How to interpret posterior probability of an regression variable?

Question

I am using the following model in WINBUGS to run a hierarchical Bayesian regression where the beta are my covariates:

If I modify this model by adding the following code:

# posterior probabilities of Positive beta's
p.beta0 <- step( beta0 )
p.beta1 <- step( beta1 )

Then I can assess the posterior probability (PP) of the (positive or negative) association of the beta covariates.

My beta values are:

beta0 = 0.23434
beta1 = -0.4582

With this code, the PP of beta0 is 0.959033, while the PP of beta1 is 0.015043. My interpretation for beta0 is that there is a 95.9033% positive association for this covaraite. However, I am not sure how to interpret for beta1 since this has a negative association and a low posterior probability. I am not sure if it is an issue with my code for computing the Posterior Probabilities.

Any insight is welcome.

Answer 1

beta are your regression coefficients, not the covariates. To understand beta0 and beta1 you have to look at the model. Part of it says that log(mu[i]) = beta0 + beta1*aff[i]/10, where mu[i] is the cancer rate of area i. exp(beta0) shows you the average cancer rate over all areas with aff = 0. Because beta1 = -0.4582 is negative, cancer rate reduces when aff increases: every time you increase aff by 10, the log cancer rate decreases by 0.4582.