Benchmarking of local genetic correlation estimation methods using summary statistics from genome-wide association studies

Abstract

Local genetic correlation evaluates the correlation of additive genetic effects between different traits across the same genetic variants at a genomic locus. It has been proven informative for understanding the genetic similarities of complex traits beyond that captured by global genetic correlation calculated across the whole genome. Several summary-statistics-based approaches have been developed for estimating local genetic correlation, including -hess, SUPERGNOVA and LAVA. However, there has not been a comprehensive evaluation of these methods to offer practical guidelines on the choices of these methods. In this study, we conduct benchmark comparisons of the performance of these three methods through extensive simulation and real data analyses. We focus on two technical difficulties in estimating local genetic correlation: sample overlaps across traits and local linkage disequilibrium (LD) estimates when only the external reference panels are available. Our simulations suggest the likelihood of incorrectly identifying correlated regions and local correlation estimation accuracy are highly dependent on the estimation of the local LD matrix. These observations are corroborated by real data analyses of 31 complex traits. Overall, our findings illuminate the distinct results yielded by different methods applied in post-genome-wide association studies (post-GWAS) local correlation studies. We underscore the sensitivity of local genetic correlation estimates and inferences to the precision of local LD estimation. These observations accentuate the vital need for ongoing refinement in methodologies.

INTRODUCTION

In recent years, genome-wide association studies (GWASs) have identified tens of thousands of genetic variants associated with a wide range of complex traits and diseases [1–4]. The summary statistics from GWAS offer opportunities to enhance our understanding of genetic architecture underlying human physiology and disease pathogenesis. In the wake of advancements in analytical methodologies, various post-GWAS approaches [5], such as fine mapping, genetic correlation, functional enrichment and polygenic risk score (PRS), are routinely conducted to delve deeper into the nature of genetic variants and elucidate the biological mechanisms behind the observed statistical associations.

Multi-trait modeling allows for the simultaneous analysis of multiple traits or diseases, providing an integrative viewpoint that single-trait analyses may fail to capture. Genetic correlation estimation between two traits using GWASs is one of these crucial methods that can quantify genetic similarities and uncover the shared genetic basis of complex traits and disorders. Leveraging genetic correlation across complex traits contributes to the development of methods for multi-trait analysis in GWAS, which not only improves statistical power for identifying associated variants but also yields more insight into the biological mechanisms of trait inheritance [6, 7]. Besides, due to pleiotropy, a phenomenon where a single gene or genetic variant impacts multiple traits, genetic correlation can be utilized to integrate information across complex traits to enhance the accuracy of polygenic risk score predictions across these traits [8–12]. Genetic correlations can be characterized globally, summarizing the average correlation of genetic effects across the genome, or locally highlighting specific regions that have correlated effects on different traits.

Global genetic correlation, which refers to the genome-wide correlation of the additive genetic effects between two traits, can be estimated using various methods and has been extensively studied [13]. Cross-trait linkage disequilibrium (LD) score regression (LDSC) was the first method that utilized GWAS summary statistics as its input [14]. Following this, GNOVA was introduced to estimate annotation-stratified genetic correlation [15], and a full likelihood-based method, high-definition likelihood (HDL) [16], was also developed by fully accounting for LD across the genome. By integrating these methods and bioinformatics servers like LD Hub [17], we can uncover insights into the comprehensive genetic architecture and the shared genetic underpinnings across various traits. However, they may not capture local genetic correlations that may be distinct across different regions [18–22]. These include opposing correlations in different regions, which can lead to a non-significant global genetic correlation. Furthermore, global genetic correlations provide limited insight into the shared biological mechanisms when different genomic regions have different correlation levels. For example, when investigating the shared genetic architecture between COVID-19 severity and idiopathic pulmonary fibrosis, the global genetic correlation between these two diseases was 0.35 (P = 0.001). However, the variants at MUC5B and ATP11A had opposing effects for these two diseases [23]. Thus, local genetic correlation is also an important method to provide a more detailed picture of genetic architecture and reveal the complex pattern of pleiotropy that may vary across different genome regions. To capture local correlation patterns, several methods have been developed for estimating or detecting local genetic correlation including -hess [24], SUPERGNOVA [20] and LAVA [21], which are methods that can estimate genetic correlation in pre-defined fixed regions.

-Hess [19] and SUPERGNOVA [20] focus on evaluating bivariate local genetic correlations, whereas LAVA [21] uses partial correlation and multiple regression to estimate bivariate and multivariate genetic correlations. These methods also differ in model assumptions relating genetic variants to their effects on traits. Whereas -hess [19] and LAVA [21] are based on fixed-effects models, SUPERGNOVA [20] is based on a random-effects model. Although simulations and real data analysis comparing some of these three methods were performed in each original publication, it is essential to note that those analyses were primarily designed to highlight the strengths of each method and there is no study comparing all three methods simultaneously. Moreover, certain studies rely only on internal reference panels for estimating linkage disequilibrium (LD), neglecting the realistic scenario where these internal reference panels may not always be available. Therefore, assessing the stability of -hess, SUPERGNOVA and LAVA when using both internal and external reference panels is crucial for their practical application in real data analysis. Given the importance of inferring local genetic correlations, there is a need for objectively benchmarking the performance of these three methods with user-defined genome partitions in realistic settings. As all three methods infer shared genetic effects and estimate local heritability in a local region, we evaluate their performances for both tasks, i.e. local genetic covariance/correlation estimation between two traits and local heritability estimation.

We conducted simulations using the observed genotype data from the UK Biobank (UKB) [25] and compared different methods using both in-sample and external reference panels to estimate local LD structure. We used genotype data from 1000 Genomes (1KG) Phase 3 [26] as the external reference panel. We considered binary and continuous traits with varying sample overlaps and region sizes. We assessed the robustness of each method against both infinitesimal and non-infinitesimal models and evaluated the reliability of SUPERGNOVA when the underlying assumption of the random effects does not hold. In addition, we investigated the stability of -hess and LAVA with different reference panels. After simulations, we applied these methods to analyze 31 complex traits with publicly available GWAS summary statistics. To validate the accuracy of these methods in real data, we used LDSC [14] to estimate heritability and global genetic covariances. We compared these estimates with the sum of local heritability and local genetic covariance. We also assessed the stability of the point estimates and inferences using different reference panels for these real data. We conducted polygenic risk score analyses using individual-level data from both UKB [25] and Simons Foundation Powering Autism Research (SPARK) [27] with markers from regions with significant positive and negative correlations between the two traits. The observations from our simulation and real data analyses offer valuable insights into each method’s statistical properties, advantages and limitations.

METHODS

Genome partition

Both -hess and SUPERGNOVA use LDetect [28] to partition the genome into non-overlapping blocks with an average width of 1.6  centimorgan (cM) per block. However, LDetect is a heuristic method that may not always produce optimal results. In contrast, LAVA divides the genome by recursively splitting the largest block into two smaller blocks, selecting a new breakpoint that minimizes local LD between the resulting blocks. To compare the performance of different partitions fairly, we used the R function snp_ldsplit [29] to partition the genome, which uses dynamic programming to minimize the sum of squared correlations between variants in different blocks. We compared the performance of different genome partitions and found that the partitions generated from snp_ldsplit led to a smaller sum of squared correlations between Single nucleotide polymorphisms (SNPs) in different blocks (Supplementary Material, Appendix A).

We used the 1KG Phase 3 [26] reference panels for our analysis. We selected the European (EUR) samples using the SuperPopulation information provided by 1KG and excluded all duplicated and ambiguous SNPs. We applied quality control to the 1KG data for EUR ancestry using PLINK1.90 [30] (—geno 0.05 —hwe 1e-10 —mind 0.05 —maf 0.05) and generated a genetic map using the https://plink.readthedocs.io/en/latest/plink_mani/ website. In addition, we conducted quality control on the data in UKB, a large, prospective study that examines complex traits and diseases in middle-aged adults. We created two UKB reference panels with 503 randomly selected non-overlapping samples from unrelated White British individuals (the same sample size as the EUR 1KG Phase 3 reference panel).

We excluded the major histocompatibility complex (MHC) block on chromosome 6 (30–31 Mb) and applied snp_ldsplit to each chromosome in parallel to partition the genome. To avoid LD leakage and biased estimates, we set the minimum size of each block to be at least 0.5 cM. We adaptively searched for the optimal values of max_r2 (the maximum squared correlation allowed for one pair of variants in two different blocks) and max_size (the maximum number of variants in each block) for each chromosome to make the LD blocks as independent as possible. It is important to note that a larger block may result in increased computational and memory requirements and obscure local signals. To find the minimal combination of max_r2 and max_size that can generate partitions with a mean block size smaller than 1.6 cM, we searched for values of max_r2 from 0.3 to 0.72 and max_size from 5 to 13 cM. From the partitions found, we selected the partitions that resulted in the minimal cost (the sum of squared correlations between SNPs at different blocks). The final max_r2 and max_size values for each chromosome are shown in Supplementary Table 1 and the partitions used in this analysis are in Supplementary Table 2. (Note that genomic coordinates for this paper are in reference to the human genome build 37.)

Methods for genetic correlation estimation

We compared the performance of three local genetic correlation estimation methods: -hess, SUPERGNOVA and LAVA. These three approaches are based on the analysis of summary statistics, with -hess and LAVA using fixed-effects models and SUPERGNOVA adopting a random-effects model. In the following, we first briefly introduce the concept of local genetic covariance and then describe the underlying statistical framework for these methods.

Let denote the standardized genotype vector of size in block , where is the number of markers in this block, and are the effect size vectors of the markers within block for two traits, then the local genetic contributions for the two traits in block are and , respectively. Local genetic covariance is defined as . The local genetic correlation can be estimated as where and are the local heritability in block . Furthermore, the global genetic covariance matches the sum of local genetic covariance when the genetic components in different partitions are independent:

(1)

where and are the genetic contributions for traits 1 and 2, respectively.

In the outputs of these three methods, -hess and SUPERGNOVA give estimates of local genetic covariance, whereas LAVA provides estimates of local genetic correlation. All three methods yield estimates of local heritability with P-values provided by -hess and LAVA, so we can obtain both local genetic correlation and covariance for all three methods.

ρ-Hess

Based on the definition of local genetic covariance introduced above, -hess defines local genetic covariance in block as , where is the local LD matrix in this block, and and are the fixed effect size vectors for two traits in block .

When there are two GWASs, with samples for trait 1 () and samples for trait 2 (), -hess assumes that and , where is the vector of trait 1 for samples and is the vector of trait 2 for samples, and are the standardized genotypes in block for and individuals, respectively, and and are the vectors of noises with and . Assume the first samples overlap and the correlation of the non-genetic effects of the shared samples for the two traits is . The marginal effect size estimates of SNPs in block from GWAS, and , follow the normal distribution. So, in the absence of sample overlap, -hess estimates the local genetic covariance in block by

(2)

However, due to sample overlap, the estimation based on (2) using and will have a bias term. Besides, in practice -hess uses truncated SVD to address rank deficiency of the LD matrix to improve stability, especially when only the external reference panel is available. By defining , where and are the th top eigenvalue and its corresponding eigenvector of the local LD matrix in block from the external reference panel, and , -hess estimates local genetic covariance after correcting for the bias by

(3)

where k is the number of top eigenvalues and their corresponding eigenvectors used, which can be input by the user and is the same for different blocks. For testing significance, -hess assumes that the local genetic correlation and covariance sampling distributions are normal and uses a parametric bootstrap approach to estimate the standard errors. This software -hess can be downloaded at https://huwenboshi.github.io/hess/local_rhog/.

SUPERGNOVA method

SUPERGNOVA also assumes traits follow the same linear models shown in -hess. However, SUPERGNOVA models genetic effects and as random rather than fixed. More specifically, and in block follow a multivariate normal distribution:

(4)

where and are the local heritability of traits 1 and 2 in block ; is the local genetic covariance between traits 1 and 2; is the identity matrix of size and is the number of SNPs in block as defined before.

The estimator used in SUPERGNOVA is defined in terms of the marginal z-statistics of a single SNP j in block , which is given by and . SUPERGNOVA performs eigen decomposition of the local LD matrix () and chooses the first eigenvectors to transform and decorrelate association statistics in a given block , where is determined adaptively. After decorrelation, local genetic covariance is estimated by modeling the expected value of the products of the projected z-statistics,

(5)

where is the th eigenvalue, 1 ≤ j ≤ , and is the sum of genetic covariances and non-genetic covariance, i.e. Besides, is the estimation of using the intercept of cross-trait LDSC [31]. Then, a weighted least squares regression is used to regress on predictor with the weight as the reciprocal of , where and are the global heritability for traits 1 and 2, respectively. SUPERGNOVA adopts an adaptive procedure to determine the number of eigenvalues/eigenvectors for each block. It is accomplished by choosing which minimizes the maximum between the theoretical variance and the empirical variance of local genetic covariance. SUPERGNOVA is available on GitHub: https://github.com/YiliangTracyZhang/SUPERGNOVA.

LAVA method

Same as -hess, LAVA also assumes that the genetic effect sizes are fixed and denotes local genetic covariance in block as . LAVA first applies singular value decomposition to the local LD matrix in block which is , and then defines as the by pruned eigenvector matrix and as the corresponding by diagonal singular value matrix, where is the number of SNPs in block and is the number of top eigenvalues that could explain 99% variance of the local LD matrix in block . Thus, the inverse of can be approximated as .

Furthermore, LAVA defines the scaled principal component (PC) matrix and the corresponding PC effects and such that and . Thus, the local genetic covariance can be represented by the covariance of the PC effects:

(6)

Based on the distribution of marginal effect sizes, PC effects can be estimated as and , and follows the distribution , where and represents the sampling covariance matrix.

The method of moments can be used to estimate

(7)

In the absence of sample overlap, is defined as the diagonal matrix with diagonal elements as the sampling variances of trait 1 and trait 2. When accounting for sample overlap, LAVA first applies LDSC to create a covariance matrix with the intercepts for the global genetic covariance for the off-diagonal elements and each trait’s univariate LDSC intercept as the diagonal elements. Then, LAVA converts this covariance matrix to a correlation matrix, C, and computes the sampling correlation matrix as , where is a vector with the sampling variances of the traits.

Once estimated, the significance of is evaluated using simulation-based P-values. Based on the definition of non-central Wishart distribution and follows the statistic has a non-central Wishart distribution with degrees of freedom, scale matrix and non-centrality matrix LAVA is also available on GitHub: https://github.com/josefin-werme/LAVA.

Simulation settings

We performed simulations using imputed genotype data from UKB [25] and selected samples from genetically unrelated participants of White British ancestry (n = 276 731). The reference panel is either the in-sample reference panel from UKB or the external reference panel from EUR 1KG Phase 3 data across the simulation settings. We first conducted extensive simulations on blocks with different sizes to evaluate the performance of -hess, SUPERGNOVA and LAVA under (i) varying sample overlaps between two GWASs, (ii) both continuous and binary traits, (iii) both infinitesimal and non-infinitesimal models, (iv) the presence or absence of correlations between effect sizes and LD, and (v) different reference panels. Then, based on the above simulation results, we further investigated the impact of the number of eigenvalues and eigenvectors on the stability and inference of -hess and LAVA (Table 1).

Table 1.

Simulation Settings in this study

Evaluation factors	Overlapping level	Traits	Reference panels	Sample sizes	Genetic correlation	Non-genetic effect	Prevelence	Figures
Sample overlapping level	No sample overlap (set1 and set2)	continuous	1. UKB from set1 2. EUR 1KG	10,000 individuals for each set	0, 0.3, 0.6, 0.9	0	/	Figure 1    Supplementary Figures 2–4
	Partial sample overlap (set1 and set3)					0.2		Supplementary Figures 5–7
	Complete sample overlap (set1 and set1)					0.2		Supplementary Figures 8–10
Infinitesimal model (20% causual SNPs)	No sample overlap (set1 and set2)	continuous	1. UKB from set1 2. EUR 1KG			0	/	Supplementary Figures 11–13
Binary traits	No sample overlap (set1 and set2)	binary	1. UKB from set1 2. EUR 1KG				0.2	Supplementary Figure 14
							0.5	Supplementary Figure 15
Effect sizes correlated with LD	No sample overlap (set1 and set2)	continuous	1. UKB from set1 2. EUR 1KG				/	Supplementary Figure 16
Stability of LAVA on reference panels	No sample overlap (set1 and set2)	continuous	1. UKB 500 from set1 2. UKB 5000 from set1 3. UKB 20 000 from set1 and set2 4. UKB 20 000 non-overlap with set1 and set2 5. EUR 1KG 6. CEU 20 000 using HAPGEN2					Figure 2
# eigenvalues and eigenvectors in p-hess	No sample overlap (set1 and set2)	continuous	1. UKB 20,000 non-overlap with set1 and set2 2. EUR 1KG 3. CEU 20,000 using HAPGEN2					Supplementary Figures 17–19
Eastern population	No sample overlap	continuous	1. EAS 1KG 2. EAS 18,000 using HAPGEN2	9000 individuals for each set				Supplementary Figure 21
Sample sizes	No sample overlap	continuous	1. EUR 1KG 2. UKB 20,000 non-overlap with set1 and set2	1. 5000 individuals for each set 2. 20,000 individuals for each set				Supplementary Figure 20

In each simulation setting, our benchmarking measures assessed the accuracy of estimating local genetic correlation, local genetic covariance and local heritability to investigate whether these three methods provide unbiased estimation. We quantified the bias by comparing the difference between the estimates of local genetic correlation, local genetic covariance and local heritability, and their true values. In addition, we conducted 100 repetitions of each simulation setting and examined the proportion of simulations yielding a P-value smaller than 0.05 to estimate the type-I error and power. When the true genetic correlation is zero, the proportion of simulations with a P-value smaller than 0.05 represents the type-I error. In contrast, it indicates power for non-zero true genetic correlation values. These measurements allowed us to assess if the methods could effectively detect more correlation regions while maintaining a low false-positive rate. It is considered unstable if a method exhibits varying levels of estimation bias, type-I errors or power when altering a specific element, such as using different reference panels. Since external reference panels are usually used to estimate LD, evaluating whether these three methods are stable using different reference panels is crucial.

We selected overlapping SNPs from chromosome 1 in UKB, EUR 1KG Phase 3 and HapMap3 [32] datasets for efficient simulations and to ensure sufficient SNP coverage. We then selected SNPs with minor allele frequency (MAF) >5%, genotype missing rate <5% and Hardy–Weinberg equilibrium P-value >1e−10. After removing SNPs with ambiguous alleles, 71 609 SNPs remained for our simulation. We randomly selected 20 000 unrelated white British individuals from UKB and divided them into two subgroups of 10 000 individuals each, labeled as set1 and set2, respectively. We formed another set, set3, with 5000 individuals from set1 and 5000 individuals from set2. We randomly selected four blocks on chromosome 1, having 525 SNPs (POS: 60197393–61754126), 743 SNPs (POS: 3264297–5311384), 1033 SNPs (POS: 245966297–249239303) and 2315 SNPs (POS: 113753415–146215362), respectively. We treated one block as the local region of interest in each simulation and the other SNPs as the background SNPs. We simulated two traits whose SNP effects followed the multivariate normal distribution, with correlation only for SNPs within the chosen region of interest. The correlation of the local genetic effects was set to be 0, 0.3, 0.6 or 0.9, respectively. The remaining SNPs on chromosome 1 were considered background SNPs without genetic correlation. We set the heritability of two traits to be 0.5, which was evenly distributed to all SNPs (71 609 SNPs), so the local heritability of the above four blocks was 0.0037, 0.0052, 0.0072 and 0.0162, respectively. We then expanded our simulations to include an East Asian (EAS) sample. We simulated genotype data using HAPGEN2 [33] based on phased haplotypes from 504 EAS individuals in the 1KG Project, thus retaining population-specific MAF and LD patterns. We created two non-overlapping sets of 9000 EAS-like samples and generated GWASs for two traits. A region of interest on chromosome 1, encompassing 939 SNPs, was selected. We varied the genetic correlation between traits within this region from 0 to 0.9 while keeping the heritability for these two traits at 0.5. The EAS 1KG Phase 3 served as our external reference panel, with the simulated genotype acting as our internal reference panel. Genome-wide Complex Trait Analysis [34] was applied to simulate continuous traits and . We used PLINK1.90 [30] to analyze the simulated traits and generate GWAS summary statistics.

We considered no sample overlap, partial sample overlap and complete sample overlap. When there was no sample overlap, two continuous traits, and , were simulated on set1 and set2, respectively. For and with partial sample overlap, set1 and set3 were used and the covariance of non-genetic effects was set to 0.2. As for the case where the samples were completely overlapping, we used set1 to simulate both and and the covariance of non-genetic effects was still set to 0.2. In addition, we considered the simulation setting that 20% of the SNPs were causal SNPs, where we randomly chose 20% of the SNPs in the regions of interest and 20% of SNPs in the background to be causal. This simulation setting is under a no-sample overlap study design with two continuous traits. We also conducted a simulation where the two traits were binary with no sample overlap. We considered the same local regions of interest, heritability and genetic correlation for continuous traits. We used a liability model to simulate the binary traits. We first simulated continuous traits and and then the binary traits were set to be and , where was the quantile of standard normal distribution. Since we considered two simulation settings with to be 80% or 50%, the prevalence of the binary traits in the two simulations was 0.2 or 0.5. Besides, we also considered the situation where the effect sizes were correlated with local LD, which was the baseline in the LAVA simulation and was also mentioned in SUPERGNOVA. In the simulation setting of LAVA, it first decomposed the local LD matrix of the reference panel as for block and obtained the subset of eigenvalues and eigenvectors that explained 99% of the variance. We denote the number of eigenvalues thus selected as . LAVA defined the projected genotype matrix in its simulation setting for block as where is the standardized genotype of the reference panel in block . It then generated , a matrix with 0 means and identity variance, and decomposed the variance–covariance matrix of the genetic components as and set It simulated the genotype component for two traits as . Thus, the effects for standardized genotype in the simulation settings of LAVA were where the effect sizes were correlated with local LD. SUPERGNOVA also conducted simulations when the effect sizes were associated with the ldscore. Thus, we also considered the simulation setting where the effect sizes were generated similarly from the simulation setting in LAVA [21], so that the effect sizes were related to local LD. For each simulation setting described above, we used both the in-sample reference panel from the UKB set1 samples and one external reference panel from the 1KG Phase 3 data.

To further investigate the stability of LAVA, we applied LAVA using six different reference panels: (i) EUR 1KG Phase 3 reference panel, (ii) UKB reference panel with 500 randomly selected individuals from set1, (iii) UKB reference panel with 5000 randomly selected individuals from set1, (iv) UKB reference panel with all 200 00 individuals from set1 and set2, (v) UKB reference panel with 20 000 individuals randomly selected from unrelated white British populations in UKB which do not overlap with set1 and set2, and (vi) 20 000 CEU individuals simulated using HAPGEN2 [33] (CEU refers to Northern Europeans from Utah).

As -hess allows users to change the number of eigenvalues, we considered different reference panels and varied the number of eigenvalues to investigate further the performances of -hess (the EUR 1KG Phase 3 reference panel; the UKB reference panel with samples from set1 and set2; and the CEU reference panel using HAPGEN2 [33] with 20 000 individuals). We varied the number of eigenvalues for each reference panel to explain 99%, 95%, 90%, 85%, 80% and 70% variance in the above-selected blocks.

We also conducted additional simulations with different sample sizes beyond our original setup of 10 000 individuals per set. We created one simulation with a reduced count of 5000 individuals per set and another expanded to 20 000 individuals per set. In these simulations, UKB genotype data were utilized to generate GWASs for two traits and assign a genetic correlation to the region of interest encompassing 743 SNPs (POS: 3264297–5311384). The local genetic correlation was set for this region at 0, 0.3, 0.6 and 0.9, with the heritability maintained at 0.5. We chose EUR 1KG Phase 3 as our external reference panel and used UKB data with 20 000 individuals for our internal reference panels.

GWAS summary statistics

We analyzed 31 complex traits whose GWAS summary statistics are publicly available. These GWASs were primarily generated using individuals of European ancestry. The sources, sample sizes and global heritability for these traits are listed in Table 2. To prepare data for analysis, we employed the munge_sumstats.py script from LDSC [31] to reformat and conducted quality control on the datasets, including the elimination of strand-ambiguous SNPs and the intersection of the remaining SNPs with those from the 1KG Project. Our analysis considered only autosomal SNPs with MAF >5% and excluded the MHC block on chromosome 6 (30–31 Mb).

Table 2.

Overview of the traits included in this study

Phenotype	Abbreviation	Sample size	Global heritability (s.e.)	PubMed ID
Alcoholism	DrnkWk	537 349	0.0492 (0.0022)	30643251 [35]
Anorexia nervosa	AN	72 517	0.1802 (0.0124)	31308545 [36]
Anxiety disorder	AXD	31 880	0.2073 (0.0204)	31116379 [37]
Asthma	Asthma	385 822	0.0494 (0.0042)	31427789 [38]
Attention-deficit/hyperactivity disorder	ADHD	53 293	0.2362 (0.0158)	30478444 [39]
Autism spectrum disorder	ASD	46 351	0.2028 (0.0164)	30804558 [40]
Bipolar disorder	BD	413 466	0.0708 (0.0028)	34002096 [41]
Body mass index	BMI	806 834	0.1880 (0.0055)	30239722 [42]
Breast cancer	BC	247 173	0.1253 (0.0100)	32424353 [43]
Cognitive performance	CP	257 828	0.1909 (0.0069)	30038396 [44]
Coronary artery disease (including angina)	CAD	547 261	0.0578 (0.0035)	29212778 [45]
Crohn’s disease	Crohn	40 266	0.4215 (0.0438)	28067908 [46]
Educational attainment	EA	765 283	0.1285 (0.0032)	35361970 [47]
Heart rate	HR	361 411	0.1337 (0.0081)	31427789 [38]
Height	Height	385 748	0.4376 (0.0194)	31427789 [38]
Hypertension	Hypertension	298 307	0.1318 (0.0056)	31427789 [38]
Inflammatory bowel disease	IBD	59 957	0.2981 (0.0263)	28067908 [46]
Low-density lipoprotein cholesterol	LDL	431 167	0.1990 (0.0225)	32493714 [48]
Lung cancer	LC	85 716	0.0795 (0.0134)	28604730 [49]
Lupus	Lupus	12 615	0.0954 (0.0459)	33536424 [50]
Major depressive disorder	MDD	500 199	0.0611 (0.0026)	30718901 [51]
Neuroticism	NSM	329 821	0.1073 (0.0046)	29255261 [52]
Obsessive compulsive disorder	OCD	9725	0.2918 (0.0516)	28761083 [53]
Rheumatoid arthritis	RA	58 284	0.1378 (0.0189)	24390342 [54]
Schizophrenia	SCZ	130 644	0.3637 (0.0126)	35396580 [55]
Sleep duration	SD	446 118	0.0676 (0.0031)	30846698 [56]
Smoking behavior	SmkInit	632 802	0.0678 (0.0022)	30643251 [35]
Type 1 diabetes	T1D	25 063	0.1824 (0.0364)	32005708 [57]
Type 2 diabetes	T2D	933 970	0.0467 (0.0022)	35551307 [58]
Ulcerative colitis	UC	45 975	0.2487 (0.0247)	28067908 [46]
Waist/hip ratio	WHR	697 734	0.1325 (0.0045)	30239722 [42]

Besides, before conducting -hess, we followed its suggestion [24] to estimate from GWAS data. We then used the estimated to re-inflate the effect sizes before estimating the local SNP heritability and genetic correlation. To improve the accuracy of -hess, we used the global heritability and its standard error from LDSC as extra inputs. The number of shared samples used in our analysis was based on the consortium from which each GWAS was generated. As indicated in Supplementary Table 4, when two traits have samples from the same consortium, we fixed the shared sample size to the minimum sample size of the common consortium. We set the shared sample size to zero when two traits came from completely different consortia. All other parameters were kept at their default values.

Polygenic risk score analysis

For real data analysis, we used phenotype and genotype data from UKB to perform polygenic risk score (PRS) analysis for four traits: coronary artery disease (CAD), type 2 diabetes (T2D), low-density lipoprotein (LDL) and body mass index (BMI). Of the participants included in the analysis, 4765 individuals were diagnosed with CAD, and 40 361 were diagnosed with T2D. The mean LDL level was 3.37 (mmol/L) with a standard deviation of 1 and the mean BMI was 31.23 (kg/m²) with a standard deviation of 5.8.

In addition to the UKB dataset, we also used data from SPARK (Simons Foundation Powering Autism Research) [27] for autism spectrum disorder (ASD) patients. We accessed the first release of the combined multi-batch SPARK WES dataset, which includes phenotype data for the SPARK Collection Version 7. The details of these samples are available on the SFARI website, https://www.sfari.org/resource/spark/. This dataset includes 69 592 samples processed on the Illumina Global Screening Array and is provided in PLINK [30] format. After removing samples with estimated ancestry other than European (EUR) and missing genotype data, 51 658 samples remained for further analysis. We applied pre-imputation quality control using PLINK1.90 [30], including the removal of SNPs with low genotype call rates (<0.95), minor allele frequencies (<0.01) or deviations from Hardy–Weinberg equilibrium (<1e−06), as well as samples with high missing genotype rates (>0.05). This left us with 455 444 SNPs and 43 891 samples. The genotype data were then phased and imputed to the Haplotype Reference Consortium (HRC) reference panel using the Michigan Imputation Server [59]. After imputation, we applied additional quality control, including the removal of SNPs with low imputation quality (<0.8) or minor allele frequency (<0.01). Finally, the SPARK study data contained 7 194 844 SNPs on the GRCh37/hg19 build, of which 5 948 083 SNPs were also included in the EUR 1KG Phase 3 data. We then retained 12 264 individuals who were ASD probands and had intelligence quotient (IQ) scores to assess the association between PRSs and IQ scores in ASD probands. Our analysis quantified cognitive performance using full-scale, verbal and non-verbal IQ. There were 1026, 785 and 830 ASD probands in SPARK who had both these IQ scores and qualify-controlled genotype data, respectively.

We used the positively correlated and negatively correlated blocks from -hess, SUPERGNOVA and LAVA (with FDR < 0.1) between ASD and CP, CAD and LDL, and T2D and BMI to construct PRS+ and PRS− for ASD, CAD and T2D, respectively. These SNPs were clumped using PLINK1.90 [30], with a significance threshold of 1 for index SNPs, an LD threshold of 0.1 for clumping and a physical distance threshold of 250 kb. PRSs were generated for ASD probands in the SPARK cohort and CAD and BMI cases in the UKB dataset. In addition, we compared CP (measured by IQ), LDL and BMI between patients with high PRS+ and those with high PRS− for relevant disorders.

RESULTS

Simulation results

Basic simulation analysis

We compared the performance of -hess, SUPERGNOVA and LAVA by point estimation of local genetic correlation, local genetic covariance, local heritability, type-I error and statistical power. Since all three methods can use customized reference panels, we performed simulations on both the in-sample reference panel and the external reference panel with matched ancestry to investigate the robustness of these methods to the choice of LD reference panels.

We considered the simulation settings described in the Methods section to simulate traits based on the genotype data in the UKB and used the EUR 1KG Phase 3 data and sample set1 from UKB as reference panels. For the continuous traits generated from non-overlapping samples (set1 and set2), SUPERGNOVA provided unbiased estimates for local genetic correlation, local genetic covariance and local heritability (Figure 1A and Supplementary Figures 2 and 3) in all settings, and the results were robust to the choice of reference panel. However, SUPERGNOVA sometimes had inflated type-I errors (Figure 1B and Supplementary Figure 4). When the EUR 1KG Phase 3 reference panel was used, which did not match the GWAS samples, LAVA overestimated local genetic covariance and local heritability and underestimated local genetic correlation (Figure 1A and Supplementary Figures 2 and 3), and the higher the local genetic covariance, the less accurate the point estimates obtained from LAVA. LAVA had a higher inflated type-I error (~20%) (Figure 1B and Supplementary Figure 4) than SUPERGNOVA and -hess. On the other hand, if the in-sample UKB reference panel was used, LAVA yielded unbiased estimates for local genetic covariance and local heritability and more accurate local genetic correlation estimates with well-controlled type-I error (Figure 1 and Supplementary Figures 2–4). Regardless of the reference panels used, -hess consistently underestimated local genetic covariance and local heritability, particularly when local genetic covariance and local heritability were high (Supplementary Figures 2 and 3). However, it provided unbiased local genetic correlation estimation, possibly due to compensation for both underestimated local genetic covariance and local heritability. The statistical test based on -hess was overly conservative, leading to reduced statistical power (Figure 1B, Supplementary Figure 4).

Figure 1.

Evaluation of local genetic correlation/covariance methods on continuous traits from non-overlapping datasets (set1 and set2) using EUR 1KG Phase 3 and UKB reference panel. (A) Local genetic correlation estimates. The dashed lines represent the true value of local genetic correlation. (B) Type-I error and statistical power. The solid line represents 5% P-values below 0.05 in 100 repeats, and the dashed line represents 10% P-values below 0.05 in 100 repeats.

Since the shared sample size between two traits needs to be provided to -hess, we used both the correct shared sample size and incorrect overlapping sample size (1000) to investigate the impact of this parameter on -hess for the partial and complete overlapping scenarios. In this case, the performance for point estimate and inference by SUPERGNOVA was the same (Supplementary Figures 5–10). However, LAVA did not have well-controlled type-I error with overlapping samples, even when the in-sample reference panel was used (Supplementary Figures 5 and 8). With an incorrect overlapping sample size, -hess had much reduced statistical power (Supplementary Figures 5 and 8).

When only some SNPs were causal, the performance of different methods was similar when all the SNPs were set to be causal, except for LAVA having some inflated type-I error even using the in-sample reference panel. This suggests that the sparsity of causal SNPs does not have much impact on the performances of local genetic correlation/covariance estimation (Supplementary Figures 11–13).

Since all three methods can be applied to binary traits, we also considered binary traits in our simulation. Even though the genetic covariance is estimated on the observed scale, there is no distinction between observed- and liability-scale genetic correlation [14]. The estimates for the genetic correlation of binary traits had similar performances to that for continuous traits except with larger variations across the 100 repeats (Supplementary Figures 14A and 15A). However, the statistical power for binary traits was lower for all methods compared to continuous traits, especially -hess which barely detected any significant blocks in our simulations (Supplementary Figures 14B and 15B). As depicted in Supplementary Figure 20, the increase in sample size reduces the estimation variance but does not influence the biases. As sample sizes grow, the power of the three methods increases. Of note is that LAVA consistently presents large type-I errors when using the external reference panel. This type-I error for LAVA also inflates even when an internal reference panel is used as the sample size increases. Notably, SUPERGNOVA’s type-I error rate also increases with larger sample sizes. Besides, we expanded our simulation also on the Eastern population. The results from this EAS simulation mirrored those from the European ancestry analysis, providing further support to our initial conclusions (Supplementary Figure 21).

LD-related effect sizes

For simulations where the effect sizes were associated with the local LD structure, similar to the simulation setting in LAVA, there was a substantial underestimation of local genetic covariance (Supplementary Figure 16) with nearly no significant block detected by SUPERGNOVA and -hess with the default settings. When we gave -hess the number of eigenvalues that could explain 99% variance and used the UKB reference panel, the performance of -hess improved but still had lower power than LAVA. In this setting, LAVA had the best performance in this simulation setting except for the inflated type-I error when using EUR 1KG Phase 3 data as the reference panel.

Robustness to reference panels

Our simulation findings indicate that the output from LAVA can be unstable. Specifically, when utilizing different reference panels, LAVA does not consistently maintain the same levels of estimation bias, type-I errors and power. Thus, we further investigated the choice of the reference panels on LAVA. We considered the EUR 1KG Phase 3 reference panel, four different UKB reference panels and one CEU reference panel (Method). As shown in Figure 2, when we used two UKB reference panels with 20 000 participants, LAVA had unbiased estimations and well-controlled type-I errors. With smaller UKB reference samples, LAVA could not provide reliable local genetic correlation estimates and well-controlled type-I errors. By comparing the performance between using the CEU reference panel and the UKB reference panel with the same sample size (20 000), the sample size of the reference panel was not a key factor for the performance of LAVA. Our results suggest that LAVA performs well with enough individuals from the genotype data cohorts used as the reference panel.

Figure 2.

Evaluation of LAVA on continuous traits from non-overlapping datasets (set1 and set2) using difference reference panels. (A) Local genetic correlation estimates. The dashed lines represent the true value of local genetic correlation. (B) Type-I error and statistical power. The gray line represents 5% P-values below 0.05 in 100 repeats, and the dashed line represents 10% P-values below 0.05 in 100 repeats. For reference panels, LAVA_1KG represents using the 1KG Phase 3, LAVA_UKB_500 represents using 500 samples from UKB, LAVA_UKB_5K represents using 5000 samples from UKB, LAVA_UKB1_20K and LAVA_UKB2_20K represent using two different sets of 20 000 samples from UKB, and LAVA_CEU_20K represents using 20 000 CEU samples simulated from HAPGEN2.

Number of eigenvalue inputs

By using the EUR 1KG Phase 3, UKB samples from set1 and set2 and 20 000 CEU individuals (Method) as reference panels, we investigated whether the optimal number of eigenvalues used in -hess stays more or less the same for different blocks and different reference panels. The optimal number of eigenvalues here is the one that could result in well-controlled type-I errors and higher powers, although the point estimates derived using these numbers could still be biased. When the reference panel was the same as that from the GWAS samples, the more eigenvalues used, the better the performance for -hess, although it still had limited statistical power (Supplementary Figure 18 and Supplementary Figure 4). With an external reference panel different from the GWAS samples, no consistent pattern was observed, and the optimal number of eigenvalues varied for blocks and reference panels (Supplementary Figures 17 and 19 and Supplementary Table 3). When using the EUR 1KG Phase 3 reference panel, the optimal number of eigenvalues was 93 (95%), 153 (90%), 146 (95%) and 85 (70%), respectively. For the CEU reference panel, the optimal number was 93 (95%), 310 (99%), 42 (70%) and 572 (99%), respectively. These observations could explain the poorer performance of -hess in Figure 1 when using the in-sample UKB reference panel than the external 1KG Phase 3 reference panel. This is because when the in-sample reference panels were used, the larger the number of eigenvalues used, the better the performance, while in Figure 1, the default number was 50.

Local genetic correlation/covariance of 31 complex traits

We considered 31 complex disorders or traits to compare the performance of the three methods. Table 2 summarizes these traits, abbreviations, sample sizes (the number of cases and controls for binary traits), global heritability and its standard error derived from LDSC [14], and the original papers.

Stability based on different reference panels

When using only EUR 1KG Phase 3 genotype data as our reference panel to estimate the local genetic correlation for the above 31 complex traits, there were substantial differences in point estimates and inferences by -hess, SUPERGNOVA and LAVA (Supplementary Material, Appendix C). We focused on comparing the point estimates and detecting significant blocks using different reference panels for the same method in this section because the simulation results showed the importance of the reference panels for both -hess and LAVA. Since the heritability of Height is the largest among all the traits, for a more straightforward and efficient comparison, we decreased the number of trait pairs estimated and compared the results of the genetic correlation between Height and other traits using two other reference panels which were generated using different randomly selected white British UKB samples (Methods) and EUR 1KG Phase 3 reference panel. We used the same methods but different reference panels to compare the local heritability and local genetic correlations in the same block for the same trait or trait pairs. As seen in Supplementary Figures 28–33, SUPERGNOVA displayed the most stable point estimates for local genetic correlation and local heritability using different reference panels, and the estimates from -hess were more stable than LAVA. Even with two different references from the same cohort, i.e. the two UKB reference panels, LAVA resulted in different estimates for the same block for the same pair of traits (Supplementary Figures 32 and 33).

Since the sum of local heritability should equal global heritability and the local genetic covariance should equal global genetic covariance (Methods), we further compared the sum of local heritability and local genetic covariance with global heritability and global genetic covariance with different reference panels. As shown in Supplementary Figure 34, LAVA tended to overestimate local heritability which is consistent with simulation results when the samples of the reference panel and the GWAS did not match. The sum of local heritability was highly concordant with the estimated global heritability for -hess. For SUPERGNOVA, except for three traits, Lupus, OCD and T1D which have the smallest sample sizes among all the traits, the sum of local heritability was also highly concordant with the global heritability. The sum of local genetic covariance was highly correlated with the global genetic covariance for SUPERGNOVA and LAVA but was lower for -hess (Figure 3). As shown in Figure 4 and Supplementary Figure 36, the significant blocks found by SUPERGNOVA and -hess were consistently detected using different reference panels, while the results differed substantially for LAVA with different reference panels.

Figure 3.

Comparisons of the sum of local genetic covariance for 465 trait pairs with the global genetic covariance derived from LDSC. Comparisons of the sum of local genetic covariances estimated from (A) ρ-hess, (B) SUPERGNOVA or (C) LAVA with the global genetic covariances using three different reference panels. Each point represents a trait pair. The color and shape of each data point denote the significance status in global–local correlation analyses. ‘local +’ denotes that there are significant blocks detected between that trait pair and ‘local −’ denotes that there are no corrected blocks detected. ‘global +’ denotes that the global genetic correlation is significant, while ‘global −’ denotes that the global genetic correlation is not significant. The figures are divided into multiple panels with each panel corresponding to different reference panels (EUR 1KG reference panel and UKB reference panels with different samples). The dashed, gray reference line with a slope of 1 represents the line of perfect correlation in each panel. The strength of the relationship is indicated by Pearson correlation coefficients, which are displayed at the bottom of each panel.

Figure 4.

Comparisons of blocks with significant local genetic correlations when using different reference panels. These plots used bars to break down the Venn diagram, illustrating the overlapping significant blocks with different reference panels at a significance level of 0.1 FDR. These blocks were detected by (A) ρ-hess, (B) SUPERGNOVA and (C) LAVA.

PRS analysis

Several studies [20, 60] including SUPERGNOVA have investigated the shared genetics among Autism spectrum disorder (ASD), attention-deficit/hyperactivity disorder (ADHD) and cognitive ability (CP) by utilizing local genetic information. To further compare the results of -hess, SUPERGNOVA and LAVA, we applied these methods to ASD, ADHD and CP. By using a false discovery rate (FDR) cutoff of 0.1, we identified one block by -hess, 55 blocks by SUPERGNOVA and 126 blocks by LAVA with significant local genetic covariances between ADHD, ASD and CP (Supplementary Table 8), respectively. The only block identified by all three methods was on chromosome 6, where ASD and CP were positively correlated (POS: 97094444–98938023). The global genetic correlation between ASD and CP was 0.2 (P = 1.8e−10), between ASD and ADHD was 0.36 (P = 1.14e−11), and between ADHD and CP was −0.38 (P < 1e−11) revealing that the local correlations of CP with ASD and ADHD were bidirectional. As in Supplementary Figure 35, there was no significant block with a negative correlation between ASD and ADHD identified using LAVA and there were only two such blocks detected by SUPERGNOVA. Besides, there was no block where ASD and ADHD showed opposite correlations with CP. SUPERGNOVA identified 12 blocks with positive correlations and 4 blocks with negative correlations between ASD and CP. LAVA identified 14 positively correlated blocks and 14 negatively correlated blocks between the two traits. We constructed positive and negative polygenic risk scores (Methods), referred to as PRS+ and PRS−, of ASD based on independent SNPs from blocks with significant positive or negative local correlations between ASD and CP detected by SUPERGNOVA or LAVA, respectively, for 1026 ASD probands who had both genotypes and IQ scores in SPARK (Methods).

We observed probands with high PRS+ had higher IQ than probands with high PRS− only in PRSs generated utilizing SUPERGNOVA (Figure 5A–I). No negative blocks were detected by -hess, resulting in only PRS+ constructed based on -hess (Figure 5C, F, I). When using PRS+ and PRS− based on SUPERGNOVA, there was a sharp change in the right tails of the PRS distribution analysis of the average full-scale IQ, from 84.7 and 83.1 in the 75th percentile to 89.9 and 75.0 in the 99th percentile for PRS+ and PRS−, respectively (Figure 5A). Similarly, the average non-verbal IQ (Figure 5D) and verbal IQ (Figure 5G) also showed a sharp change in the right tail of the PRS distribution, with respective changes from 93.2 and 92.5 in the 75th percentile to 101.7 and 84.0 in the 99th percentile, and from 94.9 and 91.4 in the 75th percentile to 102.1 and 80.4 in the 99th percentile for PRS+ and PRS−, respectively.

Figure 5.

Phenotype heterogeneity of ASD probands, CAD and T2D patients with PRS+ and PRS−. Average full-scale IQ is computed for different groups defined by PRS based on the significant blocks found by (A) SUPERGNOVA, (B) LAVA and (C) ρ-hess. Average non-verbal IQ is computed for different groups defined by PRS based on the significant blocks found by (D) SUPERGNOVA, (E) LAVA and (F) ρ-hess. Average verbal IQ is computed for different groups defined by PRS based on the significant blocks found by (G) SUPERGNOVA, (H) LAVA and (I) ρ-hess. Average LDL is computed for different groups defined by PRS based on the significant blocks found by (J) SUPERGNOVA and (K) LAVA. Average BMI is computed for different groups defined by PRS based on the significant blocks found by (L) SUPERGNOVA, (M) LAVA and (N) ρ-hess. Each interval indicated the standard error of the average values.

The LAVA study [21] explored the relationship between LDL and CAD, and between BMI and T2D from the angle of multivariate correlation. Here, we conducted a PRS analysis using the bivariate local genetic correlation results between LDL and CAD, and BMI and T2D. The global genetic correlation between LDL and CAD was 0.3 (P < 1e−15). SUPERGNOVA identified 36 positive blocks and 5 negative blocks with significant local genetic correlations between LDL and CAD at an FDR level of 0.1, and LAVA identified 108 positive blocks and 30 negative blocks (Supplementary Table 9). No significant block was identified using -hess. SUPERGNOVA and LAVA identified 22 common blocks with consistent correlation directions, including 21 positive blocks and one negative block on chromosome 5. As displayed in Figure 5J–K, CAD cases with high PRS+ had higher LDL than cases with high PRS− for both SUPERGNOVA and LAVA, with an average LDL changing from 3.41 and 3.32 for the 75th percentile to 3.54 and 3.10 for the 99th percentile when using SUPERGNOVA. However, the trend in LAVA was less apparent, with the average LDL moving from 3.39 and 3.38 for the 75th percentile to 3.38 and 3.21 for the 99th percentile.

When analyzing the local correlations between T2D and BMI, whose global genetic correlation was 0.57 (P < 1e−15), -hess identified 279 significant blocks, SUPERGNOVA identified 176 ones and LAVA identified 589 blocks (Supplementary Table 10). A total of 93 blocks were found by all three methods with just one block on chromosome 3 showing a negative correlation between T2D and BMI, and all these 93 blocks had consistent correlation direction. Among the significant blocks, -hess found 271 that were positively correlated and eight that were negatively correlated, SUPERGNOVA identified 170 that were positively correlated and 6 that were negatively correlated, and LAVA identified 66 that were positively correlated and 23 that were negatively correlated. As demonstrated in Figure 5L–N, T2D cases with high PRS+ had a greater BMI than cases with high PRS− for all three methods. For SUPERGNOVA, the average LDL changed from 32.1 to 31.2 for the 75th percentile to 32.9 and 31.6 for the 99th percentile. For LAVA, the average LDL changes from 32.5 to 30.9 for the 75th percentile to 33.1 and 30.2 for the 99th percentile. For -hess, the average LDL changes from 32.4 and 30.9 for the 75th percentile to 33.4 and 30.7 for the 99th percentile.

DISCUSSION

In recent years, there has been an increasing interest in inferring local genetic correlation in post-GWAS analyses in addition to global genetic correlation. This trend can be attributed to advancements in methodologies for estimating local genetic correlation and detecting locally significant blocks, as well as a growing knowledge of the limitations of global genetic correlation for revealing the underlying genetic similarity between complex traits. Local genetic correlation has also improved association studies and PRS prediction.

The first step for local genetic correlation is determining how to partition the whole genome into approximately independent blocks. The larger the blocks, the more independent the partitions, but larger blocks may mask local information in the same way that global genetic correlation does. On the other hand, smaller blocks may result in LD leakage and biased estimates. The three methods compared in this paper (Table 3) all provide partitions, but also allow users to use user-defined partitions. Another issue that needs to be addressed for local genetic correlation is also considered for global correlation, i.e. how to deal with pervasive sample overlap across GWASs. The common solution for these three methods is to utilize the cross-trait LDSC intercept to calculate the phenotypic correlation. -Hess is the only method that requires the shared sample size between two GWASs as input. However, as the number of GWASs generated by meta-analysis grows, the exact number of overlapping sample sizes is difficult to obtain. Our simulation results suggest that the power of -hess will decrease if an incorrect number of shared sample sizes is given. The other two methods have more stable performances in terms of the sample-overlapping level. The third and most crucial challenge with these three methods is estimating the local LD structure using external reference panels. Ideally, the external reference panels applied should have the same LD structure as the genotype data used to calculate summary statistics. In the real world, because access to individual-level data from the GWAS dataset is typically limited due to practical constraints, it is common to choose an external reference panel. Through extensive simulations and real data analysis, we have demonstrated that the estimation of the local LD matrix is critical for both estimation and inference. SUPERGNOVA is the most robust method for choosing reference panels because it has an adaptive procedure to select the number of eigenvalues and eigenvectors used for different blocks and reference panels. However, the type-I error of SUPERGNOVA is still inflated in some simulation settings. So, developing a more stable adaptive procedure in SUPERGNOVA that can select a more reliable number of eigenvalues may be one potential solution to solve this problem. LAVA recommends using the number of eigenvalues and eigenvectors that explain 99% of the variances. Thus, it performed the best when the genotype data and the reference panel matched perfectly. Therefore, if an in-sample reference panel is available, LAVA would be a suitable choice. However, with different reference panels, LAVA could provide different estimations, and the significant blocks detected were also inconsistent. -Hess needs to be given the number of eigenvalues as input, and the default number is set to be 50. However, the optimal number of eigenvalues and eigenvectors depends on both the local LD structure of the reference panels and the LD structure of the blocks in the genotype data. In summary, -hess can provide unbiased estimates if the proper number of eigenvalues is selected based on different reference panels, while SUPERGNOVA yields unbiased estimates assuming the underlying assumption holds. In addition, LAVA produces unbiased estimates when an in-sample reference panel with sufficient sample sizes is utilized. While -hess generally has well-controlled type-I error rates, it may have lower power. SUPERGNOVA is generally more stable across different reference panels but may sometimes have slightly inflated type-I errors. LAVA only produces well-controlled type-I error rates when an in-sample reference panel with sufficient sample sizes is used. In conclusion, there is a clear need for developing novel methodologies or refining existing ones.

Table 3.

Overview of local genetic correlation methods analyzed in this study

Methods	Assumption	Output	Regions analyzed	Number of eigenvalues/vectors used	Advantages	Disadvantages
ρ -Hess	Fixed-effects models	Bivariate local genetic correlations (with P-values)	User-defined regions	User-defined (default = 50)	1. Well-controlled type-I error	1. Low power
		Local heritability (with P-values)				2. Power drops with wrong shared sample size given
						3. The optimal number of eigenvalues varies based on LD structure, with no standard rule
SUPERGNOVA	Random-effects model	Bivariate local genetic correlations (with P-values)	User-defined regions	Adaptive procedure	1. Robust to the level of sample overlap	1. Inflated type-I error
		Local heritability (without P-values)			2. Robust to the choice of reference panels	2. Poor performance when LD is linked to effect sizes
LAVA	Fixed-effects models	Bivariate and multivariate genetic correlations (with P-values)	User-defined regions	Explaining 99% variance of LD	1. Robust to the level of sample overlap	1. Only using enough in-sample genotypes as reference panels, can provide an unbiased estimator and well-controlled type-I error
		Local heritability (without P-values)			2. The Best performance when using enough in-sample genotypes as reference panels.

Despite extensive simulation settings and real data sets considered, there are limitations in our study. First, all these three methods can provide both estimates and references with user-defined partitions. However, there are also other methods that evaluate the concordance of two traits. For example, LOGODetect [22] uses scan statistics without pre-specified candidate regions to autonomously identify regions with shared genetic components. Although our limited simulation studies suggest the low statistical power of LOGODetect (Supplementary Material, Appendix C; Supplementary Figure 37), a more comprehensive comparison may be warranted in the future. Second, the methods compared in this study can reveal correlated blocks between two traits within a single population (e.g. European). However, there are other methods that could detect corrected blocks between different populations (e.g. European and African) for the same trait [61]. Thus, a more general comparison or review is needed. Thirdly, there is no gold standard to compare these methods in real-world data applications since true local genetic correlations or significantly correlated blocks between phenotypic pairs are unknown. Even though we have conducted PRS analysis to help assess the performances of these methods, other downstream analyses can be done to compare the performance of different methods.

Key Points

• The power of -hess will decrease if an incorrect number of shared sample sizes is given, while SUPERGNOVA and LAVA are robust to the level of sample overlap.

• -hess needs to be given the optimal number of eigenvalues as input to have a good performance, while the optimal number of eigenvalues depends on the local LD structure of the reference panels and the genotype data, which does not have a common rule.

• SUPERGNOVA is robust for the choice of reference panels, while in some situations, the type-I error of SUPERGNOVA can be inflated, and when the effect sizes are associated with LD, SUPERGNOVA shows very poor performance.

• When using an in-sample reference panel with enough sample size, LAVA can provide unbiased estimation and well-controlled type-I error.

• Our paper indicates a need for more advanced methods to calculate local genetic correlation/covariance that can provide unbiased estimation, well-controlled type-I error, and high power in terms of different reference panels and shared sample sizes.

Author Biographies

Chi Zhang is a doctoral student in the Department of Biostatistics, Yale School of Public Health, New Haven, CT, USA.

Yiliang Zhang is a doctoral student in the Department of Biostatistics, Yale School of Public Health, New Haven, CT, USA.

Yunxuan Zhang is a master student in the Department of Biostatistics, Yale School of Public Health, New Haven, CT, USA.

Hongyu Zhao is Ira V. Hiscock Professor of Biostatistics, Professor of Genetics, and Professor of Statistics and Data Science at Yale University, New Haven, CT, USA.

FUNDING

The National Institutes of Health (NIH) Research Project Grant (R01) [GM134005, HG012735].

DATA AVAILABILITY

Details of the GWAS summary data of the 31 complex traits are summarized in Table 2 and can be downloaded publicly. Genotype data used in the simulation and real data analysis were downloaded from UK Biobank under approved data requests (access ref: 29900). Data on ASD probands and siblings were accessed from the SPARK study through the Simons Foundation Autism Research Initiative (SFARI).