Last updated: 2020-01-14

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File Version Author Date Message
Rmd 0c53a5b hakyimlab 2020-01-15 null p winners curse
html 0c53a5b hakyimlab 2020-01-15 null p winners curse 
library(tidyverse)
── Attaching packages ─────────────────────────────────────────────────── tidyverse 1.2.1 ──
✔ ggplot2 3.2.1     ✔ purrr   0.3.3
✔ tibble  2.1.3     ✔ dplyr   0.8.3
✔ tidyr   1.0.0     ✔ stringr 1.4.0
✔ readr   1.3.1     ✔ forcats 0.4.0
── Conflicts ────────────────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag()    masks stats::lag()
fastlm = function(xx,yy)
{
  ## compute betahat (regression coef) and pvalue with Ftest
  ## for now it does not take covariates

  df1 = 2
  df0 = 1
  ind = !is.na(xx) & !is.na(yy)
  xx = xx[ind]
  yy = yy[ind]
  n = sum(ind)
  xbar = mean(xx)
  ybar = mean(yy)
  xx = xx - xbar
  yy = yy - ybar

  SXX = sum( xx^2 )
  SYY = sum( yy^2 )
  SXY = sum( xx * yy )

  betahat = SXY / SXX

  RSS1 = sum( ( yy - xx * betahat )^2 )
  RSS0 = SYY

  fstat = ( ( RSS0 - RSS1 ) / ( df1 - df0 ) )  / ( RSS1 / ( n - df1 ) )
  pval = 1 - pf(fstat, df1 = ( df1 - df0 ), df2 = ( n - df1 ))
  res = list(betahat = betahat, pval = pval)

  return(res)
}

calculate probability of at least one false positive (reject null when null is true)

alpha = 0.05

Patleastonemistake = function(m) {1 - (1-alpha)^m}

curve(Patleastonemistake,from = 1, to=100, ylab="Prob at least one wrong", xlab="m = number of tests")
grid()
abline(h=1,col='gray')

Version Author Date
0c53a5b hakyimlab 2020-01-15

show p(null) has uniform distribution when phenotype Y is unrelated to genotype X

nsim = 1000000


nsam = 100
maf = 0.30
Xfather = matrix( rbinom(nsam * nsim,1,maf), nsam, nsim )
Xmother = matrix( rbinom(nsam * nsim,1,maf), nsam, nsim )
Xboth = Xfather+ Xmother

Y = matrix( rnorm(nsam*nsim), nsam, nsim)

pvec = rep(NA,nsim)
bvec = rep(NA,nsim)

for(ss in 1:nsim)
{
  if(round(ss/100)==ss) print(ss)
  fit = fastlm(Xboth[,ss], Y[,ss])
  pvec[ss] = fit$pval  
  bvec[ss] = fit$betahat
}


hist(pvec,xlab="p-value",main="Histogram of p-values under Null")

example of winner’s curse (even when effect size is 0, we get larger when we select significant SNPs)

ind = which(pvec < 0.0001)

hist(abs(bvec[ind]),main='Selected estimated effect sizes')

hist(abs(bvec),main='Unselected estimated effect sizes')

df = tibble(effect = c(bvec[ind],bvec), type = c(rep("signif",length(ind)),rep("all",length(bvec)) ) )

ggplot(df, aes(abs(effect), fill=type)) + geom_density(alpha = 0.6, color=NA) + theme_bw(base_size = 15) + ggtitle("Winner's curse")


sessionInfo()
R version 3.6.1 (2019-07-05)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Catalina 10.15.2

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.6/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] forcats_0.4.0   stringr_1.4.0   dplyr_0.8.3     purrr_0.3.3    
[5] readr_1.3.1     tidyr_1.0.0     tibble_2.1.3    ggplot2_3.2.1  
[9] tidyverse_1.2.1

loaded via a namespace (and not attached):
 [1] tidyselect_0.2.5 xfun_0.11        haven_2.2.0      lattice_0.20-38 
 [5] colorspace_1.4-1 vctrs_0.2.0      generics_0.0.2   htmltools_0.4.0 
 [9] yaml_2.2.0       rlang_0.4.1      later_1.0.0      pillar_1.4.2    
[13] withr_2.1.2      glue_1.3.1       modelr_0.1.5     readxl_1.3.1    
[17] lifecycle_0.1.0  munsell_0.5.0    gtable_0.3.0     workflowr_1.5.0 
[21] cellranger_1.1.0 rvest_0.3.5      evaluate_0.14    labeling_0.3    
[25] knitr_1.26       httpuv_1.5.2     broom_0.5.2      Rcpp_1.0.3      
[29] promises_1.1.0   backports_1.1.5  scales_1.1.0     jsonlite_1.6    
[33] farver_2.0.1     fs_1.3.1         hms_0.5.2        digest_0.6.22   
[37] stringi_1.4.3    grid_3.6.1       rprojroot_1.3-2  cli_1.1.0       
[41] tools_3.6.1      magrittr_1.5     lazyeval_0.2.2   crayon_1.3.4    
[45] whisker_0.4      pkgconfig_2.0.3  zeallot_0.1.0    xml2_1.2.2      
[49] lubridate_1.7.4  assertthat_0.2.1 rmarkdown_1.17   httr_1.4.1      
[53] rstudioapi_0.10  R6_2.4.1         nlme_3.1-142     git2r_0.26.1    
[57] compiler_3.6.1