The PHREG Procedure

Confidence Limits for a Hazard Ratio

Point Estimate
Wald’s Confidence Limits
Profile-Likelihood Confidence Limits

Let $\mb {e}_ j$ 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 $\beta _ j=\mb {e}_ j’\bbeta$ is defined as $\textrm{exp}(\beta _ j)$ . In general, a log-hazard ratio can be written as $\mb {h}’\bbeta$ , a linear combination of the regression coefficients, and the hazard ratio $\textrm{exp}({\mb {h}’\bbeta })$ is obtained by replacing $\mb {e}_ j$ with $\mb {h}$ .

Point Estimate

The hazard ratio $\textrm{exp}(\mb {e}_ j’\bbeta )$ is estimated by $\textrm{exp}(\mb {e}_ j’\hat{\beta })$ , where $\hat{\bbeta }$ is the maximum likelihood estimate of the $\bbeta$ .

Wald’s Confidence Limits

The $100(1-{\alpha })\%$ confidence limits for the hazard ratio are calculated as

$\mr {exp} \left(\mb {e}_ j’ \hat{\beta } \pm z_{\alpha /2} \sqrt {\mb {e}_ j’\hat{\mb {V}}(\hat{\bbeta })\mb {e}_ j} \right)$

where $\hat{\bV }(\hat{\bbeta })$ is estimated covariance matrix, and $z_{\alpha /2}$ is the $100(1-{\alpha }/2)$ 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 $\chi ^2$ distribution of the generalized likelihood ratio test of Venzon and Moolgavkar (1988). Suppose that the parameter vector is $\bbeta = (\beta _{1},\ldots ,\beta _{k})’$ and you want to compute a confidence interval for $\beta _{j}$ . The profile-likelihood function for $\beta _{j}=\gamma$ is defined as

$l_ j^*(\gamma ) = \max _{\bbeta \in \mc {B}_ j(\gamma )} l(\bbeta )$

where $\mc {B}_ j(\gamma )$ is the set of all $\bbeta$ with the jth element fixed at $\gamma$ , and $l(\bbeta )$ is the log-likelihood function for $\bbeta$ . If $l_{\max } = l(\hat{\bbeta })$ is the log likelihood evaluated at the maximum likelihood estimate $\hat{\bbeta }$ , then $2( l_{\max } - l_ j^{*}(\beta _{j} ))$ has a limiting chi-square distribution with one degree of freedom if $\beta _{j}$ is the true parameter value. Let $l_0=l_{\max } -0.5\chi ^{2}_{1}(1-\alpha )$ , where $\chi ^{2}_{1}(1-\alpha )$ is the $100(1-\alpha )$ percentile of the chi-square distribution with one degree of freedom. A $100(1-\alpha )$ % confidence interval for $\beta _{j}$ is

$\{ \gamma : l_ j^*(\gamma ) \geq l_{0} \}$

The endpoints of the confidence interval are found by solving numerically for values of $\beta _{j}$ 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 $\bbeta$ is approximated by the quadratic function

$\tilde{l}(\bbeta + \bdelta ) = l(\bbeta ) + \bdelta ’\mb {g} + \frac{1}{2}\bdelta ’ \mb {V} \bdelta$

where $\mb {g}=\mb {g}(\bbeta )$ is the gradient vector and $\bV =\bV (\bbeta )$ is the Hessian matrix. The increment $\bdelta$ for the next iteration is obtained by solving the likelihood equations

$\frac{d}{d\bdelta }\{ \tilde{l}(\bbeta + \bdelta ) + \lambda ( \mb {e}_ j’\bdelta - \gamma )\} = \mb {0}$

where $\lambda$ is the Lagrange multiplier, $\mb {e}_ j$ is the jth unit vector, and $\gamma$ is an unknown constant. The solution is

$\bdelta = -\mb {V}^{-1}(\mb {g} + \lambda \mb {e}_ j)$

By substituting this $\bdelta$ into the equation $\tilde{l}(\bbeta + \bdelta ) = l_0$ , you can estimate $\lambda$ as

$\lambda = \pm \biggl (\frac{2(l_0 - l(\bbeta ) + \frac{1}{2}\mb {g}\mb {V}^{-1}\mb {g})}{\mb {e}_ j\mb {V}^{-1}\mb {e}_ j}\biggr )^{ \frac{1}{2}}$

The upper confidence limit for $\beta _ j$ is computed by starting at the maximum likelihood estimate of $\bbeta$ and iterating with positive values of $\lambda$ until convergence is attained. The process is repeated for the lower confidence limit, using negative values of $\lambda$ .

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

$|l(\bbeta )-l_{0}| \leq \epsilon$

and

$({\mb {g}} + \lambda {\mb {e}_ j})’{\mb {V}}^{-1}(\mb {g} + \lambda {\mb {e}_ j}) \leq \epsilon$

The profile-likelihood confidence limits for the hazard ratio $\textrm{exp}(\mb {e}_ j’\bbeta )$ are obtained by exponentiating these confidence limits.