Understanding stan_glmer output for meta-analysis

Modeling
1

Hi there,

probably this is a very basic question. I would like to do a meta-analysis with rstanarm. I get the output below.
I was wondering why the population intercept of 5.4 (Median equals Mean in this case) does not equal the mean of the posterior predictive distribution, which is 5.6, and how to interpret the population level intercept of 5.4?

stan_lmer
 family:       gaussian [identity]
 formula:      data ~ 1 + (1 | WWTP)
 observations: 307
------
            Median MAD_SD
(Intercept) 5.4    0.1   
sigma       1.2    0.1   

Error terms:
 Groups   Name        Std.Dev.
 WWTP     (Intercept) 0.68    
 Residual             1.20    
Num. levels: WWTP 45 

Sample avg. posterior predictive distribution of y:
         Median MAD_SD
mean_PPD 5.6    0.1   

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For info on the priors used see help('prior_summary.stanreg').
2

The deviations in the WWTP-specific intercepts are not exactly zero (apparently they average to about 0.2). In other words, the “Sample avg. posterior predictive distribution of y” is conditional on the levels of WWTP rather than marginalizing over them.