Intervals for the Predicted Response

Confidence intervals for predicted responses in nonlinear models are exceedingly difficult to calculate with the current state of statistical software. The reason being that the calculation requires decomposition of the Jacobian using the QR decomposition with further matrix manipulations. For simple models with p-estim-able parameters and n observations, an approximate (1 — a)100% confidence interval for a single predicted response, x0, can be developed from 

2 Note The p-norm of a matrix is defined as (X) x pJ. Unless p is explicitly defined, p is assumed to equal 2. 1  

As an example, consider the data in Table 3.5 wherein a 1-compartment model with absorption [Eq. (3.115)] was fit to the data. The model has partial derivatives 

The F-value using 3, 9 degrees of freedom was 3.86, Student s two-tailed t-distribution with 9 degrees of freedom was 2.262, and the mean square error was 2427.8. Suppose the 95% confidence interval for the predicted value at Tmax was needed. At a Tmar of 3 h, the predicted value was 740.03 ng/mL. The Jacobian matrix evaluated at 3 h under the final parameter estimates was 

The norm of JXoR 1 was 0.45449. Hence the 95% confidence interval at Tmax was 689.37 ng/mL, 790.68 ng/mL. A 95% confidence band across all predicted responses is shown in Fig. 3.12. As can be seen, the standard error of prediction was not a constant, but varied from 0 ng/mL at t = 0 h to about 104.6 ng/mL at t = 22.8 h. At no point along the curve was the standard error of prediction constant. 

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Table IX. Confidence Intervals for the Predicted Response from Inverse Transformed Data. a 0.025.

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