Is it possible to calculate Bayes Factors for regression coefficients, e.g. to test if a regression coefficient is equal to 0?
Is it possible for arbitrarily complex models (e.g. Tobit), also when including covariates in the model, and are there R packages for that? Or does this still need to be developed?
Yes, testing point hypotheses is possible.
The brms package provides both censoring (?resp_cens) and bayes factors (?hypothesis), so that should be reasonably straightforward. For more complex models, you could use stan directly.
The MCMCtobit function in package MCMCpack might also help.
There probably are other options too.
I forgot to specify my question: Is this also possible if the regression model is not Bayesian, but estimated via lm(y~x) or do I need to run the regression in a Bayesian framework, i.e. including priors, etc.?
I am asking because I am not experienced in running and interpreting Bayesian regression outcomes, and would rather like to calculate a Bayes Factor for a point estimate of a "normal" regression coefficient of a Tobit model, in order to be able to say something about the relative evidence of H0 (coefficient = 0), vs. Ha (coefficient not 0) for a coefficient that is not statistically significantly different from 0 in the "Frequentist" framework.
Does that make sense?
Bayes factors do indeed require priors, so you cannot use them in a frequentist framework. They are not even defined for improper priors. The "closest" thing to Bayes factors in a frequentist framework would be likelihood ratio tests.
Though I do not really see how one would not feel comfortable running and interpreting a Bayesian regression, but be fine with interpreting Bayes factors. The other way around would be quite common though, plenty of Bayesians do not like to use Bayes factors.
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