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I am having trouble interpreting the output of an MCMC Logistic Regression run using R from the MCMCpack. Unfortunately I have had very little luck in finding sources on the web.

I am assuming that the means (below in Output) are mean estimates of the coefficients / intercept. How do I verify the significance of each? Do I take the mean value from the sample and compare it?

Is there an R^2 equivalent? How do I compute it / obtain it?

How do I verify that the model is significant (typically this is done with chi-square in "normal" logistic regression)?

Code

Base <- MCMClogit(zTurnoutBin ~ Rating, data=DebateYear, user.prior.density=NULL,burnin = 1000, mcmc = 10000, thin = 1, tune = 1.1,
logfun=TRUE)

head(Base)

Output

Iterations = 1001:11000
Thinning interval = 1 
Number of chains = 1 
Sample size per chain = 10000 

1. Empirical mean and standard deviation for each variable,
   plus standard error of the mean:

                 Mean      SD  Naive SE Time-series SE
(Intercept) -0.259743 2.78933 0.0278933       0.091107
Rating       0.006701 0.06538 0.0006538       0.002145

2. Quantiles for each variable:

               2.5%      25%       50%     75%  97.5%
(Intercept) -5.9529 -2.08606 -0.213151 1.55037 5.2016
Rating      -0.1201 -0.03571  0.005031 0.04899 0.1416



Markov Chain Monte Carlo (MCMC) output:
Start = 1001 
End = 1007 
Thinning interval = 1 
     (Intercept)      Rating

[1,]  -0.6755339  0.01569481
[2,]   2.9115262 -0.04856103
[3,]   4.0022365 -0.07819378
[4,]  -1.8353887  0.03590127
[5,]   0.6784049 -0.02264779
[6,]   0.6784049 -0.02264779
[7,]   0.6784049 -0.02264779
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To determine significance you need to look at the whole distribution for a particular coefficient and look for high-density regions that exclude 0. For example, if the $\beta > 0$ for 95% of the samples then you can say that $\beta > 0$ with 95% probability (under the model and prior assumptions, etc.). Usually, if this distribution crosses zero near its mode, we would consider it to be insignificant, but since Bayesian methods allow for exact statements of probability, you should state your results within this framework (i.e. there are no p-values). Also note that these regions don't have to be contiguous; $\beta \in (-\infty, -1] \cup [1, \infty)$ could also be an interval that excludes 0 with high probability, but makes inference on the exact direction of a relationship difficult.

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