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Re: st: xtlogit vs gllamm


From   jlinhart@stata.com (Jean Marie Linhart, StataCorp LP)
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: xtlogit vs gllamm
Date   Wed, 17 May 2006 09:15:25 -0500

Karine Begin <k.begin at umontreal dot ca> asks about performing both
an -xtlogit- and -gllamm-.  Different results are obtained, though
both indicate very high sigma values -- 3.46 from -xtlogit- and 15.1
(square root of 227.29) for -gllamm-.

Bryan Griffin <bwgriffin at GeorgiaSouthern dot edu> points out the
large random-effect variance and notes that it creates problems, and
suggests there might be something odd with the data.

Bryan notes that he has run simulations and reported that -xtlogit-'s
adaptive quadrature gives biased estimates when large random effects
are present, but that -gllamm- is more accurate, and that the
non-adaptive quadrature of -xtlogit- can also be more accurate in this
case.

Bryan also notes that a large number of integration points are needed
whether with -xtlogit- or with -gllamm- in cases of large random
effects.

                        >>>>>><<<<<<


There are 8 observations per panel and this data is almost completely
correlated.  Using the 227.29026 estimate of the within-panel variance
from -gllamm-, this implies a within panel correlation of 227.29026 /
(227.29026 + 3.2898684) = .986.  This is well into the region where
random-effects estimation breaks down.  With this high of a
correlation, it is more like having one observation per panel rather
than 8.

I'd be really interested in seeing this data, if Karine would like to
send it to me privately.

I would recommend using -quadchk- with this data.  I suspect it will
tell Karine that the quadrature is not accurately representing the
likelihood.  -quadchk- was created for the purpose of identifying when
changing the number of quadrature points varies the estimates of
parameters, and I'd be very surprised if it didn't show large
variations.

As Bryan says, using a large number of quadrature points can give
better estimates in these difficult cases with a large random effect,
though it is also more difficult to be certain in this case that the
model is fully converged.  -quadchk- varies the number of quadrature
points by 4 -- this is a substantial fraction of a total of 12
quadrature points, but only a small fraction of 100 or 150.  The
increase in the sampling of the likelihood function is responsible for
the improvement with increasing the quadrature points.

Bryan notes that -xtlogit- and -gllamm- do not use the same method of
adaptive quadrature, which explains, in part, the difference in the
estimates.
 
The adaptive quadrature method used by -xtlogit- is that of

	Liu, Qing and Pierce, D. A. (1994) A note on Gauss Hermite
	quadrature, Biometrika 81:624-629.

-gllamm- uses the method of 

	Naylor, J.C. and Smith, A. F. M.  (1982) Applications of a
	method for the efficient computation of posterior
	distributions, Applied Statistics 31:214-225.

This method is perhaps best explained in 

	Skrondal, Anders and Rabe-Hesketh, Sophia (2004) Generalized
	latent variable modelling, CRC Press Boca Raton, FL 
	pp. 167-170.

-xtlogit- adapts the quadrature via the mode and curvature of the mode
of the likelihood function, whereas -gllamm- uses an estimate of the
mean and variance of the likelihood to adapt.

In these difficult cases of large random effects (large sigma), I have
observed that the mode and curvature of the mode underestimate the
spread of the likelihood function and consequently underestimate the
likelihood, even when a large number of quadrature points are used.

The method used by -gllamm- is to estimate the mean and variance of
the likelihood function and use this information analogously to the
method above to adapt the quadrature points to the log-likelihood
function.  My experimentation indicates this works better in the
difficult case of large random effects.

Non-adaptive quadrature does not attempt to adapt the quadrature
points to the likelihood function, but because of the general form of
the non-adaptive likelihood there are cases of high random effects
where it produces efficacious results -- there are also cases where it
performs poorly, and one must beware.

We are in the process of evaluating these methods on these difficult
cases, and we have had some good results.  When we are satisfied we
will let you know.

--Jean Marie 
jlinhart@stata.com
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