The LIFEREG Procedure

Confidence Intervals

Subsections:

Confidence intervals are computed for all model parameters and are reported in the "Analysis of Parameter Estimates" table. The confidence coefficient can be specified with the ALPHA=$\alpha $ MODEL statement option, resulting in a $(1-\alpha )\times 100\% $ two-sided confidence coefficient. The default confidence coefficient is 95%, corresponding to $\alpha =0.05$.

Regression Parameters

A two-sided $(1-\alpha )\times 100\% $ confidence interval $[\beta _{iL},\; \; \beta _{iU}]$ for the regression parameter $\beta _ i$ is based on the asymptotic normality of the maximum likelihood estimator $\hat{\beta _ i}$ and is computed by

\[ \beta _{iL}=\hat{\beta }_{i} - z_{1-\alpha /2}(\mr{SE}_{\hat{\beta }_{i}}) \]
\[ \beta _{iU}=\hat{\beta }_{i} + z_{1-\alpha /2}(\mr{SE}_{\hat{\beta }_{i}}) \]

where $\mr{SE}_{\hat{\beta }_{i}}$ is the estimated standard error of $\hat{\beta _ i}$, and $z_ p$ is the $p\times 100\% $ percentile of the standard normal distribution.

Scale Parameter

A two-sided $(1-\alpha )\times 100\% $ confidence interval $[\sigma _ L,\; \; \sigma _ U]$ for the scale parameter $\sigma $ in the location-scale model is based on the asymptotic normality of the logarithm of the maximum likelihood estimator $\log (\hat{\sigma })$, and is computed by

\[ \sigma _{L}=\hat{\sigma }/\exp [z_{1-\alpha /2}\mr{(SE}_{\hat{\sigma }})/\hat{\sigma }] \]
\[ \sigma _{U}=\hat{\sigma }\exp [z_{1-\alpha /2}\mr{(SE}_{\hat{\sigma }})/\hat{\sigma }] \]

See Meeker and Escobar (1998) for more information.

Weibull Scale and Shape Parameters

The Weibull distribution scale parameter $\eta $ and shape parameter $\beta $ are obtained by transforming the extreme-value location parameter $\mu $ and scale parameter $\sigma $:

\[ \eta = \exp (\mu ) \]
\[ \beta = 1 / \sigma \]

Consequently, two-sided $(1-\alpha )\times 100\% $ confidence intervals for the Weibull scale and shape parameters are computed as

\[ [\eta _ L,\; \; \eta _ U ] = [\exp (\mu _ L),\; \; \exp (\mu _ U)] \]
\[ [\beta _ L,\; \; \beta _ U] = [1/\sigma _ U,\; \; 1/\sigma _ L] \]

Gamma Shape Parameter

A two-sided $(1-\alpha )\times 100\% $ confidence interval for the three-parameter gamma shape parameter $\delta $ is computed by

\[ [\delta _ L,\; \; \delta _ U] = [ \hat{\delta } - z_{1-\alpha /2}(\mr{SE}_{\hat{\delta }}), \; \; \hat{\delta } + z_{1-\alpha /2}(\mr{SE}_{\hat{\delta }}) ] \]