How to perform a t test for two variables have a positive effect on the dependent?

Question

I thought I should perform 2 separate t tests for H0: beta = 0. Thus checking that they are statistically significant. I need to; Perform a t-test for the hypotheses that X1 and X2 have a positive impact on the dependent. Can I do this with one test?

Answer 1

I am assuming you are talking about a function of the form $y=\beta_1x_1+\beta_2x_2+\alpha+\epsilon$. The traditional form to prevent family-wise errors is to do a master F test with a t-test for each parameter. Normally the test would be that $\beta_1=\beta_2=0$ with the F-test and $\beta_1=0$ and $\beta_2=0$ with the t-tests.

Except that you would have to do the work by hand, there is no reason you cannot test $\beta_1\le{0}$ and $\beta_2\le{0}$ in the t-tests as your null hypothesis. In lieu of the F-test, you could do the Holm-Bonferroni correction to the t-tests to test them one at a time. It would be a bit of work, but there is no conceptual reason you cannot test to see if they are positive by assuming they are non-positive and testing to see if that is false.

The only way to do this as one test is to do a Bayesian test. Bayesian methods allow compound hypotheses and do not restrict you to two hypothesis. You would test $\Pr(\beta_1>0\wedge\beta_2>0)$ versus $\Pr(\beta_1\le{0}\vee\beta_2\le{0})$. Because this is a binary hypothesis that is also exhaustive and mutually exclusive, you could use Bayes factors without running into the difficulties that p-values run into.

Bayes factors are similar to the likelihood ratio test, except they consider all possible values that could occur and not merely the most likely value that could occur. There is a good paper on Bayes factors at http://www.stat.cmu.edu/~kass/papers/bayesfactors.pdf

While I am aware that if you are asking if you can escape two t-tests, for one, then that implies you have not used Bayesian methods, I provided it for completeness. You should expand your skills in life and the world is bigger than t-tests and F-tests.