Confidence Intervals for Parameters |
There are two methods of computing confidence intervals for the regression parameters. One is based on the profile-likelihood function, and the other is based on the asymptotic normality of the parameter estimators. The latter is not as time-consuming as the former, since it does not involve an iterative scheme; however, it is not thought to be as accurate as the former, especially with small sample size. You use the CLPARM= option to request confidence intervals for the parameters.
The likelihood ratio-based confidence interval is also known as the profile-likelihood confidence interval. The construction of this interval is derived from the asymptotic distribution of the generalized likelihood ratio test (Venzon and Moolgavkar; 1988). Suppose that the parameter vector is and you want to compute a confidence interval for . The profile-likelihood function for is defined as
where is the set of all with the th element fixed at , and is the log-likelihood function for . If is the log likelihood evaluated at the maximum likelihood estimate , then has a limiting chi-square distribution with one degree of freedom if is the true parameter value. Let , where is the percentile of the chi-square distribution with one degree of freedom. A % confidence interval for is
The endpoints of the confidence interval are found by solving numerically for values of that satisfy equality in the preceding relation. To obtain an iterative algorithm for computing the confidence limits, the log-likelihood function in a neighborhood of is approximated by the quadratic function
where is the gradient vector and is the Hessian matrix. The increment for the next iteration is obtained by solving the likelihood equations
where is the Lagrange multiplier, is the th unit vector, and is an unknown constant. The solution is
By substituting this into the equation , you can estimate as
The upper confidence limit for is computed by starting at the maximum likelihood estimate of and iterating with positive values of until convergence is attained. The process is repeated for the lower confidence limit by using negative values of .
Convergence is controlled by the value specified with the PLCONV= option in the MODEL statement (the default value of is 1E–4). Convergence is declared on the current iteration if the following two conditions are satisfied:
and
Wald confidence intervals are sometimes called the normal confidence intervals. They are based on the asymptotic normality of the parameter estimators. The % Wald confidence interval for is given by
where is the th percentile of the standard normal distribution, is the maximum likelihood estimate of , and is the standard error estimate of .