## Prediction

The nonlinear mixed model is a useful tool for statistical prediction.
Assuming a prediction is to be made regarding the *i*th subject,
suppose that is a differentiable function predicting
some quantity of interest. Recall that denotes the vector of
unknown parameters and *u*_{i} denotes the vector of random effects for
the *i*th subject. A natural point prediction is
, where is the maximum
likelihood estimate of and is the empirical Bayes
estimate of *u*_{i} described previously in "Integral
Approximations."
An approximate prediction variance matrix for
is

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 is computed as
follows using the delta method (Billingsley, 1986). The derivative of
is computed with respect to each element of
and evaluated at . If *a*_{i}
is the resulting vector, then the prediction variance is *a*^{T}_{i} *P*
*a*_{i}.

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.