The RELIABILITY Procedure 
Regression Model ObservationWise Statistics
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 30.21 and Table 30.22 for the syntax for requesting
these statistics.
Predicted Values
The linear predictor is
where x_{i} is the vector of explanatory variables for the ith
observation.
Percentiles
An estimator of the p×100% percentile x_{p} for the ith
observation for the extreme value, normal, and logistic
distributions is
where z_{p}=G^{1}(p), G is the standardized CDF,
and is the distribution scale parameter.
An estimator of the p×100% percentile t_{p} for the ith
observation for the Weibull, lognormal, and loglogistic
distributions is
where G is the standardized CDF of the
extreme value, normal, or logistic distribution that corresponds to
the logarithm of the lifetime,
and is the distribution scale parameter.
The percentile of the lognormal (base 10) distribution is
where G is the CDF of the standard normal distribution.
An estimator of the p×100% percentile t_{p} for the ith
observation for the generalized gamma
distribution is
where
and is the p×100% percentile of the
chisquared 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×100%
percentile is computed as
where
and is the covariance matrix of .For the Weibull, lognormal, and loglogistic distributions, the
standard error is computed as
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
The standard error for the generalized gamma distribution percentile is
computed as
where
is the covariance matrix of
, is the
vector of regression parameters, is the scale parameter, and
is the shape parameter.
Confidence Limits for Percentiles
Twosided approximate confidence limits for x_{i,p}
for the extreme value, normal, and logistic distributions are
computed as
where represents the percentile of the standard normal distribution.
Limits for the Weibull, lognormal, and loglogistic percentiles are computed as
where x_{L} and x_{U} are computed from the corresponding distributions
for the logarithms of the lifetimes.
For the lognormal (base 10) distribution,
Limits for the generalized gamma distribution percentiles are computed as
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
where G(x) is the standardized CDF of the distribution
from Table 30.47.
Estimates of the reliability function evaluated at the response
t_{i} for the Weibull, lognormal,
loglogistic, and generalized gamma distributions are computed
as
where G(x) is the standardized CDF of the corresponding
extreme value, normal, logistic, or generalized loggamma
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, loglogistic,
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
Standardized Residuals
The standardized residual is computed as
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 percentile using the
RESIDALPHA= option in MODEL statement.
The percentile residual life is computed as in
Joe and Proschan (1984).
The adjusted residual is computed as
where G is the standard CDF,

S(u)=1G(u)
is the reliability
function, and
If the generalized gamma distribution is specified, the standardized
CDF and reliability functions include the estimated shape
parameter .Modified CoxSnell Residuals
Let
The CoxSnell residual is defined as

r_{Ci} = log(R(y_{i}))
where
is the reliability function.
The modified CoxSnell residual is computed as in Collett (1994, p.152):
where is an adjustment factor. If the fitted model is
correct, the CoxSnell 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 adjusts the
censored residuals upward to account for the censoring.
The default is , the median of the
standard exponential distribution. You can, optionally, specify any
adjustment factor by using the MODEL
statement option RESIDADJ=. Another commonly used value
is the mean of the standard exponential distribution, .Deviance Residuals
Deviance residuals are a zeromean, symmetrized version of
modified CoxSnell residuals.
Deviance residuals are computed as in Collett (1994, p.153):
where
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.