Inference in regression: F-test Jan 25 2019

Homework #1

Solutions on canvas

I graded the initial data analysis.

Homework #3

I’ve posted an example with some guidelines as 01-initial-data-analysis-report, but I started from 01-initial-data-analysis-draft.

Key things I’ll be looking for in HW #3:



t-tests on individual parameters only allow us to ask a limited number of questions.

To ask questions about more than one coefficient we need something more complicted.

F-tests do this by comparing nested models. In practice, the hard part is translating a scientific question in a comparison of two models.


Let \(\Omega\) denote a larger model of interest with \(p\) parameters
and \(\omega\) a smaller model that represents some simplification of \(\Omega\) with \(q\) parameters.

Intuition: If both models “fit” as well as each other, we should prefer the simpler model, \(\omega\). If \(\Omega\) shows substantially better fit than \(\omega\), that suggests the simplification is not justified.

How do we measure fit? What is substantially better fit?


\[ F = \frac{(\RSS{\omega} - \RSS{\Omega})/(p - q)}{\RSS{\Omega}/(n - p)} \]

Null hypothesis: the simplification to \(\Omega\) implied by the simpler model, \(\omega\).

Under the null hypothesis, the F-statistic has an F-distribution with \(p-q\) and \(n-p\) degrees of freedom.

Leads to tests of the form: reject \(H_0\) for \(F > F_{p-q, n-p}^{(\alpha)}\).

Deriving this fact is beyond this class (take Linear Models).

Example: Overall regression F-test

The overall regression F-test asks if any predictors are related to the response.

Full model: \(Y = X\beta + \epsilon, \quad \epsilon \sim N(0, \sigma^2 I)\)
Reduced model: \(Y = \beta_0 + \epsilon\)

Null hypothesis: \(H_0: \beta_1 = \beta_2 = \ldots = \beta_{p-1} = 0\)
All the parameters (other than the intercept) are zero.

Alternative hypothesis: At least one parameter is non-zero.

Exercise: question #1 on handout

If there is evidence against the null hypothesis:

If there is no evidence against the null hypothesis:

Example: One predictor

Null hypothesis: \(\beta_j = 0\)

Equivalent to the t-test, reject null if \[ |t_j| = \left|\frac{\hat{\beta_j}}{\SE{\hat{\beta_j}}}\right| > t_{n-p}^{\alpha/2} \]

In fact, in this case, \(F = t_j^2\).

Exercise: questions #2 & #3 on handout

Other examples

Exercise: questions #4 & #5 on handout

We can’t do F-tests when