Accelerating Heritability, Genetic Correlation, and Genome‐Wide Association Imaging Genetic Analyses in Complex Pedigrees

ABSTRACT

National and international biobanking efforts led to the collection of large and inclusive imaging genetics datasets that enable examination of the contribution of genetic and environmental factors to human brains in illness and health. High‐resolution neuroimaging (~10^4–6 voxels) and genetic (10^6–8 single nucleotide polymorphic [SNP] variants) data are available in statistically powerful (N = 10^3–5) epidemiological and disorder‐focused samples. Performing imaging genetics analyses at full resolution afforded in these datasets is a formidable computational task even under the assumption of unrelatedness among the subjects. The computational complexity rises as ~N ^2–3 (where N is the sample size), when accounting for relatedness among subjects. We describe fast, non‐iterative simplifications to accelerate classical variance component (VC) methods including heritability, genetic correlation, and genome‐wide association in dense and complex empirical pedigrees. These approaches linearize (from N ^2–3 to N ^~1) computational effort while maintaining fidelity (r ~ 0.95) with the VC results and take advantage of parallel computing provided by central and graphics processing units (CPU and GPU). We show that the new approaches lead to a 10⁴‐ to 10⁶‐fold reduction in computational complexity—making voxel‐wise heritability, genetic correlation, and genome‐wide association studies (GWAS) analysis practical for large and complex samples such as those provided by the Amish and Human Connectome Projects (N = 406 and 1052 subjects, respectively) and UK Biobank (N = 31,681). These developments are shared in open‐source, SOLAR‐Eclipse software.

The heatmaps show kinship matrices, coefficients of relatedness (CR), for individuals in ACP, HCP, and UKBB pedigrees. The diagonal composed of CR = 1 that corresponds to the same individual.

1. Introduction

The focus of collaborative science has expanded toward biobanking of large and multidimensional datasets to understand complex biological mechanisms that govern human health and illnesses (Thompson et al. 2017, 2020). It includes ascertainment of large cohorts of participants and the collection of multidimensional data, including genotyping and neuroimaging. These datasets provide a vehicle for identifying common and rare genetic variants in subjects with complex disorders such as schizophrenia, major depressive disorder, and Alzheimer's disease, among other conditions. Interrogation of these datasets will also help with understanding of effects of environment and gene by environment interactions on health and disease throughout human development and aging (Warrier et al. 2023). Yet, performing these analyses at full imaging and genetic resolution presents a daunting computational challenge, and furthermore, it becomes herculean when correcting for underlying population and family structures even in seemingly unrelated samples. Failure to correct for this can bias and inflate the outcomes and reduce replicability of the findings (Thomson and McWhirter 2017). Previously, we reported that calculation of the likelihood ratio statistic that powers classical genetic analyses can be accelerated using an eigen simplification of the underlying kinship structure (Blangero et al. 2013). This algorithmic approach performs analyses of datasets with complex underlying relatedness, commensurate with the analysis of the same number of unrelated individuals. In this study, we describe further algorithmic improvements that use non‐iterative approximations and take advantage of modern computational hardware to enable the use of these classical genetic analyses in large samples. We focus our developments to enable the genetic interrogation of neuroimaging datasets, but the underlying methods are appropriate for all quantitative phenotypes.

We showcase the acceleration of heritability, genetic correlation, and association analysis of neuroimaging data. Neuroimaging data is an ideal phenotype for genetic analyses because it is quantitative, informative of the underlying structure and function, informative of illness progression and has high precision and reproducibility (Acheson et al. 2017; Adhikari et al. 2019; Agartz et al. 2001; Kim et al. 2005; Kochunov and Duff 2009; Lerch and Evans 2005). Previously, genetic interrogation of these datasets has been performed primarily using the whole‐brain or regionally averaged measurements, to overcome the computational challenges associated with voxel‐wise analyses. For example, voxel‐wise heritability or genetic correlation analyses of the brain's structural or functional measures requires ~10⁵ variance component (VC) calculations (one for each voxel‐wise trait), while a voxel‐wise genome association study (vGWAS) using 10⁶ genetic markers correspondingly requires ~10⁵ × 10⁶ or ~10¹¹ VC calculations (Hibar et al. 2011; Jahanshad, Rajagopalan, et al. 2013; Medland et al. 2014). These computational challenges forced imaging genetics analyses to be performed under the assumption of genetic unrelatedness among the subjects (Grasby et al. 2020; Stein et al. 2011; Zhao et al. 2021). However, even unrelated samples have sufficient hidden relatedness that can bias the results of association analyses, leading to reduced replicability (Thomson and McWhirter 2017; Uffelmann et al. 2021). Alternative approaches included dimensionality reduction of the imaging data through the use of multivariate varying coefficient model, variational autoencoders, or low‐dimensional embeddings (Huang et al. 2017; Shen and Thompson 2020; Yun et al. 2024).

VC method, including linear mixed models (LLMs), has supported statistical genetics for over a century (Fisher 1919). These approaches, are implemented in SOLAR‐Eclipse, OpenMX, GCTA, and other software and work by partitioning the variance into additive and dominant genetic components and shared and unique environmental components and quantify the degree of shared genetic and environmental variance between traits and degree of variance contributed by specific polymorphisms (Boker et al. 2011; Speed et al. 2017). VC analyses use linear mixed effect modeling and iterative likelihood estimation (ILE) to fit the fixed (relatedness within the pedigree and genetic variants on the trait) and mixed (environmental) effects on the variance in a trait. Their chief limitation is the computational complexity that rises as a power function of the size of the sample (~N ^2–3). This becomes even more formidable if the genetic relatedness among subjects is characterized as dense empirical kinship matrices versus the sparse self‐reported relatedness (Kochunov, Donohue, et al. 2019). Self‐reported relatedness is defined as the length of the shortest ancestral path (kinship) between two individuals. Each path coefficient codes the expected degree of relatedness: 1 for the self or a monozygotic twin; ½ for parents, full siblings, and dizygotic twins; ¼ for grandparents or half‐siblings; 1/8 for cousins; and null for unrelated individuals. The resulting kinship matrix is usually sparse because participants from different families are coded as unrelated and algebraic simplification approaches can be used to reduce the size of the matrix, simplifying its inversion (Kochunov, Donohue, et al. 2019). However, nearly all unrelated individuals are “deeply related” and share a nonzero degree of genetic relatedness, resulting in empirical kinship matrices that are densely coded (Visscher et al. 2006, 2007). The use of empirical relatedness obtained from genetic markers provides more accurate estimates of genetic variance and can be used for performing classical genetic analyses in epidemiological samples but at greatly increased computational costs (Kochunov, Donohue, et al. 2019; Ramstetter et al. 2017; Toro et al. 2014; Wood et al. 2014; Yang et al. 2010).

Fast approximation for LMM‐based genetic analyses has been developed by several groups to address the computational burden associated with correcting for population structure, primarily for the genome‐wide association studies (GWAS). This included FaST‐LMM, GRAMMAR‐Gamma, BOLT‐LMM, and REGENIE (Lippert et al. 2011; Loh et al. 2018; Mbatchou et al. 2021; Svishcheva et al. 2012). All these approaches perform a data transformation or decomposition to linearize the underlying population structure and employ an approximation for the likelihood function with certain assumptions. For example, the FaST‐LMM uses a spectral decomposition to “rotate” the phenotype, genotypes, and covariates in such a way that the data for related subjects become uncorrelated. This approach is valid under the assumptions that the number of single nucleotide polymorphism (SNP) that code for sharing of genetic relatedness is smaller than the number of people in a dataset. Under this assumption, the likelihood calculation can be approximated as the function of a ratio of variance explained by SNP versus residual variance (Lippert et al. 2011). FaST‐LMM and other approaches employ these data transforms and algorithmic approximation and to optimize a specific analysis for a specific need case, for example, large‐N GWAS. Here, we present a more universal approach that starts with the classical VC model and detail the steps we took to optimize common genetic analyses for pedigrees of all sizes and with arbitrary relatedness that is either self‐reported or derived empirically.

In our efforts to develop universal accelerated VC analyses, we proposed eigenvalue decomposition (EVD) to simplify the calculation of likelihood in arbitrary pedigrees (Blangero et al. 2013). We further proposed non‐iterative solutions for mixed linear effect models for heritability and association studies in empirical pedigrees (Ganjgahi et al. 2015, 2018; Kochunov, Patel, et al. 2019). Here, we summarize the adaptation of VC algorithms for three types of genetic analyses: heritability, genetic correlation, and genome‐wide association. We describe algorithmic implementations that take advantage of the parallel nature of voxel‐wise imaging genetic analysis that opens it for parallel computing provided by modern multicore central and graphics processors (Table 1). We demonstrate that the combination of algorithmic and hardware approaches enables full voxel‐wise and genetic resolution analyses. We performed these analyses using publicly available datasets consisting of fractional anisotropy (FA) measurements collected in complex pedigrees from the Amish Connectome Project (N = 406), twins and siblings from the in Human Connectome Project (N = 1052), and unrelated subjects of European ancestry collected by the UK Biobank (N = 31,681). We chose FA as an exemplar because it is the most analyzed scalar parameter extracted from DTI (Basser and Pierpaoli 1996, 2011) and is a sensitive index of fiber coherence, myelination levels, and axonal integrity (Thomason and Thompson 2011), linked with cognitive function, and is under a strong genetic control (Bartzokis 2004; Bartzokis et al. 2008; Charlton et al. 2009; Geng et al. 2012; Glahn et al. 2013; Jahanshad, Kochunov, et al. 2013; Karbasforoushan et al. 2015; Kennedy and Raz 2009; Kochunov, Coyle, et al. 2009; Kochunov, Robin, et al. 2009; Konrad et al. 2009; Muetzel et al. 2008; Penke et al. 2010; Schiavone et al. 2009; Shen et al. 2014; Vernooij et al. 2009; Wright et al. 2015).

TABLE 1.

Processors per Computer Cluster Node.

Type of processor	Processing card	Number of processors	Cores per processor
GPU	RTX A4500	1	7168
CPU	Xeon Gold 6442Y	96	24

2. Methods

2.1. Algorithmic Acceleration of Genetic Analysis

In the Supplement section, we describe the steps we took to develop a general model of algorithmically accelerated genetic analyses. We start with description of a standard polygenic model (S1, see Supporting Information), next we describe the EVD that linearizes the model by reducing the effort associated with inversion of the covariance matrix (S2, see Supporting Information). We describe the fast and powerful heritability inference (FPHI) that provides a solution for accelerating heritability estimates (S3, see Supporting Information). We next build on FPHI method to accelerate bivariate genetic correlation analysis (S4, see Supporting Information). Finally, we describe the fast and powerful genome‐wide association (FPGA) approach that expands FPHI for association analysis (S5, see Supporting Information).

2.2. Acceleration Using Parallel Computing

The proposed approximation algorithms call for efficient implementation using parallel computing abilities of modern computational clusters. Computational clusters are built from nodes equipped with multiple central processing or graphics processing units (CPU/GPU) that each offers multiple computational cores (up to 128–256 per CPUs and 6000–12,000 per GPU). GPUs are specifically designed to reduce the cost of parallel computing by offering thousands of computational cores on a single board that is equipped with dedicated high‐speed memory. Each CPU/GPU core can act as an independent computational unit that accesses memory and performs calculations in parallel with other cores. However, the CPU and GPU implementation approaches vary due to technological differences between the two environments. Here, we implemented both CPU and GPU versions of the aforementioned algorithms in SOLAR.

2.3. Implementation Using Parallel CPU Computing

The proposed algorithms were implemented for parallel analysis using OpenMP (https://www.openmp.org) software library and thread‐level parallelization. Each thread is tasked to perform a single FPHI calculation or a single bivariate genetic correlation or one SNP versus all traits in the FPGA/FPGA‐Wald analysis of association. This strategy allows for improved computational efficiency because nearby voxel‐wise traits share 40%–80% of the variance, and therefore, the converged model parameters from a cluster of neighboring voxel can be used as the starting model parameters for both Newton and approximation approaches. Within a thread, linear algebra operations such as vector–vector, matrix–vector, and matrix–matrix operations were coded using Basic Linear Algebra Subprograms implemented in the Intel Math Kernel Library (https://software.intel.com/en‐us/mkl). FPHI analysis is used to precompute the variance parameters for the traits in FPGA.

2.4. Implementation Using Parallel GPU Computing

GPU implementations of the three algorithms were coded using cuBLAS (https://developer.nvidia.com/cublas) linear algebra libraries for GPU computing. The parallelization of algorithmic calculations for GPU differed from those used for the CPU. GPU cores are optimized for single‐instruction, multiple thread (SIMT) calculations. SIMT examples include ray tracing and texture mapping that are simple algebraic vector operations. SIMT algorithms prioritize single operation parallelization such as summation or multiplication of two vector elements. This contrasts with CPUs, where thread‐parallelization involves a complex sequence of operations per thread. Scientific algorithms are often redesigned from the ground up to adhere to the SIMT architecture (Lee et al. 2012). The cuBLAS library provides for porting code written for Intel Math Kernel Library for execution in the GPU environment. It handles parallelization across multiple GPUs and manages the allocation of global (accessible across GPU), shared (accessible to all threads within a thread block), and register (accessible only to one thread) memory for the developer. The disadvantage of this approach is that it is specific to the devices that support Compute Unified Device Architecture (CUDA). Another disadvantage of GPU computing is that methods that do not readily adapt to linear approximations and genetic correlation analyses do not readily benefit from this approach. In fact, cuBLAS implementation of genetic correlation was significantly slower than the CPU version.

2.5. Experimental Data

The novel computational approaches were evaluated in three independent datasets representing common recruitment designs: family‐based (ACP), twin‐and‐sibling (HCP), and epidemiological survey (UKBB). The purpose of this evaluation was (A) to demonstrate the cross‐study applicability of new tools and (B) to evaluate the computational efforts with respect to the sample size. The focus of the study was on the analysis of FA data using a consistent preprocessing protocol.

2.6. Amish Connectome Project (ACP)

Participants. We analyzed diffusion imaging data for N = 406 (173 M/233 F; age = 39.6 ± 17.4 years) ACP participants. Individuals were recruited from 17 nuclear families from Pennsylvania and Maryland, who could be combined into a single extended family that connected them across eight generations based on genealogical records maintained by the old order Amish/Mennonite community and incorporated into the NIH Anabaptist Genealogy Database, which traces back to the founders (Agarwala et al. 1999). Exclusion criteria included major medical and neurological conditions that might affect gross brain structures such as developmental disability, head trauma, seizure, stroke, or transient ischemic attack. All subjects provided written informed consent on forms approved by the Institutional Review Board (IRB) of University of Maryland, Baltimore.

Imaging. The diffusion imaging data were collected at the Maryland Psychiatric Research Center using a Siemens Prisma 3 Tesla scanner using a 64‐channel coil. DWI data were collected using an expansion of the HCP protocol that consisted of 6 shells of b values (b = 600, 900, 1200, 1500, 1800, and 3000 s/mm²) with 98 isotropically distributed diffusion‐weighted directions per shell collected twice with the reversal of the phase encoding and readout gradients (anterior‐to‐posterior AP and posterior‐to‐anterior PA) to correct for spatial distortions, including 20 b = 0 images interleaved within the acquisition. The data were collected using a multiband, echo‐planar, spin‐echo, T2‐weighted sequence (TE/TR/multiband factor = 97/4000 ms/4 with the FOV = 200 mm) with isotropic spatial resolution of 1.6 mm.

Data Preprocessing. Diffusion data were preprocessed using the HCP Diffusion pipeline (Glasser et al. 2013; Sotiropoulos et al. 2013), which was combined with DESIGNER diffusion preprocessing tools including advanced denoising, Gibbs ringing correction and correction of EPI distortions (Ades‐Aron et al. 2018). FA maps were obtained by fitting diffusion tensor model using the FSL‐FDT tool kit (Behrens et al. 2003).

Genotyping and Imputation. All subjects were genotyped using Illumina Infinium Global Screening Array v2.0 SNP‐array that provides extended coverage for 613,599 polymorphic markers selected to provide high imputation accuracy for population‐scale genetics studies. The 407,171 SNPs that satisfied the quality control criteria of ENIGMA protocol were used for quantification of the degree of relatedness and imputation. We used the ENIGMA Genetics Imputation protocol (http://enigma.ini.usc.edu/protocols/genetics‐protocols/) to prepare the genotype data, using the 1000 Genomes Project (phase 1 v3) reference set. The imputation was performed using the Michigan Imputation Server (https://imputationserver.sph.umich.edu) (Das et al. 2016). Following the imputation, the dataset contained approximately 27.51 × 10⁶ SNPs. Additional quality control was applied by filtering SNPs at MAF of 0.01, Hardy–Weinberg equilibrium of 1 × 10⁻⁶, r‐squared of greater than 0.9, and a call rate of greater than 0.95. This resulted in a dataset with around 10 × 10⁶ SNPs.

2.7. Human Connectome Project (HCP)

Participants. We analyzed diffusion imaging data for N = 1052 (483/569 M/F; age = 28.7 ± 3.7 years) healthy participants from the Human Connectome Project (HCP) for whom (1) the scans and data were released in June 2014 (humanconnectome.org) and that (2) passed the HCP and ENIGMA quality control and assurance standards (Marcus et al. 2013). The participants in the HCP study were recruited from the Missouri Family and Twin Registry, a large population‐based study (Van Essen et al. 2013).

Imaging. The diffusion imaging data were collected at Washington University in St. Louis using a customized Siemens Magnetom Connectome 3 Tesla scanner and a 32‐channel head coil. Details on the scanner, image acquisition and reconstruction are provided in (Ugurbil et al. 2013). Diffusion data were collected using a single‐shot, single refocusing spin‐echo, echo‐planar imaging sequence with 1.25‐mm isotropic spatial resolution (TE/TR = 89.5/5520 ms, FOV = 210 × 180 mm). DWI data were collected using 90 directions with right‐to‐left and left‐to‐right phase encoding polarities for each of the four diffusion weightings (b = 1000, 2000, and 3000 s/mm²).

Data Preprocessing. Diffusion data were preprocessed using the HCP Diffusion pipeline (Glasser et al. 2013; Sotiropoulos et al. 2013) that included normalization of b ₀ image intensity across runs and correction for EPI susceptibility and eddy‐current‐induced distortions, gradient‐nonlinearities, subject motion, and application of a brain mask. FA maps were obtained by fitting a diffusion tensor model using the FSL‐FDT tool kit (Behrens et al. 2003).

Genotyping and Imputation. The genotyping data for N = 1141 subjects available through the dbGAP database (https://www.ncbi.nlm.nih.gov/projects/gap/cgi‐bin/study.cgi?study_id=phs001364.v1.p1) were used. Briefly, all subjects were genotyped using Illumina Multi‐Ethnic Global Array (MEGA) SNP‐array that provides extended coverage in European, East Asian, and South Asian populations. The imputation and preprocessing of the genetic data were identical to that of the ACP and yielded approximately the same number (~10 × 10⁶) of SNPs.

2.8. UK Biobank (UKBB)

Participants. The UK Biobank (UKBB) dataset included N = 31,681 European ancestry individuals (14,956 M/16,725 F; age = 63.8 ± 7.5 years) at the baseline. The full set of inclusion and exclusion criteria are detailed elsewhere (Manolio et al. 2012). All participants provided written informed consent.

Imaging. Diffusion data were acquired with the following parameters: a resolution = 2 × 2 × 2 mm and two diffusion‐weighted shells with all 100 distinct diffusion‐encoding directions, eight b = 0 (+3 b = 0 blip‐reversed) images, 50 × b = 1000 and 2000 s/mm², FOV = 104 × 104 × 72, and a 7‐min duration.

Data Preprocessing. The FA maps were provided by the UKBB workflow (https://git.fmrib.ox.ac.uk/falmagro/UK_biobank_pipeline_v_1) and used the same FSL‐FDT tool kit (Behrens et al. 2003).

Genotyping and Imputation. Genotyping data for the UKBB were downloaded as version 3 imputed data from the UKBB showcase website. The protocol for genotyping, imputation, and quality control is described in sections of the UK Biobank documentation (https://biobank.ndph.ox.ac.uk/showcase/showcase/docs/genotyping_qc.pdf) and (https://biobank.ndph.ox.ac.uk/showcase/showcase/docs/impute_ukb_v1.pdf) in summary; all participants were genotyped using the UKBB Axiom array from Affymetrix and imputed using Haplotype Reference Consortium (HRC) and UK10K haplotype resource. The UKBB data were already imputed with 8,521,984 SNPs satisfying the consortium‐wide quality control criteria.

2.9. Simulated Data

Genotyping data were simulated with N = 2000 subjects (1000 cases and 1000 controls) and 1,000,000 SNPs (allele frequency range: 0–1) using plink (https://www.cog‐genomics.org/plink/). The phenotype data including 10 traits were simulated at heritability 0.5 using GCTA (https://yanglab.westlake.edu.cn/software/gcta/).

2.10. Quantification of the Degree of Relationship

SOLAR‐Eclipse uses coefficients of relationship (r _i,j ) (equal to twice the coefficients of kinship) to code the probability that two alleles from individuals i and j are identical by descent. The r _i,j were calculated directly from genotyping data using methods implemented in SOLAR‐Eclipse software (www.solar‐eclipse‐genetics.org). The pedifromsnps function takes the allelic data and produces a pedigree file (Kochunov, Donohue, et al. 2019). The r _i,j were inferred based on the average identity by state statistics while weighting the result by sample‐level allele frequency at each SNP using weighted allelic correlation (WAC) approaches (Hayes, Visscher, and Goddard 2009; Kochunov, Donohue, et al. 2019).

2.11. ENIGMA‐DTI Processing

We used the ENIGMA‐DTI protocol to extract voxel‐wise FA values for all three datasets. These protocols are detailed elsewhere (Jahanshad, Kochunov, et al. 2013) and are available online at http://enigma.ini.usc.edu/protocols/dti‐protocols/. In brief, FA images from all subjects were non‐linearly registered to the ENIGMA‐DTI target FA image using FSL's FNIRT (Smith et al. 2006). This target was created as a minimal deformation target based on images from the participating studies as previously described (Jahanshad, Kochunov, et al. 2013; Kochunov et al. 2002). The data were then processed using FSL's tract‐based spatial statistics (TBSS) analytic method (Smith et al. 2006) modified to project individual FA values onto the ENIGMA‐DTI skeleton mask. After extracting, the skeletonized white matter was used to project individual FA values on the ENIGMA skeleton mask. The protocol, target brain, ENIGMA‐DTI skeleton mask, source code, and executables are all publicly available (https://www.nitrc.org/projects/enigma_dti). This protocol was shown to provide highly replicable measurements based on test‐rest analyses in human subjects (Acheson et al. 2017; McGuire et al. 2017).

2.12. Preparation of Voxel‐Wise Traits for Genetic Analyses

Each individual image had ~117 × 10³ voxel‐wise traits. The nifti_to_csv tool in Solar‐Eclipse was used to extract skeletonized FA values and store neighboring voxel‐wise data in the SOLAR‐eclipse file format. The voxel‐wise traits were prepared for genetic analyses using the following preprocessing steps as implemented in the sporadic_normalize function: (A) regression of covariates including age, sex, and scanner, and (B) inverse normal transformation of the residuals to ensure the normality of the traits. The quantitative trait models used for genetic analyses are sensitive to deviations from normal distribution including outliers, skewness, kurtosis, and others. The rank‐based inverse normal transformation was used to ensure a normal distribution for the quantitative traits and to improve agreement in heritability values across different datasets (Kochunov, Patel, et al. 2019). For each phenotype, values are replaced with the expected ranked values of a standard normal distribution with the same number of observations, ensuring that the univariate distribution is normal and reducing the impact of non‐Gaussian data‐related artifacts; for more discussion on this transformation see Beasley et al. (2009).

2.13. Cognitive Processing Speed Assessment

Information processing speed measures were available for all three samples. The processing speed in ACP and HCP samples were assessed using the NIH Toolbox Pattern Comparison Processing Speed (PCPS) Test (http://www.nihtoolbox.org) (Carlozzi et al. 2013). This test asks participants to discern whether two side‐by‐side pictures are the same or not and measures the number of items correct in a 90‐s period. The PCPS is appropriate for use across the lifespan (ages: 3–85 years) and has high construct validity (Carlozzi et al. 2014). The processing speed in the UKBB sample was assessed with the Digit Symbol Coding subtest of the WAIS‐3, which is a test for speed of information processing and psychomotor response (Wechsler 1997). The task reports the number of correctly coded symbols within a 2‐min interval. Besides processing speed, the task requires elements of attention, visuoperceptual processing, and working memory. Raw neuropsychological assessment scores were used, and corrections for age and sex were performed as part of the statistical modeling by including age, age², age × sex, and age² × sex as covariates.

2.14. Measuring the Statistical Power of Pedigree Using Expected Likelihood Ratio Test (ELRT)

The ELRT method is used by SOLAR‐Eclipse software to evaluate the statistical power of a pedigree for heritability analysis and to compare power between two pedigrees. This function is based on the functionality proposed by Blangero et al. (2013) and further generalized by Raffa and Thompson (2016). The ELRT is defined as the expectation of twice the difference of the log‐likelihoods evaluated at the true parameter and at several different null‐parameter values, respectively (Raffa and Thompson 2016). It uses Taylor series approximations to summarize the relatedness in a pedigree to accurately approximate the expectation of the likelihood ratio test and expected confidence interval widths (Raffa and Thompson 2016).

2.15. Voxel‐Wise Heritability Analysis

Heritability analyses were performed using empirical kinship coefficients. Heritability (h ² ) is the proportion of the total phenotypic variance (σ _P ²) that can be explained by the additive genetic factors (σ _A ² ). We quantified the fidelity of the h ² values of FPHI versus VC approaches for the two samples that included related subjects (ACP and HCP) as well as the computation times with respect to the number of subjects.

2.16. Bivariate Voxel‐Wise Genetic Correlation Analysis

We estimated the extent to which the relationship between two traits is due to genetic factors in brain processing speed and FA values jointly using voxel‐wise bivariate genetic correlation. The accelerated approach was compared to the classical VC methods. Both methods provided estimates for ρ _G and ρ _E and their standard errors. The significance of these coefficients was determined by a z‐test of difference from zero. If ρ _G was significantly different from zero, then a significant proportion of the traits' covariance was influenced by shared genetic factors (Almasy and Blangero 2010; Almasy, Dyer, and Blangero 1997).

2.17. Voxel‐Wise GWAS Analysis

GWAS analysis of the voxel‐wise FA values was performed on the traits that demonstrated significant (p < 0.05) heritability across all three datasets. We filtered the voxel‐wise traits by making a mask of significant voxels in the dataset and repeating the preparation step to generate new trait files. GWAS analyses in the UKBB sample were performed with and without correction for underlying relatedness. Some GWAS analyses in UKBB are performed with the assumption of genetic independence while others use SAIGE to model relatedness. We compared the outcome of our analysis with results of the SAIGE and then repeated it by using the identity matrix instead of the empirical kinship matrix, to evaluate potential biases.

3. Results

3.1. Comparisons of the Three Samples

The kinship matrices and statistical power of discovery for a significant additive genetic contribution to a simulated trait for three pedigrees are shown in Figure 1A,B. The twin‐sibling design of HCP had the highest average coefficient of relatedness (CR = 0.030), followed by the family study design of ACP (CR = 0.028) and “unrelated” sample UKBB (CR = 0.0003). Despite the lowest average CR, the UKBB showed the highest power to discover the genetic contribution to a trait, followed by HCP and then ACP (Figure 1B).

FIGURE 1.

(A) The heatmaps show kinship matrices, coefficients of relatedness (CR), for individuals in ACP, HCP, and UKBB pedigrees. The diagonal composed of CR = 1 that corresponds to the same individual. (B) The red, blue, and green dots indicate expected likelihood ratio test (ELRT) at null‐heritability values for the ACP, HCP, and UKBB, respectively.

3.2. Voxel‐Wise Heritability Analyses

Significant heritability estimates (h² ) (p < 0.05) were reported by FPHI for n = 25,666; 82,287; and 125,144 voxel‐wise traits for the ACP, HCP, and UKBB (Figure 2). The number of significant voxels per cohort agreed well with the power estimates (Figure 1B), where higher powered pedigree allowed for quantification of smaller genetic contribution to traits' variance. A VC approach was used in ACP and HCP, but not in UKBB, where the full voxel‐wise heritability analyses were estimated to take ~10 years of compute hours on our cluster. Overall, there was an excellent agreement with estimates obtained using a fully iterative polygenic approach for two related datasets (r = 0.99 and 0.96 for ACP and HCP). The FPHI slightly under‐estimated h² values versus the VC approach (slope = 0.97 and 0.92 for ACP and HCP, respectively) with near zero intercepts (intercept = −0.004 and − 0.019 for ACP and HCP). The absolute bias was defined as estimated h² minus VC's h² and was small (0.006 and 0.051 for ACP and HCP datasets, respectively). The scatter plots of heritability estimates for FPHI versus VC are shown in Figure S1 (see Supporting Information). The GPU and CPU versions of FPHI produced identical estimates of heritability for both datasets. The scatter plots of FPHI heritability of voxels for UKBB versus HCP showed good agreement (r > 0.6) (Figure S2).

FIGURE 2.

Heritability maps of 25,666 and 82,287 and 125,144 significant (p < 0.05) voxel traits in the ACP (A), HCP (B), and UKBB (C). Heritability analyses were performed on 117,140 voxels for ACP and HCP and 137,634 voxels for UKBB, respectively.

Timing Analysis. GPU‐FPHI showed the lowest computational time, followed by CPU‐FPHI and VC approach (Figure 3 and Table 2) in the UKBB.

FIGURE 3.

Computation times of heritability analyses for 1000 voxel traits using polygenic, CPU‐FPHI, and GPU‐FPHI approaches on the ACP, HCP, and UKBB datasets. The polygenic, FPHI, and GPU FPHI were performed on 10 voxels of each of ACP, HCP, and UKBB dataset, respectively. The x‐axis represented heritability approaches and y‐axis represented computation time on logarithmic scale with base 10.

TABLE 2.

Run time of heritability in seconds for 1000 traits of ACP, HCP, and UKBB datasets on RTX A4500 and Xeon Gold CPU.

		ACP	HCP	UKBB
XEON GOLD	Polygenic	2037.1	15,861.5	670,129,000.0
	FPHI	6.4	16.4	1630.0
RTX A4500	FPHI	2.1	2.9	217.2

3.3. Genetic Correlation Analyses

Voxel‐wise genetic correlation maps between FA and processing speed were calculated for the n = 29,627 voxel‐wise traits that were significantly heritable (p < 0.05) across all three datasets (Figure 4). Voxel‐wise genetic correlation analyses were performed to compare the accuracy and performance of the approximation approach. The average FA values and processing speed measures showed significant heritability in ACP (h ² = 0.61 ± 0.09 and 0.29 ± 0.11, p < 10⁻³, for FA and processing speed, respectively), HCP (h ² = 0.87 ± 0.02 and 0.33 ± 0.06, p < 10⁻⁵) and UKBB (h ² = 0.40 ± 0.06 and 0.52 ± 0.05, p < 10⁻¹⁶). The genetic correlation values were negative indicating that higher FA values corresponded to shorter times for completion of the task. The computation took 42, 160, and 1588 CPU compute hours or 0.55, 1.0, and 225 s per trait versus 304 and 2553 s for classical VC approach for ACP and HCP. The ρG values closely approximated those obtained using the standard model (b = 0.98 and 0.97) with the minimal intercepts 0.01 and − 0.02 and bias 0.007 and 0.06 for ACP and HCP, respectively. The standard model analysis could not be performed for UKBB.

FIGURE 4.

Genetic correlation maps between FA and processing speed in the ACP (A), HCP (B), and UKBB (C), respectively.

3.4. GWAS Analysis

Voxel‐wise GWAS analysis was performed on 29,627 voxel‐wise traits that were significantly heritable (p < 0.05) across three datasets. Analysis used each of the voxel‐wise location as a quantitative trait for genetic association analysis of 7 × 10⁵ polymorphic markers. The 3D statistical maps of the number of genetic polymorphisms that showed significant association with the voxel‐wise FA values are shown in Figure 5. We benchmarked the FPGA computational times and compared estimates with these obtained using the classical VC and another accelerated technique—the Scalable and Accurate Implementation of Generalized mixed model (SAIGE) (Zhou et al. 2020). We did not compare it to FaST‐LMM because its requirements on the number of subjects prevented its use in ACP and HCP datasets. The GPU‐FPGA using Wald test showed the fastest calculation time, followed by GPU‐FPGA (Figure 6 and Table 3). The voxel‐wise GPU‐FPGA:Wald GPU analysis showed an ~884 improvement over SAIGE in the ACP dataset, ~400 and 713 improvement over SAIGE in HCP and UKBB datasets, and ~10⁷ over the classical polygenic models. The slopes in the scatter plots of FPGA versus SAIGE were near unity (slope = 0.94–0.96) with a near zero intercepts (intercept ~0.01; Figure 7). The z‐scores between voxel‐wise GPU‐FPGA and GPU‐FPGA:Wald were compared in ACP, HCP, and UKBB. The slopes in the scatter plots of GPU‐FPGA versus GPU‐FPGA:Wald were also near unity (slope = 0.997–1) with a near zero intercepts (intercept = < 0.002; Figure 8).

FIGURE 5.

3D surface of total number of significant (p < 0.05) SNPs per voxel in the corpus callosum in the ACP (A), HCP (B), and UKBB (C).

FIGURE 6.

Computation times of GWAS were estimated for 29,627 voxel traits using GPU‐FPGA:Wald, GPU‐FPGA, CPU‐FPGA:Wald, CPU‐FPGA, and SAIGE approaches on the ACP, HCP, and UKBB datasets.

TABLE 3.

Run time in hours for SAIGE, FPGA, and FPGA:Wald approaches to GWAS analysis of 29,627 heritably significant voxels.

		ACP	HCP	UKBB
Xeon Gold	SAIGE	26,549	31,972	11,104,200
	FPGA	74	592	4,533,516
	FPGA:Wald	40	107	4,475,815
RTX A4500	FPGA	56	97	16,215
	FPGA:Wald	30	80	15,575

FIGURE 7.

The scatter plots of p values for FPGA versus SAIGE in the ACP and HCP, respectively. Linear regression models were fitted to the negative log p values derived from the FPGA and SAIGE approaches in the ACP, HCP, and UKBB. The linear regression fits include fit lines (red dashed lines), equations, and coefficients of determination (R² ). The black dashed lines are identity lines. The R² estimates are 0.99, 0.95, and 0.97 in (A), (B), and (C), respectively.

FIGURE 9.

(A) Average z‐scores between GWAS and GWAS adjusting for relatedness. (B) Mean differences in z‐scores across all GCC voxel between GWAS and GWAS adjusting for relatedness.

In addition, we evaluated the FPGA computational times and estimates with SAIGE (Zhou et al. 2020) and FaST‐LMM in a simulated data. The GPU‐FPGA using Wald test also showed the fastest calculation time, followed by GPU‐FPGA (Figure S3, see Supporting Information). The GPU‐FPGA:Wald GPU analysis showed an ~185‐fold improvement over SAIGE and was approximately 10 times faster than FaST‐LMM. The slopes in the scatter plots of FPGA versus SAIGE/FaST‐LMM were near unity (slope = 1) with a near zero intercept (intercept ~0.01; Figure S4, see Supporting Information).

3.5. Impact of Hidden Relatedness on GWAS Analysis in UKBB

We compared the outcomes of the GWAS analysis in UKBB following correction for hidden relatedness versus assuming genetic independence. The z‐scores of GWAS without adjustment for relatedness were ~3.1% higher than these obtained after correction for hidden relatedness, suggesting inflation of p value estimates (Figure 9), replicating simulation studies (Thomson and McWhirter 2017).

FIGURE 8.

The scatter plots of z‐scores between FPGA:Wald and FPGA in the ACP, HCP, and UKBB datasets. The linear regression fits include fit lines (black lines), equations and R².

4. Discussion

This is an update on a decade‐long effort to develop universal accelerated classical VC approach to support imaging genetics (Blangero et al. 2013). In the original manuscripts, we described the eigen decomposition for linearization of likelihood calculations and the use of specific computational approximations (Blangero et al. 2013; Ganjgahi et al. 2015, 2018; Kochunov, Patel, et al. 2019). Since then, the field has shifted toward collecting these data in even large and richer population‐level studies. Here, we report methodological developments to accommodate the increases of pedigree size (up to 10^5‐6) and the use of dense empirical kinship structure. Furthermore, we report on the linearization of genetic correlation calculation and propose further development to accelerate calculations by taking advantage of parallel computing. We evaluated these approaches in three independent datasets and compared the outcomes and computational times to classical VC and other computational approaches such as the Scalable and Accurate Implementation of Generalized mixed model (SAIGE) (Zhou et al. 2020) and factored spectrally transformed linear mixed models (FaST‐LMMs) (Lippert et al. 2011). These approaches were implemented using central and graphics processing units (CPU and GPU). We showed significant advantages of GPU computing for larger datasets such as UKBB. These developments are now available as a part of the open‐source SOLAR‐Eclipse distribution.

Imaging genetics combines two disciplines that have advanced clinical and neurological sciences in recent decades. Modern neuroimaging modalities offer phenotypic measures that provide more detailed and quantitative descriptions than disorder‐based diagnostic status or clinical symptoms. In parallel, advancement in genotyping technology led to the collection of GWAS and whole genome sequence (WGS) datasets in large and diverse populations. The multivariate nature of the neuroimaging data provides comprehensive characterization of the integrity of cerebral tissue, including microscopic integrity of the cerebral gray and white matter. Genetics and neuroimaging were used to derive disorder‐specific patterns in severe mental and neurological illnesses such as Alzheimer's disease, schizophrenia, and others that are stable and replicable and have led to novel biomarkers to characterize individual similarity to these illnesses (Kochunov et al. 2020, 2022, 2023; Ma et al. 2023). The full resolution (~10⁵ traits and ~10⁶ genotypes) imaging genetics analyses of complex disease‐related phenotypes can also inform us about the mechanistic nature of these illnesses and provide a causal evaluation of hypotheses and interventions.

The VC approach to quantitative genetics has served the scientific community for over seven decades. The chief limitation of this model is the computational burden associated with iterative maximization of the likelihood function that requires inversion of the kinship matrix at every iteration with about 3–50 iterations per trait (Blangero et al. 2001, 2013). The computational effort becomes especially burdensome with dense empirical kinship matrices where fast matrix inversion approaches cannot provide the efficiency they show in sparsely coded self‐reported pedigrees (Kochunov, Donohue, et al. 2019), and the computation burden grows as a power function (N ^2‐3) with the sample size. The focus of our development has been to linearize the dependency on the sample size. The non‐iterative approximations, described in this manuscript, take advantage of the highly parallel nature of imaging traits and high covariance among neighboring voxel‐wise traits to further accelerate calculations using the neighborhood results as the starting points. The testing in three samples shows the linear scaling of effort and time with the size of sample, making UKBB‐size analyses practical even for modest computational clusters. Future studies in even larger samples can be accommodated through a linear increase in the computational capacity, for example, number of multicore nodes or GPUs per node.

All analyses in this study were performed using a dense empirical pedigree structure. Inferring relatedness is an essential component of imaging genetic analyses. Genetic panels provide the opportunity to directly measure the genetic similarity between any two individuals in a study and calculate the relatedness matrix using empirical, rather than self‐reported kinship. Our previous findings show that genetic inference derived using the empirical kinship had better confidence intervals and lower p values as compared to those from analyses using self‐reported kinship and recommend this approach for genetic analyses in related samples (Kochunov, Donohue, et al. 2019). The empirical kinship approaches were first developed to estimate SNP‐h ² from unrelated individuals (Visscher et al. 2006, 2007). However, the SNP‐h ² estimates are accurate for related samples (Zaitlen et al. 2013), and the patterns of additive genetic variance for regional neuroimaging traits showed high correlation regardless of the computational approach or sample composition (Gao et al. 2021; Kochunov, Donohue, et al. 2019). Here, we further confirmed the excellent agreement between estimates produced by the approximation approaches and the classical VC outcomes.

An important achievement of this study is the acceleration of GWAS for pedigrees of arbitrary size and density. Traditionally, GWAS assumed that the underlying cohort consists of unrelated individuals that share the same population background. However, the confounding problem of the population structure became widely appriciated (Kang et al. 2010). The presence of known and hidden relatedness in samples such as ACP, HCP, and UKBB can bias the results of association analyses. The failure to account for the relatedness renders statistical tests invalid: the lack of true independence between individuals leads to underestimation of standard error and inflation of the test statistics (Dyer et al. 2001). Even seemly unrelated samples such as UKBB have enough hidden relativeness to power the accurate estimate additive genetic control of quantitative traits (Gao et al. 2021). Here, we show that the impact of ignoring empirical relationships can inflate the significance scores in the UKBB sample on average by 3.1%, which can lead to false positive associations. The effects we observed in UKBB were modest compared to 5%–20% inflation in ACP and HCP samples where there is a higher degree of subject relationship (Newman et al. 2001). This inflation of significance is important for imaging genetic analyses where it can systematically increase the type‐1‐errors for voxel‐wise analyses that can lead to an unrealistically high number of associations. The inclusion of the kinship structure in the association analyses doesn't “guarantee” the true associations. However, we observed an excellent agreement between FPGA and other accelerated GWAS approaches such as SAGE and FaST‐LMM.

The FPGA approach is conceptually similar to other algorithmically accelerated GWAS approaches such as SAIGE and FaST‐LMM. All three methods use data transformation to rotate or transform the covariance matrix and employ approximation of likelihood function. Not surprisingly, all three approaches provided very good agreement in the outcomes when a similar empirical kinship approach was used to derive the underlying relatedness matrix. One advantage of FPGA is that it was coded to work with an arbitrary pedigree/relatedness matrix using different data such as self‐reported values and SNP data. The FaST‐LMM uses SNP panel data to decrypt relationship matrix as part of GWAS analysis. This may be important because there are multiple and sometimes contrasting approaches to derive the sample interrelatedness. Several methods have been proposed based on the weighting of low versus high allelic SNP frequency in a sample. Our group reported that higher h² values for brain volume and other neuroimaging measurements were seen when the kinship was derived from low allelic frequency (AF < 5%) SNPs (Kochunov, Donohue, et al. 2019), suggesting that the variance in complex traits may be linked to lower rather than higher frequency variants (Speed et al. 2017). Likewise, single‐chromosome kinship matrices or matrices based on specific genetic regions can be derived for specialized analyses focused on the sharing of specific factors (Kochunov, Donohue, et al. 2019). FPGA has the flexibility to allow for such specialized applications, all the while providing faster computation speed.

5. Conclusion

We have presented methodological developments to optimize computational effort for VC approaches to make them practical in the era of big data imaging genetics analyses. The novel algorithms linearize calculation of likelihood for heritability, genetic correlation, and genetic association analyses and take advantage of covariance among voxel‐wise traits and modern hardware to greatly improve (10^3–6) efficiency compared to VC approaches and other methods.

Conflicts of Interest

The authors declare no conflicts of interest.

Supporting information

Data S1: Supporting information.

Data Availability Statement

The Amish Connectome Project data that support the findings of this study are available from the corresponding author upon reasonable request. Other data that support the findings of this study are available with permission from the Human Connectome Project and UK BioBank.