| Regression Model Statistics Computed for Each Observation |
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 14.28 and Table 14.29 for the syntax for requesting these statistics.
Predicted Values
The linear predictor is
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where 
is the vector of explanatory variables for the 
th observation.
Percentiles
An estimator of the 
percentile 
for the th observation for the extreme value, normal, and logistic distributions is
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where 
, 
is the standardized CDF, and 
is the distribution scale parameter.
An estimator of the percentile 
for the th observation for the Weibull, lognormal, and log-logistic distributions is
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where 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
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where is the CDF of the standard normal distribution.
An estimator of the percentile for the th observation for the generalized gamma distribution is
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where
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and 
is the percentile of the chi-square distribution with 
degrees of freedom.
Standard Errors of Percentile Estimator
For the extreme value, normal, and logistic distributions, the standard error of the estimator of the percentile is computed as
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where
 |
and 
is the covariance matrix of 
.
For the Weibull, lognormal, and log-logistic distributions, the standard error is computed as
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where 
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
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The standard error for the generalized gamma distribution percentile is computed as
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where
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is the covariance matrix of 
, 
is the vector of regression parameters, is the scale parameter, and 
is the shape parameter.
Confidence Limits for Percentiles
Two-sided approximate 
confidence limits for for the extreme value, normal, and logistic distributions are computed as
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where 
represents the 
percentile of the standard normal distribution.
Limits for the Weibull, lognormal, and log-logistic percentiles are computed as
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|
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where 
and 
are computed from the corresponding distributions for the logarithms of the lifetimes. For the lognormal (base 10) distribution,
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Limits for the generalized gamma distribution percentiles are computed as
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Reliability Function
For the extreme value, normal, and logistic distributions, an estimate of the reliability function evaluated at the response 
is computed as
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where 
is the standardized CDF of the distribution from Table 14.63.
Estimates of the reliability function evaluated at the response 
for the Weibull, lognormal, log-logistic, and generalized gamma distributions are computed as
 |
where 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, represents the th response value if the extreme value, normal, or logistic distributions are specified. If is the th response and if the Weibull, lognormal, log-logistic, or generalized gamma distributions are specified, then represents the logarithm of the response 
. If the lognormal (base 10) distribution is specified, then 
.
Raw Residuals
The raw residual is computed as
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Standardized Residuals
The standardized residual is computed as
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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 by using the RESIDALPHA=
option in the MODEL statement. The 
percentile residual life is computed as in Joe and Proschan (1984). The adjusted residual is computed as
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where is the standard CDF,
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is the reliability function, and
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If the generalized gamma distribution is specified, the standardized CDF and reliability functions include the estimated shape parameter 
.
Modified Cox-Snell Residuals
Let
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The Cox-Snell residual is defined as
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where
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is the reliability function. The modified Cox-Snell residual is computed as in Collett (1994, p. 152):
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where 
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 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, 
.
