equivalence of t-test and F-test in ANOVA

#1
Something about ANOVA is confusing me. Say you have a regression model:

y = b0 + b1*x1 (where b0 and b1 are coefficients and x1 is an explanatory variable)

and want to do an F-test on the null hypothesis, b2 = 0 (ie "should I include explanatory variable x2 in my model?")

Is this F-test not equivalent to actually calculating the coefficients for the model:

y = b0 + b1*x1 + b2*x2

and performing a t-test based on the null hypothesis b2 = 0 ?
 
Last edited:

Dragan

Super Moderator
#2
Something about ANOVA is confusing me. Say you have a regression model:

y = b0 + b1*x1 (where b0 and b1 are coefficients and x1 is an explanatory variable)

and want to do an F-test on the null hypothesis, b2 = 0 (ie "should I include explanatory variable x2 in my model?")

Is this F-test not equivalent to actually calculating the coefficients for the model:

y = b0 + b1*x1 + b2*x2

and performing a t-test based on the null hypothesis b2 = 0 ?
There are two F-tests: One for the full model (that includes both X1 and X2 that tests the null hypothesis that b1 = b2 = 0) and another one that could be computed to test for the incremental increase in R^2 associated with including X2 (b2 =0).

The latter is as you state: the F test is the square of the t-test for the null hypothesis that b2 = 0.
 
#3
There are two F-tests: One for the full model (that includes both X1 and X2 that tests the null hypothesis that b1 = b2 = 0) and another one that could be computed to test for the incremental increase in R^2 associated with including X2 (b2 =0).

The latter is as you state: the F test is the square of the t-test for the null hypothesis that b2 = 0.
Hi Dragon,

Thank you for your help!

Yes I meant the incremental F test of including x2 in the model. So if this F-test and the t-test for the coefficient b2 are equivalent, which approach is best to take?

I have a pre-existing "most satisfactory" model, but have thus far excluded a potentially important dummy variable from it. I thought I would incrementally include the dummy in my model and perform an F-test and then incrementally include interactions and perform F-tests on those.

I found the first F-test not to be significant, ie strong evidence against the dummy being included. I am presuming this means I need not bother including interactions?