### Prediction

The nonlinear mixed model is a useful tool for statistical prediction. Assuming a prediction is to be made regarding the ith 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 ith 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

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).