Confidence Intervals

The LIFEREG Procedure

Confidence Intervals

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 $(1-\alpha)x100\%$ two-sided confidence coefficient. The default confidence coefficient is 95%, corresponding to $\alpha=.05$ .

Regression Parameters

A two-sided $(1-\alpha)x100\%$ 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

$\beta_{iL}=\hat{\beta}_{i} - z_{1-\alpha/2}({\rm SE}_{\hat{\beta}_{i}})$

$\beta_{iU}=\hat{\beta}_{i} + z_{1-\alpha/2}({\rm SE}_{\hat{\beta}_{i}})$

where ${\rm SE}_{\hat{\beta}_{i}}$ is the estimated standard error of $\hat{\beta_i}$ , and z_p is the p×100% percentile of the standard normal distribution.

Scale Parameter

A two-sided $(1-\alpha)x100\%$ 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

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

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

Refer to 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 $(1-\alpha)x100\%$ 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 $(1-\alpha)x100\%$ confidence interval for the 3-parameter gamma shape parameter $\delta$ is computed by

$[\delta_L,\;\;\delta_U]=[ \hat{\delta} - z_{1-\alpha/2}({\rm SE}_{\hat{\delta}}), \;\; \hat{\delta} + z_{1-\alpha/2}({\rm SE}_{\hat{\delta}}) ]$

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