2

Suppose we have the linear model for Y, with X1, X2 and X3 being continuous independent variables:

Y = b0 + b1 * X1 + b2 * X2 + b3 * X3 + e

To test H0: b1=b2=0 against H1: b1 or b2 or both are different from zero, the F-test can simply be used. If the right tail probability for F is lower than 0.05, say, H0 can be rejected. Here, both "sub-hypotheses" about b1 and b2 are tested two-sided.

But what if the alternative hypothesis H1 would be: b1>0 or b2>0 or both > 0. In that case, both "sub-hypotheses" about b1 and b2 are directional. In a t-test for each b separately, a two-sided p-value (if the software used would only produce two-sided p-values) should be halved. But what about the F-value? Is it possible to use the (or some) F statistic for such simultaneous directional alternative hypotheses? Any thoughts or references would be great, thanks!

Ben.

BenP
  • 1,124
  • The F statistic cannot do that as it relies on squares of the differences of estimates and null values, so negative and positive deviations contribute in the same way to the statistic. See e.g. https://stats.stackexchange.com/questions/497035/f-test-for-linear-regression-why-only-upper-tail https://stats.stackexchange.com/questions/258461/proof-that-f-statistic-follows-f-distribution – Christoph Hanck Oct 05 '22 at 14:20
  • @ChristophHanck That’s a fair point, but what if we want to test the one-sided hypothesis? (I saw an example last week where it would have made sense to use a one-sided test, and the presentation settled for a two-sided test that could have missed important inferences about the parameters being tested that a one-sided test would have caught.) – Dave Oct 05 '22 at 14:23
  • Fair question that I do not answer :-). I feel like I have come across solutions in this regard, but do not recall a reference right now. As a (as usual, conservative) solution, one could do Bonferroni on the separate t-statistics. – Christoph Hanck Oct 05 '22 at 14:26
  • Just for what it's worth: in case of testing only a single b coefficient one-sided, an F-test can be used, and the right-tail probability of the F distribution can be halved, because halve of the extremest F values come from squaring the negative t-values, which are not relevant for the one-sided test. But in case of two or more b-coefficients I don't believe it is as simple as that. – BenP Oct 05 '22 at 14:51
  • It will be pretty straightforward to do Bayesian inference (using brms for example) and compute the direct evidence (ie. the posterior probability) for the complex alternative. – dipetkov Oct 05 '22 at 18:19

0 Answers0