The NLIN Procedure

specifies the level of significance used in the construction of % confidence intervals. The value must be strictly between 0 and 1; the default value of results in 95% intervals. This value is used as the default confidence level for limits computed in the "Parameter Estimates" table and with the LCLM, LCL, UCLM, and UCL options in the OUTPUT statement.

requests that PROC NLIN display the residual sums of squares only for the best combinations of possible starting values from the grid. If you do not specify the BEST= option, PROC NLIN displays the residual sum of squares for every combination of possible parameter starting values.

adds Box’s bias and percentage bias measures to the "Parameter Estimates" table (Box 1971). Box’s bias measure, along with Hougaard’s measure of skweness, is used for assessing a parameter estimator’s close-to-linear behavior (Ratkowsky 1983, 1990). Hence, it is useful for identifying problematic parameters (Seber and Wild 1989, sec. 4.7.1). When you specify the BIAS option, Box’s bias measure (Box 1971) and the percentage bias (the bias expressed as a percentage of the least-squares estimator) are added for each parameter to the "Parameter Estimates" table. Ratkowsky (1983, p. 21) takes a percentage bias in excess of 1% to be a good rule of thumb for indicating nonlinear behavior.

See the section Box’s Measure of Bias for further details. Example 62.4 shows how to use this measure, along with Hougaard’s measure of skewness, to evaluate changes in the parameterization of a nonlinear model. Computation of the Box’s bias measure requires first and second derivatives. If you do not provide derivatives with the DER statement—and it is recommended that you do not—the analytic derivatives are computed for you.

specifies the convergence criterion for PROC NLIN. For all iterative methods the relative offset convergence measure of Bates and Watts is used by default to determine convergence. This measure is labeled "R" in the "Estimation Summary" table. The iterations are said to have converged for CONVERGE= if

where is the residual vector and is the matrix of first derivatives with respect to the parameters. The default LOSS function is the sum of squared errors (SSE), and denotes the value of the loss function at the th iteration. By default, CONVERGE=. The R convergence measure cannot be computed accurately in the special case of a perfect fit (residuals close to zero). When the SSE is less than the value of the SINGULAR= criterion, convergence is assumed.

uses the change in the LOSS function as the convergence criterion and tunes the criterion. The iterations are said to have converged for CONVERGEOBJ= if

where LOSS is the LOSS for the th iteration. The default LOSS function is the sum of squared errors (SSE), the residual sum of squares. The constant should be a small positive number. For more details about the LOSS function, see the section Special Variable Used to Determine Convergence Criteria. For more details about the computational methods in the NIN procedure, see the section Computational Methods.

Note that in SAS 6 the CONVERGE= and CONVERGEOBJ= options both requested that convergence be tracked by the relative change in the loss function. If you specify the CONVERGEOBJ= option in newer releases, the CONVERGE= option is disabled. This enables you to track convergence as in SAS 6.

uses the maximum change among parameter estimates as the convergence criterion and tunes the criterion. The iterations are said to have converged for CONVERGEPARM= if

where is the value of the th parameter at the th iteration.

The default convergence criterion is CONVERGE. If you specify CONVERGEPARM=, the maximum change in parameters is used as the convergence criterion. If you specify both the CONVERGEOBJ= and CONVERGEPARM= options, PROC NLIN continues to iterate until the decrease in LOSS is sufficiently small (as determined by the CONVERGEOBJ= option) and the maximum change among the parameters is sufficiently small (as determined by the CONVERGEPARM= option).

specifies the input SAS data set to be analyzed by PROC NLIN. If you omit the DATA= option, the most recently created SAS data set is used.

displays a message for each statement in the model program as it is executed. This debugging option is rarely needed, and it produces large amounts of output.

uses a Moore-Penrose inverse (-inverse) in parameter estimation. See Kennedy and Gentle (1980) for details.

adds Hougaard’s measure of skewness to the "Parameter Estimates" table (Hougaard 1982, 1985). The skewness measure is one method of assessing a parameter estimator’s close-to-linear behavior in the sense of Ratkowsky (1983, 1990). The behavior of estimators that are close to linear approaches that of least squares estimators in linear models, which are unbiased and have minimum variance. When you specify the HOUGAARD option, the standardized skewness measure of Hougaard (1985) is added for each parameter to the "Parameter Estimates" table. Because of the linkage between nonlinear behavior of a parameter estimator in nonlinear regression and the nonnormality of the estimator’s sampling distribution, Ratkowsky (1990, p. 28) provides the following rules to interpret the (standardized) Hougaard skewness measure:

• Values less than 0.1 in absolute value indicate very close-to-linear behavior.

• Values between 0.1 and 0.25 in absolute value indicate reasonably close-to-linear behavior.

• The nonlinear behavior is apparent for absolute values above 0.25 and is considerable for absolute values above 1.

See the section Hougaard’s Measure of Skewness for further details. Example 62.4 shows how to use this measure to evaluate changes in the parameterization of a nonlinear model. Computation of the Hougaard skewness measure requires first and second derivatives. If you do not provide derivatives with the DER statement—and it is recommended that you do not—the analytic derivatives are computed for you.

displays the model program and variable lists, including the statements added by macros. Note that the expressions displayed by the LIST option do not necessarily represent the way the expression is actually calculated—because intermediate results for common subexpressions can be reused—but are shown in expanded form. To see how the expression is actually evaluated, use the LISTCODE option.

selects the LIST, LISTDEP, LISTDER, and LISTCODE options.

displays the derivative tables and the compiled model program code. The LISTCODE option is a debugging feature and is not normally needed.

produces a report that lists, for each variable in the model program, the variables that depend on it and the variables on which it depends.

displays a table of derivatives. The derivatives table lists each nonzero derivative computed for the problem. The derivative listed can be a constant, a variable in the model program, or a special derivative variable created to hold the result of an expression.

specifies the maximum number of iterations in the optimization process. The default is .

places a limit on the number of step halvings. The value of MAXSUBIT must be a positive integer and the default value is .

specifies the iterative method employed by the NLIN procedure in solving the nonlinear least squares problem. The GAUSS, MARQUARDT, and NEWTON methods are more robust than the GRADIENT method. If you omit the METHOD= option, METHOD=GAUSS is used. See the section Computational Methods for more information.

displays the global nonlinearity measures table. These measures include the maximum intrinsic and parameter-effects curvatures (Bates and Watts 1980), the root mean square (RMS) intrinsic and parameter-effects curvatures and the critical curvature value (Bates and Watts 1980). In addition, the variances of the ordinary and projected residuals are included. According to Bates and Watts (1980), both intrinisic and parameter-effects curvatures are deemed negligible if they are less than the critical curvature value. This critical value is given by where . The value can be considered as the radius of curvature of the percent confidence region (Bates and Watts 1980).

suppresses the display of the "Iteration History" table.

removes the restriction that the objective value must decrease at every iteration. Step halving is still used to satisfy BOUNDS and to ensure that the number of observations that can be evaluated does not decrease. The NOHALVE option can be useful in weighted nonlinear least squares problems where the weights depend on the parameters, such as in iteratively reweighted least squares (IRLS) fitting. See Example 62.2 for an application of IRLS fitting.

suppresses the display of the output. Note that this option temporarily disables the Output Delivery System (ODS). For more information, see Chapter 20, Using the Output Delivery System.

specifies an output data set that contains the parameter estimates produced at each iteration. See the section Output Data Sets for details. If you want to create a permanent SAS data set, you must specify a two-level name. See the chapter "SAS Files" in SAS Language Reference: Concepts for more information about permanent SAS data sets.

controls the plots produced through ODS Graphics. When you specify only one plot request, you can omit the parentheses around the plot request. Here are some examples:

  plots
  plots        = none
  plots        = diagnostics(unpack)
  plots        = fit(stats=none)
  plots        = residuals(residualtype=proj unpack smooth)     
  plots(stats=all)  = (diagnostics(stats=(maxincurv maxpecurv)) fit)

ODS Graphics must be enabled before requesting plots. For example:

ods graphics on;

proc nlin plots=diagnostics(stats=all);
   model y = alpha - beta*(gamma**x);
run;

ods graphics off;

For more information about enabling and disabling ODS Graphics, see the section Enabling and Disabling ODS Graphics in Chapter 21, Statistical Graphics Using ODS.

To produce graphs, ODS Graphics must be enabled. If you do not specify the PLOTS option, PROC NLIN produces no graphs. PROC NLIN produces the plots listed in Table 62.2 with the default set of statistics and options if you specify the PLOTS option simply as PLOTS.

You can request additional graphs by specifying the PLOTS=plot-request option. For a listing of all the plots that PROC NLIN produces, see the section ODS Graphics. Each global-plot-option applies to all plots generated by the NLIN procedure, unless it is altered by a specific option after a plot-request. The following global-plot-options are available:

You can specify the following plot-requests in the PLOTS= option:

displays the result of each statement in the program as it is executed. This option is a debugging feature that produces large amounts of output and is normally not needed.

specifies a value that controls the step-size search. By default RHO=0.1, except when METHOD=MARQUARDT. In that case, RHO=10. See the section Step-Size Search for more details.

specifies that, when the iteration limit is exceeded, the parameter estimates from the final iteration be output to the OUTEST= data set. These parameter estimates are associated with the observation for which _TYPE_="FINAL". If you omit the SAVE option, the parameter estimates from the final iteration are not output to the data set unless convergence has been attained.

specifies a value to use as the estimate of the residual variance in lieu of the estimated mean-squared error. This value is used in computing the standard errors of the estimates. Fixing the value of the residual variance can be useful, for example, in maximum likelihood estimation.

specifies the singularity criterion, , which is the absolute magnitude of the smallest pivot value allowed when inverting the Hessian or the approximation to the Hessian. The default value is times the machine epsilon; this product is approximately on most computers.

specifies the step-size search method. The default is SMETHOD=HALVE. See the section Step-Size Search for details.

specifies a value that is used to control the step-size search. The default is TAU=1, except when METHOD=MARQUARDT. In that case the default is TAU=0.01. See the section Step-Size Search for details.

adds to the analysis of variance table the uncorrected total sum of squares in models that have an (implied) intercept, and adds the corrected total sum of squares in models that do not have an (implied) intercept.

displays the result of each operation in each statement in the model program as it is executed, in addition to the information displayed by the FLOW and PRINT options. This debugging option is needed very rarely, and it produces even more output than the FLOW and PRINT options.

displays a cross-reference of the variables in the model program showing where each variable is referenced or given a value. The XREF listing does not include derivative variables.

specifies that no degrees of freedom be lost when a bound is active. When the UNCORRECTEDDF option is not specified, an active bound is treated as if a restriction were applied to the set of parameters, so one parameter degree of freedom is deducted.