t-tests

TODO

  • link to resamplingPerm

One-sample \(t\)-test

Test

set.seed(123)
N    <- 100
DV   <- rnorm(N, 5, 20)
muH0 <- 0
t.test(DV, alternative="two.sided", mu=muH0)

    One Sample t-test

data:  DV 
t = 3.729, df = 99, p-value = 0.0003203
alternative hypothesis: true mean is not equal to 0 
95 percent confidence interval:
  3.186 10.431 
sample estimates:
mean of x 
    6.808 

Effect size estimate (Cohen's \(d\))

(d <- (mean(DV) - muH0) / sd(DV))
[1] 0.3729

Two-sample \(t\)-test for independent samples

\(t\)-Test

Nj     <- c(18, 21)
DVm    <- rnorm(Nj[1], 180, 10)
DVf    <- rnorm(Nj[2], 175, 6)
tIndDf <- data.frame(DV=c(DVm, DVf),
                     IV=factor(rep(c("f", "m"), Nj)))
t.test(DVf, DVm, alternative="less", var.equal=TRUE)
t.test(DV ~ IV, alternative="greater", var.equal=TRUE, data=tIndDf)

    Two Sample t-test

data:  DV by IV 
t = 1.114, df = 37, p-value = 0.1363
alternative hypothesis: true difference in means is greater than 0 
95 percent confidence interval:
 -1.23   Inf 
sample estimates:
mean in group f mean in group m 
          177.0           174.7 

Welch \(t\)-Test

t.test(DV ~ IV, alternative="greater", var.equal=FALSE, data=tIndDf)

    Welch Two Sample t-test

data:  DV by IV 
t = 1.103, df = 34.36, p-value = 0.1388
alternative hypothesis: true difference in means is greater than 0 
95 percent confidence interval:
 -1.272    Inf 
sample estimates:
mean in group f mean in group m 
          177.0           174.7 

Effect size estimate (Cohen's \(d\))

n1 <- Nj[1]
n2 <- Nj[2]
sdPool <- sqrt(((n1-1)*var(DVm) + (n2-1)*var(DVf)) / (n1+n2-2))
(d     <- (mean(DVm) - mean(DVf)) / sdPool)
[1] 0.3577

Two-sample \(t\)-test for dependent samples

Test

N      <- 20
DVpre  <- rnorm(N, mean=90,  sd=15)
DVpost <- rnorm(N, mean=100, sd=15)
tDepDf <- data.frame(DV=c(DVpre, DVpost),
                     IV=factor(rep(0:1, each=N), labels=c("pre", "post")))
t.test(DV ~ IV, alternative="less", paired=TRUE, data=tDepDf)

    Paired t-test

data:  DV by IV 
t = -2.992, df = 19, p-value = 0.003748
alternative hypothesis: true difference in means is less than 0 
95 percent confidence interval:
   -Inf -6.739 
sample estimates:
mean of the differences 
                 -15.97 
DVdiff <- DVpre - DVpost
t.test(DVdiff, alternative="less")

Effect size estimate (Cohen's \(d\))

(d <- mean(DVdiff) / sd(DVdiff))
[1] -0.669

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