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The MODEL Procedure |
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 can be constructed by inverting a Wald-based or a likelihood ratio-based test.
The approximate % Wald confidence interval for a parameter is
where z_{p} is the 100pth percentile of the standard normal distribution, is the maximum likelihood estimate of , and is the standard error estimate of .
A likelihood ratio-based confidence interval is derived from the distribution of the generalized likelihood ratio test. The approximate confidence interval for a parameter is
To request confidence intervals on estimated parameters, specify the following option in the FIT statement:
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;
Note that the likelihood ratio test reported the probability that zo = 40.47543 is 5% but zo = 40.47543 is the upper bound of a 95% confidence interval. To understand this conundrum, note that the TEST statement is using the likelihood ratio statistic to test the null hypothesis H_{0} : zo = 40.47543 with the alternate that . The upper confidence interval can be viewed as a test with the null hypothesis H_{0} : zo < = 40.47543.
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