Testing the Overall Utility of the Model

Test the model for significancy on all predictor variables simultaneously using an $F$ test.

Motivation

To test if the model is useful in explaining the predictor variable, we can use t-test on each ${\beta }_i$ of the model.

However, this method is generally not recommended, because it for each t-test, the probability of getting a Type I error increases.

To properly test the utility of a multiple linear regression model, we will need to do a global test (test that encompasses all the $\beta$ parameters)

Warning

A rejection in the $H_0$ of the test means that the model is “statistically useful”. However, this does not mean the “best”.

Instead, there might be other models which is more reliable.

Assumptions

See Assumptions for the Error Component

Hypotheses

• $H_0:{\beta }_1={\beta }_2=\cdots ={\beta }_k=0$
• $H_1:$ Atleast one ${\beta }_i\ne 0$

Test statistic

$$\begin{array}{rl}F& =\frac{(S{S}_{yy}-SSE)/k}{SSE/(n-(k+1))}\\ & =\frac{MS(\text{Model})}{MSE}\end{array}$$

Critical region

Reject $H_0$, if

• $F>F_{\alpha ;k;n-(k+1)}$
• $\alpha >\text{p-value}$