PROC PHREG: Confidence Limits for a Hazard Ratio :: SAS/STAT(R) 9.2 User's Guide, Second Edition

The PHREG Procedure

Confidence Limits for a Hazard Ratio

Let $\text{[math]}$ be the $\text{[math]}$ th unit vector—that is, the $\text{[math]}$ th entry of the vector is 1 and all other entries are 0. The hazards ratio for the explanatory variable with regression coefficient $\text{[math]}$ is defined as $\text{[math]}$ . In general, a log-hazard ratio can be written as $\text{[math]}$ , a linear combination of the regression coefficients, and the hazard ratio $\text{[math]}$ is obtained by replacing $\text{[math]}$ with $\text{[math]}$ .

Point Estimate

The hazards ratio $\text{[math]}$ is estimated by $\text{[math]}$ , where $\text{[math]}$ is the maximum likelihood estimate of the $\text{[math]}$ .

Wald’s Confidence Limits

The $\text{[math]}$ confidence limits for the hazard ratio are calculated as

$\text{[math]}$

where $\text{[math]}$ is estimated covariance matrix, and $\text{[math]}$ is the $\text{[math]}$ percentile point of the standard normal distribution.

Profile-Likelihood Confidence Limits

The construction of the profile-likelihood-based confidence interval is derived from the asymptotic $\text{[math]}$ distribution of the generalized likelihood ratio test of 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, using negative values of $\text{[math]}$ .

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

$\text{[math]}$

and

$\text{[math]}$

The profile-likelihood confidence limits for the hazard ratio $\text{[math]}$ are obtained by exponentiating these confidence limits.

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