TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Levajkovic)/Übungen 2023W/HW09.5

Simulation of test-power

Simulate the distribution of the $p$-value in the two-sided two-sample $t$-test: Let $X_{1},\dots ,X_{20}$, $Y_{1},\dots ,Y_{20}$ be independent random variables with $X_{i}\sim N(0,1)$ $X_{i}\sim N(d,1)$ for all $i=1,2,\dots 20$. For each $d\in \{0,0.25,0.5\}$, derive $p$-values in 10000 simulations $(H_{0}:d=0)$ and plot them in a histogram of unit area. Comment on your three histograms.

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Hilfreiche Tipps

Fast identisches Beispiel: Übungen 2020W - HW 10.6

Lösungsvorschlag von Lessi

--Lessi 2024-02-07T13:04:11Z

# Simulate the distribution of the p-value in the two-sided two-sample t-test: Let X1, . . . , X20,
# Y1, . . . , Y20 be independent random variables with Xi ∼ N(0, 1) and Yi ∼ 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.

i <- 20

par(mfrow=c(3,1))

for (d in c(0, 0.25, 0.5)) {
  results <- c()
  for (m in 0:10000) {
    sampleX <- rnorm(i, 0, 1)
    sampleY <- rnorm(i, d, 1)
    
    p <- t.test(sampleY, sampleX, var.equal = TRUE)$p.value
    results[m] <- p
  }
  
  hist(results, main = paste("Histogram for d =", d),
       xlab = "P-Value", col = "lightblue", border = "black", probability = TRUE)
}

par(mfrow=c(1,1))