Correct $p$ value for a t-test

Question

I have two groups and I computed the t-test (both sides, $\alpha$=0.05) on them. The result shows significance difference between the means of two groups.

Now I want to know which group has a higher mean, hence I want to repeat my t-test but this time only one side (e.g. right tail).

First of all is this way correct to find out which mean is higher? Second, do I have to use $\alpha/2$,i.e., $\alpha$=0.025 this time?

Answer 1

You have the estimates of the population means (or can just compute them from the data again if the stats software you used doesn't report them). By definition the group with the larger/higher population mean is the group with the higher mean in the test.

You shouldn't repeat the test but using a one-sided hypothesis once you know the outcome of the two-sided test. Just compute the sample means for the groups from the data and you have all you need to know with the first $t$ test.

For example (using R and some dummy data with known different means):

set.seed(1)
df <- data.frame(x = rnorm(100, mean = c(-5, 5)),
                 grp =  factor(rep(c("A", "B"), times = 50)))
t.test(x ~ grp, data = df)

In R we get:

> t.test(x ~ grp, data = df)

    Welch Two Sample t-test

data:  x by grp
t = -54.683, df = 97.885, p-value < 2.2e-16
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -10.183345  -9.470102
sample estimates:
mean in group A mean in group B 
      -4.804474        5.022249  <------ Group sample means

So we can clearly see which group has the larger mean. But if we didn't get that in the output in whatever stats software we're are using, just proceed by computing the groups means. Here I do it the slow way (better R code is available)

with(subset(df, grp == "A"), mean(x))
with(subset(df, grp == "B"), mean(x))

> with(subset(df, grp == "A"), mean(x))
[1] -4.804474
> with(subset(df, grp == "B"), mean(x))
[1] 5.022249

So even if we don't know from the test which group had the higher mean we can always find it from the sample means (estimates of the population means).

---

(Nicer R code might include:

> aggregate(x ~ grp, data = df, FUN = mean)
  grp         x
1   A -4.804474
2   B  5.022249

.)