Let be the jth unit vector—that is, the jth entry of the vector is 1 and all other entries are 0. The hazard ratio for the explanatory variable with regression coefficient is defined as . In general, a log-hazard ratio can be written as , a linear combination of the regression coefficients, and the hazard ratio is obtained by replacing with .
The hazard ratio is estimated by , where is the maximum likelihood estimate of the .
The confidence limits for the hazard ratio are calculated as
where is estimated covariance matrix, and is the th percentile point of the standard normal distribution.
The construction of the profile-likelihood-based confidence interval is derived from the asymptotic distribution of the generalized likelihood ratio test of 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 jth 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 th 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 jth 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, using negative values of .
Convergence is controlled by 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
The profile-likelihood confidence limits for the hazard ratio are obtained by exponentiating these confidence limits.