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The NLMIXED Procedure |
Prediction |
The nonlinear mixed model is a useful tool for statistical prediction. Assuming a prediction is to be made regarding the th subject, suppose that
is a differentiable function predicting some quantity of interest. Recall that
denotes the vector of unknown parameters and
denotes the vector of random effects for the
th subject. A natural point prediction is
, where
is the maximum likelihood estimate of
and
is the empirical Bayes estimate of
described previously in the section Integral Approximations.
An approximate prediction variance matrix for is
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where is the approximate Hessian matrix from the optimization for
,
is the approximate Hessian matrix from the optimization for
, and
is the derivative of
with respect to
, evaluated at
. The approximate variance matrix for
is the standard one discussed in the previous section, and that for
is an approximation to the conditional mean squared error of prediction described by Booth and Hobert (1998).
The prediction variance for a general scalar function is defined as the expected squared difference
PROC NLMIXED computes an approximation to it as follows. The derivative of
is computed with respect to each element of
and evaluated at
. If
is the resulting vector, then the approximate prediction variance is
. This approximation is known as the delta method (Billingsley 1986, Cox 1998).
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