What is the relationship between the standard deviation of a random intercept and the value of the integer scalar? I fit a mixed-effects logistic regression model in R with the following formula:
glmer.traditional <- glmer(AGENT.EXPONENCE ~ ASPECT + (1 | LEMMA), data = hdtpassive, family = binomial(link="logit")) 
The standard deviation for the random intercept is really high:
Random effects:
Groups Name        Variance Std.Dev.
LEMMA  (Intercept) 400.4    20.01   
Number of obs: 438, groups:  LEMMA, 174

When, however, I use the following formula, the standard deviation plummets:
glmer.traditional <- glmer(AGENT.EXPONENCE ~ ASPECT + (1 | LEMMA), data = hdtpassive, family = binomial(link="logit"), control = glmerControl(optimizer = "bobyqa"), nAGQ = 25)
Random effects:
Groups Name        Variance Std.Dev.
LEMMA  (Intercept) 27.28    5.223   
Number of obs: 438, groups:  LEMMA, 174

The nAGQ is the scalar that is used for approximating the log-likelihood. Higher values for this argument produce more accurate approximations, but come at the expense of speed. 
I have two questions about this:


*

*How does the value of the integer scalar affect the standard deviation of the random intercept? I don't know how the Gauss-Hermite quadrature works. 

*Are there guidelines on the interpretation of standard deviations for random intercepts? E.g., is a really high standard deviation a warning sign of some kind? 
 A: This answer is more of a jumping-off point for those more experienced than I.  I just did a bit of preliminary research, for those who may be interested.  To answer question (1), by pulling from the glmer documentation...

nAGQ: integer scalar - the number of points per axis for evaluating the adaptive Gauss-Hermite approximation to the log-likelihood. Defaults to 1, corresponding to the Laplace approximation. Values greater than 1 produce greater accuracy in the evaluation of the log-likelihood at the expense of speed. A value of zero uses a faster but less exact form of parameter estimation for GLMMs by optimizing the random effects and the fixed-effects coefficients in the penalized iteratively reweighted least squares step.

So, increasing nAGQ will (i) take longer, and (ii) increase accuracy of the log-likelihood evaluation.  
An answer to part (2) can be seen in this thread.  An explanation of the intercept term itself is seen here.  In the comments it is noted that:

"If all the variables, both predictors and response, are centered, then you don't need the intercept term. Instead, you take away 1 df for residual because of centering the response variable. Once you have done all that, it is equivalent to including the intercept in the model where the variables are not centered."

