Profile Likelihood Confidence Intervals

Wald-based and likelihood-ratio-based confidence intervals are available in the MODEL procedure for computing a confidence interval on an estimated parameter. A confidence interval on a parameter ${\theta }$ can be constructed by inverting a Wald-based or a likelihood-ratio-based test.

The approximate ${100(1-{\alpha })}$ % Wald confidence interval for a parameter ${\theta }$ is

\[  \hat{\theta } {\pm } z_{1-{\alpha }/2}\hat{{\sigma }}  \]

where ${z_{p}}$ is the ${100p}$th percentile of the standard normal distribution, ${\hat{\theta }}$ is the maximum likelihood estimate of ${\theta }$, and ${\hat{{\sigma }}}$ is the standard error estimate of ${\hat{\theta }}$.

A likelihood-ratio-based confidence interval is derived from the ${{\chi }^{2}}$ distribution of the generalized likelihood ratio test. The approximate ${1-{\alpha }}$ confidence interval for a parameter ${\theta }$ is

\[  \theta : 2[{l(\hat{\theta }) - l(\theta )}] {\leq } q_{1,1-{\alpha }} = 2 l^{{\ast }}  \]

where ${q_{1,1-{\alpha }}}$ is the ${(1-{\alpha })}$ quantile of the ${{\chi }^{2}}$ with one degree of freedom, and ${l(\theta )}$ is the log likelihood as a function of one parameter. The endpoints of a confidence interval are the zeros of the function ${l(\theta ) - l^{{\ast }}}$. Computing a likelihood-ratio-based 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 );

proc model data=exp;
   parm zo 35 b;
      dert.z = b * z;
   fit y init=(z=zo) / prl=both;
   test zo = 40.475437 ,/ lr;

The output from the requested confidence intervals and the TEST statement are shown in Figure 19.58

Figure 19.58: Confidence Interval Estimation

The MODEL Procedure

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, $\textit{zo}=40.475437$, is close to the upper bound computed for the likelihood ratio confidence interval, $\textit{zo}\le 40.4921$. This coincidence is not germane to the analysis however, since the likelihood ratio test is a test of the null hypothesis $\mi {H}_0 : \textit{zo}=40.475437$ and the confidence interval can be viewed as a test of the null hypothesis $\mi {H}_0 : 32.8381\le \textit{zo}\le 40.4921$.