# How does one calculate the F-value-threshold used to evaluate an F-test?

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.

## 1 Answer

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.