TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Bura)/Übungen 2020W/HW10.6

Distribution of the p-value in simulations in the (two-sided) two-sample t-test

Let X₁,...,X₂₀,Y₁,...,Y₂₀ be independent random variables with X_i ∼ N(0,1) and Y_i ∼ N (d, 1) for all i = 1, 2, . . . , 20. For each d ∈ {0, 0.25, 0.5}, derive p-values in 10000 simulations (H0 : d = 0) and plot them in a histogram of unit area. Comment on your three histograms.

Lösungsvorschlag von Friday

--Friday Sa 30 Jan 2021 17:50:00 CET

# Statistics and Probability - HW #10
# Friday 
# Duedate: 14.12.2020

create_plot <- function(d, col) {
  pvalues <- c()
  for (i in 1:10000){
    x <- rnorm(20,0,1)
    y <- rnorm(20,d,1)
    pvalues <- c(pvalues, t.test(x, y)$p.value)
  }
  hist(pvalues, 
       ylab="Density", 
       xlab="p-value", 
       main = sprintf("Distribution (d=%.2f)", d),
       col=col,
       prob=TRUE)
}

(par(mfrow=c(3,1)))
col = rainbow(3, alpha = 0.3)
create_plot(0, col[1])
create_plot(0.25, col[2])
create_plot(0.5, col[3])