
AGGREGATE= (variablelist)  variable
AGGREGATE

specifies the subpopulations
on which the Pearson chisquare
and the deviance
are calculated. This option applies only to the multinomial distribution or the binomial distribution with binary (single
trial syntax) response. It is ignored if specified for other cases. Observations with common values in the given list of variables
are regarded as coming from the same subpopulation. This affects the computation of the deviance and Pearson chisquare statistics.
Variables in the list can be any variables in the input data set. Specifying the AGGREGATE option is equivalent to specifying
the AGGREGATE= option with a variable list that includes all explanatory variables in the MODEL statement. Pearson chisquare
and deviance statistics are not computed for multinomial models unless this option is specified.

ALPHA=number
ALPH=number
A=number

sets the confidence coefficient
for parameter confidence intervals to 1–number. The value of number must be between 0 and 1. The default value of number is 0.05.

CICONV=number

sets the convergence criterion for
profile likelihood confidence intervals. See the section Confidence Intervals for Parameters for the definition of convergence. The value of number must be between 0 and 1. By default, CICONV=1E–4.

CL

requests that confidence limits for predicted values
be displayed (see the OBSTATS option).

CODING=EFFECT  FULLRANK

specifies that effect coding be used for all classification variables in the model.
This is the same as specifying PARAM=EFFECT as a CLASS statement option.

CONVERGE=number

sets the convergence criterion
.
The value of number must be between 0 and 1. The iterations are considered to have converged when the maximum change in the parameter estimates
between iteration steps is less than the value specified. The change is a relative change if the parameter is greater than
0.01 in absolute value; otherwise, it is an absolute change. By default, CONVERGE=1E–4. This convergence criterion is used
in parameter estimation for a single model fit, Type 1 statistics, and likelihood ratio statistics for Type 3 analyses and
CONTRAST statements.

CONVH=number

sets the relative Hessian convergence criterion.
The value of number must be between 0 and 1. After convergence is determined with the change in parameter criterion specified with the CONVERGE=
option, the quantity is computed and compared to number, where g is the gradient vector, H is the Hessian matrix for the model parameters, and f is the loglikelihood function. If tc is greater than number, a warning that the relative Hessian convergence criterion has been exceeded is printed. This criterion detects the occasional
case where the change in parameter convergence criterion is satisfied, but a maximum in the loglikelihood function has not
been attained. By default, CONVH=1E–4.

CORRB

requests that the parameter estimate correlation
matrix be displayed.

COVB

requests that the parameter estimate covariance matrix
be displayed.

DIAGNOSTICS
INFLUENCE

requests that case deletion diagnostic statistics be displayed
(see the OBSTATS option).

DIST=keyword
D=keyword
ERROR=keyword
ERR=keyword

specifies the builtin probability distribution
to use in the model. If you specify the DIST= option and you omit a userdefined link function, a default link function is
chosen as displayed in the following table. If you specify no distribution and no link function, then the GENMOD procedure
defaults to the normal distribution with the identity link function. Models for data with correlated responses fit by the
GEE method are not available for the zeroinflated distributions.
DIST=

Distribution

Default Link Function

BINOMIAL  BIN  B

Binomial

Logit

GAMMA  GAM  G

Gamma

Inverse ( power(–1) )

GEOMETRIC  GEOM

Geometric

Log

IGAUSSIAN  IG

Inverse Gaussian

Inverse squared ( power(–2) )

MULTINOMIAL  MULT

Multinomial

Cumulative logit

NEGBIN  NB

Negative binomial

Log

NORMAL  NOR  N

Normal

Identity

POISSON  POI  P

Poisson

Log

TWEEDIE<(Tweedieoptions)>

Tweedie

Log

ZIP

Zeroinflated Poisson

Log/logit

ZINB

Zeroinflated negative binomial

Log/logit

You can specify the following Tweedieoptions when you specify DIST=TWEEDIE.

INITIALP=startingvalue

specifies a starting value for iterative estimation of the Tweedie power parameter.

P=powerparameter

specifies a fixed Tweedie power parameter.

EPSILON=tolerance

specifies the tolerance for series approximation of the Tweedie density function.

OFFSET=constantvalue

specifies a constant value to be added to the response variable for evaluating the extended quasilikelihood. By default,
OFFSET=0.5.

NTHREADS=number

specifies the number of threads to be used in computation.

EXACTMAX<=variable>

names a variable to be used for performing an exact Poisson regression. For
each observation, the integer part of the EXACTMAX value should be nonnegative and at least as large as the response value.
If the EXACTMAX option is specified without a variable, then default values are computed. See the section Exact Logistic and Exact Poisson Regression for information about using this option.

EXPECTED

requests that the expected Fisher information
matrix be used to compute parameter estimate covariances and the associated statistics. The default action is to use the observed
Fisher information matrix. This option does not affect the model fitting, only the way in which the covariance matrix is computed
(see the SCORING= option.)

ID=variable

causes the values of variable in the input data set to be displayed in the OBSTATS table. If an explicit format for variable has been defined, the formatted values are displayed. If the OBSTATS option is not specified, this option has no effect.

INITIAL=numbers

sets initial values
for parameter estimates in the model. The default initial parameter values are weighted least squares estimates based on using
the response data as the initial mean estimate. This option can be useful in case of convergence difficulty. The intercept
parameter is initialized with the INTERCEPT= option and is not included here. The values are assigned to the variables in
the MODEL statement in the same order in which they appear in the MODEL statement. The order of levels for CLASS variables
is determined by the ORDER= option. Note that some levels of classification variables can be aliased; that is, they correspond
to linearly dependent parameters that are not estimated by the procedure. Initial values must be assigned to all levels of
classification variables, regardless of whether they are aliased or not. The procedure ignores initial values corresponding
to parameters not being estimated. If you specify a BY statement, all classification variables must take on the same number
of levels in each BY group. Otherwise, classification variables in some of the BY groups are assigned incorrect initial values.
Types of INITIAL= specifications are illustrated in the following table.
Type of List

Specification

List separated by blanks

INITIAL = 3 4 5

List separated by commas

INITIAL = 3, 4, 5

x to y

INITIAL = 3 to 5

x to y by z

INITIAL = 3 to 5 by 1

Combination of list types

INITIAL = 1, 3 to 5, 9


INTERCEPT=number  numberlist

initializes the intercept term to
number for parameter estimation. If you specify both the INTERCEPT= and the NOINT options, the intercept term is not estimated,
but an intercept term of number is included in the model. If you specify a multinomial model for ordinal data, you can specify a numberlist for the multiple intercepts in the model.

ITPRINT

displays the iteration history for all iterative processes:
parameter estimation, fitting constrained models for contrasts and Type 3 analyses, and profile likelihood confidence intervals.
The last evaluation of the gradient and the negative of the Hessian (second derivative) matrix are also displayed for parameter
estimation. If you perform a Bayesian analysis by specifying the BAYES statement, the iteration history for computing the
mode of the posterior distribution is also displayed.
This option might result in a large amount of displayed output, especially if some of the optional iterative processes are
selected.

LINK=keyword

specifies the link function
to use in the model. The keywords and their associated builtin link functions are as follows.
LINK=

Link Function

CUMCLL


CCLL

Cumulative complementary loglog

CUMLOGIT


CLOGIT

Cumulative logit

CUMPROBIT


CPROBIT

Cumulative probit

CLOGLOG


CLL

Complementary loglog

IDENTITY


ID

Identity

LOG

Log

LOGIT

Logit

PROBIT

Probit

POWER(number)  POW(number)

Power with = number

If no LINK= option is supplied and there is a userdefined link function, the userdefined link function is used. If you specify
neither the LINK= option nor a userdefined link function, then the default canonical link function is used if you specify
the DIST= option. Otherwise, if you omit the DIST= option, the identity link function is used.
The cumulative link functions are appropriate only for the multinomial distribution.

LOGNB

specifies that the maximum likelihood estimate and confidence limits of the
negative binomial dispersion parameter k be computed based . This is the default method used for the negative binomial dispersion parameter, so that specifying no option or specifying
the LOGNB option have the same effect. The GENMOD procedure computes the maximum likelihood estimate of and computes confidence limits based on the asymptotic normality of rather than of k. The results are always reported in terms of k rather than of . This method ensures that the estimate and confidence limits for k are positive. See Meeker and Escobar (1998, p. 163) for details about this method of computing confidence limits.

LRCI

requests that twosided confidence intervals for all model
parameters be computed based on the profile likelihood function.
This is sometimes called the partially maximized likelihood function. See the section Confidence Intervals for Parameters for more information about the profile likelihood function. This computation is iterative and can consume a relatively large
amount of CPU time. The confidence coefficient can be selected with the ALPHA=number option. The resulting confidence coefficient is 1–number. The default confidence coefficient is 0.95.

MAXITER=number
MAXIT=number

sets the maximum allowable number of iterations for
all iterative computation processes in PROC GENMOD. By default, MAXITER=50.

NOINT

requests that no intercept term
be included in the model. An intercept is included unless this option is specified.

NOLOGNB

specifies that the maximum likelihood estimate and confidence limits of the negative binomial dispersion parameter k be computed based on k rather than . If this option is not specified, then the GENMOD procedure computes the maximum likelihood estimate of and computes confidence limits based on the asymptotic normality of rather than of k. The results are always reported in terms of k rather than of . This method ensures that the estimate and confidence limits for k are positive. See Meeker and Escobar (1998, p. 163) for details about this method of computing confidence limits.

NOSCALE

holds the scale parameter fixed.
Otherwise, for the normal, inverse Gaussian, and gamma distributions, the scale parameter is estimated by maximum likelihood.
If you omit the SCALE= option, the scale parameter is fixed at the value 1.

OBSTATS

specifies that an additional table of statistics be displayed.
Formulas for the statistics are given in the section Predicted Values of the Mean, the section Residuals, and the section Case Deletion Diagnostic Statistics. Residuals and fit diagnostics are not computed for multinomial models.
For each observation, the following items are displayed:

the value of the response variable (variables if the data are binomial), frequency, and weight variables

the values of the regression variables

predicted mean, , where is the linear predictor and g is the link function. If there is an offset, it is included in .

estimate of the linear predictor . If there is an offset, it is included in .

standard error of the linear predictor

the value of the Hessian weight at the final iteration

lower confidence limit of the predicted value of the mean.
The confidence coefficient is specified with the ALPHA= option. See the section Confidence Intervals on Predicted Values for the computational method.

upper confidence limit of the predicted value of the mean

raw residual, defined as

Pearson, or chi residual, defined as the square root of the contribution for the observation to the Pearson chisquare—that
is,
where Y is the response, is the predicted mean, w is the value of the prior weight variable specified in a WEIGHT statement, and V() is the variance function evaluated at .

the standardized Pearson residual

deviance residual, defined as the square root of the deviance contribution for the observation, with sign equal to the sign
of the raw residual

the standardized deviance residual

the likelihood residual

a Cook distance type statistic for assessing the influence of individual observations on overall model fit

observation leverage

DFBETA, defined as an approximation to for each parameter estimate , where is the parameter estimate with the ith observation deleted

standardized DFBETA, defined as DFBETA, normalized by its standard deviation

zero inflation probability for zeroinflated models

the mean of a zeroinflated response
The following additional cluster deletion diagnostic statistics are created and displayed for each cluster if a REPEATED statement
is specified:

a Cook distance type statistic for assessing the influence of entire clusters on overall model fit

a studentized Cook distance for assessing influence of clusters

cluster leverage

cluster DFBETA for assessing the influence of entire clusters on individual parameter estimates

cluster DFBETA normalized by its standard deviation
If you specify the multinomial distribution, only regression variable values, response values, predicted values, confidence
limits for the predicted values, and the linear predictor are displayed in the table. Residuals and other diagnostic statistics
are not available for the multinomial distribution.
The RESIDUALS, DIAGNOSTICS  INFLUENCE, PREDICTED, XVARS, and CL options cause only subgroups of the observation statistics
to be displayed. You can specify more than one of these options to include different subgroups of statistics.
The ID=variable option causes the values of variable in the input data set to be displayed in the table. If an explicit format for variable has been defined, the formatted values are displayed.
If a REPEATED statement is present, a table is displayed for the GEE model specified in the REPEATED statement. Regression
variables, response values, predicted values, confidence limits for the predicted values, linear predictor, raw residuals,
Pearson residuals for each observation in the input data set are available. Case deletion diagnostic statistics are available
for each observation and for each cluster.

OFFSET=variable

specifies a variable in the input data set to be used as an offset
variable. This variable cannot be a CLASS variable, and it cannot be the response variable or one of the explanatory variables.
When you perform an exact Poisson regression with an OFFSET= variable but the EXACTMAX= option is not specified, then if is the offset for the ith observation, floor(exp()) should be greater than or equal to the response value. See the section Exact Logistic and Exact Poisson Regression for information about the use of the offset in the exact Poisson model.

PREDICTED
PRED
P

requests that predicted values, the linear predictor, its
standard error, and the Hessian weight be displayed (see the OBSTATS option).

RESIDUALS
R

requests that residuals and standardized residuals be displayed.
Residuals and other diagnostic statistics are not available for the multinomial distribution (see the OBSTATS option).

SCALE=number
SCALE=PEARSON  P
PSCALE
SCALE=DEVIANCE  D
DSCALE

sets the value used for the scale
parameter where the NOSCALE option is used. For the binomial and Poisson distributions, which have no free scale parameter,
this can be used to specify an overdispersed model. In this case, the parameter covariance matrix and the likelihood function are adjusted by the scale parameter. See
the section Dispersion Parameter and the section Overdispersion for more information. If the NOSCALE option is not specified, then number is used as an initial estimate of the scale parameter.
Specifying SCALE=PEARSON or SCALE=P is the same as specifying the PSCALE option. This fixes the scale parameter at the value
1 in the estimation procedure. After the parameter estimates are determined, the exponential family dispersion parameter is
assumed to be given by Pearson’s chisquare statistic divided by the degrees of freedom, and all statistics such as standard
errors and likelihood ratio statistics are adjusted appropriately.
Specifying SCALE=DEVIANCE or SCALE=D is the same as specifying the DSCALE option. This fixes the scale parameter at a value
of 1 in the estimation procedure.
After the parameter estimates are determined, the exponential family dispersion parameter is assumed to be given by the deviance
divided by the degrees of freedom. All statistics such as standard errors and likelihood ratio statistics are adjusted appropriately.

SCORING=number

requests that on iterations up to number, the Hessian
matrix be computed using the Fisher scoring method. For further iterations, the full Hessian matrix is computed. The default
value is 1. A value of 0 causes all iterations to use the full Hessian matrix, and a value greater than or equal to the value
of the MAXITER option causes all iterations to use Fisher scoring. The value of the SCORING= option must be 0 or a positive
integer.

SINGULAR=number

sets the tolerance for testing singularity
of the information matrix and the crossproducts matrix. Roughly, the test requires that a pivot be at least this number times
the original diagonal value. By default, number is times the machine epsilon. The default number is approximately on most machines. This value also controls the check on estimability for ESTIMATE and CONTRAST statements.

TYPE1

requests that a Type 1, or sequential, analysis be performed.
This consists of sequentially fitting models, beginning with the null (intercept term only) model and continuing up to the
model specified in the MODEL statement. The likelihood ratio statistic between each successive pair of models is computed
and displayed in a table.
A Type 1 analysis is not available for GEE models, since there is no associated likelihood.

TYPE3

requests that statistics for Type 3 contrasts be computed
for each effect specified in the MODEL statement. The default analysis is to compute likelihood ratio statistics for the contrasts
or score statistics for GEEs. Wald statistics are computed if the WALD option is also specified.

WALD

requests Wald statistics for Type 3 contrasts.
You must also specify the TYPE3 option in order to compute Type 3 Wald statistics.

WALDCI

requests that twosided Wald confidence intervals
for all model parameters be computed based on the asymptotic normality of the parameter estimators. This computation is not
as timeconsuming as the LRCI method, since it does not involve an iterative procedure. However, it is thought to be less
accurate, especially for small sample sizes. The confidence coefficient can be selected with the ALPHA= option in the same
way as for the LRCI option.

XVARS

requests that the regression variables be included in the
OBSTATS table.