Hazard Ratios

The hazard ratio for a quantitative effect with regression coefficient $\beta _ j = \mb {e}_ j’ \bbeta $ is defined as exp$(\beta _ j)$, where $\mb {e}_ j$ denotes the jth unit vector. In general, a log-hazard ratio can be written as $\mb {l}’\bbeta $, a linear combination of the regression coefficients, and the hazard ratio exp$(\mb {l}’\bbeta )$ is obtained by replacing $e_ j$ with $\mb {l}$.

The confidence intervals for hazard ratios are obtained by exponentiating the confidence limits of the corresponding linear combination. Thus, the $100(1-\alpha )$ confidence limits are

\[  \exp \left( \mb {e}_ j’ \hat{\bbeta } \pm t_{\mi {df},\alpha /2} \sqrt {\mb {e}_ j’\hat{\mb {V}}(\hat{\bbeta })\mb {e}_ j} \right)  \]

where $t_{\mi {df},\alpha /2}$ is the $100(1-\alpha /2)$ percentile point of the t distribution with df degrees of freedom. See the section Degrees of Freedom for more information about df. If you use the DF=NONE option in the MODEL statement, then the procedure uses the $100(1-\alpha /2)$ percentile point of the standard normal distribution.