Overview: NLIN Procedure

The NLIN procedure fits nonlinear regression models and estimates the parameters by nonlinear least squares or weighted nonlinear least squares. You specify the model with programming statements. This gives you great flexibility in modeling the relationship between the response variable and independent (regressor) variables. It does, however, require additional coding compared to model specifications in linear modeling procedures such as the REG, GLM, and MIXED procedures.

Estimating parameters in a nonlinear model is an iterative process that commences from starting values. You need to declare the parameters in your model and supply their initial values for the NLIN procedure. You do not need to specify derivatives of the model equation with respect to the parameters. Although facilities for specifying first and second derivatives exist in the NLIN procedure, it is not recommended that you specify derivatives this way. Obtaining derivatives from user-specified expressions predates the high-quality automatic differentiator that is now used by the NLIN procedure.

Nonlinear least-squares estimation involves finding those values in the parameter space that minimize the (weighted) residual sum of squares. In a sense, this is a "distribution-free" estimation criterion since the distribution of the data does not need to be fully specified. Instead, the assumption of homoscedastic and uncorrelated model errors with zero mean is sufficient. You can relax the homoscedasticity assumption by using a weighted residual sum of squares criterion. The assumption of uncorrelated errors (independent observations) cannot be relaxed in the NLIN procedure. In summary, the primary assumptions for analyses with the NLIN procedure are as follows:

• The structure in the response variable can be decomposed additively into a mean function and an error component.

• The model errors are uncorrelated and have zero mean. Unless a weighted analysis is performed, the errors are also assumed to be homoscedastic (have equal variance).

• The mean function consists of known regressor (independent) variables and unknown constants (the parameters).

Fitting nonlinear models can be a difficult undertaking. There is no closed-form solution for the parameter estimates, and the process is iterative. There can be one or more local minima in the residual sum of squares surface, and the process depends on the starting values supplied by the user. You can reduce the dependence on the starting values and reduce the chance to arrive at a local minimum by specifying a grid of starting values. The NLIN procedure then computes the residual sum of squares at each point on the grid and starts the iterative process from the point that yields the lowest sum of squares. Even in this case, however, convergence does not guarantee that a global minimum has been found.

The numerical behavior of a model and a model–data combination can depend on the way in which you parameterize the model—for example, whether parameters are expressed on the logarithmic scale or not. Parameterization also has bearing on the interpretation of the estimated quantities and the statistical properties of the parameter estimators. Inferential procedures in nonlinear regression models are typically approximate in that they rely on the asymptotic properties of the parameter estimators that are obtained as the sample size grows without bound. Such asymptotic inference can be questionable in small samples, especially if the behavior of the parameter estimators is "far-from-linear." Reparameterization of the model can yield parameters whose behavior is akin to that of estimators in linear models. These parameters exhibit close-to-linear behavior.

The NLIN procedure solves the nonlinear least squares problem by one of the following four algorithms (methods):

• steepest-descent or gradient method

• Newton method

• modified Gauss-Newton method

• Marquardt method

These methods use derivatives or approximations to derivatives of the SSE with respect to the parameters to guide the search for the parameters producing the smallest SSE. Derivatives computed automatically by the NLIN procedure are analytic, unless the model contains functions for which an analytic derivative is not available.

Using PROC NLIN, you can also do the following:

• confine the estimation procedure to a certain range of values of the parameters by imposing bounds on the estimates

• produce new SAS data sets containing predicted values, parameter estimates, residuals and other model diagnostics, estimates at each iteration, and so forth.

You can use the NLIN procedure for segmented models (see Example 60.1) or robust regression (see Example 60.2). You can also use it to compute maximum-likelihood estimates for certain models (see Jennrich and Moore 1975; Charnes, Frome, and Yu 1976). For maximum likelihood estimation in a model with a linear predictor and binomial error distribution, see the LOGISTIC, PROBIT, GENMOD, GLIMMIX, and CATMOD procedures. For a linear model with a Poisson, gamma, or inverse gaussian error distribution, see the GENMOD and GLIMMIX procedures. For likelihood estimation in a linear model with a normal error distribution, see the MIXED, GENMOD, and GLIMMIX procedures. The PHREG and LIFEREG procedures fit survival models by maximum likelihood. For general maximum likelihood estimation, see the NLP procedure in the SAS/OR User’s Guide and the NLMIXED procedure in the SAS/STAT User’s Guide. These procedures are recommended over the NLIN procedure for solving maximum likelihood problems.

PROC NLIN uses the Output Delivery System (ODS), a SAS subsystem that provides capabilities for displaying and controlling the output from SAS procedures. ODS enables you to convert any of the output from PROC NLIN into a SAS data set. See the section ODS Table Names for a listing of the ODS tables that are produced by the NLIN procedure.