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 = 3.7292, df = 99, p-value = 0.0003203
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
  3.185669 10.430568
sample estimates:
mean of x 
 6.808118 

Effect size estimate (Cohen’s \(d\))

(d <- (mean(DV) - muH0) / sd(DV))

[1] 0.3729185

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 = 1.1137, df = 37, p-value = 0.1363
alternative hypothesis: true difference in means is greater than 0
95 percent confidence interval:
 -1.230298       Inf
sample estimates:
mean in group f mean in group m 
       177.0479        174.6580 

Welch \(t\)-Test

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

    Welch Two Sample t-test

data:  DV by IV
t = 1.1032, df = 34.359, p-value = 0.1388
alternative hypothesis: true difference in means is greater than 0
95 percent confidence interval:
 -1.27206      Inf
sample estimates:
mean in group f mean in group m 
       177.0479        174.6580 

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] 0.3577436

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 = -2.9918, df = 19, p-value = 0.003748
alternative hypothesis: true difference in means is less than 0
95 percent confidence interval:
      -Inf -6.739295
sample estimates:
mean of the differences 
              -15.96821 

DVdiff <- DVpre - DVpost
t.test(DVdiff, alternative="less")

Effect size estimate (Cohen’s \(d\))

(d <- mean(DVdiff) / sd(DVdiff))

[1] -0.6689888

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