The GENMOD Procedure
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Overview
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Getting Started
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Syntax
PROC GENMOD Statement ASSESS Statement BAYES Statement BY Statement CLASS Statement CONTRAST Statement DEVIANCE Statement EFFECTPLOT Statement ESTIMATE Statement EXACT Statement EXACTOPTIONS Statement FREQ Statement FWDLINK Statement INVLINK Statement LSMEANS Statement LSMESTIMATE Statement MODEL Statement OUTPUT Statement Programming Statements REPEATED Statement SLICE Statement STORE Statement STRATA Statement VARIANCE Statement WEIGHT Statement ZEROMODEL Statement
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Details
Generalized Linear Models Theory Specification of Effects Parameterization Used in PROC GENMOD Type 1 Analysis Type 3 Analysis Confidence Intervals for Parameters F Statistics Lagrange Multiplier Statistics Predicted Values of the Mean Residuals Multinomial Models Zero-Inflated Models Generalized Estimating Equations Assessment of Models Based on Aggregates of Residuals Case Deletion Diagnostic Statistics Bayesian Analysis Exact Logistic and Exact Poisson Regression Missing Values Displayed Output for Classical Analysis Displayed Output for Bayesian Analysis Displayed Output for Exact Analysis ODS Table Names ODS Graphics
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Examples
Logistic Regression Normal Regression, Log Link Gamma Distribution Applied to Life Data Ordinal Model for Multinomial Data GEE for Binary Data with Logit Link Function Log Odds Ratios and the ALR Algorithm Log-Linear Model for Count Data Model Assessment of Multiple Regression Using Aggregates of Residuals Assessment of a Marginal Model for Dependent Data Bayesian Analysis of a Poisson Regression Model Exact Poisson Regression
- References
| Case Deletion Diagnostic Statistics |
For ordinary generalized linear models, regression diagnostic statistics developed by Williams (1987) can be requested in an output data set or in the OBSTATS table by specifying the DIAGNOSTICS | INFLUENCE option in the MODEL statement. These diagnostics measure the influence of an individual observation on model fit, and generalize the one-step diagnostics developed by Pregibon (1981) for the logistic regression model for binary data.
Preisser and Qaqish (1996) further generalized regression diagnostics to apply to models for correlated data fit by generalized estimating equations (GEEs), where the influence of entire clusters of correlated observations, or the influence of individual observations within a cluster, is measured. These diagnostic statistics can be requested in an output data set or in the OBSTATS table if a model for correlated data is specified with a REPEATED statement.
The next two sections use the following notation:
is the maximum likelihood estimate of the regression parameters
, or, in the case of correlated data, the solution of the GEEs. 
is the corresponding estimate evaluated with the
th observation deleted, or, in the case of correlated data, with the th cluster deleted. 
is the dimension of the regression parameter vector
. 
is the standardized Pearson residual
, where 
is the variance of the th response and 
is the leverage defined in the section H | LEVERAGE. is the variance of response
, 
, where 
is the variance function and 
is the dispersion parameter. 
is the prior weight of the
th observation specified with the WEIGHT statement. If there is no WEIGHT statement, 
for all .
All unknown quantities are replaced by their estimated values in the following two sections.
Diagnostics for Ordinary Generalized Linear Models
The following statistics are available for generalized linear models.
DFBETA
The DFBETA statistic for measuring the influence of the th observation is defined as the one-step approximation to the difference in the MLE of the regression parameter vector and the MLE of the regression parameter vector without the th observation. This one-step approximation assumes a Fisher scoring step, and is given by
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where is the leverage defined in the section H | LEVERAGE.
DFBETAS
The standardized DFBETA statistic for assessing the influence of the th observation on the 
th regression parameter is defined as the DFBETA statistic for the th parameter divided by its estimated standard deviation, where the standard deviation is estimated from all the data.
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DOBS | COOKD | COOKSD
In normal linear regression, the influence of observation can be measured by Cook’s distance (Cook and Weisberg; 1982). A measure of influence of observation for generalized linear models that is equivalent to Cook’s distance for normal linear regression is given by
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where is the leverage defined in the section H | LEVERAGE. This measure is the one-step approximation to 
, where 
is the log likelihood evaluated at .
H | LEVERAGE
The Fisher scores, or expected, weight for observation is 
. Let 
be the diagonal matrix with 
as the th diagonal. The leverage of the th observation is defined as the th diagonal element of the hat matrix
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Diagnostics for Models Fit by Generalized Estimating Equations (GEEs)
The diagnostic statistics in this section were developed by Preisser and Qaqish (1996). See the section Generalized Estimating Equations for further information and notation for generalized estimating equations (GEEs). The following additional notation is used in this section.
Partition the design matrix 
and response vector 
by cluster; that is, let 
, and 
corresponding to the 
clusters.
Let 
be the number of responses for cluster , and denote by 
the total number of observations. Denote by 
the 
diagonal matrix with 
as the th diagonal element. If there is a WEIGHT statement, the diagonal element of is 
, where 
is the specified weight of the th observation in the th cluster. Let 
the 
diagonal matrix with 
as diagonal elements, 
, 
. Let 
the diagonal matrix corresponding to cluster with as the th diagonal element.
Let be the 
block diagonal weight matrix whose th block, corresponding to the th cluster, is the matrix
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where 
is the working correlation matrix for cluster .
Let
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where 
is the 
design matrix corresponding to cluster .
Define the adjusted residual vector as
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and 
, the estimated residual for the th cluster.
Let the subscript 
denote estimates evaluated without the th cluster, 
estimates evaluated using all the data except the 
th observation of the th cluster, and let 
denote matrices corresponding to the th cluster without the th observation.
The following statistics are available for generalized estimating equation models.
CH | CLUSTERH | CLEVERAGE
The leverage of cluster is contained in the matrix 
, and is summarized by the trace of 
,
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The leverage of the th observation in the th cluster is the th diagonal element of .
DFBETAC
The effect of deleting cluster on the estimated parameter vector is given by the following one-step approximation for 
:
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DFBETACS
The cluster deletion statistic DFBETAC can be standardized using the variances of based on the complete data. The standardized one-step approximation for the change in 
due to deletion of cluster is
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DFBETAO
Partition the matrices 
and 
as
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 |
and let 
and 
.
The effect of deleting the th observation from the th cluster is given by the following one-step approximation to 
:
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where 
, 
, and 
. Note that 
, 
, and 
are scalars.
DFBETAOS
The observation deletion statistic DFBETAO can be standardized using the variances of based on the complete data. The standardized one-step approximation for the change in due to deletion of observation in cluster is
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DCLS | CLUSTERCOOKD | CLUSTERCOOKSD
A measure of the standardized influence of the subset 
of observations on the overall fit is 
. For deletion of cluster , this is approximated by
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DOBS | COOKD | COOKSD
The measure of overall fit in the section DCLS | CLUSTERCOOKD | CLUSTERCOOKSD for the deletion of the th observation in the th cluster is approximated by
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where , , and are defined in the section DFBETAO. In the case of the independence working correlation, this is equal to the measure for ordinary generalized linear models defined in the section DOBS | COOKD | COOKSD.
MCLS | CLUSTERDFIT
A studentized distance measure of the type defined in the section DCLS | CLUSTERCOOKD | CLUSTERCOOKSD of the influence of the th cluster is given by
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