L3-Multiple-Testing

Last updated: 2020-01-15

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File	Version	Author	Date	Message
Rmd	e4490eb	hakyimlab	2020-01-14	winners
html	e4490eb	hakyimlab	2020-01-14	winners
Rmd	b5bd1ad	hakyimlab	2020-01-14	winners
html	b5bd1ad	hakyimlab	2020-01-14	winners
html	0734fad	hakyimlab	2020-01-14	null p winners curse
Rmd	0c53a5b	hakyimlab	2020-01-14	null p winners curse
html	0c53a5b	hakyimlab	2020-01-14	null p winners curse

library(tidyverse)

Registered S3 methods overwritten by 'ggplot2':
  method         from 
  [.quosures     rlang
  c.quosures     rlang
  print.quosures rlang

── Attaching packages ───────────────────────────── tidyverse 1.2.1 ──

✔ ggplot2 3.1.1     ✔ purrr   0.3.2
✔ tibble  2.1.2     ✔ dplyr   0.8.1
✔ tidyr   0.8.3     ✔ 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)
}

## pvalue vs uniform

qqunif = 
function(p,BH=T,CI=T,...)
{
  p=na.omit(p)
  nn = length(p)
  xx =  -log10((1:nn)/(nn+1))
  plot( xx,  -sort(log10(p)),
     xlab=expression(Expected~~-log[10](italic(p))),
        ylab=expression(Observed~~-log[10](italic(p))),
       cex.lab=1.4,mgp=c(2,1,0),
       ... )
  abline(0,1,col='gray')
  if(BH)
    {
      abline(-log10(0.05),1, col='red',lty=1)
      abline(-log10(0.10),1, col='orange',lty=2)
      abline(-log10(0.25),1, col='yellow',lty=3)
      legend('bottomright', c("FDR = 0.05","FDR = 0.10","FDR = 0.25"),
             col=c('red','orange','yellow'),lty=1:3, cex=1)
      abline(h=-log10(0.05/nn)) ## bonferroni
    }
  if(CI)
  {
    ## create the confidence intervals
    c95 <- rep(0,nn)
    c05 <- rep(0,nn)
    ## the jth order statistic from a
    ## uniform(0,1) sample
    ## has a beta(j,n-j+1) distribution
    ## (Casella & Berger, 2002,
    ## 2nd edition, pg 230, Duxbury)
    ## this portion was posted by anonymous on
    ## http://gettinggeneticsdone.blogspot.com/2009/11/qq-plots-of-p-values-in-r-using-ggplot2.html
    
    for(i in 1:nn)
    {
      c95[i] <- qbeta(0.95,i,nn-i+1)
      c05[i] <- qbeta(0.05,i,nn-i+1)
    }

    lines(xx,-log10(c95),col='gray')
    lines(xx,-log10(c05),col='gray')
  }
}

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
0734fad	hakyimlab	2020-01-14
0c53a5b	hakyimlab	2020-01-14

simulate a GWAS data under null (no assoc between X and Y)

nsnp = 10000
nsam = 1000
maf = 0.30

## to simplify, we use the same maf for all SNPs in the GWAS

simGWASnull = function(nsnp,nsam,maf)
{
  Xfather = matrix( rbinom(nsam * nsnp,1,maf), nsam, nsnp )
  Xmother = matrix( rbinom(nsam * nsnp,1,maf), nsam, nsnp )
  Xboth = Xfather+ Xmother

  Y = matrix( rnorm(nsam))
  return(list(Y=Y, Xmat=Xboth))
}

simu = simGWASnull(nsnp, nsam, maf)

run GWAS by regression and show that the p-values are uniformly distributed

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

for(ss in 1:nsnp)
{
  fit = fastlm(simu$X[,ss], simu$Y)
  pvec[ss] = fit$pval  
  bvec[ss] = fit$betahat
}


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

show qqplot against expected null

qqunif(pvec)

what does the most significant association look like

ind = which( pvec == min(pvec) )

boxplot(simu$Y ~ simu$X[,ind])

points(jitter(simu$X[,ind]), simu$Y, type="p")

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

ind = which(pvec < 0.01)

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 3.6.0 (2019-04-26)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Mojave 10.14.6

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_0.8.1     purrr_0.3.2    
[5] readr_1.3.1     tidyr_0.8.3     tibble_2.1.2    ggplot2_3.1.1  
[9] tidyverse_1.2.1

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.2       cellranger_1.1.0 plyr_1.8.4       pillar_1.4.1    
 [5] compiler_3.6.0   git2r_0.25.2     workflowr_1.3.0  tools_3.6.0     
 [9] digest_0.6.19    lubridate_1.7.4  jsonlite_1.6     evaluate_0.14   
[13] nlme_3.1-139     gtable_0.3.0     lattice_0.20-38  pkgconfig_2.0.2 
[17] rlang_0.4.1      cli_1.1.0        rstudioapi_0.10  yaml_2.2.0      
[21] haven_2.1.0      xfun_0.7         withr_2.1.2      xml2_1.2.0      
[25] httr_1.4.0       knitr_1.23       hms_0.4.2        generics_0.0.2  
[29] fs_1.3.1         rprojroot_1.3-2  grid_3.6.0       tidyselect_0.2.5
[33] glue_1.3.1       R6_2.4.0         readxl_1.3.1     rmarkdown_1.13  
[37] modelr_0.1.4     magrittr_1.5     whisker_0.3-2    backports_1.1.4 
[41] scales_1.0.0     htmltools_0.4.0  rvest_0.3.4      assertthat_0.2.1
[45] colorspace_1.4-1 labeling_0.3     stringi_1.4.3    lazyeval_0.2.2  
[49] munsell_0.5.0    broom_0.5.2      crayon_1.3.4