Last updated: 2020-10-07
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B. K. Bulik-Sullivan et al. (2015) introduce LD Score regression, which quantifies the contribution of a true polygenic signal or bias by examining the relationship between test statistics and linkage disequilibrium (LD). The LD Score regression intercept can be used to estimate a more powerful and accurate correction factor than genomic control. Variants in LD with a causal variant show an elevation in test statistics in association analysis proportional to their LD (measured by r2) with the causal variant.
Under a polygenic model, in which effect sizes for variants are drawn independently from distributions with variance proportional to \(1/(p(1 – p))\), where \(p\) is the minor allele frequency (MAF), the expected \(\chi^2\) statistic of variant \(j\) is: \[ E(\chi^2|l_j) = Nh^2l_j/M + Na + 1, \] where \(N\) is sample size, \(M\) is the number of SNPs, \(h^2/M\) is the average heritability explained per SNP, \(a\) measures the contribution of confounding biases, such as cryptic relatedness and population stratification, \(l_j=\sum_k r_{jk}^2\) is the for variant \(j\).
Regressing the \(\chi^2\) statistics against LD Scores, the intercept minus one is an estimator of the mean contribution of confounding bias to the inflation in the test statistics.
If there is a mismatch between the LD Scores from the reference population and the target population used for GWAS, then LD Score regression can be biased.
B. Bulik-Sullivan et al. (2015) provide a method to estimate genetic correlation using cross-trait LD Score regression.
For a polygenic trait, SNPs with high LD will have higher \(\chi^2\) statistics on average than SNPs with low LD.
Under a polygenic model, the expected value of \(z_{1j} z_{2j}\) for a SNP \(j\) is \[ E(z_{1j} z_{2j} l_j) = \frac{\sqrt{N_1 N_2} \varrho_g}{M} l_j + \frac{\varrho N_s}{\sqrt{N_1 N_2}}, \] where \(N_i\) is the sample size for study \(i\), \(\varrho_g\) is the genetic covariance, \(l_j\) is the LD Score, \(N_S\) is the number of individuals included in both studies and \(\varrho\) is the phenotypic correlation among the \(N_s\) overlapping samples.
Normalizing genetic covariance by SNP heritabilities yields genetic correlation \[ r_g = \frac{\varrho}{\sqrt{h_1^2 h_2^2}} \] where \(h_i^2\) denotes the SNP heritability from study \(i\).
Bulik-Sullivan, Brendan K, Po-Ru Loh, Hilary K Finucane, Stephan Ripke, Jian Yang, Nick Patterson, Mark J Daly, Alkes L Price, and Benjamin M Neale. 2015. “LD Score Regression Distinguishes Confounding from Polygenicity in Genome-Wide Association Studies.” Nature Genetics 47 (3). Nature Publishing Group: 291–95.
Bulik-Sullivan, Brendan, Hilary K Finucane, Verneri Anttila, Alexander Gusev, Felix R Day, Po-Ru Loh, Laramie Duncan, et al. 2015. “An Atlas of Genetic Correlations Across Human Diseases and Traits.” Nature Genetics 47 (11). Nature Publishing Group: 1236.