Another way of determining whether a regression model is a good fit is to look at whether the population multiple correlation coefficient R between y and ŷ is zero (the null hypothesis). As noted in Multiple Correlation a sample’s adjusted R is an unbiased estimate of the population R. Instead of R, we generally test R2, and use the following property, which is an extension of Theorem 1 of One Sample Testing of Correlation.
Property 1: If the population R = 0, then
Observation: If k = 1, then
Example 1: Show that the regression model in Example 2 of Multiple Regression Analysis is a good fit by using Property 1.
We test the null hypothesis H0: R = 0 (see Figure 1).
Figure 1 – F-test of data in Example 1 using Property 1
As we can see from the above analysis, we reject the null hypothesis, and conclude that the fit of the regression model with the data is not due simply to chance.