Finding critical value for F statistic

Question

I am performing a hypothesis test and therefore intersted in finding the F critical value for my F statistic. Below is the anova output when I compare two models.

Analysis of Variance Table

Model 1: Y ~ X1 + X2 + X3 + X4
Model 2: Y ~ X1 + X2 + X3 + X4 + (X2 * X4) + (X3 * X4)
  Res.Df    RSS Df Sum of Sq      F Pr(>F)
1    108 127.24                           
2    106 122.05  2    5.1964 2.2566 0.1097

Could someone please help me find the F critical Value to compare against the F statistic (2.2566) ANOVA is showing? I am confused when it comes to numerator and denominator values since different sources state different rules for finding those. The data used has total 113 observations for both models.

Answer 1

Let's assume that $\beta_0$ is the intercept, $\beta_1$ is the regression coefficient associated to $X_1$, $\beta_2$ associated to $X_2$, $\ldots$, $\beta_5$ the coefficient associated to $X_2*X_4$ and $\beta_6$ the coefficient associated to $X_3*X_4$.

The test you are applying is for $$H_0:\beta_{5}=\beta_6=0$$ against $$H_1: \beta_5\neq 0 \text{ or }\beta_6\neq 0 \text{ or both}.$$

In this case, under $H_0$, the $F$ statistic has distribution $F_{2,106}$, where 2 is the degrees of freedom in the numerator due to the number of parameters you are fixing and $106 = n-p$ is the degrees of freedom in the denominator with $p=7$ and $n=113$.

Thus, for a level $\alpha= 5\%$ test, the threshold is 3.082, whereas the p-value is $$P(F_{2,106} > 2.2566) = 0.1097.$$