I am still learning about Bayesian Data Analysis, so forgive me if this is a simple or misguided question.
I am estimating a multilevel model in brms for the purpose of testing hypotheses about specific regression coefficients (i.e., fixed effects). I have the model estimated and am comfortable interpreting the point and interval estimates, as well as the posterior distribution plots.
My question is about interpreting the results overall. In the Bayesian framework, I understand that thresholds like p-values and significance are downplayed. However, it is not entirely clear to me yet what this is meant to be replaced with.
- How do I discuss the results in an appropriate Bayesian way?
To make things more concrete, here are example results for two coefficients:
b_x1 = -0.25, 95% HDI: [-0.30, -0.21]
b_x2 = -0.02, 95% HDI: [-0.04, 0.01]
My frequentist-trained brain looks at this and sees that x1 is significant (the HDI does not include 0) and x2 is not. Thus, I would say that this is support for the hypothesis that x1 is negative and a lack of support for the hypothesis that x2 is negative. However, most of the posterior density on x2 is negative, so perhaps these results should be considered somewhat supportive of the second hypothesis as well.
I found the hypothesis() function and tried that, but am unsure if I am using it as intended. Given my frequentist training, I first tried testing that x1 and x2 are equal to 0, but that did not yield an evidence ratio or posterior probability. Then I tried making it directional: testing if they are less than 0.
- Is this an appropriate way to test my hypothesis that the regression coefficient for x1 and x2 are negative?
Hypothesis (x1 = 0), Evid.Ratio = NA, Post.Prob = NA
Hypothesis (x2 = 0), Evid.Ratio = NA, Post.Prob = NA
Hypothesis (x1 < 0), Evid.Ratio = Inf, Post.Prob = 1.00
Hypothesis (x2 < 0), Evid.Ratio = 16.32, Post.Prob = 0.94
- How do I interpret the evidence ratios and posterior probabilities?
My sense is that the evidence ratios are Bayes Factors and that values higher than 1 indicate relative support for the stated hypothesis over its alternative. I think the infinite ratio on x1 could be interpreted as being some value greater than 2000 (the default number of samples). Is that right?
The posterior probabilities seem more intuitive and kind of like a Bayesian alternative to a p-value (when subtracted from 1). Basically the probability of the hypothesis, given the data. Is that right?
- What numbers are typically presented in a results write-up in Bayesian Data Analysis?
I was planning to put the estimate, HDI, and maybe the evidence ratio and/or posterior probability for each into a table. What is typical or recommended?
Answers and recommended readings would be appreciated. Thanks in advance.