The NLP Procedure

PROFILE Statement

PROFILE parms [ / [ ALPHA= values ] [ options ] ] ;

where parms is given in the format pnam_1 pnam_2 …pnam_n, and values is the list of $\alpha$ values in (0,1).

The PROFILE statement

writes the $(x,y)$ coordinates of profile points for each of the listed parameters to the OUTEST= data set
displays, or writes to the OUTEST= data set, the profile likelihood confidence limits (PL CLs) for the listed parameters for the specified $\alpha$ values. If the approximate standard errors are available, the corresponding Wald confidence limits can be computed.

When computing the profile points or likelihood profile confidence intervals, PROC NLP assumes that a maximization of the log likelihood function is desired. Each point of the profile and each endpoint of the confidence interval is computed by solving corresponding nonlinear optimization problems.

The keyword PROFILE must be followed by the names of parameters for which the profile or the PL CLs should be computed. If the parameter name list is empty, the profiles and PL CLs for all parameters are computed. Then, optionally, the $\alpha$ values follow. The list of $\alpha$ values may contain TO and BY keywords. Each element must satisfy $0 < \alpha < 1$ . The following is an example:

   profile l11-l15 u1-u5 c /
           alpha= .9 to .1 by -.1 .09 to .01 by -.01;

Duplicate $\alpha$ values or values outside $(0,1)$ are automatically eliminated from the list.

A number of additional options can be specified.

FFACTOR= $r$: specifies the factor relating the discrepancy function $f(\theta )$ to the $\chi ^2$ quantile. The default value is $r=2$ .
FORCHI= F | CHI: defines the scale for the $y$ values written to the OUTEST= data set. For FORCHI=F, the $y$ values are scaled to the values of the log likelihood function $f=f(\theta )$ ; for FORCHI=CHI, the $y$ values are scaled so that $\hat{y}=\chi ^2$ . The default value is FORCHI=F.
FEASRATIO= $r$: specifies a factor of the Wald confidence limit (or an approximation of it if standard errors are not computed) defining an upper bound for the search for confidence limits. In general, the range of $x$ values in the profile graph is between $r=1$ and $r=2$ times the length of the corresponding Wald interval. For many examples, the $\chi ^2$ quantiles corresponding to small $\alpha$ values define a $y$ level $\hat{y} - \frac{1}{2}q_1(1-\alpha )$ , which is too far away from $\hat{y}$ to be reached by $y(x)$ for $x$ within the range of twice the Wald confidence limit. The search for an intersection with such a $y$ level at a practically infinite value of $x$ can be computationally expensive. A smaller value for $r$ can speed up computation time by restricting the search for confidence limits to a region closer to $\hat{x}$ . The default value of $r=1000$ practically disables the FEASRATIO= option.
OUTTABLE: specifies that the complete set $\theta$ of parameter estimates rather than only $x=\theta _ j$ for each confidence limit is written to the OUTEST= data set. This output can be helpful for further analyses on how small changes in $x=\theta _ j$ affect the changes in the $\theta _ i, i \neq j$ .

For some applications, it may be computationally less expensive to compute the PL confidence limits for a few parameters than to compute the approximate covariance matrix of many parameters, which is the basis for the Wald confidence limits. However, the computation of the profile of the discrepancy function and the corresponding CLs in general will be much more time-consuming than that of the Wald CLs.