The LOGISTIC Procedure

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.

Likelihood Ratio-Based Confidence Intervals

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 $\text{[math]}$ distribution of the generalized likelihood ratio test (Venzon and Moolgavkar; 1988). Suppose that the parameter vector is $\text{[math]}$ and you want to compute a confidence interval for $\text{[math]}$ . The profile-likelihood function for $\text{[math]}$ is defined as

$\text{[math]}$

where $\text{[math]}$ is the set of all $\text{[math]}$ with the $\text{[math]}$ th element fixed at $\text{[math]}$ , and $\text{[math]}$ is the log-likelihood function for $\text{[math]}$ . If $\text{[math]}$ is the log likelihood evaluated at the maximum likelihood estimate $\text{[math]}$ , then $\text{[math]}$ has a limiting chi-square distribution with one degree of freedom if $\text{[math]}$ is the true parameter value. Let $\text{[math]}$ , where $\text{[math]}$ is the $\text{[math]}$ percentile of the chi-square distribution with one degree of freedom. A $\text{[math]}$ % confidence interval for $\text{[math]}$ is

$\text{[math]}$

The endpoints of the confidence interval are found by solving numerically for values of $\text{[math]}$ 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 $\text{[math]}$ is approximated by the quadratic function

$\text{[math]}$

where $\text{[math]}$ is the gradient vector and $\text{[math]}$ is the Hessian matrix. The increment $\text{[math]}$ for the next iteration is obtained by solving the likelihood equations

$\text{[math]}$

where $\text{[math]}$ is the Lagrange multiplier, $\text{[math]}$ is the $\text{[math]}$ th unit vector, and $\text{[math]}$ is an unknown constant. The solution is

$\text{[math]}$

By substituting this $\text{[math]}$ into the equation $\text{[math]}$ , you can estimate $\text{[math]}$ as

$\text{[math]}$

The upper confidence limit for $\text{[math]}$ is computed by starting at the maximum likelihood estimate of $\text{[math]}$ and iterating with positive values of $\text{[math]}$ until convergence is attained. The process is repeated for the lower confidence limit by using negative values of $\text{[math]}$ .

Convergence is controlled by the value $\text{[math]}$ specified with the PLCONV= option in the MODEL statement (the default value of $\text{[math]}$ is 1E–4). Convergence is declared on the current iteration if the following two conditions are satisfied:

$\text{[math]}$

and

$\text{[math]}$

Wald Confidence Intervals

Wald confidence intervals are sometimes called the normal confidence intervals. They are based on the asymptotic normality of the parameter estimators. The $\text{[math]}$ % Wald confidence interval for $\text{[math]}$ is given by

$\text{[math]}$

where $\text{[math]}$ is the $\text{[math]}$ th percentile of the standard normal distribution, $\text{[math]}$ is the maximum likelihood estimate of $\text{[math]}$ , and $\text{[math]}$ is the standard error estimate of $\text{[math]}$ .