specificities, precisions and NPVs (standard errors across 500 iterations) for BCRA, subsample-BCRA and GWAS under the cross scenario. For SNP-level results, because some SNPs are in LD with others, if any SNP are within the 50kb windows size with the true signal SNP is selected, this signal SNP is also considered as being identified.

Method	Weight	SEN	SPE	PREC	NPV
SNP-set
BCRA	constant	0.9958(0.0372)	0.9982(0.0079)	0.988(0.0535)	0.9995(0.004)
	probability	0.9907(0.0548)	0.9985(0.0073)	0.9899(0.0492)	0.999(0.0059)
	chis-quared	0.9983(0.0236)	0.9987(0.0068)	0.9912(0.0462)	0.9998(0.0025)
subsample-BCRA	constant	0.8912(0.1622)	0.9995(0.0045)	0.9956(0.0363)	0.9884(0.0173)
	probability	0.8072(0.2165)	0.9995(0.0042)	0.9961(0.0345)	0.9796(0.0227)
	chi-squared	0.8573(0.1839)	0.9995(0.0045)	0.9963(0.0302)	0.9848(0.0195)
GWAS	NA	0.513(0.037)	0.9869(4e-04)	0.052(0.0035)	0.9993(1e-04)
SNP
BCRA	constant	0.3265(0.0618)	0.9976(0.0034)	0.9147(0.083)	0.9553(0.0039)
	probability	0.2357(0.0593)	0.9981(0.0032)	0.9075(0.0954)	0.9496(0.0037)
	chi-squared	0.3629(0.0598)	0.9975(0.0026)	0.9151(0.0608)	0.9576(0.0038)
subsample-BCRA	constant	0.6902(0.1802)	0.9779(0.0111)	0.6982(0.0828)	0.9787(0.012)
	probability	0.6917(0.1781)	0.9779(0.0108)	0.6974(0.0815)	0.9788(0.0118)
	chi-squared	0.6949(0.1773)	0.9777(0.0108)	0.6953(0.0793)	0.979(0.0118)
GWAS	NA	0.513(0.037)	0.9869(4e-04)	0.052(0.0035)	0.9993(1e-04)

Though BCRA and subsample-BCRA aim to perform inference from the SNP-set level, they can also select individual SNPs. Thus, we also evaluated individual SNP identification through sensitivity, specificity, positive predictive value and NPV (Table 2 and Table S6–S8). Because some SNPs are in LD with others, if any selected SNP is within the 50kb windows size with the true SNP, it was considered as correct identification. Across all scenarios, subsample-BCRA had the highest sensitivity which showed its power of detecting true signals. For BCRA, it achieved lower sensitivity but higher precision, indicating that it was more conservative in detecting individual SNPs. The SNPs it selected were almost true signals. On the contrary, low precision values of traditional single-SNP GWAS approach demonstrated that most identified signals were false discoveries. Since there were high correlations among SNPs, the sensitivities were not as high as those in the SNP-set identification.

Finally, we reported the average running time (in seconds) for performing BCRA, and subsample BCRA across 30 SNPs in Table 1(b). Notice that the average running time for BCRA was 700 times of that for subsample-BCRA. The substantial amount of reduced time by subsample-BCRA with comparable performance to BCRA justifies the use of subsample-BCRA in applications.

4. UKB Data Applications

4.1. Data

We applied our proposed methods to the UKB dataset (Sudlow et al. 2015) with imputed genotype data (Field ID: 22828) and the resting-state partial correlation matrix from the UKB (Field ID: 25753). For imaging preprocessing, brain regions that were considered to be artificial in the matrix (i.e., relating either to scanning artifacts, or to non-neuronal biophysical processes such as cardiac fluctuations and head motion) were dropped during connectivity construction, resulting in a functional connectivity matrix of 55 × 55. The complete list of brain regions, 3D spatial maps of them and more resources about preprocessing steps can be found at https://www.fmrib.ox.ac.uk/datasets/ukbiobank/index.html.

For genotype processing, we used 41,502,298 SNPs on autosomes that satisfied the same quality control criteria as in our simulation studies above. The whole genome was further divided into 2,732 local SNP-sets, each with physical length 1Mb. We used the chromosome number and set number to denote SNP-sets. For instance, SNP-set chr1 _ 1 consisted of SNPs on chromosome 1 with base-pair position value falling in 1 to 999,999, and chr1 _ 2 composed of those with base-pair position value between 1,000,000 to 1,999,999.

We followed a common practice (Zhao et al. 2022) to adjust for some potential confounding factors including age (centered, at imaging), age-squared (of centered age), sex, age-sex interaction, age-squared-sex interaction, imaging site, genotype array, and the top 10 genetic principal components (PCs). Specifically, we first regressed out the covariates with resting-state functional connectivity and take the residuals as new response.

We utilize data from UK Biobank (UKB) participants of White British ancestry as the discovery set to mitigate the effects of population stratification, based on self-reported ethnic background (data-field 21000). Our robust discovery and internal validation procedure is outlined in Fig.1, where we split the complete dataset into two equal subsets (each containing 16,131 individuals). The first subset is used to construct and rank SNP-sets using our ranking and aggregation method to identify significant associations with functional connectivity. The second subset validates these associations using a BCov permutation test with a Bonferroniadjusted threshold of $1.84\times {10}^{-5}$, accounting for multiple comparisons across the 2723 SNP-sets. This process is repeated 10 times, retaining SNP-sets validated in at least one iteration for further analysis. For external validation, we apply the SNP-sets to a separate dataset comprising UKB participants of European (non-British White) ancestry (n = 1927). We aggregate SNPs that contribute to the SNP-sets at least once in the 10 iterations and test these sets for their association with functional connectivity using a similar BCov permutation procedure, controlling for the same covariates. A SNP-set is considered externally validated if it achieves a significance level of 0.05. This methodology ensures that the SNP-sets we report are not only statistically significant after rigorous multiple testing adjustments but also reproducible across different population subsets, thus enhancing the validity of our associations.

4.2. Results

Ten super-variants were consistently identified in at least one iteration within the discovery set of UKB participants of White British ancestry and met the significance threshold of 0.05 in the validation set consisting of UKB participants of European (non-British White) ancestry. We have presented the p-values for each identified super-variant during the discovery phase in Supplementary Fig.S5. This figure provides a clear visual representation of the negative log10-transformed p-values for each super-variant across ten iterations, effectively illustrating the reproducibility and significance of our findings. The lowest p-values observed across these iterations, along with their corresponding validation p-values from the European (non-British White) UKB participants, are detailed in Table 3.

Table 3:

Summary of verified super-variants, showing the lowest p-values recorded across 10 iterations from the discovery set of UKB participants of White British ancestry, along with their corresponding p-values during verification among UKB participants of European (non-British White) ancestry.

	Super-variant
	chr1_144	chr1_149	chr1_160	chr1_223	chr4_48	chr5_22	chr5_23	chr8_145	chr21_11	chr22_39
p (discovery)	1.30e-05	8.00e-06	1.00e-06	3.00e-06	1.00e-06	1.00e-06	2.00e-06	1.00e-06	5.00e-06	1.00e-05
p (verification)	0.0470	0.0117	0.0437	0.0387	0.0357	0.0190	0.0210	0.0387	0.0160	0.0457

There were 29 SNPs that were included in these super-variants. They were further annotated to the cytogenic region and nearest gene (Table S9). Of them, 24 SNPs were located in functional regions of 14 genes. For example, rs9286338 in the super-variant chr1_149 was an intronic variant in gene NBPF15. Two SNPs for the super-variant chr4 _ 48 (rs76814271 and rs115442203) were located within genes ATP10D and CORIN. The SNP rs189661811 in the super-variant chr5_22 was an intronic variant in gene GUSBP1. Additionally, 11 SNPs of the super-variant chr5_23 lied in the intronic region of gene CDH12. Another two SNPs for the super-variant chr22_39 (rs565047646 and rs145125237) were located within genes GGA1 and DMC1.

To evaluate the shared genetic components between intrinsic brain activity and other complex traits, we carried out association lookups for selected SNPs within verified super-variants in the NHGRI-EBI GWAS catalog (Buniello et al. 2019). Our results tagged many variants reported for a wide range of complex traits in different trait domains, including 12 associations for neurological and psychiatric disorders (Table S10). Below, we highlight colocalizations in a few selected genomic regions. For super-variant chr5_23, 10 SNPs in the region 5p14.3 and gene CDH12 were found to be associated with bipolar disorder in a previous study (Hamshere et al. 2009). For SNPs in the super-variant chr22_39, 3 SNPs in the region 22q13.1 were found to be associated with attention deficit disorder with hyperactivity (ADHD) (Lasky-Su et al. 2008).

Next, we examined whether SNPs (or SNPs with LD score $r^2\ge 0.6$) affected gene expressions through previously published human brain expression quantitative trait loci (eQTL) datasets (Consortium 2020). It was hypothesized that in some tissues, a SNP affected expression at a nearby gene, and both the gene and tissue could play a role in the disease mechanism. Through integration of eQTL, we expected to reveal potential biological mechanisms of the identified SNPs. Our analysis discovered several eQTLs. For example, SNP rs9286338 was an eQLT of gene NBPF15 through modifying the gene expression levels in multiple brain tissues (Fig.5). Additionally, there were 18 SNPs in the same LD block among which SNPs rs10158015 and rs58312111 were eQTLs of the same gene NBPF15. Previous studies reported that eQLTs of gene NBPF15 was associated with functional connectivity (Smith et al. 2021). Moreoever, gene copy number variations in this gene have been implicated in a number of developmental and neurogenetic diseases such as microcephaly, macrocephaly, autism, schizophrenia and cognitive disability (Zhu & Su 2021).

Figure 5:

The expression levels of gene NBPF15 across the eQTL rs9286338. It showed that these eQTLs influenced the expression of NBPF15 in nine brain tissues.

Finally, MAGMA (de Leeuw et al. 2015) gene set analysis was performed to prioritize the enriched biological pathways. We found 9 significantly enriched gene sets at the 5% level (Table S11). Some of these pathways are related to cell-cell communication, cell cycle and transport of small molecules.

To visualize the influence of the 10 super-variants on functional connectivity, we compare the average functional connectivity matrices by their presence (i.e., super-variant =1 vs. super-variant = 0). There were some clear patterns of connectivity alterations across regions as shown in Fig. 6 and Fig.S6–S14. For instance, the connections between regions 48 and 18, regions 24 and 25 were decreased if the super-variant on chromosome 1 set 149 was present while the connections between regions 10 and 49 and regions 24 and 29 were enhanced if this super-variant was present. The connectivity alterations of the other super-variants are presented in Figs.S5–S13. We refer to the UKB website for more details on all regions that we reported here.

Figure 6:

The influence of the super-variant on Chromosome 1 set 149 on brain connectivity. We standardize the elements of the connectivity matrices to have mean 0 and variance 1. Individuals in the combined set are separated into two groups according to the minor and major variants of the super-variant on Chromosome 1 set 149. The difference matrix is calculated by subtracting the average connectivity matrix of the group with the major variant from the average connectivity matrix of the group with the minor variant. For visualization, only differences with absolute values in top 5% are plotted in the chord diagram. Red (blue) bands indicate the positive (negative) differences, and the widths of the bands indicate the magnitudes of the differences. The numbers in the outer circle indicate specific regions in the brain. We provide the axial/sagittal/coronal view of the brain regions with stronger differences in connectivity, including regions indexed as 10, 18, 24, 25, 29, 48 and 49

5. Discussion

In this paper, we propose a novel SNP-set hypothesis test procedure BCRA and its computational-efficiency version subsample-BCRA that include a ranking and aggregation method based on the Ball Covariance statistic to identify sets of interactive genetic variants with matrix response. It can capture nonlinear relationships between SNPs and an outcome of interest with complex structure, such as matrix structure of functional connectivity. To further consider the interactive effects of SNPs and tackle the challenge of weak genetic signals, we adopted the concept of super-variant, similar to but different from the biological concept of the gene, to group any number of SNPs together as the basis of genetic risk factor for SNP-set level inference.

Despite many existing methods for screening genetic variants with matrix response (Kong et al. 2019, Li et al. 2021), our work is the first one to propose a hypothesis test procedure that can identify interactive genetic signals with a matrix response meanwhile controlling for false discoveries and enjoys great advantages. As demonstrated in simulations, both BCRA and subsample-BCRA achieved much higher rates in detecting true interactive SNPs than the traditional single SNP-based association method regardless of the sparse or dense signals distribution across the genome. Furthermore, our approach can also pinpoint individual SNPs that might be associated with the outcome.

For the comparison between BCRA and subsample-BCRA, both methods performed similarly across different cases, indicating reliability in detecting true signals despite complex structures. Subsample-BCRA, with a 700-fold reduction in running time, is preferable for practical applications. Sensitivity analyses (Table S1–S2, Fig. S1–S3) showed that BCRA excels at detecting nonlinear patterns within SNP-sets but struggles with SNP-set interactions. In contrast, subsample-BCRA maintains moderate power across models, except with promiscuous interactions, where it focuses on intraset patterns. While BCRA performs well overall, subsample-BCRA shows a slight decrease in detection rates, especially with geodesic distance, due to data distortion from subsampling. However, ensuring a subset size above 10% of total subjects preserves power and balances efficiency with reliability

In our analysis of the UKB resting-state fMRI dataset, we replicated the previously reported functional connectivity-associated genomic region and a documented gene in the literature as well as identified several novel genes for further investigation. For SNPs of the super-variant chr1_144, they were located in the cytogenetic region of 1q21.1, which was reported to be associated with edge-level functional connectivity in a previous study (Smith et al. 2021). Furthermore, cytogenic regions 5p14.3 (Elsheikh et al. 2020), 8q24.3 (Sha et al. 2023), and 22q13.1 (Sha et al. 2023) were also reported to be related to brain connectivity. However, our findings did not overlap with those reported by Zhao et al. (2022). This observation underscores a crucial point: the genetic landscapes revealed by GWAS and our approach are complementary rather than contradictory. Each method elucidates different aspects of genetic architecture, with GWAS focusing on strong, independent effects, and our approach illuminating the interactions among weaker signals. This complementary result is essential for advancing our understanding of complex traits and should be viewed as a significant enhancement to the genetic analysis toolkit, not as a substitute for traditional methods. Future efforts will aim to refine our approach to better detect significant effects, particularly when they are involved in interactions, thereby ensuring a more comprehensive assessment of genetic influences on phenotypes.

We discovered several novel genes for further investigation, including PPIAL4G, NBPF15, COMMD8, ATP10D, CORIN, GUSBP1, CDH12, GGA1 and DMC1. Gene CDH12 was associated with schizophrenia (Trubetskoy et al. 2022) and educational attainment (Okbay et al. 2022). Gene GGA1 may be involved in the response to anti-depressant treatment in major depressive disorder (Clark et al. 2012). However, the detailed mechanism and relationship to functional connectivity and cognitive behavior warrant further investigation.

We should note that although the proposed method enjoys important advantages and shows promising results, it has some limitations for future improvement. First, our study focused on individuals of European ancestry to control population structure with genetic principal components. Although this method helps manage broad stratification, it might not completely capture the nuanced stratification in polygenic traits. We targeted this group to minimize heterogeneity and clarify genetic effects, as the greater diversity in non-white ancestries introduces complexities that exceed the adjustment capacity of principal components. Future research should include more diverse populations to broaden these findings. (Ruan et al. 2022). Statistically, scalar responses allow for straightforward computation of p-values and effect sizes, but matrix responses pose ongoing challenges for statistical inference. Although we assess the significance of associations between super-variants and the functional connectivity matrix, causality within the matrix cannot be inferred. Furthermore, while our study utilized the high-dimensional structure of data to preserve phenotypic information and relied on the ball correlation method suitable for such data, we acknowledge the potential advantages of dimensionality reduction techniques for certain applications. Future investigations could explore the integration of dimensionality reduction to assess its impact on the interpretability and computational efficiency of the methodology, especially as computational resources continue to evolve. This would also allow us to evaluate the trade-offs between maintaining high-dimensional integrity and the potential benefits of simplifying the data structure. Additionally, a focused analysis on chr5 _ 22 using LD blocks compared to physical distance partitioning showed no significant p-values (Fig.S15), but the smallest p-value $(5.7\times {10}^{-5})$ indicated similar trends. Choosing between LD block and physical distance partitioning requires balancing study goals and resource constraints. LD blocks represent non-random allele associations across loci, often spanning multiple genes, and are computationally challenging to define due to varying LD patterns across populations and regions. For large datasets like UKB, calculating LD involves billions of pairwise comparisons, making traditional methods impractical without advanced algorithms or approximations. The population-specific variability also complicates the LD block boundary detection (Berisa & Pickrell 2016, Salehi Nowbandegani et al. 2023). Future research should integrate LD-based partitions into method development while balancing computational efficiency and biological relevance. Another limitation is the sensitivity to genome partition with different number of SNP sets. Large-size SNP-sets dilute signals with noise, while fine-grained partitions disrupt interactive structures, reducing power and increasing computational cost. Careful selection of partition, ideally based on biological relevance (e.g., genes), can give relatively fixed number of SNP-sets and improve interpretability and mitigate these challenges. Finally, our study focuses on the resting-state brain activity, but combining rsfMRI and task fMRI could boost GWAS power (Elliott et al. 2019), offering further insights into genetic influences on brain function.

Data Availability:

This research has been conducted using the UK Biobank Resource under Application Number 42009. All data used in this study are publicly accessible from UK Biobank via their standard data access procedure at https://www.ukbiobank.ac.uk/. Researchers can apply for access to the UK Biobank data via the Access Management System (AMS) (https://www.ukbiobank.ac.uk/enable-your-research/apply-for-access). The genetic data used in this study are the genotypes with imputation from the UKB (Field ID: 22828). The resting-state functional MRI (rfMRI) partial correlation matrix (dimension 100) used in this study is downloaded from the UKB (Field ID: 25753). Detailed acquisition and preprocessing procedures for imaging data have been described in the brain imaging documentation Casey et al. (2018) at https://biobank.ctsu.ox.ac.uk/crystal/crystal/docs/brain_mri.pdf.