Prediction

The NLMIXED Procedure

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 $f(\theta,u_i)$ is a differentiable function predicting some quantity of interest. Recall that $\theta$ denotes the vector of unknown parameters and u_i denotes the vector of random effects for the ith subject. A natural point prediction is $f(\hat{\theta},\hat{u}_i)$ , where $\hat{\theta}$ is the maximum likelihood estimate of $\theta$ and $\hat{u_i}$ is the empirical Bayes estimate of u_i described previously in "Integral Approximations."

An approximate prediction variance matrix for $(\hat{\theta},\hat{u}_i)$ is

$P = [ \hat{H}^{-1} & \hat{H}^{-1} ( \frac{\partial \hat{u}_i}{\partial \theta}... ...l \theta} ) \hat{H}^{-1} ( \frac{\partial \hat{u}_i}{\partial \theta} )^T ]$

where $\hat{H}$ is the approximate Hessian matrix from the optimization for $\hat{\theta}$ , $\hat{\Gamma}$ is the approximate Hessian matrix from the optimization for $\hat{u_i}$ ,and $(\partial \hat{u}_i/\partial \theta)$ is the derivative of $\hat{u_i}$ with respect to $\theta$ , evaluated at $(\hat{\theta},\hat{u}_i)$ . The approximate variance matrix for $\hat{\theta}$ is the standard one discussed in the previous section, and that for $\hat{u_i}$ is an approximation to the conditional mean squared error of prediction described by Booth and Hobert (1998).

The prediction variance for $f(\hat{\theta},\hat{u}_i)$ is computed as follows using the delta method (Billingsley, 1986). The derivative of $f(\theta,u_i)$ is computed with respect to each element of $(\theta,u_i)$ and evaluated at $(\hat{\theta},\hat{u}_i)$ . If a_i is the resulting vector, then the prediction variance is a^T_i P a_i.

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