Why ever use F-statistic?

Question

We can use F-statistic for determining whether at least one of the predictors has an effect on the response. But why just not take minimal p-value across all predictors? It doesn't require introducing a new concept.

Answer 1

Here I assume you intend to compare some test based on the minimum p-value with an overall F-test.

1. By choosing the smallest p-value and comparing with some particular significance level $\alpha$, your p-value no longer has the same meaning as it would do for a single test. In effect you're testing all individual variables (and if any lead to rejection, you conclude at least one of the predictors has an effect on the response). The overall type I error rate is much higher than a single test. (The multiple testing problem.)

2. If you reduce your significance level on the individual tests, so you make fewer type I errors (to make it more comparable with the overall test at significance level $\alpha$), you're less likely to find a significant effect (you give up power by reducing the type I error rate).

3. Even if you don't account for the multiple testing by shifting the significance level, the univariate statistics can easily all be insignificant when there's a clear joint relationship involving both of them.

Here's an example. First, some data:

y:
 4.941, 4.459, 4.116, 3.759, 5.171, 5.101, 5.454, 5.277, 5.402, 
 4.68, 3.433, 5.508, 4.122, 3.355, 3.622, 4.45, 4.872, 4.202, 
 5.276, 4.415, 5.311, 4.105, 3.282, 4.152, 5.416, 4.615, 3.804, 
 5.299, 4.603, 4.868

x1:
 42.305, 16.828, 46.515, 32.567, 40.827, 45.755, 34.227, 43.799, 
 54.659, 34.991, 15.134, 29.115, 20.617, 1.252, 25.844, 19.563, 
 21.53, 22.989, 38.993, 44.955, 30.799, 32.639, 8.707, 46.945, 
 38.992, 25.717, 40.875, 26.049, 36.121, 39.868

x2:
 24.279, 8.844, 27.888, 19.099, 23.732, 28.648, 19.26, 26.578, 
 32.764, 21.302, 8.583, 17.026, 12.047, 0.085, 16.636, 10.021, 
 12.487, 13.745, 23.557, 26.67, 19.881, 20.23, 4.377, 27.865, 
 23.359, 15.006, 25.909, 14.772, 21.5, 23.002

Regression output (from R:)

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  3.47760    0.32703  10.634 3.74e-11 
x1           0.14999    0.09194   1.631    0.114    
x2          -0.19524    0.14741  -1.324    0.196    
---    
Residual standard error: 0.5884 on 27 degrees of freedom
Multiple R-squared:  0.3167,    Adjusted R-squared:  0.2661 
F-statistic: 6.257 on 2 and 27 DF,  p-value: 0.005851

The smallest p-value is 0.114 -- you wouldn't have rejected the null hypothesis of no association even at the 10% significance level, but the overall regression would lead to rejection even if your significance level were 1%. This is without even dealing with the multiple testing problem.

It's also no help to run separate regressions and check p-values there, because (in a different sort of example to the one above) its quite possible for the to be no relationship in the univariate regressions while there's a strong relationship in the bivariate regression.