11.7 F-Distribution and t-Distribution

Shapes of the F-Distribution

The shape of the F-distribution varies quite a bit depending on the degrees of freedom (df1 and df2). To illustrate, look at the plots below. On the left, we have depicted three F-distributions that have the same df1 (that is df1 = 2) but differ on df2 (2, 12, 1000). On the right, we have held df2 constant at 1000 and varied df1 (1, 5, 30).

When df1 (i.e., df Model) is held constant (left panel of the figure), that means that the number of parameters estimated for the model is held constant. For the three-group model, df1 = 2 – 2 being the number of parameters estimated beyond the one for the empty model. We can see that changing the sample size, and thus the values of df2 (i.e., df Error), has only a slight effect on the shape of the F-distribution when df1 is held constant. Even at a df2 of 12 (the blue line), it’s very similar to the F-distribution where df2 is 1000 (black line). Once df2 gets above 30 or so, it barely changes at all.

Changing the number of parameters estimated for the model (df Model), on the other hand, has a more profound influence on the shape of the F-distribution. In the right panel of the figure above, where we hold the sample size constant at a fairly large df2 of 1000, increasing the number of parameters (df1) from 1 to 5 to 30 produces a big difference in shape. As the number of parameters goes up, e.g., as high as 30, the F-distribution starts to look almost normal in shape.

The F-Distribution and T-Distribution are Actually the Same

We have now used one mathematical model for the sampling distribution of \(b_1\) (the t-distribution) and another for the sampling distribution of PRE and F (the F-distribution). But we found that in the tipping study, whether we use t or F, the p-value comes out exactly the same (.0762).

The reason is that fundamentally, the F-distribution and the t-distribution are actually one and the same! If you randomly sample values from a t-distribution, and then square each one, you will get exactly an F-distribution!

In the graph below on the left we show the distribution of 1000 \(b_1\)s that we created using shuffle(). We know from the prior chapter that this distribution is well modelled by the t-distribution. We then squared each of the 1000 \(b_1\)s and graphed the distribution of 1000 b1_squareds. As you can see, it now looks like the F distribution.

In the case of the Condition model of Tip, we can calculate the t statistic using the t.test function, and the F statistic using supernova().

t.test(Tip ~ Condition, data = TipExperiment, var.equal=TRUE)
supernova(Tip ~ Condition, data = TipExperiment)

data:  Tip by Condition
t = -1.818, df = 42, p-value = 0.0762

Analysis of Variance Table (Type III SS)
Model: Tip ~ Condition

                              SS df      MS     F    PRE     p
----- --------------- | -------- -- ------- ----- ------ -----
Model (error reduced) |  402.023  1 402.023 3.305 0.0729 .0762
Error (from model)    | 5108.955 42 121.642
----- --------------- | -------- -- ------- ----- ------ -----
Total (empty model)   | 5510.977 43 128.162

Notice two things. First, the p-value is exactly the same for the two-sample t-test as it is for the model comparisons using F: .0762. Second, notice the values of t (-1.818) and F (3.305). Guess what you would get if you square -1.818? Yep, 3.305.

Instead of trying to think about how these methods are different from each other (e.g., F-test versus t-test, or the permutation test versus mathematical functions), we want you, for now, to appreciate just how similar they are to each other. They all help us locate our parameter estimates in distributions of other estimates that could have been generated by the empty model.