
ALPHA=

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.

BEST=n

requests that PROC NLIN display the residual sums of squares only for the best n 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.

BIAS

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 skewness, is used for assessing a parameter estimator’s closetolinear
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 leastsquares 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 69.4 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.

CONVERGE=c

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=c 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 ith 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.

CONVERGEOBJ=c

uses the change in the LOSS function as the convergence criterion and tunes the criterion. The iterations are said to have
converged for CONVERGEOBJ=c if
where LOSS is the LOSS for the ith iteration. The default LOSS function is the sum of squared errors (SSE), the residual sum of squares. The constant c 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 NLIN 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.

CONVERGEPARM=c

uses the maximum change among parameter estimates as the convergence criterion and tunes the criterion. The iterations are
said to have converged for CONVERGEPARM=c if
where is the value of the jth parameter at the ith iteration.
The default convergence criterion is CONVERGE
. If you specify CONVERGEPARM=c, 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).

DATA=SASdataset

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.

FLOW

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.

G4

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

HOUGAARD

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 closetolinear 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 closetolinear behavior.

Values between 0.1 and 0.25 in absolute value indicate reasonably closetolinear 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 69.4 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. For weighted least squares,
the NLIN procedure ignores the weights for computing the Hougaard skewness measure. This can be a strong assumption as the
formulation in Hougaard (1985) assumes homoscedastic errors.

LIST

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.

LISTALL

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

LISTCODE

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

LISTDEP

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.

LISTDER

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.

MAXITER=n

specifies the maximum number n of iterations in the optimization process. The default is n = 100.

MAXSUBIT=n

places a limit on the number of step halvings. The value of MAXSUBIT must be a positive integer and the default value is n = 30.

METHOD=GAUSS  MARQUARDT  NEWTON  GRADIENT

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.

NLINMEASURES

displays the global nonlinearity measures table. These measures include the maximum intrinsic and parametereffects curvatures
(Bates and Watts, 1980), the root mean square (RMS) intrinsic and parametereffects 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 intrinsic and parametereffects 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). For weighted least squares, the NLIN procedure ignores the weights for computing the curvature measures. This can be a
strong assumption as the original derivation in Bates and Watts (1980) assumes homoscedastic errors.

NOITPRINT

suppresses the display of the "Iteration History" table.

NOHALVE

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 69.2 for an application of IRLS fitting.

NOPRINT

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.

OUTEST=SASdataset

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 SAS data set in a permanent library, you must specify a twolevel name. For more information
about permanent libraries and SAS data sets, see
SAS Language Reference: Concepts.

PLOTS <(globalplotoption)> <= (plotrequest<(options)> <... plotrequest<(options)>>)>

controls most of the plots that are produced through ODS Graphics (other plots are controlled by the BOOTSTRAP
and PROFILE
statements). When you specify only one plotrequest, you can omit the parentheses around it. 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 plots can be requested. 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.
If ODS Graphics is enabled and if you specify the PLOTS option without any globalplotoption or plotrequests, PROC NLIN produces the plots listed in Table 69.2 with the default set of statistics and options. If you do not specify the PLOTS option, PROC NLIN does not produce any of
these graphs.
Table 69.2: Graphs Produced When the PLOTS Option Is Specified
Plot

Conditional On

ContourFitPlot

Model with two regressors

FitDiagnosticsPanel

Unconditional

FitPlot

Model with one regressor

LeveragePlot

Unconditional

LocalInfluencePlot

Unconditional

ResidualPanel

Unconditional

You can request additional plots by specifying plotrequests. For a listing of all the plots that PROC NLIN produces, see the section ODS Graphics. Each globalplotoption applies to all plots that are generated by the NLIN procedure except for plots that are controlled by the BOOTSTRAP
and PROFILE
statements. The globalplotoption can be overridden by a specific option after a plotrequest.
The following globalplotoptions are available:

RESIDUALTYPE=RAW  PROJ  BOTH

specifies the residual type to be plotted in the fit diagnostics and residual plots. RESIDUALTYPE=RAW requests that only the
ordinary residuals be included in the plots; RESIDUALTYPE=PROJ sets the choice to projected residuals. By default, both residual
types are included, which can also be effected by setting RESIDUALTYPE=BOTH. See the section Residuals in Nonlinear Regression for details about the properties of ordinary and projected residuals in nonlinear regression.

STATS=ALL  DEFAULT  NONE  (plotstatistics)

requests the statistics to be included in all plots, except the ResidualPlots and the unpacked diagnostics plots. Table 69.3 lists the statistics that you can request. STATS=ALL requests all these statistics, STATS=NONE suppresses all statistics,
and STATS=DEFAULT selects the default statistics. You request statistics in addition to the default set by including the keyword
DEFAULT in the plotstatistics list.
Table 69.3: Statistics Available in Plots
Keyword

Default

Description

DEFAULT


All default statistics

MAXINCURV


Maximum intrinsic curvature

MAXPECURV


Maximum parametereffects curvature

MSE

x

Mean squared error, estimated or set by the SIGSQ
option

NOBS

x

Number of observations used

NPARM

x

Number of parameters in the model

PVAR

x

Estimated variance of the projected residuals

RMSINCURV


Root mean square intrinsic curvature

RMSPECURV


Root mean square parametereffects curvature

VAR

x

Estimated variance of the ordinary residuals

Along with the maximum intrinsic and parametereffects curvatures, the critical curvature (CURVCRIT) value, where , is also displayed. You do not need to specify any option for it. See the section Relative Curvature Measures of Nonlinearity for details about curvature measures of nonlinearity.

UNPACK

suppresses paneling.
You can specify the following plotrequests in the PLOTS= option:

ALL

produces all appropriate plots.

NONE

suppresses all plots.

DIAGNOSTICS <(diagnosticsoptions)>

produces a summary panel of fit diagnostics, leverage plots, and localinfluence plots. The fit diagnostics panel includes:

histogram of the ordinary residuals

histogram of the projected residuals

response variable values versus the predicted values

expectation or mean of the ordinary residuals versus the predicted values

ordinary and projected residuals versus the predicted values

standardized ordinary and projected residuals versus the predicted values

standardized ordinary and projected residuals versus the tangential leverage

standardized ordinary and projected residuals versus the Jacobian leverage

box plot of the ordinary and projected residuals if you specify the STATS=NONE suboption
The leverage and local influence plots are produced separately. The leverage plot is an index plot of the tangential and
Jacobian leverages (by observation), and the localinfluence plot contains the local influence by observation for a perturbation
of the response variable. See the sections Leverage in Nonlinear Regression and Local Influence in Nonlinear Regression for a some details about leverages and localinfluence in nonlinear regression.
You can specify the following diagnosticsoptions:

RESIDUALTYPE=RAW  PROJ  BOTH

specifies the residual type to be plotted in the panel. See the RESIDUALTYPE= globalplotoption for details. This diagnosticsoption overrides the PLOTS RESIDUALTYPE globalplotoption. Only the plots that overlay both ordinary and projected residuals in the same plot are affected by this option.

LEVERAGETYPE=TAN  JAC  BOTH

specifies the leverage type to be plotted in the leverage plot. LEVERAGETYPE=TAN specifies that only the tangential leverage
be included in the leverage plot, and LEVERAGETYPE=JAC specifies that only the Jacobian leverage be included. By default,
both are displayed in the leverage plot. The same result can be effected by setting LEVERAGETYPE=BOTH. Only the leverage plot
is affected by this option.

LABELOBS

specifies that the leverage and localinfluence plots be labeled with the observation number. Only these two plots are affected
by this option.

STATS=statsoptions

determines which statistics are included in the panel. See the STATS= globalplotoption for details. This diagnosticsoption overrides the PLOTS STATS globalplotoption.

UNPACK

produces the plots in the diagnostics panel as individual plots. The statistics panel is not included in the individual plots,
even if STATS= globalplotoption or STATS= diagnosticsoption or both are specified.

FITPLOT  FIT <(fitoptions)>

produces, depending on the number of regressors, a scatter or contour fit plot. For a singleregressor model, a scatter plot
of the data overlaid with the regression curve, confidence, and prediction bands is produced. For tworegressor models, a
contour fit plot of the model with overlaid data is produced. If the model contains more than two regressors, no fit plot
is produced.
You can specify the following fitoptions:

NOCLI

suppresses the prediction limits for singleregressor models.

NOCLM

suppresses the confidence limits for singleregressor models.

NOLIMITS

suppresses the confidence and prediction limits for singleregressor models.

OBS=GRADIENT  NONE  OUTLINE  OUTLINEGRADIENT

controls how the observations are displayed. The suboptions are as follows:
 GRADIENT

specifies that observations be displayed as circles colored by the observed response. The same color gradient is used to display
the fitted surface and the observations. Observations for which the predicted response is close to the observed response have
similar colors—the greater the contrast between the color of an observation and the surface, the larger the residual is at
that point. OBS=GRADIENT is the default.
 NONE

suppresses the observations.
 OUTLINE

specifies that observations be displayed as circles with a border but with a completely transparent fill.
 OUTLINEGRADIENT

is the same as OBS=GRADIENT except that a border is shown around each observation. This option is useful for identifying the
location observations for which the residuals are small, because at these points the color of the observations and the color
of the surface are indistinguishable.

CONTLEG

specifies that a continuous legend be included in the contour fit plot of a tworegressor model.

STATS=statsoptions

determines which model fit statistics are included in the panel. See the STATS= globalplotoption for details. This fitoption overrides the PLOTS STATS globalplotoption.

RESIDUALS <(residualoptions)>

produces panels of the ordinary and projected residuals versus the regressors in the model. Each panel contains at most six
plots, and multiple panels are used in the case where there are more than six regressors in the model.
The following residualoptions are available:

RESIDUALTYPE=RAW  PROJ  BOTH

specifies the residual type to be plotted in the panel. See the RESIDUALTYPE= globalplotoption for details. This residualoption overrides the PLOTS RESIDUALTYPE globalplotoption.

SMOOTH

requests a nonparametric smooth of the residuals for each regressor. Each nonparametric fit is a loess fit that uses local
linear polynomials, linear interpolation, and a smoothing parameter selected that yields a local minimum of the corrected
Akaike information criterion (AICC). See Chapter 59: The LOESS Procedure, for details.

UNPACK

suppresses paneling.

PRINT

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.

RHO=value

specifies a value that controls the stepsize search. By default RHO=0.1, except when METHOD=
MARQUARDT. In that case, RHO=10. See the section StepSize Search for more details.

SAVE

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.

SIGSQ=value

specifies a value to use as the estimate of the residual variance in lieu of the estimated meansquared 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.

SINGULAR=s

specifies the singularity criterion, s, 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 1E4 times the machine epsilon; this product is approximately 1E12 on most computers.

SMETHOD=HALVE  GOLDEN  CUBIC

specifies the stepsize search method. The default is SMETHOD=HALVE. See the section StepSize Search for details.

TAU=value

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

TOTALSS

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.

TRACE

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.

XREF

displays a crossreference 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.

UNCORRECTEDDF

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.