# 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))
 0.3729

## Two-sample $t$-test for independent samples

### $t$-Test

Nj     <- c(18, 21)
DVm    <- rnorm(Nj, 180, 10)
DVf    <- rnorm(Nj, 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
n2 <- Nj
sdPool <- sqrt(((n1-1)*var(DVm) + (n2-1)*var(DVf)) / (n1+n2-2))
(d     <- (mean(DVm) - mean(DVf)) / sdPool)
 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))
 -0.669