nAGQ

I agree with Dimitris.  Adaptive Gauss-Hermite quadrature is used to
approximate the integral of the conditional density of a random effect
given the observed data.  We go into some detail about the model and the
derivation of the integral in question in
https://embraceuncertaintybook.com/aGHQ.html.

With 500 binary observations for each of 5000 groups, the integral in
question will be very close to a scaled Gaussian density, and the Laplace
approximation will be more than adequate. I am not surprised that there are
almost no differences between the results from the Laplace approximation
and AGQ of different orders.  Bear in mind that, for high orders, the
weights drop dramatically for evaluations far from the mode of the
conditional distribution (see Fig. C1 and C2 in the above-mentioned book
for the case of nAGQ = 9).  For very large order, the Golub-Welsch
algorithm, which IIRC is the way the weights and abscissae for the
Gauss-Hermite rule are calculated in lme4, the weights for the remote
evaluations are actually zero.

The table of abscissae and weights for nAGQ=31 is enclosed.  You can see
that when you get beyond three or four standard deviations from the mean
(or "six sigma" for the Quality Control crowd) the additional evaluations
have very little effects on the value of the integral.

Is your simulation based on observed data or an actual study or
experiment?  I have never seen cases of that many observations per group,
especially over that number of groups.

nAGQ

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