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Before the class install R package `pwr`, `tidyverse`

install.packages("pwr")
install.packages("tidyverse")

Workflow overview

In this vignette, we’d like to:

Write a simulator to simulate genotype and phenotype under some pre-specified model.
Simulate genotype and phenotype using the simulator under the null and alternative.
Perform certain hypothesis test.
Calculate the power of the test.

Consider a single locus, its genotype is \(X\). We pre-specify the model for continuous trait \(Y\) and genotype \(X\) as

\[Y = \beta X + \epsilon, \epsilon ~ N(0, \sigma_\epsilon^2)\] where we assume

genotype of this locus follows Hardy-Weinberg equilibrium with a pre-specified minor allele frequency.
residual variance 1.

Phenotype-genotype simulator

To simulate genotype, we assume the locus is bialleilic and each individual is diploid. So that \(X \sim Binomial(2, f)\) with \(f\) as minor allele frequency (here we encode minor allele as 1 and major allele as 0). with the linear model for simulate \(Y\) above, this basically means we are simulating \(Y\) under the additive model.

Given genotype, to simulate phenotype, we need to know \(\beta\) and \(\sigma_\epsilon^2\).

library(tidyverse)

── Attaching packages ─────────────────────────────────────── tidyverse 1.3.1 ──

✔ ggplot2 3.3.5     ✔ purrr   0.3.4
✔ tibble  3.1.2     ✔ dplyr   1.0.7
✔ tidyr   1.1.3     ✔ stringr 1.4.0
✔ readr   1.4.0     ✔ forcats 0.5.1

── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag()    masks stats::lag()

simulate_genotype = function(maf, num_individuals, num_replicates) {
  # maf: minor allele frequency
  # num_individuals: the number of individuals in each replicates
  # num_replicates: the number of replicates
  # it returns a matrix with num_individuals rows and num_replicates columns
  genotype = matrix( 
    rbinom(num_individuals * num_replicates, 2, maf), 
    nrow = num_individuals, 
    ncol = num_replicates 
  )
  return(genotype)
}
simulate_phenotype = function(genotype, beta, sig2epsi) {
  # genotype: each column is one replicate 
  # beta: effect size of the linear model
  # sig2epsi: the variance of the noise term
  num_individuals = nrow(genotype)
  num_replicates = ncol(genotype)
  epsilon = matrix( 
    rnorm(num_individuals * num_replicates, mean = 0, sd = sqrt(sig2epsi)), 
    nrow = num_individuals, 
    ncol = num_replicates 
  )
  phenotype = genotype * beta + epsilon
  return(phenotype)
}

linear_model_simulator = function(num_replicates, num_individuals, maf, beta, sig2epsi) {
  # simulate genotype
  X = simulate_genotype(maf, num_individuals, num_replicates)
  
  # simulate phenotype given genotype and model parameters
  Y = simulate_phenotype(X, beta, sig2epsi)
  return(list(Y = Y, X = X))
}

Run the simulator under the null and alternative

Here we simulate 1000 individuals per replicate and 100 replicates in total. With parameters:

Minor allele frequency is 0.3.
effect size (\(\beta\)) of the minor allele 0.05. Effect size is the coefficient in a regression model which measures the regression effect of the locus per copy of the variant allele.
Variance of residual (\(\sigma_\epsilon^2\)) = 1.

# specify paramters
nindiv = 1000
nreplicate = 5000
maf = 0.30
b = 0.05
sig2e = 1

# run simulator 
## under the alternative
data_alt = linear_model_simulator(nreplicate, nindiv, maf, 0.05, sig2e)  
## under the null
data_null = linear_model_simulator(nreplicate, nindiv, maf, 0, sig2e)

Perform hypothesis test

The following chunk of R code implement hypothesis test procedure based on linear regression. Essentially, the R function calcz takes genotype X and Y and returns test statistic z-score.

runassoc = function(X,Y)
{
  pvec = rep(NA,ncol(X))
  bvec = rep(NA,ncol(X))
  for(ss in 1:ncol(X))
  {
    x = X[,ss]
    y = Y[,ss]
    fit = lm(y~x)
    pvec[ss] = summary(fit)$coefficients[2,4]  
    bvec[ss] =  summary(fit)$coefficients[2,1]  
  }
  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)
}

Now that we can calculate test statistics under the null and alternative.

Zalt = calcz(data_alt$X, data_alt$Y)
Znull = calcz(data_null$X, data_null$Y)

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
1cb2d1e	simingz	2024-01-07

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] 0.048

check with pwr.r.test function

library(pwr)
calc_r = function(b,maf,sdy) {sdx = sqrt(2 * maf * (1-maf)); sdx * b * sdy}
r = calc_r(b=b,maf=maf,sdy= sqrt(sig2e + b**2*2*maf*(1-maf)))
pwr.r.test(n = nindiv, r= r, sig.level = alpha)


     approximate correlation power calculation (arctangh transformation) 

              n = 1000
              r = 0.03242071
      sig.level = 0.01
          power = 0.06048404
    alternative = two.sided

Reference

Credits go to https://hakyimlab.github.io/hgen471/L4-power.html.

sessionInfo()

R version 4.1.0 (2021-05-18)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: CentOS Linux 7 (Core)

Matrix products: default
BLAS:   /software/R-4.1.0-no-openblas-el7-x86_64/lib64/R/lib/libRblas.so
LAPACK: /software/R-4.1.0-no-openblas-el7-x86_64/lib64/R/lib/libRlapack.so

locale:
 [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C         LC_TIME=C           
 [4] LC_COLLATE=C         LC_MONETARY=C        LC_MESSAGES=C       
 [7] LC_PAPER=C           LC_NAME=C            LC_ADDRESS=C        
[10] LC_TELEPHONE=C       LC_MEASUREMENT=C     LC_IDENTIFICATION=C 

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

other attached packages:
 [1] pwr_1.3-0       forcats_0.5.1   stringr_1.4.0   dplyr_1.0.7    
 [5] purrr_0.3.4     readr_1.4.0     tidyr_1.1.3     tibble_3.1.2   
 [9] ggplot2_3.3.5   tidyverse_1.3.1

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.9       lubridate_1.7.10 assertthat_0.2.1 rprojroot_2.0.2 
 [5] digest_0.6.27    utf8_1.2.1       R6_2.5.0         cellranger_1.1.0
 [9] backports_1.2.1  reprex_2.0.0     evaluate_0.20    highr_0.9       
[13] httr_1.4.2       pillar_1.6.1     rlang_1.1.0      readxl_1.3.1    
[17] rstudioapi_0.13  whisker_0.4      jquerylib_0.1.4  rmarkdown_2.21  
[21] labeling_0.4.2   munsell_0.5.0    broom_0.7.8      compiler_4.1.0  
[25] httpuv_1.6.1     modelr_0.1.8     xfun_0.38        pkgconfig_2.0.3 
[29] htmltools_0.5.5  tidyselect_1.1.1 workflowr_1.6.2  fansi_0.5.0     
[33] crayon_1.5.2     dbplyr_2.1.1     withr_2.5.0      later_1.2.0     
[37] grid_4.1.0       jsonlite_1.7.2   gtable_0.3.0     lifecycle_1.0.3 
[41] DBI_1.1.1        git2r_0.28.0     magrittr_2.0.1   scales_1.1.1    
[45] cli_3.6.1        stringi_1.6.2    cachem_1.0.5     farver_2.1.0    
[49] fs_1.6.1         promises_1.2.0.1 xml2_1.3.2       bslib_0.4.2     
[53] ellipsis_0.3.2   generics_0.1.0   vctrs_0.3.8      tools_4.1.0     
[57] glue_1.4.2       hms_1.1.0        fastmap_1.1.0    yaml_2.2.1      
[61] colorspace_2.0-2 rvest_1.0.0      knitr_1.42       haven_2.4.1     
[65] sass_0.4.0

e1-GWAS-power

Siming Zhao

2024-01-07

Before the class install R package `pwr`, `tidyverse`

Workflow overview

Phenotype-genotype simulator

Run the simulator under the null and alternative

Perform hypothesis test

Calculate power

check with pwr.r.test function

Reference

e1-GWAS-power

Siming Zhao

2024-01-07

Before the class install R package pwr, tidyverse

Workflow overview

Phenotype-genotype simulator

Run the simulator under the null and alternative

Perform hypothesis test

Calculate power

check with pwr.r.test function

Reference

Before the class install R package `pwr`, `tidyverse`