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

Interfaces brms
1

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?

2

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.

3

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?

4

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.

5

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.

6

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?

7

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

8

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.

9

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?

10

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.

11

That makes sense!

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

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