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LDSC (DECODE (Seropositive), RA)

Linkage Disequilibrium Score Regression1 was applied to the DECODE meta-GWAS of seropositive rheumatoid arthritis2. The aim was to estimate heritability and look for signs of stratification and confounding.

The results are below:

Parameter Value
h2_liab 0.1003
h2_liab_se 0.02075
Lambda_gc 1.254
Mean_chi2 1.372
Intercept 1.081
Intercept_se 0.01097
Ratio 0.2174
Ratio_se 0.02949

We get a liability-scale heritability of 0.1003, and an intercept indicating a relatively low risk of population stratification.

The original DECODE paper reports an observed-scale heritability of 0.19. To check whether our heritability result is consistent with this, we can convert from liability- to observed-scale heritability. Let \(h^2_O\) denote observed-scale heritability, \(h^2_L\) liability-scale heritability, \(K\) population prevalence, \(P\) sample prevalence, \(\Phi^{-1}\) the inverse of the normal cumulative distribution function, and \(\phi\) the normal density function.

We apply the standard formula and substitute the same prevalence parameters we used in our LDSC run.

\[ \begin{align} h^2_O&= h^2_L \frac{P(1-P) }{ K^2 (1-K)^2 } \phi(\Phi^{-1}(1-K))^2\\ &=0.1003 \frac{0.5^2}{0.005(1-0.005)} \phi(\Phi^{-1}(1-0.005))^2\\ &\approx .1003 (9900) (0.000209)\\ &\approx .1003 (2.07)\\ &\approx 0.21 \end{align} \]

As expected, this result approximates the observed-scale heritability from the DECODE paper (0.19). The remaining discrepancy is likely due to differences in SNP filtering.

To illustrate the fit of the LDSC model to the RA GWAS, I generated the LDSC diagnostic plot below

LDSC diagnostic plot. Genetic variants are binned by LD Score. For each bin, we plot the mean LD score of variants in the bin (x axis) against Wald chi squared statistic (y axis 1) and scaled Wald chi squared statistic (y axis 2). Each plotted bin includes error bars indicating the standard error of bin chi squared mean. We also plot the LDSC regression line (solid) and the line corresponding to chi squared =1 (dashed). If the LDSC model fits the data perfectly, we should expect that the intercept of the regression line is at chi squared=1

Especially at lower LD scores, the LDSC model appears to fit well.

  1. Brendan K Bulik-Sullivan, Po-Ru Loh, Hilary K Finucane, Stephan Ripke, Jian Yang, Nick Patterson, Mark J Daly, Alkes L Price, and Benjamin M Neale. LD Score regression distinguishes confounding from polygenicity in genome-wide association studies. Nature Genetics, 47(3):291–295, 2015. URL: https://pmc.ncbi.nlm.nih.gov/articles/PMC4495769/. ↩

  2. Saedis Saevarsdottir, Lilja Stefansdottir, Patrick Sulem, Gudmar Thorleifsson, Egil Ferkingstad, Gudrun Rutsdottir, Bente Glintborg, Helga Westerlind, Gerdur Grondal, Isabella C Loft, and others. Multiomics analysis of rheumatoid arthritis yields sequence variants that have large effects on risk of the seropositive subset. Annals of the rheumatic diseases, 81(8):1085–1095, 2022. URL: https://www.sciencedirect.com/science/article/pii/S0003496724209766. ↩