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The NLMIXED Procedure |
Suppose (zj, wj; j = 1, ... ,p) denote the standard Gauss-Hermite
abscissas and weights (Golub and Welsch 1969, or Table 25.10 of
Abramowitz and Stegun 1972). The adaptive Gaussian quadrature
integral approximation is as follows.
where r is the dimension of ui, is the
Hessian matrix from the empirical Bayes minimization,
zj1, ... ,jr is a vector with elements
(zj1, ... ,zjr), and
PROC NLMIXED selects the number of quadrature points adaptively by evaluating the log likelihood function at the starting values of the parameters until two successive evaluations have a relative difference less than the value of the QTOL= option. The specific search sequence is described under the QFAC= option. Using the QPOINTS= option, you can adjust the number of quadrature points p to obtain different levels of accuracy. Setting p=1 results in the Laplacian approximation as described in Beal and Sheiner (1992), Wolfinger (1993), Vonesh (1992, 1996), Vonesh and Chinchilli (1997), and Wolfinger and Lin (1997).
The NOAD option in the PROC NLMIXED statement requests nonadaptive
Gaussian quadrature. Here all are set equal to zero, and
the Cholesky root of the estimated variance matrix of the random
effects is substituted for
in the
preceding expression for aj1, ... ,jr. The NOADSCALE option
requests the same scaling substitution but with the empirical Bayes
.
PROC NLMIXED computes the derivatives of the adaptive Gaussian quadrature approximation when carrying out the default dual quasi-Newton optimization.
The first-order approximation is obtained by expanding
with a one-term Taylor series expansion about
ui = 0, resulting in the approximation
Assuming that is normal with mean 0 and variance
matrix
, the first-order integral approximation is computable
in closed form after completing the square:
where . The resulting approximation for
is then
minimized over
to obtain the first-order
estimates. PROC NLMIXED uses finite-difference derivatives of the
first-order integral approximation when carrying out the default
dual quasi-Newton optimization.
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