L4-power

Last updated: 2020-01-15

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File	Version	Author	Date	Message
Rmd	ca9b472	Hae Kyung Im	2020-01-15	wflow_publish(files = “analysis/L4-power.Rmd”)
Rmd	8eeb085	Hae Kyung Im	2020-01-15	power calc
html	8eeb085	Hae Kyung Im	2020-01-15	power calc

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)
}

power calculation - simulate Y=phenotype under null and alternative

nsim = 10000
nsam = 1000
maf = 0.30
r2 = 0.01 ## effect size is calculated as r2 = beta^2 *2*maf*(1-maf)

sig2Y = 1
beta = sqrt( r2 * sig2Y / (2*maf*(1-maf)) )
sig2epsi = sig2Y * (1 - r2)

simpower = function(nsim,nsam,maf,beta)
{
  Xfather = matrix( rbinom(nsam * nsim,1,maf), nsam, nsim )
  Xmother = matrix( rbinom(nsam * nsim,1,maf), nsam, nsim )
  Xboth = Xfather+ Xmother
  Yalt = matrix( rnorm(nsam*nsim), nsam, nsim)*sig2epsi + Xboth * beta
  Ynull = matrix( rnorm(nsam*nsim), nsam, nsim)*sig2Y
  return(list(Yalt=Yalt, Ynull=Ynull, Xmat=Xboth))
}

simp = simpower(nsim,nsam,maf,beta)

define some functions

runassoc = function(X,Y)
{
  pvec = rep(NA,ncol(X))
  bvec = rep(NA,ncol(X))
  for(ss in 1:ncol(X))
  {
    fit = fastlm(X[,ss], Y[,ss])
    pvec[ss] = fit$pval  
    bvec[ss] = fit$betahat
  }
  list(pvec=pvec, bvec=bvec)
}

p2z = function(b,p)
{
  ## calculate zscore from p-value and sign of effect size
  sign(b) * abs(qnorm(p/2))
}

calcz = function(X,Y)
{
  tempo = runassoc(X,Y)
  p2z(tempo$bvec,tempo$pvec)
}

calculate test statistics under the null and alternative

Zalt = calcz(simp$Xmat, simp$Yalt)
Znull = calcz(simp$Xmat, simp$Ynull)

tibble(Y = c(Zalt,Znull), type=c(rep("alt",length(Zalt)),rep("null",length(Znull))) ) %>% ggplot(aes(Y,fill=type)) + geom_density(color=NA,alpha=0.6) + theme_bw(base_size = 15)

Version	Author	Date
8eeb085	Hae Kyung Im	2020-01-15

calculate power

## define significance level

alpha = 0.01

## find threshold for rejection; we want P(Znull > alpha/2) two-sided

threshold = quantile(Znull, 1 - alpha/2)

## calculate proportion of Zalt above threshold

mean(Zalt > threshold)

[1] 0.7341

check with pwr.r.test function

## install.packages("pwr")
 library(pwr)
pwr.r.test(n=nsam, r= sqrt(r2), sig.level = alpha)


     approximate correlation power calculation (arctangh transformation) 

              n = 1000
              r = 0.1
      sig.level = 0.01
          power = 0.7234392
    alternative = two.sided

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] pwr_1.2-2       forcats_0.4.0   stringr_1.4.0   dplyr_0.8.1    
 [5] purrr_0.3.2     readr_1.3.1     tidyr_0.8.3     tibble_2.1.2   
 [9] ggplot2_3.1.1   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