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Extremely sparse models of linkage disequilibrium in ancestrally diverse association studies

Nature Genetics volume 55, pages 1494–1502 (2023)Cite this article

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Abstract

Linkage disequilibrium (LD) is the correlation among nearby genetic variants. In genetic association studies, LD is often modeled using large correlation matrices, but this approach is inefficient, especially in ancestrally diverse studies. In the present study, we introduce LD graphical models (LDGMs), which are an extremely sparse and efficient representation of LD. LDGMs are derived from genome-wide genealogies; statistical relationships among alleles in the LDGM correspond to genealogical relationships among haplotypes. We published LDGMs and ancestry-specific LDGM precision matrices for 18 million common variants (minor allele frequency >1%) in five ancestry groups, validated their accuracy and demonstrated order-of-magnitude improvements in runtime for commonly used LD matrix computations. We implemented an extremely fast multiancestry polygenic prediction method, BLUPx-ldgm, which performs better than a similar method based on the reference LD correlation matrix. LDGMs will enable sophisticated methods that scale to ancestrally diverse genetic association data across millions of variants and individuals.

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Fig. 1: Deriving LDGMs from genome-wide genealogies.
Fig. 2: Accuracy and sparsity of the LDGM precision matrix for a representative LD block.
Fig. 3: Accuracy and sparsity of LDGM precision matrices across the genome.
Fig. 4: Computational advantages of LDGMs.
Fig. 5: Best linear unbiased prediction with LDGMs.

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Data availability

LDGMs, LDGM precision matrices and tree sequences are available from Zenodo (ref. 84; https://doi.org/10.5281/zenodo.8157131). High-coverage phased 1000 Genomes genotype data are available at http://ftp.1000genomes.ebi.ac.uk/vol1/ftp/data_collections/1000G_2504_high_coverage/working/20201028_3202_phased. LD-independent blocks are available at https://github.com/jmacdon/LDblocks_GRCh38. UK Biobank summary statistics and LD are available at s3://broad-alkesgroup-ukbb-ld/UKBB_LD/. Ancestral states are available via Ensembl release 100 and can be downloaded from ftp://ftp.ensembl.org/pub/release-100/fasta/ancestral_alleles (ref. 83).

Code availability

We have released an open-source software package, ldgm v.0.1, implemented in python and MATLAB. ldgm allows inference of LDGMs and LDGM precision matrices, as well as computationally efficient analyses of GWAS summary statistics using LDGMs. It is available at https://github.com/awohns/ldgm and is deposited to Zenodo85 (https://doi.org/10.5281/zenodo.8161389). All the functions for analyzing GWAS summary statistics with LDGMs, including BLUPx-ldgm, are currently implemented in MATLAB; a Python implementation is planned. BLUPx-ldgm is also implemented in bcftools, available at https://github.com/freeseek/score; tskit is available at https://github.com/tskit-dev/tskit. Scripts to reproduce the results of this manuscript are available at https://github.com/awohns/ldgm_paper.

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Acknowledgements

We thank D. J. Weiner, A. Nadig, A. L. Price, R. Walters, H. Finucane, X. Lin, H. Li, B. Lehmann, P. Ralph, G. Gorjanc, J. Kelleher and R. Mazumder for helpful discussions. We also thank G. Genovese for his implementation of BLUPx-ldgm in bcftools.

Author information

Author notes

  1. These authors contributed equally: Pouria Salehi Nowbandegani, Anthony Wilder Wohns.

Authors and Affiliations

  1. Program in Medical and Population Genetics, Broad Institute of MIT and Harvard, Cambridge, MA, USA

    Pouria Salehi Nowbandegani, Anthony Wilder Wohns, Jenna L. Ballard, Eric S. Lander, Benjamin M. Neale & Luke J. O’Connor

  2. Department of Biomedical Informatics, Harvard Medical School, Boston, MA, USA

    Pouria Salehi Nowbandegani, Anthony Wilder Wohns & Luke J. O’Connor

  3. Stanford University School of Medicine, Stanford, CA, USA

    Anthony Wilder Wohns

  4. Graduate Group in Genomics and Computational Biology, University of Pennsylvania, Philadelphia, PA, USA

    Jenna L. Ballard

  5. Department of Biology, MIT, Cambridge, MA, USA

    Eric S. Lander

  6. Department of Systems Biology, Harvard Medical School, Boston, MA, USA

    Eric S. Lander

  7. Analytic and Translational Genetics Unit, Department of Medicine, Massachusetts General Hospital, Boston, MA, USA

    Alex Bloemendal & Benjamin M. Neale

  8. Stanley Center for Psychiatric Research, Broad Institute of MIT and Harvard, Cambridge, MA, USA

    Alex Bloemendal & Benjamin M. Neale

  9. The Novo Nordisk Foundation Center for Genomic Mechanisms of Disease, Broad Institute of MIT and Harvard, Cambridge, MA, USA

    Alex Bloemendal & Benjamin M. Neale

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Contributions

A.W.W., P.S.N. and L.J.O. developed the methods. E.S.L., B.M.N. and A.B. suggested analyses. A.W.W., P.S.N., J.L.B. and L.J.O. performed the experiments. A.W.W., P.S.N., J.L.B., B.M.N. and L.J.O. wrote the paper. L.J.O. supervised the research.

Corresponding authors

Correspondence to Pouria Salehi Nowbandegani, Anthony Wilder Wohns or Luke J. O’Connor.

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Competing interests

B.M.N. is a member of the scientific advisory board at Deep Genomics and Neumora. The remaining authors declare no competing interests.

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Extended data

Extended Data Fig. 1 LD between adjacent pairs of LD blocks.

For all blocks of chromosomes 1 to 22, we evaluated the mean r2 between every pair of SNPs (‘LDsum’) in consecutive LD blocks (n = 1,360 pairs), within each ancestry group. The expected mean r2 is around 1 × 10−3, that is 1/2n. The lower whisker, lower hinge, center, upper hinge and upper whisker correspond to (lower hinge − 1.5× interquartile range (IQR)) and the 25th percentile, median, 75th percentile, and (upper hinge + 1.5× IQR), respectively.

Extended Data Fig. 2 Within-sample accuracy of LDGM precision matrices.

a-c, Boxplots showing the error of the LDGM precision matrix. d-f, Boxplots showing the error of the identity matrix. Three different error metrics were used. Boxplots indicate the median, quartiles and range for the 20 LD blocks on chromosome 22, for each 1000 Genomes ancestry group. a and d show the mean-squared error (see Methods). b and e show the mean-squared error after restricting to SNP pairs with a correlation of r2 > 0.01. c and f show the alternative mean-squared error, defined as m−2 Tr((I−PR)(I−RP)). This measures the difference between PR, the product of the LD correlation matrix and the LDGM precision matrix, and the identity matrix (see Supplementary Note, section 4). Compared with the MSE, the alternative MSE is less sensitive to large eigenvalues of R, probably explaining why it is not elevated for AMR. For the identity matrix, the alternative MSE and the MSE are identical. In all plots, the lower whisker, lower hinge, center, upper hinge and upper whisker correspond to (lower hinge − 1.5× interquartile range (IQR)) and the 25th percentile, median, 75th percentile, and (upper hinge + 1.5× IQR), respectively.

Extended Data Fig. 3 Comparison of LDGM precision matrices with Wen-Stephens shrinkage estimator.

The comparison was performed in EUR, on chromosome 22 only (n = 20 LD blocks). To vary the amount of shrinkage, we changed the sample size parameter in the Wen-Stephens estimator (actual sample size: 1,006). a, Mean-squared error between the Wen-Stephens estimator and the LDGM precision matrix inverse. Dotted line denotes the median MSE between the LD sample correlation matrix and the LDGM precision matrix inverse. b, Mean-squared error between the Wen-Stephens estimator and the sample correlation matrix. Values are larger than the corresponding numbers in a for sample size parameters up to 40, and smaller for sample size parameters of 201 or higher. c, Number of nonzero entries per SNP in the Wen-Stephens estimator. Correlations with absolute value less than 1 × 10−8 are set to zero (consistent with the original paper), resulting in slightly increased sparsity for small values of the sample size parameter. At larger parameter values, no SNP pairs are below the threshold within LD blocks, but this approach can still be used to produce a sparse, banded diagonal matrix when it is not desired to use discrete blocks. Somewhat more sparsity can be achieved by relaxing the 1 × 10−8 threshold, but not without causing increased error. In all plots, the lower whisker, lower hinge, center, upper hinge and upper whisker correspond to (lower hinge − 1.5× interquartile range (IQR)) and the 25th percentile, median, 75th percentile, and (upper hinge + 1.5× IQR), respectively.

Extended Data Fig. 4 Comparison of LDGM precision matrices vs. rank-k approximations for the local LD matrix.

The comparison was performed in EUR, on chromosome 22 only (20 LD blocks), and we considered different values of k. a, To quantify the accuracy of the rank-k approximation to the LD correlation matrix, we computed its MSE at different values of k. k = 10 corresponds most closely to the density of the LDGM precision matrix, which is a symmetric matrix with 20 nonzero entries per SNP (10 per SNP in its upper triangle). The MSE at k = 10 was about 3 times higher than that of the LDGM precision matrix; it was most similar at k = 50. MSE is always zero when k is greater than or equal to the sample size (that is, when k = 1,006). b, To quantify the accuracy of the rank-k approximation to the LD precision matrix, we computed the alternative MSE ratio, which quantifies whether the approximate precision matrix multiplied by the correlation matrix is close to the identity (see Extended Data Fig. 2 and Supplementary Note). By this metric, the LDGM performs much better than a rank-k approximation even at k = 500. c, We calculated the MSE between the rank-k approximation and the inverse of the LDGM precision matrix. This was never significantly smaller than the MSE of the LDGM precision matrix inverse with the sample correlation matrix. In all plots, the lower whisker, lower hinge, center, upper hinge and upper whisker correspond to (lower hinge − 1.5× interquartile range (IQR)) and the 25th percentile, median, 75th percentile, and (upper hinge + 1.5× IQR), respectively.

Extended Data Fig. 5 Performance of LDGM precision matrices in cross-validation.

For each LD block on chromosome 22 (n = 20 LD blocks), we randomly split the 1000 Genomes EUR haploid samples into two subsets of equal size. We computed an LDGM precision matrix from one of the two subsets (the LDGM was constructed from all samples in 1000 Genomes). We computed the MSE for three comparisons: the precision matrix vs. the correlation matrix from the same sample; the precision matrix vs. the correlation matrix from the opposite sample; and the correlation matrix from one sample vs. the correlation matrix from the opposite sample. The lower whisker, lower hinge, center, upper hinge and upper whisker correspond to (lower hinge − 1.5× interquartile range (IQR)) and the 25th percentile, median, 75th percentile, and (upper hinge + 1.5× IQR), respectively.

Extended Data Fig. 6 Accuracy and sparsity of precision matrices constructed from naive LDGMs.

a,b, In 1000 Genomes EUR data from chromosome 22, we compare our inferred LDGM (derived from tree sequences) with a banded-diagonal LDGM, an r2 threshold LDGM, and a banded-diagonal LDGM with a large band for 20 LD blocks. a, MSE versus control LDGMs. b, Density versus control LDGMS. For the first banded LDGM, the band size was chosen to match the number of edges with path weight less than 4 in our tree-sequence based LDGM for each LD block (approximately 50 edges per SNP). For the r2-threshold LDGM, the threshold was chosen to produce the same number of edges. For the large banded-diagonal LDGM, we used a band size that corresponded to the number of edges with path weight less than 8 (approximately 300 edges per SNP). For each LDGM, we computed precision matrices with an L1 penalty of 0.1, and we calculated the mean squared error (a) and the number of edges per SNP in the precision matrix (b). In all plots, the lower whisker, lower hinge, center, upper hinge and upper whisker correspond to (lower hinge − 1.5× interquartile range (IQR)) and the 25th percentile, median, 75th percentile, and (upper hinge + 1.5× IQR), respectively.

Extended Data Fig. 7 Accuracy and density of LDGM precision matrices at different allele frequencies.

For the 20 EUR LDGM precision matrices on chromosome 22, we partitioned SNPs into three bins by their minor allele frequency in EUR. Each bin contained a similar number of SNPs. a, MSE across pairs of SNPs (i, j) where SNP i has the specified allele frequency (and SNP j may or may not). b, Average number of neighbors per SNP in each MAF bin (including edges with SNPs not in the bin). In all plots the lower whisker, lower hinge, center, upper hinge and upper whisker correspond to (lower hinge − 1.5× interquartile range (IQR)) and the 25th percentile, median, 75th percentile, and (upper hinge + 1.5× IQR), respectively.

Extended Data Fig. 8 Tradeoff between accuracy and sparsity with different parameter settings.

a-d, On the 20 LD blocks of chromosome 22 EUR data, we varied the path distance threshold (a, b) and the L1 penalty (c, d). Our default parameter settings are a distance threshold of 4 and an L1 penalty of 0.1. Precision matrix inference runtime also varies with parameter settings, with greater runtime for settings that produce higher density. In all plots, the lower whisker, lower hinge, center, upper hinge and upper whisker correspond to (lower hinge − 1.5× interquartile range (IQR)) and the 25th percentile, median, 75th percentile, and (upper hinge + 1.5× IQR), respectively.

Extended Data Fig. 9 MSE of CpGs compared to non CpG sites on chromosome 21 and 22.

The mutation rate at CpG sites is more than an order of magnitude greater than non-CpG sites, and thus have a higher rate of recurrent mutations (Wohns, 2022). To evaluate how disregarding recurrent mutation effects overall accuracy (see Methods), we examined the MSE of CpG sites vs. non-CpG sites on the 40 LD blocks of chromosomes 21 and 22 between the LDGM precision matrix and sample correlation matrices. The lower whisker, lower hinge, center, upper hinge and upper whisker correspond to (lower hinge − 1.5× interquartile range (IQR)) and the 25th percentile, median, 75th percentile, and (upper hinge + 1.5× IQR), respectively. For numerical results, see Supplementary Table 13.

Extended Data Fig. 10 Runtime for LDGM and precision matrix inference across chromosome 22.

The first five boxplots indicate the runtime for precision matrix inference for each ancestry group; the sixth indicates the runtime to derive the LDGM from the original tree sequence. For the LDGM inference step, we used 5 compute threads (1 for the precision matrix inference step). Runtime varies across LD blocks and ancestry groups due to variation in the number of SNPs. The lower whisker, lower hinge, center, upper hinge and upper whisker correspond to (lower hinge − 1.5× interquartile range (IQR)) and the 25th percentile, median, 75th percentile, and (upper hinge + 1.5× IQR), respectively. For numerical results, see Supplementary Table 4.

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Salehi Nowbandegani, P., Wohns, A.W., Ballard, J.L. et al. Extremely sparse models of linkage disequilibrium in ancestrally diverse association studies. Nat Genet 55, 1494–1502 (2023). https://doi.org/10.1038/s41588-023-01487-8

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