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 The GENMOD Procedure

## Residuals

The GENMOD procedure computes three kinds of residuals. The raw residual is defined as
where yi 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 chi-square.

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, wi is a user-specified prior weight (if not specified, wi=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 We be the diagonal matrix with wei denoting the ith diagonal element. The weight matrix We is used in computing the expected information matrix.

Define hi as the ith diagonal element of the matrix

We(1/2) X (X' We X)-1 X' We(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 di 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

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