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.