3.2.6.4 F Test

The test of significance of the fitted model is performed with the so called F Test. This test provides a measurement for the probability that the variances of two independent samples of the size n₁ and n₂ are identical.

A ratio F is calculated out of the estimators of the variance s₁² and s₂²

$$F = \frac{{s_1}^2}{{s_2}^2} .$$ (3.24)

The estimator of the variance can be calculated by

$${s_j}^2 = \sum_{i}^{n_j} \frac{x_{i,j}}{n_j - 1}$$ (3.25)

with n_j -1 degree of freedom (x_i,j is the i-th value of the vector $\vec{x}_j$).

This ratio F is compared with values $F_\alpha(p-1,n-p)$, the F-Distribution^3.6 at the significance level $\alpha$. A value $F < F_\alpha(p-1,n-p)$ confirms the hypothesis that the variances s₁ and s₂ are identical. The values of the F-Distributions are listed in tables and can be found in [6,7].

For the values from the analysis of variance table the ratio is calculated using

$$F(p-1,n-p) = \frac{\frac{SSR}{p-1}}{\frac{SSE}{n-p}} .$$ (3.26)

If $F(p-1,n-p) < F_\alpha(p-1,n-p)$, the fitted model adequately describes the behavior of the response over the experimental region.

A large and significant value of F discredits the fitted model. In most cases this would initiate a search for a more adequate model, or for transformations of the variables.

Footnotes

...-Distribution^3.6

It is also called the Fisher's F-Distribution or Snedecor's F-Distribution or the Fisher-Snedecor Distribution. Details about the F-Distribution can be found in Appendix B.8.