f-test

F-test for regression

Q1 of the four “important questions”: is at least one of the predictors useful? Generalizes the t-test to multiple coefficients; reduces to the t-test when $p=1$ ($F_{1,n-p-1}=t_{n-p-1}^2$). The prof was emphatic: “won’t ask you to compute it”, only why you’d use it and what null it tests.

Definition (prof’s framing)

Test the joint null that all slopes are zero:

$H_0:{\beta }_1={\beta }_2=\cdots ={\beta }_p=0\text{vs.}{H}_1:\text{at least one }{\beta }_j\ne 0.$

“It’s like we’re assuming innocence so that we can prove guilty… We don’t prove anything in statistics… at least I’ve never proved anything with data.” - L06-linreg-2

Test statistic:

$F=\frac{(\mathrm{TSS}-\mathrm{RSS})/p}{\mathrm{RSS}/(n-p-1)}\sim F_{p,n-p-1}\text{ under }{H}_0.$

Subset variant: test that a group of $q$ coefficients are simultaneously zero (e.g. all dummies of a $K$-level factor):

$F=\frac{({\mathrm{RSS}}_0-\mathrm{RSS})/q}{\mathrm{RSS}/(n-p-1)}\sim F_{q,n-p-1}.$

Notation & setup

• Large model: $p+1$ regression parameters, RSS.
• Small model: $p+1-q$ parameters (drop $q$ predictors), ${\mathrm{RSS}}_0$.
• Numerator df = $p$ (all slopes) or $q$ (subset variant).
• Denominator df = $n-p-1$.
• Two equivalent computations: from $R^2$ ($F=R^2/(1-R^2)\cdot (n-p-1)/p$), or from the RSS difference.

Insights & mental models

Reduces to t when $p=1$

“The F-statistic generalizes [the t-test] and degrades back to the t-statistic when p = 1.” - L06-linreg-2

Specifically, $F_{1,n-p-1}=t_{n-p-1}^2$. So F isn’t new machinery, it’s the multivariate generalization. Both come out of the same Gaussian-error sampling distribution.

Why ask Q1 first

Slide flag from L06-linreg-2: “only checking individual p-values is dangerous.” Why?

“The variables can actually be correlated, and then none of them actually look significant, but overall the test is very significant.” - L06-linreg-2

So sequence: F-test (“is anything good at all?”) → drill into individual t’s. Skipping to t-tests can hide a collinearity-masked signal. This is the case-in-point connection between collinearity and t-test-and-significance.

The subset / partial F

Useful for testing whether a block of predictors matters, most importantly, all $K-1$ dummies for a categorical factor. The R idiom is anova(small, large), comparing nested models. The denominator uses the larger model’s RSS / df.

Where it doesn’t generalize

The prof’s bigger reservation:

“I’m more interested in things that can generalize to other distributions. This thing’s rather specific to the case of linear regression.” - L06-linreg-2

Outside Gaussian linear regression (logistic, GLM, GAM), the F machinery doesn’t transfer cleanly. Use likelihood-ratio tests / deviance comparisons instead, taught later.

”It matters for ANOVA”

The prof noted F is the workhorse of ANalysis Of VAriance, heavily used in design of experiments / engineering. That’s a different course; the machinery is the same idea (block-of-coefficients null hypothesis).

Exam signals

F-test scope flag - verbatim

“I’m going to say right now I probably won’t ask any questions about an F-test, which is what I’m going to show you in a minute. I think it’s too boring for this class, to be honest… I don’t think I’m going to say anything about an F-test. I’m more interested in things that can generalize to other distributions. This thing’s rather specific to the case of linear regression.” - L06-linreg-2

“If I was going to ask you something about this, it’d be more about the null hypothesis. And then you could say, ‘I would use an F-test.’ But I’m not going to make you compute it because I honestly don’t care.” - L06-linreg-2

“I might say in the multivariate linear regression case, how would I test if at least one of the predictors is useful in predicting the response, or I might ask: why would I want to know this, what’s the point?” - L06-linreg-2

2025 Q6a reformulation: “Is there evidence that variable Z is relevant?” → he’d say what test would you use (F-test for ANOVA), without making you compute it numerically. - L27-summary

Pitfalls

• Computing the statistic. Don’t bother memorizing the formula, prof said he won’t ask. Memorize the null hypothesis and the qualitative reasoning instead.
• Confusing the partial F with the global F. The global F tests all slopes = 0. The partial F tests a subset (e.g., all dummies of a factor). Same machinery, different RSS comparison.
• Skipping F. Only checking individual t’s misses correlated-predictor cases. See collinearity.
• Generalizing to GLMs. Doesn’t transfer; use likelihood-ratio tests in module 4 onward.
• F-statistic ≈ 1 under H₀. A common conceptual checkpoint: if the model explains nothing, numerator and denominator are both estimating the noise, so $F\approx 1$. Big F → real signal.

Scope vs ISLP

• In scope: the null hypothesis of the global F-test, the conceptual reason to use it (avoid the t-test trap under collinearity / many predictors), the partial F for testing categorical predictors, the equivalence with t when $p=1$.
• Look up in ISLP: §3.2.2 (pp. 75-77, Is There a Relationship?): concise treatment with the formula and the partial F via anova().
• Skip in ISLP (book-only / prof excluded): F-test mechanics, p-value computation from the F-distribution, ANOVA tables in detail. Walpole’s “good for classic statistics” reference is what the prof points at if you want the full classical treatment.

Exercise instances

None directly tagged. The F-test appears implicitly in Exercise 3.1d (testing the 3-level factor origin with anova()), but the learning target there is categorical encoding, not the F-test itself.

How it might appear on the exam

• State the null hypothesis. Most likely: “What test would you use to check whether any predictor is useful?” Answer: F-test on $H_0:{\beta }_1=\cdots ={\beta }_p=0$.
• Why not just use t-tests? Because under collinearity, individual t’s can both be insignificant while the joint F is highly significant. State the trap.
• Test a categorical predictor. “How would you test whether a 3-level factor matters?” → F-test (or anova()) on the joint null that all $K-1$ dummies are zero.
• F = t² when p = 1. Conceptual: the F-test reduces to the t-test for a single coefficient.
• Don’t compute. Prof has explicitly said he won’t make you compute the F-statistic. If asked anyway, write the formula and stop.

Related

• linear-regression: the parent model
• t-test-and-significance: same machinery for single coefficients; F=t² when p=1
• sampling-distribution-of-beta: derivation foundation
• collinearity: why you ask Q1 before drilling into t’s
• categorical-encoding-and-interactions: partial F is how you test factor predictors