This section describes the displayed output from PROC NLMIXED. See
the section "ODS Table Names" for details about how this output interfaces with
the Output Delivery System.
The NLMIXED procedure first displays the "Specifications"
table, listing basic information about the nonlinear mixed model that
you have specified. It includes the principal variables and
The "Dimensions" table lists counts of important quantities in
your nonlinear mixed model, including the number of observations,
subjects, parameters, and quadrature points.
The "Parameters" table displays the information you provided
with the PARMS statement and the value of the negative log likelihood
function evaluated at the starting values.
Starting Gradient and Hessian
The START option in the PROC NLMIXED statement displays the
gradient of the negative log likelihood function at the starting
values of the parameters. If you also specify the HESS option,
then the starting Hessian is displayed as well.
The iteration history consists of one line of output for each
iteration in the optimization process. The iteration history is
displayed by default because it is important that you check for
possible convergence problems. The default iteration history includes
the following variables:
- Iter, the iteration number
- Calls, the number of function calls
- NegLogLike, the value of the objective function
- Diff, the difference between adjacent function values
- MaxGrad, the maximum of the absolute (projected) gradient
components (except NMSIMP)
- Slope, the slope gTs of the search direction s at the current
parameter iterate (QUANEW only)
- Rho, the ratio between the achieved and predicted value of Diff
- Radius, the radius of the trust region (TRUREG only)
- StdDev, the standard deviation of the simplex values (NMSIMP only)
- Delta, the vertex length of the simplex (NMSIMP only)
- Size, the size of the simplex (NMSIMP only)
For the QUANEW method, the value of Slope should be significantly
negative. Otherwise, the line-search algorithm has difficulty
reducing the function value sufficiently. If this difficulty is
encountered, an asterisk (*) appears after the iteration number. If
there is a tilde after the iteration
number, the BFGS update is skipped, and very high values of the
Lagrange function are produced. A backslash (\ ) after
the iteration number indicates that Powell's correction for the BFGS
update is used.
For methods using second derivatives, an asterisk (*) after the
iteration number means that the computed Hessian approximation was
singular and had to be ridged with a positive value.
For the NMSIMP method, only one line is displayed for several internal
iterations. This technique skips the output for some iterations
because some of the termination tests (StdDev and Size) are rather
time consuming compared to the simplex operations, and they are
performed only every five simplex operations.
The ITDETAILS option in the PROC NLMIXED statement provides a
more detailed iteration history. Besides listing the current
values of the parameters and their gradients, the following
values are provided in addition to the default output:
- Restart, the number of iteration restarts
- Active, the number of active constraints
- Lambda, the value of the Lagrange multiplier (TRUREG and DBLDOG
- Ridge, the ridge value (NRRIDG only)
- Alpha, the line-search step size (QUANEW only)
An apostrophe (') trailing the number of active constraints indicates
that at least one of the active constraints was released from the
active set due to a significant Lagrange multiplier.
The "Fitting Information" table lists the final maximized value
of the log likelihood as well as the information criteria of Akaike
and Schwarz in two different forms. The smaller-is-better forms are
where is the vector of parameter estimates and s is
the number of subjects. The larger-is-better forms for AIC and BIC
multiply the preceding values by -0.5.
The "Parameter Estimates" table lists the estimates of the
parameter values after successful convergence of the optimization
problem or the final values of the parameters under nonconvergence.
If the problem did converge, standard errors are computed from the
final Hessian matrix. The ratio of the estimate with its standard
error produces a t value, with approximate degrees of freedom
computed as the number of subjects minus the number of random effects.
A p-value and confidence limits based on this t distribution are
also provided. Finally, the gradient of the negative log likelihood
function is displayed for each parameter, and you should verify that
they each are sufficiently small for non-constrained parameters.
Covariance and Correlation Matrices
Following standard maximum likelihood theory (for example, Serfling
1980), the asymptotic variance-covariance matrix of the parameter
estimates equals the inverse of the Hessian matrix. You can display
this matrix with the COV option in the PROC NLMIXED statement. The
corresponding correlation form is available with the CORR option.
The "Additional Estimates" table displays the results of all
ESTIMATE statements that you specify, with the same columns as the
"Parameter Estimates" table. The ECOV and ECORR options in the
PROC NLMIXED statement produce tables displaying the approximate
covariance and correlation matrices of the additional estimates. They
are computed using the delta method (Billingsley 1986). The EDER
option in the PROC NLMIXED statement produces a table displaying
the derivatives of the additional estimates with respect to the
model parameters evaluated at their final estimated values.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.