Simulate data

Simulate three groups of data by sampling from three normal distributions with the same standard deviation (1) but different means. The set.seed(123) line makes this data reproducible.

set.seed(123)
a <- rnorm(n = 20, mean = 3, sd = 1)
b <- rnorm(n = 20, mean = 3.5, sd = 1)
c <- rnorm(n = 20, mean = 4, sd = 1)
d <- data.frame(grp = rep(c("a", "b", "c"), each = 20),
                y = c(a, b, c))

Calculate the means for each group:

means <- aggregate(y ~ grp, data = d, mean)
means
##   grp        y
## 1   a 3.141624
## 2   b 3.448743
## 3   c 4.106485

Calculate difference in means:

ba <- means$y[means$grp == "b"] - means$y[means$grp == "a"]
ca <- means$y[means$grp == "c"] - means$y[means$grp == "a"]
cb <- means$y[means$grp == "c"] - means$y[means$grp == "b"]
results <- data.frame(comparison = c("b-a", "c-a", "c-b"), 
           diff = c(ba, ca, cb))
results
##   comparison      diff
## 1        b-a 0.3071190
## 2        c-a 0.9648614
## 3        c-b 0.6577424

T-tests

Run t-tests for all three comparisons:

ba <- t.test(y ~ grp, data = d, subset = grp != "c")
ca <- t.test(y ~ grp, data = d, subset = grp != "b")
cb <- t.test(y ~ grp, data = d, subset = grp != "a")

Put p-values in a table:

results$pvalue <- c(ba$p.value, ca$p.value, cb$p.value)
results
##   comparison      diff      pvalue
## 1        b-a 0.3071190 0.289679919
## 2        c-a 0.9648614 0.003078154
## 3        c-b 0.6577424 0.025826189

Notice two are “significant” at the 0.05 level.

ANOVA and Tukey’s HSD

Now analyzing using a one-way ANOVA and conduct pairwise comparisons using Tukey’s HSD. Notice all p-values are higher than their t-test counterparts and only one is “significant” at 0.05 level.

m <- aov(y ~ grp, data = d)
TukeyHSD(m)
##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = y ~ grp, data = d)
## 
## $grp
##          diff         lwr      upr     p adj
## b-a 0.3071190 -0.39465579 1.008894 0.5468419
## c-a 0.9648614  0.26308661 1.666636 0.0045735
## c-b 0.6577424 -0.04403243 1.359517 0.0705975