Can F-test be used when comparing two groups? Why not use T-test?

[Jun Park]

I recently read a paper and the authors run a survey experiment and has 4 groups including a control group.
The author writes the following: the treatment about better responding to economic crises has no systematic effect on exchange rate policy preferences. In addition, the estimated effect from T1 is larger than T2 (F-test βT1 > βT2, p-value = .06).

What I don't understand is, why would the authors use an F-test instead of a one-tailed T-test? Is the result going to be the same? If not, what are some advantages on doing this? Lastly, when they say F-test, does it mean ANOVA?

[Clyde Schechter]

When doing a two-group comparison, the F-test and the t-test are completely equivalent. In fact, the F statistic with 1 numerator df and n denominator df is exactly equal to the square of the corresponding t-statistic with n denominator df. So you will always get the same p-values from both statistics. Since the study you mention started out with 4 groups, their overall analysis probably involved either a regression or an ANOVA, and the F-statistic is what is typically generated there. When they followed up with a 1 df contrast, the software probably just gave them the F-statistic.

[Jun Park]

Clyde Schechter

Hello Clyde, Thank you for your reply!

1. By n denominator df for the t-statistics, that's just the variable df, right? The F-statistic is a ratio so having separate df makes sense, but for the t-statistics would that just be the number of variables?
2. But can the F-test be used to test a directional hypothesis? As the authors write: (F-test βT1 > βT2, p-value = .06). For the t-test, we can use the one-tailed test, but I wasn't sure if that's possible with an F-test.
3. When you wrote "followed up with a 1 df contrast" does that mean having 1 numerator df?

[Clyde Schechter]

1. By n denominator df for the t-statistics, that's just the variable df, right? The F-statistic is a ratio so having separate df makes sense, but for the t-statistics would that just be the number of variables?

In a t-test the df is equal to the number of observations minus 2. (The paired t-test is different, number of pairs minus 1.)

2. But can the F-test be used to test a directional hypothesis? As the authors write: (F-test βT1 > βT2, p-value = .06). For the t-test, we can use the one-tailed test, but I wasn't sure if that's possible with an F-test.

Strictly speaking, no. The F-statistic, being the square of the t-statistic, is always positive, so it is "directionless." However, remember that a one-tailed t-test is done using the same t-statistic as the two-tailed test, just applying a different critical value. You can do the same with the F-statistic--apply a different critical value (the square of the critical value for the one-tailed t-test).

3. When you wrote "followed up with a 1 df contrast" does that mean having 1 numerator df?

Yes. Precisely so.