The LIFEREG Procedure

Confidence Intervals

Regression Parameters
Scale Parameter
Weibull Scale and Shape Parameters
Gamma Shape Parameter

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 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

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