Power of discovery variant to associate with PRS

Author

Gibran Hemani

Published

March 20, 2025

Background

Given that a variant associates with a trait due to polygenic confounding, what is the minimum \(R^2\) of the PRS required to detect polygenic confounding with 80% power?

library(pwr)
library(ggplot2)

param <- expand.grid(
    N = 500000,
    Fdisc = qf(10^-c(8:30), 1, 500000, low=FALSE),
    nsig = c(1, 10, 100)
)

param$r2 = param$Fdisc / param$N
param$validate_alpha = 0.05 / param$nsig
param$validate_r2 = sapply(1:nrow(param), \(i) {
    pwr.r.test(param$N[i], sig.level=param$validate_alpha[i], power=0.8)$r^2
})
param$prs_r2 = param$validate_r2 / param$r2

ggplot(param, aes(x=-log10(pf(Fdisc, 1, 500000, low=F)), y=prs_r2)) +
geom_point(aes(colour=as.factor(nsig))) +
geom_line(aes(colour=as.factor(nsig))) +
theme_bw() +
labs(x="Discovery -log10 p-value", y = "Minimim cor(y,PRS)^2 needed for polygenic confounding to be detected", colour="Number of GWAS\nhits to test")

sessionInfo()
R version 4.4.3 (2025-02-28)
Platform: aarch64-apple-darwin20
Running under: macOS Sonoma 14.6.1

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRblas.0.dylib 
LAPACK: /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.12.0

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

time zone: Europe/London
tzcode source: internal

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

other attached packages:
[1] ggplot2_3.5.1 pwr_1.3-0    

loaded via a namespace (and not attached):
 [1] vctrs_0.6.5       cli_3.6.3         knitr_1.48        rlang_1.1.4      
 [5] xfun_0.48         generics_0.1.3    jsonlite_1.8.9    labeling_0.4.3   
 [9] glue_1.8.0        colorspace_2.1-1  htmltools_0.5.8.1 scales_1.3.0     
[13] fansi_1.0.6       rmarkdown_2.27    grid_4.4.3        evaluate_1.0.1   
[17] munsell_0.5.1     tibble_3.2.1      fastmap_1.2.0     yaml_2.3.10      
[21] lifecycle_1.0.4   compiler_4.4.3    dplyr_1.1.4       htmlwidgets_1.6.4
[25] pkgconfig_2.0.3   farver_2.1.2      digest_0.6.37     R6_2.5.1         
[29] tidyselect_1.2.1  utf8_1.2.4        pillar_1.9.0      magrittr_2.0.3   
[33] withr_3.0.2       tools_4.4.3       gtable_0.3.6