0
$\begingroup$

I am studying Response Surface Methodology (by Myers, Montgomery, and Anderson-Cook). When introducing the significance testing for regression parameters, an $F$-test is introduced but only partially explained. I am wondering if the unexplained bit should be common knowledge that I am missing.

As a specific example, to test if there is a linear relationship between the response variable $y$ and a subset of the regressor variables $x_1, x_2, \dots, x_k$ one considers the null hypothesis $$H_0: \beta_1 = \beta_2 = \cdots = \beta_k = 0 $$ where $k$ is the number of regression variables, and $\beta_i$ is the regression coefficient for the $i^{th}$ variable. To test this hypothesis, the text says to compute $$F_0=\frac{SS_R/k}{SS_E/(n-k-1)}=\frac{MS_R}{MS_E}$$ where $SS_R$ is the sum of squares due to the model (or to regression), $SS_E$ is the sum of squares due to the residual (or error), $n$ is the number of measurements, and $MS$ is used to denote the mean squared error.

After calculating $F_0$, the text explains that one should reject $H_0$ is $F_0$ exceeds $F_{\alpha, k, n-k-1}$, where $\alpha$ is the significance level (commonly set at $\alpha=0.05$). How do you calculate $F_{\alpha, k, n-k-1}$? Or is this something you usually look up in a table?

There is no mention to this in the text. This makes me think I should already know this (maybe from basic statistics?). This comes up again when using a partial $F$-test to evaluate individual or groups of regression coefficients.

$\endgroup$
1
$\begingroup$

The value $F_{\alpha, k, n-k-1}$ is called a critical $F$-value and can either be looked up in a table (example table) or calculated from virtually any statistical software. A decent applet is found here: http://www.danielsoper.com/statcalc/calculator.aspx?id=4.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.