I am fitting Bayesian multivariate generalized linear models with correlated groupspecific terms via Stan to model the binary outcomes for three outcomes; y1, y1 and y3 with a single predictor. In addition I have two grouping variables; hhid and years. The model is specified as below
model < stan_mvmer(
formula = list(
y1bin ~ wealthindex + (1  years) + (1  hhid)
, y2bin ~ wealthindex + (1  years) + (1  hhid)
, y3bin ~ wealthindex + (1  years) + (1  hhid)
)
, data = sim_dflist[[1]]
, family = list(binomial, binomial, binomial)
, …
)
Below are some of the outputs (estimates from the model fit)
Estimates:
mean sd
y1wealthindex 0.452 0.266
y1(Intercept) 0.337 0.142
y1mean_PPD 0.578 0.012
y2(Intercept) 0.322 0.092
y2wealthindex 0.803 0.203
y2mean_PPD 0.581 0.012
y3(Intercept) 0.423 0.174
y3wealthindex 0.413 0.303
y3mean_PPD 0.595 0.012
Sigma[years:y1(Intercept),y1(Intercept)] 0.352 0.154
Sigma[years:y2(Intercept),y1(Intercept)] 0.035 0.063
Sigma[years:y3(Intercept),y1(Intercept)] 0.036 0.116
Sigma[years:y2(Intercept),y2(Intercept)] 0.137 0.065
Sigma[years:y3(Intercept),y2(Intercept)] 0.103 0.082
Sigma[years:y3(Intercept),y3(Intercept)] 0.550 0.231
Sigma[hhid:y1(Intercept),y1(Intercept)] 0.311 0.069
Sigma[hhid:y2(Intercept),y1(Intercept)] 0.060 0.032
Sigma[hhid:y3(Intercept),y1(Intercept)] 0.094 0.050
Sigma[hhid:y2(Intercept),y2(Intercept)] 0.069 0.033
Sigma[hhid:y3(Intercept),y2(Intercept)] 0.106 0.038
Sigma[hhid:y3(Intercept),y3(Intercept)] 0.452 0.089
logposterior 6394.996 22.838
Groups
Groups Name Std.Dev. Corr
years y1(Intercept) 0.59344
y2(Intercept) 0.36975 0.158
y3(Intercept) 0.74185 0.083 0.376
hhid y1(Intercept) 0.55775
y2(Intercept) 0.26280 0.411
y3(Intercept) 0.67248 0.250 0.602
Question?

What’s the interpretation difference between the Sigma in Estimates (first table), e.g., Sigma[years:y1(Intercept),y1(Intercept)] = 0.352 and Std.Dev. in Group (second table), e.g.,
years y1(Intercept) = 0.59344. 
If I simulated my data with a prespecified covariance structure (Sigma) of the y1, y2 and y3 intercepts and then sampled from a MVN(intercept_betas, Sigma), how do I compare my output to the prespecified Sigma? Or rather, what represents my Sigma in the model fit output?
I will appreciate any help from you. Thanks
 Operating System: 2.18.2
 rstanarm Version: Ubuntu 18.10