Testing compound hypotheses using the F test

Abstract

The real utility of the F test stems from the fact that it can be used to test compound hypotheses. In short, it can be used to test hypotheses in which restrictions are imposed upon several regression coefficients at once. Specifically, the F test can be used to test whether or not all of the partial regression coefficients in a multiple regression equation are equal to zero. For example, we might use the F test to test the following compound hypothesis:

$$ H_0 :{\mathbf{ }}b_{y1.2} {\mathbf{ }} = {\mathbf{ }}b_{y2.1{\mathbf{ }}} = 0 $$

In this case, the null hypothesis imposes the restrictions that both of the partial regression coefficients are equal to zero. If both of these partial regression coefficients are equal to zero, it follows that the coefficient of determination for the multiple regression equation containing these two independent is equal to zero. Of course, due to sampling error, the sample coefficient of determination may not be exactly equal to zero, even though the population coefficient of determination is equal to zero. Consequently, the F test is used to determine whether or not we might have obtained a particular nonzero coefficient of determination in the sample by chance, even though the coefficient of determination in the population is zero.