Residuals
The GENMOD procedure computes three kinds of residuals.
The raw residual
is defined as
where y_{i} is the ith response and
is the corresponding predicted mean.
The Pearson residual
is the square root of the ith
contribution to the Pearson's chisquare.
Finally, the deviance residual
is defined as the square root of the contribution of the ith
observation to the deviance, with the sign of the raw residual.
The adjusted Pearson, deviance, and likelihood
residuals are defined by Agresti (1990),
Williams (1987), and Davison and Snell (1991).
These residuals are useful for outlier
detection and for assessing the influence
of single observations on the fitted model.
For the generalized linear model, the variance
of the ith individual observation is given by
where is the dispersion parameter, w_{i} is a
userspecified prior weight (if not specified, w_{i}=1),
is the mean, and is the variance function.
Let
for the ith observation, where is
the derivative of the link function, evaluated at .Let W_{e} be the diagonal matrix with
w_{ei} denoting the ith diagonal element.
The weight matrix W_{e} is used in
computing the expected information matrix.
Define h_{i} as the ith diagonal element of the matrix

W_{e}^{(1/2)} X (X' W_{e} X)^{1} X' W_{e}^{(1/2)}
The Pearson
residuals, standardized to have unit
asymptotic variance, are given by
The deviance
residuals, standardized to have unit
asymptotic variance, are given by
where d_{i} is the square root of the contribution to the total
deviance from observation i, and sign is 1 if
is positive and 1 if is negative.
The likelihood
residuals are defined by
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.