# What are the meanings of CI.Lower and CI.Upper in brms directional hypothesis test

As in the title … what do the upper and lower CIs refer to in the output table provided from a directional hypothesis in brms? Put differently, of what “thing” are they the upper and lower limits?

Credible intervals. In other words, the proportion of MCMC draws for that parameter or function of parameters between the lower and the upper bound is equal to the specified probability.

Sorry, I cannot follow the meaning of that explanation. An example of the output is:

     Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio Post.Prob Star
(Intercept) > 0     0.43      0.05     0.35      Inf        Inf      1.00    *


In the above example, what is the “thing” that has a lower CI of 0.35 and an upper CI of Inf?

The function in question is the posterior odds that the intercept is positive. The one-sided interval \left(0.35,\infty\right) contains 95% of the posterior mass for those odds. The fact that this interval includes 1 in your case is considered meaningful by some people.

Aah the posterior odds – that makes complete sense now! I don’t think this is documented in brms help, at least nowhere obvious. Thanks for clearing that up.

Wait a moment – here’s another example row:

         Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio Post.Prob Star
(IsOld:Balance) > 0     0.02      0.10    -0.14      Inf       1.53      0.60


How can odds be negative?

Is it the log-odds (logit)? I think we’re guessing now!

In your first example, you take the intercept of your regression model and ask whether (or how likely) it is greater 0. That’s bascially it. Since the upper bound of a CI is not of interest when you ask intercept > 0, it is set to infinity (that’s not my idea, it is common in classical statistics and referred to as one-sided CI). And then you know that with 95% probabilty (by default), your intercept is between 0.35 and infinity.

The same reasoning applies to your interaction term IsOld:Balance.

Still confused, I am afraid

The model summary gives this for the intercept:

Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
Intercept                           0.43      0.05     0.33     0.52        563 1.00


which I take to mean that the intercept has a 95% probability of lying between 0.33 and 0.52.

The corresponding hypothesis test is

     Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio Post.Prob Star
(Intercept) > 0     0.43      0.05     0.35      Inf        Inf      1.00    *


which you suggest above means “you know that with 95% probabilty (by default), your intercept is between 0.35 and infinity”. Neither bound matches the first. What am I missing here?

The first line tells you that 2.5% of draws are below .33 and 2.5% are above .52. The second line tells you that 5% of draws are below .35. That’s because the second line is a directional (one-tailed) hypothesis test.

That makes sense!

Paul, perhaps some additional clarification in the help for hypothesis would be useful?

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