### Nonlinear Regression

Recall from Chapter 3: Introduction to Statistical Modeling with SAS/STAT Software, that a nonlinear regression model is a statistical model in which the mean function depends on the model parameters in a nonlinear function. The SAS/STAT procedures that can fit general, nonlinear models are the NLIN and NLMIXED procedures. The procedures have the following in common:

• Nonlinear models are fit by iterative methods.

• You must provide an expression for the model through programming statements.

• Analytic derivatives of the objective function with respect to the parameters are calculated automatically.

• A grid search is available to select the best starting values for the parameters from a set of starting points that you provide.

The following items reflect some important differences between the NLIN and NLMIXED procedures:

• Parameters are estimated by nonlinear least squares with the NLIN procedure and by maximum likelihood with the NLMIXED procedure.

• The NLMIXED procedure enables you to construct nonlinear models that contain normally distributed random effects.

• The NLIN procedure requires that you declare all model parameters in the PARAMETERS statement and assign starting values. The NLMIXED procedure determines the parameters in your model based on the PARAMETER statement and the other modeling statements. It is not necessary to supply starting values for all parameters in the NLMIXED procedure, but it is highly recommended.

• The residual variance is not a parameter in models fit with the NLIN procedure, but it is in models fit with the NLMIXED procedure.

• The default iterative optimization method in the NLIN procedure is the Gauss-Newton method; the default method in the NLMIXED procedure is the quasi-Newton method. Other optimization techniques are available in both procedures.

Nonlinear models are fit with iterative techniques that begin from starting values and attempt to iteratively improve on the estimates by updating the estimates. There is no guarantee that the solution achieved when the iterative algorithm converges will correspond to a global optimum.