Waldbased and likelihoodratiobased confidence intervals are available in the MODEL procedure for computing a confidence interval on an estimated parameter. A confidence interval on a parameter can be constructed by inverting a Waldbased or a likelihoodratiobased test.
The approximate % Wald confidence interval for a parameter is

where is the th percentile of the standard normal distribution, is the maximum likelihood estimate of , and is the standard error estimate of .
A likelihoodratiobased confidence interval is derived from the distribution of the generalized likelihood ratio test. The approximate confidence interval for a parameter is

where is the quantile of the with one degree of freedom, and is the log likelihood as a function of one parameter. The endpoints of a confidence interval are the zeros of the function . Computing a likelihoodratiobased confidence interval is an iterative process. This process must be performed twice for each parameter, so the computational cost is considerable. Using a modified form of the algorithm recommended by Venzon and Moolgavkar (1988), you can determine that the cost of each endpoint computation is approximately the cost of estimating the original system.
To request confidence intervals on estimated parameters, specify the PRL= option in the FIT statement. By default, the PRL option produces 95% likelihood ratio confidence limits. The coverage of the confidence interval is controlled by the ALPHA= option in the FIT statement.
The following is an example of the use of the confidence interval options.
data exp; do time = 1 to 20; y = 35 * exp( 0.01 * time ) + 5*rannor( 123 ); output; end; run; proc model data=exp; parm zo 35 b; dert.z = b * z; y=z; fit y init=(z=zo) / prl=both; test zo = 40.475437 ,/ lr; run;
The output from the requested confidence intervals and the TEST statement are shown in Figure 19.58
Figure 19.58: Confidence Interval Estimation
Nonlinear OLS Parameter Estimates  

Parameter  Estimate  Approx Std Err  t Value  Approx Pr > t 
zo  36.58933  1.9471  18.79  <.0001 
b  0.006497  0.00464  1.40  0.1780 
Test Results  

Test  Type  Statistic  Pr > ChiSq  Label 
Test0  L.R.  3.81  0.0509  zo = 40.475437 
Parameter Wald 95% Confidence Intervals 


Parameter  Value  Lower  Upper 
zo  36.5893  32.7730  40.4056 
b  0.00650  0.00259  0.0156 
Parameter Likelihood Ratio 95% Confidence Intervals 


Parameter  Value  Lower  Upper 
zo  36.5893  32.8381  40.4921 
b  0.00650  0.00264  0.0157 
In this example the parameter value used in the likelihood ratio test, , is close to the upper bound computed for the likelihood ratio confidence interval, . This coincidence is not germane to the analysis however, since the likelihood ratio test is a test of the null hypothesis and the confidence interval can be viewed as a test of the null hypothesis .