Last updated: 2021-01-18

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Rmd 184f6ac zouyuxin 2021-01-18 wflow_publish(“analysis/amelialocus1563.Rmd”) 
dat = readRDS('data/locus1563.RDS')
z = dat$gwas$zscore
logp = -log10(pnorm(-abs(z))*2)
R = cor(dat$genotypes)
z_ld_weight = 1/500

# modify R with z
Rmod = cov2cor((1-z_ld_weight)*R + z_ld_weight * tcrossprod(z))
Rmod = (Rmod + t(Rmod))/2

Plot of z scores from GWAS and corresponding p values(the red point is the SNP in CS):

par(mfrow=c(1,2))
plot(z, main='z scores')
points(210, z[210], col='red', pch=16)
plot(logp, ylab='-log10 pvalues', main='-log10 pvalues')
points(210, logp[210], col='red', pch=16)

We reproduce susie_rss result under L=1:

library(susieR)
res = susieR::susie_rss(z, Rmod, L=1)
susie_plot(res, y='PIP')

The CS is

res$sets$cs
$L1
[1] 210

Since the LD matrix is from a reference panel, we used z_ld_weight = 1/500 to adjust LD matrix. The details are in https://stephenslab.github.io/susieR/articles/finemapping_summary_statistics.html#using-ld-from-reference-panel.

The question is why SNP 210 has PIP close to 1 while there are other SNPs in high LD and similar p values. The susie_rss model is based on the projected z scores in the column space of LD.

After projecting z into column space of LD, the z scores are

eigenR = eigen(Rmod, symmetric = T)
eigenR$values[abs(eigenR$values) < 1e-8] = 0

Rhat = eigenR$vectors %*% (eigenR$values * t(eigenR$vectors))
U = eigenR$vectors[,eigenR$values > 0]
UUz = U %*% crossprod(U, z)
par(mfrow=c(1,2))
plot(UUz, main='projected z scores')
points(210, UUz[210], col='red', pch=16)
plot(-log10(pnorm(-abs(UUz))*2), ylab = '-log10 p', main='- log10 pvalues for projected z scores')
points(210, -log10(pnorm(-abs(UUz))*2)[210], col='red', pch=16)

The 210 SNP has the strongest signal. The minimum correlation between the top SNP and the SNPs around (with p values \(< 10^{-50}\)) it is 0.88. There is no SNP in perfect correlation with SNP 210.

With L = 1, the PIP doesn’t depend on LD. The PIP is normalized Bayes Factor and the Bayes Factor is related to the projected z scores, \(BF\propto \exp(0.5* \text{(posterior variance)} * \text{(projected z)}^2)\).

par(mfrow=c(2,2))
plot(UUz^2, ylab = '(projected z)^2', main='(projected z)^2')
points(210, UUz[210]^2, col='red', pch=16)
plot(exp(UUz^2), ylab='exp((projected z)^2)', main = 'exp((projected z)^2)')
points(210, exp(UUz[210]^2), col='red', pch=16)
post_var = 1/(1+1/252)
bf = exp(0.5*post_var*UUz^2)
plot(bf, ylab='BF', main='BF')
points(210, bf[210], col='red', pch=16)
plot(bf/sum(bf), ylab='normalized BF', main='PIP = normalized BF')
points(210, bf[210]/sum(bf), col='red', pch=16)

The SNP 210 has PIP close to 1 because it has the strongest z score and there is no SNP in perfect LD with it.


sessionInfo()
R version 4.0.3 (2020-10-10)
Platform: x86_64-apple-darwin17.0 (64-bit)
Running under: macOS Big Sur 10.16

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRblas.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] susieR_0.9.57   workflowr_1.6.2

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.5       plyr_1.8.6       pillar_1.4.7     compiler_4.0.3  
 [5] later_1.1.0.1    git2r_0.27.1     tools_4.0.3      digest_0.6.27   
 [9] evaluate_0.14    lifecycle_0.2.0  tibble_3.0.4     gtable_0.3.0    
[13] lattice_0.20-41  pkgconfig_2.0.3  rlang_0.4.10     Matrix_1.2-18   
[17] rstudioapi_0.13  yaml_2.2.1       xfun_0.19        dplyr_1.0.2     
[21] stringr_1.4.0    knitr_1.30       generics_0.1.0   fs_1.5.0        
[25] vctrs_0.3.6      tidyselect_1.1.0 rprojroot_2.0.2  grid_4.0.3      
[29] reshape_0.8.8    glue_1.4.2       R6_2.5.0         rmarkdown_2.5   
[33] purrr_0.3.4      ggplot2_3.3.3    magrittr_2.0.1   whisker_0.4     
[37] scales_1.1.1     promises_1.1.1   ellipsis_0.3.1   htmltools_0.5.0 
[41] colorspace_2.0-0 httpuv_1.5.4     stringi_1.5.3    munsell_0.5.0   
[45] crayon_1.3.4