The RELIABILITY Procedure

Regression Model Statistics Computed for Each Observation for Lifetime Data

Subsections:

Predicted Values
Percentiles
Standard Errors of Percentile Estimator
Confidence Limits for Percentiles
Reliability Function
Residuals

This section describes statistics that are computed for each observation when you fit a model for lifetime data. For regression models that are fit using the MODEL statement, you can specify a variety of statistics to be computed for each observation in the input data set. This section describes the method of computation for each statistic. See Table 16.32 and Table 16.33 for the syntax to request these statistics.

Predicted Values

The linear predictor is

$\hat{\mu }_{i}=\mb {x}^\prime \hat{\bbeta }$

where $\mb {x}_{i}$ is the vector of explanatory variables for the ith observation.

Percentiles

An estimator of the $p\times 100\%$ percentile $x_{p}$ for the ith observation for the extreme value, normal, and logistic distributions is

$\hat{x}_{i,p}=\mb {x}^\prime \hat{\bbeta } + z_{p}\hat{\sigma }$

where $z_{p}=G^{-1}(p)$ , G is the standardized CDF, and $\sigma$ is the distribution scale parameter.

An estimator of the $p\times 100\%$ percentile $t_{p}$ for the ith observation for the Weibull, lognormal, and log-logistic distributions is

$\hat{t}_{i,p}=\exp [\mb {x}^\prime \hat{\bbeta } + z_{p}\hat{\sigma }]$

where G is the standardized CDF of the extreme value, normal, or logistic distribution that corresponds to the logarithm of the lifetime, and $\sigma$ is the distribution scale parameter.

The percentile of the lognormal (base 10) distribution is

$\hat{t}_{i,p}=10^{[\mb {x}^\prime \hat{\bbeta } + z_{p}\hat{\sigma }]}$

where G is the CDF of the standard normal distribution.

An estimator of the $p\times 100\%$ percentile $t_{p}$ for the ith observation for the generalized gamma distribution is

$\hat{t}_{i,p} = \exp [\mb {x}^\prime \hat{\bbeta } + w_{\lambda ,p}\hat{\sigma }]$

where

$w_{\lambda ,p} =\frac{1}{\lambda } \log \left(\frac{\lambda ^{2}}{2}\chi ^{2}_{(2/\lambda ^{2}),p}\right)$

and $\chi ^{2}_{k,p}$ is the $p\times 100\%$ percentile of the chi-square distribution with k degrees of freedom.

Standard Errors of Percentile Estimator

For the extreme value, normal, and logistic distributions, the standard error of the estimator of the $p\times 100\%$ percentile is computed as

$\sigma _{i,p}=\sqrt {\mb {z}’\bSigma \mb {z}}$

where

$\mb {z}=\left[ \begin{array}{c} \mb {x}_{i} \\ z_{p} \end{array} \right]$

and $\bSigma$ is the covariance matrix of $(\hat{\bbeta }, \hat{\sigma })$ .

For the Weibull, lognormal, and log-logistic distributions, the standard error is computed as

$\sigma _{i,p}=\exp (x_{i,p})\sqrt {\mb {z}’\bSigma \mb {z}}$

where $x_{i,p}$ is the percentile computed from the extreme value, normal, or logistic distribution that corresponds to the logarithm of the lifetime. The standard error for the lognormal (base 10) distribution is computed as

$\sigma _{i,p}=10^{x_{i,p}}\sqrt {\mb {z}’\bSigma \mb {z}}$

The standard error for the generalized gamma distribution percentile is computed as

$\sigma _{i,p} = \exp [\mb {x}^\prime \hat{\bbeta } + w_{\lambda ,p}\hat{\sigma }]\sqrt {\mb {z}’\bSigma \mb {z}}$

where

$\mb {z}=\left[ \begin{array}{c} \mb {x}_{i} \\ w_{\lambda ,p} \\ \hat{\sigma }\frac{\partial w_{\lambda ,p}}{\partial \lambda } \end{array} \right]$

$\bSigma$ is the covariance matrix of $(\hat{\bbeta }, \hat{\sigma }, \hat{\lambda })$ , $\bbeta$ is the vector of regression parameters, $\sigma$ is the scale parameter, and $\lambda$ is the shape parameter.

Confidence Limits for Percentiles

Two-sided approximate $100\gamma \%$ confidence limits for $x_{i,p}$ for the extreme value, normal, and logistic distributions are computed as

$\begin{eqnarray*} x_{L} & = & \hat{x}_{i,p} - K_{\gamma }\sigma _{i,p} \\ x_{U} & = & \hat{x}_{i,p} + K_{\gamma }\sigma _{i,p} \end{eqnarray*}$

where $K_{\gamma }$ represents the $100(1+\gamma )/2 \times 100\%$ percentile of the standard normal distribution.

Limits for the Weibull, lognormal, and log-logistic percentiles are computed as

$\begin{eqnarray*} t_{L} & = & \exp (x_{L}) \\ t_{U} & = & \exp (x_{U}) \end{eqnarray*}$

where $x_{L}$ and $x_{U}$ are computed from the corresponding distributions for the logarithms of the lifetimes. For the lognormal (base 10) distribution,

$\begin{eqnarray*} t_{L} & = & 10^{x_{L}} \\ t_{U} & = & 10^{x_{U}} \end{eqnarray*}$

Limits for the generalized gamma distribution percentiles are computed as

$\begin{eqnarray*} t_{L} = \exp \left[\mb {x}_ i^\prime \bbeta + w_{\lambda ,p}\hat{\sigma } - K_{\gamma }\sqrt {\mb {z}’\bSigma \mb {z}}\right] \\ t_{U} = \exp \left[\mb {x}_ i^\prime \bbeta + w_{\lambda ,p}\hat{\sigma } + K_{\gamma }\sqrt {\mb {z}’\bSigma \mb {z}}\right] \end{eqnarray*}$

Reliability Function

For the extreme value, normal, and logistic distributions, an estimate of the reliability function evaluated at the response $y_{i}$ is computed as

$R(y_{i}) = 1 - G\left(\frac{y_{i}-\mb {x}^\prime \hat{\bbeta }}{\hat{\sigma }}\right)$

where $G(x)$ is the standardized CDF of the distribution from Table 16.69.

Estimates of the reliability function evaluated at the response $t_{i}$ for the Weibull, lognormal, log-logistic, and generalized gamma distributions are computed as

$R(t_{i}) = 1 - G\left(\frac{\log (t_{i})-\mb {x}^\prime \hat{\bbeta }}{\hat{\sigma }}\right)$

where $G(x)$ is the standardized CDF of the corresponding extreme value, normal, logistic, or generalized log-gamma distributions.

Residuals

The RELIABILITY procedure computes several different kinds of residuals. In the following equations, $y_{i}$ represents the ith response value if the extreme value, normal, or logistic distributions are specified. If $t_{i}$ is the ith response and if the Weibull, lognormal, log-logistic, or generalized gamma distributions are specified, then $y_{i}$ represents the logarithm of the response $y_{i}=\log (t_{i})$ . If the lognormal (base 10) distribution is specified, then $y_{i}=\log _{10}(t_{i})$ .

Raw Residuals

The raw residual is computed as

$r_{Ri}=y_{i}-\mb {x}^\prime \hat{\bbeta }$

Standardized Residuals

The standardized residual is computed as

$r_{Si}=\frac{y_{i}-\mb {x}^\prime \hat{\bbeta }}{\hat{\sigma }}$

Adjusted Residuals

If an observation is right censored, then the standardized residual for that observation is also right censored. Adjusted residuals adjust censored standardized residuals upward by adding a percentile of the residual lifetime distribution, given that the standardized residual exceeds the censoring value. The default percentile is the median (50th percentile), but you can, optionally, specify a $\gamma \times 100\%$ percentile by using the RESIDALPHA= $\gamma$ option in the MODEL statement. The $\gamma \times 100$ percentile residual life is computed as in Joe and Proschan (1984). The adjusted residual is computed as

$r_{Ai}= \left\{ \begin{array}{ll} G^{-1}[1-(1-\gamma )S(u_{i})] & \mbox{ for right-censored observations } \\ u_{i} & \mbox{ for uncensored observations } \end{array} \right.$

where G is the standard CDF,

$S(u)=1-G(u)$

is the reliability function, and

$u_{i} = \frac{y_{i}-\mb {x}^\prime \hat{\bbeta }}{\hat{\sigma }}$

If the generalized gamma distribution is specified, the standardized CDF and reliability functions include the estimated shape parameter $\hat{\lambda }$ .

Modified Cox-Snell Residuals

Let

$\delta _{i}= \left\{ \begin{array}{ll} 1 & \mbox{ for uncensored observations } \\ 0 & \mbox{ for right-censored observations } \end{array} \right.$

The Cox-Snell residual is defined as

$r_{Ci} = -\log (R(y_{i}))$

where

$R(y)=1-G\left(\frac{y-\mb {x}^\prime \hat{\bbeta }}{\hat{\sigma }}\right)$

is the reliability function. The modified Cox-Snell residual is computed as in Collett (1994, p. 152):

$r_{Ci}^{} = r_{Ci} + (1-\delta _{i})\alpha$

where $\alpha$ is an adjustment factor. If the fitted model is correct, the Cox-Snell residual has approximately a standard exponential distribution for uncensored observations. If an observation is censored, the residual evaluated at the censoring time is not as large as the residual evaluated at the (unknown) failure time. The adjustment factor $\alpha$ adjusts the censored residuals upward to account for the censoring. The default is $\alpha =1.0$ , the mean of the standard exponential distribution. You can, optionally, specify any adjustment factor by using the MODEL statement option RESIDADJ= $\alpha$ . Another commonly used value is the median of the standard exponential distribution, $\alpha =0.693$ .

Deviance Residuals

Deviance residuals are a zero-mean, symmetrized version of modified Cox-Snell residuals. Deviance residuals are computed as in Collett (1994, p. 153):

$r_{Di}=\mr {sgn}(\delta _{i}-r_{Ci}) \{ -2[\delta _{i}-r_{Ci}+\delta _{i}\log (r_{Ci})]\} ^{1/2}$

where

$\mbox{sgn}(u) = \left\{ \begin{array}{rl} -1 & \mbox{ if } u < 0 \\ 1 & \mbox{ if } u \ge 0 \end{array} \right.$