Design of Experiments Guide > Nonlinear Designs > Statistical Details for Nonlinear Designs > Radial-Spherical Integration of the Optimality Criterion
Publication date: 11/29/2021

## Radial-Spherical Integration of the Optimality Criterion

The optimality criterion is the expectation of the logarithm of the determinant of the information matrix with respect to the prior distribution. Consequently, finding an optimal nonlinear design requires maximizing the integral of the log of the determinant of the Fisher information matrix with respect to the prior distribution of the parameters. This integral must be calculated numerically. The approach used in the Nonlinear Design platform is based on Gotwalt et al. (2009).

For normal distribution priors, the integral is reparameterized into a radial direction and a number of angular directions equal to the number of parameters minus one. The radial part of the integral is computed using Radau-Gauss-Laguerre quadrature with an evaluation at radius = 0. This is done by constructing a certain number of hyperoctahedra and randomly rotating each of them.

If the prior distribution is not normal, then the integral is reparameterized so that the new parameters have a normal distribution. Then the radial-spherical integration method is applied.

Note: If the prior distribution for the parameters does not lend itself to a solution and the process fails, a message is added to the window that the Fisher information matrix is singular in a region of the parameter space. When this occurs, consider changing the prior distribution or the ranges of the parameters.