# In multiple linear regression why is it best to use an $F$-statistic when evaluating predictors?

I am currently going through Hastie and Tibshirani's 'Introduction to Statistical Learning' textbook and I have come across something I don't understand on page 77. I have two questions.

The author states that if we had 100 variable predictors (with respective coefficients $\beta_i$) and supposed that the null hypothesis, $$H_0 : \beta_1 = ... = \beta_{100} = 0$$ were true, then roughly $5\%$ of the $p$-values would fall below $0.05$ by chance, and therefore we might wrongly conclude that certain predictors are related to the response. Why would this happen by chance? Is this simply a mathematical truth?

In addition, the author then goes on to state that the $F$-statistic is a better measure because "if $H_0$ were true, then there is only a $5\%$ chance that the $F$-statistic would result in a $p$-value below $0.05$". I don't understand the difference - could somebody explain a bit more clearly?

When you calculate the $p$-value of an individual coefficient, you're looking at the magnitude of the coefficient, the standard error of the coefficient, making some distributional assumptions about it, and asking the following question: "what is the probability of seeing a coefficient value this extreme if the true value is actually zero?"
If our distributional assumptions are correct, any given coefficient with a true value of 0 will report a <0.05 $p$-value approximately 5% of the time. That's not so much of a problem if we only have one predictor, but by the law of large numbers, if we have lots of predictors, we'd expect 5% of them to report a $p$-value this low. This makes the $p$-values in high-dimensional regressions hard to interpret.
The $F$-test is different. Instead of evaluating every single coefficient for statistical significance, it applies a single test to the entire regression. So instead of having 100 chances to throw up an erroneous $p$-value, it only has one chance. This makes the $F$-test useful for evaluating whether or not there is a regression effect for high-dimensional regressions.
• Thank you for your help. So, just to clarify one other thing, supposing we have $n$ predictors the $p$-value, say for $\beta_1$, is calculated via the hypothesis: $H_0: \beta_1 = 0, \ \ H_a: \beta_1 \neq 0$, correct? i.e. $H_0$ would be the case in which only $\beta_1$ is zero and the rest are non-zero? Mar 23, 2017 at 14:02
• When you calculate the $p$-value for a single predictor, you are calculating it independently of all the other predictors. So your null hypothesis in this case (call it $H_0^{\beta_1}$) would simply be the case in which $\beta_1 = 0$. It doesn't tell you anything about your other predictors. Mar 23, 2017 at 14:12