The GENMOD Procedure

213 Pages·2016·9.06 MB·English

SAS/STAT® 14.2 User’s Guide
The GENMOD Procedure
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Chapter 45
The GENMOD Procedure
Contents
Overview: GENMODProcedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3063
WhatIsaGeneralizedLinearModel? . . . . . . . . . . . . . . . . . . . . . . . . . . 3064
ExamplesofGeneralizedLinearModels . . . . . . . . . . . . . . . . . . . . . . . . 3065
TheGENMODProcedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3066
GettingStarted: GENMODProcedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3068
PoissonRegression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3068
BayesianAnalysisofaLinearRegressionModel . . . . . . . . . . . . . . . . . . . . 3073
GeneralizedEstimatingEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3085
Syntax: GENMODProcedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3088
PROCGENMODStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3089
ASSESSStatement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3094
BAYESStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3095
BYStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3105
CLASSStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3105
CODEStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3108
CONTRASTStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3109
DEVIANCEStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3112
EFFECTPLOTStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3112
ESTIMATEStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3113
EXACTStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3115
EXACTOPTIONSStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3117
FREQStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3120
FWDLINKStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3120
INVLINKStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3120
LSMEANSStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3121
LSMESTIMATEStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3122
MODELStatement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3123
OUTPUTStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3134
ProgrammingStatements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3137
REPEATEDStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3139
SLICEStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3144
STOREStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3144
STRATAStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3144
VARIANCEStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3146
WEIGHTStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3146
ZEROMODELStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3146
(cid:70)
3062 Chapter45:TheGENMODProcedure
Details: GENMODProcedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3147
GeneralizedLinearModelsTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . 3147
SpecificationofEffects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3158
ParameterizationUsedinPROCGENMOD. . . . . . . . . . . . . . . . . . . . . . . 3159
Type1Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3159
Type3Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3160
ConfidenceIntervalsforParameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 3162
FStatistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3163
LagrangeMultiplierStatistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3163
PredictedValuesoftheMean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3164
Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3164
MultinomialModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3166
Zero-InflatedModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3166
TweedieDistributionForGeneralizedLinearModels . . . . . . . . . . . . . . . . . . 3168
GeneralizedEstimatingEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3170
AssessmentofModelsBasedonAggregatesofResiduals . . . . . . . . . . . . . . . 3178
CaseDeletionDiagnosticStatistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 3182
BayesianAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3185
ExactLogisticandExactPoissonRegression . . . . . . . . . . . . . . . . . . . . . . 3190
ResponseLevelOrdering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3193
MissingValues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3194
DisplayedOutputforClassicalAnalysis . . . . . . . . . . . . . . . . . . . . . . . . 3194
DisplayedOutputforBayesianAnalysis . . . . . . . . . . . . . . . . . . . . . . . . 3202
DisplayedOutputforExactAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . 3204
ODSTableNames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3204
ODSGraphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3208
Examples: GENMODProcedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210
Example45.1:LogisticRegression . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210
Example45.2:NormalRegression,LogLink . . . . . . . . . . . . . . . . . . . . . . 3212
Example45.3:GammaDistributionAppliedtoLifeData . . . . . . . . . . . . . . . 3215
Example45.4:OrdinalModelforMultinomialData . . . . . . . . . . . . . . . . . . 3218
Example45.5:GEEforBinaryDatawithLogitLinkFunction . . . . . . . . . . . . . 3221
Example45.6:LogOddsRatiosandtheALRAlgorithm . . . . . . . . . . . . . . . . 3224
Example45.7:Log-LinearModelforCountData . . . . . . . . . . . . . . . . . . . . 3226
Example45.8:ModelAssessmentofMultipleRegressionUsingAggregatesofResiduals 3231
Example45.9:AssessmentofaMarginalModelforDependentData . . . . . . . . . 3238
Example45.10:BayesianAnalysisofaPoissonRegressionModel . . . . . . . . . . 3240
Example45.11:ExactPoissonRegression . . . . . . . . . . . . . . . . . . . . . . . 3253
Example45.12:TweedieRegression . . . . . . . . . . . . . . . . . . . . . . . . . . 3256
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3259
(cid:70)
Overview: GENMODProcedure 3063
Overview: GENMOD Procedure
The GENMOD procedure fits generalized linear models, as defined by Nelder and Wedderburn (1972).
The class of generalized linear models is an extension of traditional linear models that allows the mean
ofapopulationtodependonalinearpredictor throughanonlinearlinkfunctionandallowstheresponse
probability distribution to be any member of an exponential family of distributions. Many widely used
statisticalmodelsaregeneralizedlinearmodels. Theseincludeclassicallinearmodelswithnormalerrors,
logisticandprobitmodelsforbinarydata,andlog-linearmodelsformultinomialdata. Manyotheruseful
statistical models can be formulated as generalized linear models by the selection of an appropriate link
functionandresponseprobabilitydistribution.
SeeMcCullaghandNelder(1989)foradiscussionofstatisticalmodelingusinggeneralizedlinearmodels.
ThebooksbyAitkinetal.(1989)andDobson(1990)arealsoexcellentreferenceswithmanyexamplesof
applicationsofgeneralizedlinearmodels. Firth(1991)providesanoverviewofgeneralizedlinearmodels.
Myers,Montgomery,andVining(2002)provideapplicationsofgeneralizedlinearmodelsintheengineering
andphysicalsciences. Collett(2003)andHilbe(2009)providecomprehensiveaccountsofgeneralizedlinear
modelswhentheresponsesarebinary.
Theanalysisofcorrelateddataarisingfromrepeatedmeasurementswhenthemeasurementsareassumedto
bemultivariatenormalhasbeenstudiedextensively. However,thenormalityassumptionmightnotalwaysbe
reasonable;forexample,differentmethodologymustbeusedinthedataanalysiswhentheresponsesare
discreteandcorrelated. Generalizedestimatingequations(GEEs)provideapracticalmethodwithreasonable
statisticalefficiencytoanalyzesuchdata.
LiangandZeger(1986)introducedGEEsasamethodofdealingwithcorrelateddatawhen,exceptforthe
correlationamongresponses,thedatacanbemodeledasageneralizedlinearmodel. Forexample,correlated
binaryandcountdatainmanycasescanbemodeledinthisway.
TheGENMODprocedurecanfitmodelstocorrelatedresponsesbytheGEEmethod. YoucanusePROC
GENMODtofitmodelswithmostofthecorrelationstructuresfromLiangandZeger(1986)byusingGEEs.
For more details on GEEs, see Hardin and Hilbe (2003); Diggle, Liang, and Zeger (1994); Lipsitz et al.
(1994).
Bayesian analysis of generalized linear models can be requested by using the BAYES statement in the
GENMOD procedure. In Bayesian analysis, the model parameters are treated as random variables, and
inference about parameters is based on the posterior distribution of the parameters, given the data. The
posterior distribution is obtained using Bayes’ theorem as the likelihood function of the data weighted
with a prior distribution. The prior distribution enables you to incorporate knowledge or experience of
the likely range of values of the parameters of interest into the analysis. If you have no prior knowledge
of the parameter values, you can use a noninformative prior distribution, and the results of the Bayesian
analysis will be very similar to a classical analysis based on maximum likelihood. A closed form of the
posteriordistributionisoftennotfeasible,andaMarkovchainMonteCarlomethodbyGibbssamplingis
usedtosimulatesamplesfromtheposteriordistribution. SeeChapter7,“IntroductiontoBayesianAnalysis
Procedures,”foranintroductiontothebasicconceptsofBayesianstatistics. Alsoseethesection“Bayesian
Analysis: AdvantagesandDisadvantages”onpage128inChapter7,“IntroductiontoBayesianAnalysis
Procedures,”foradiscussionoftheadvantagesanddisadvantagesofBayesiananalysis. SeeIbrahim,Chen,
andSinha(2001)foradetaileddescriptionofBayesiananalysis.
In a Bayesian analysis, a Gibbs chain of samples from the posterior distribution is generated for the
model parameters. Summary statistics (mean, standard deviation, quartiles, HPD and credible intervals,
(cid:70)
3064 Chapter45:TheGENMODProcedure
correlationmatrix)andconvergencediagnostics(autocorrelations;Gelman-Rubin,Geweke,Raftery-Lewis,
andHeidelbergerandWelchtests;theeffectivesamplesize;andMonteCarlostandarderrors)arecomputed
foreachparameter,aswellasthecorrelationmatrixandthecovariancematrixoftheposteriorsample. Trace
plots,posteriordensityplots,andautocorrelationfunctionplotsthatarecreatedusingODSGraphicsarealso
providedforeachparameter.
The GENMOD procedure enables you to perform exact logistic regression, also called exact conditional
binarylogisticregression,andexactPoissonregression,alsocalledexactconditionalPoissonregression,by
specifyingoneormoreEXACTstatements. Youcantestindividualparametersorconductajointtestfor
severalparameters. Theprocedurecomputestwoexacttests: theexactconditionalscoretestandtheexact
conditionalprobabilitytest. Youcanrequestexactestimationofspecificparametersandcorrespondingodds
ratioswhereappropriate. Pointestimates,standarderrors,andconfidenceintervalsareprovided.
TheGENMODprocedureusesODSGraphicstocreategraphsaspartofitsoutput. Forgeneralinformation
aboutODSGraphics,seeChapter21,“StatisticalGraphicsUsingODS.”
What Is a Generalized Linear Model?
Atraditionallinearmodelisoftheform
y D x0ˇC"
i i i
wherey istheresponsevariablefortheithobservation. Thequantityx isacolumnvectorofcovariates,or
i i
explanatoryvariables,forobservationithatisknownfromtheexperimentalsettingandisconsideredtobe
fixed,ornonrandom. Thevectorofunknowncoefficientsˇ isestimatedbyaleastsquaresfittothedatay.
The" areassumedtobeindependent,normalrandomvariableswithzeromeanandconstantvariance. The
i
expectedvalueofy ,denotedby(cid:22) ,is
i i
(cid:22) D x0ˇ
i i
Whiletraditionallinearmodelsareusedextensivelyinstatisticaldataanalysis,therearetypesofproblems
suchasthefollowingforwhichtheyarenotappropriate.
(cid:15) It might not be reasonable to assume that data are normally distributed. For example, the normal
distribution(whichiscontinuous)mightnotbeadequateformodelingcountsormeasuredproportions
thatareconsideredtobediscrete.
(cid:15) Ifthemeanofthedataisnaturallyrestrictedtoarangeofvalues,thetraditionallinearmodelmight
0
notbeappropriate,sincethelinearpredictorx ˇ cantakeonanyvalue. Forexample,themeanofa
i
measured proportion is between 0 and 1, but the linear predictor of the mean in a traditional linear
modelisnotrestrictedtothisrange.
(cid:15) Itmightnotberealistictoassumethatthevarianceofthedataisconstantforallobservations. For
example,itisnotunusualtoobservedatawherethevarianceincreaseswiththemeanofthedata.
Ageneralizedlinearmodelextendsthetraditionallinearmodelandisthereforeapplicabletoawiderrange
ofdataanalysisproblems. Ageneralizedlinearmodelconsistsofthefollowingcomponents:
(cid:15) Thelinearcomponentisdefinedjustasitisfortraditionallinearmodels:
(cid:17) D x0ˇ
i i
(cid:70)
ExamplesofGeneralizedLinearModels 3065
(cid:15) A monotonic differentiable link function g describes how the expected value of y is related to the
i
linearpredictor(cid:17) :
i
g.(cid:22) / D x0ˇ
i i
(cid:15) Theresponsevariablesy areindependentfori=1,2,...andhaveaprobabilitydistributionfroman
i
exponentialfamily. Thisimpliesthatthevarianceoftheresponsedependsonthemean(cid:22)througha
variancefunctionV:
(cid:30)V.(cid:22) /
Var.y / D i
i
w
i
where(cid:30) isaconstantandw isaknownweightforeachobservation. Thedispersionparameter(cid:30) is
i
eitherknown(forexample,forthebinomialorPoissondistribution,(cid:30) D 1)ormustbeestimated.
Seethesection“ResponseProbabilityDistributions”onpage3147fortheformofaprobabilitydistribution
fromtheexponentialfamilyofdistributions.
As in the case of traditional linear models, fitted generalized linear models can be summarized through
statistics such as parameter estimates, their standard errors, and goodness-of-fit statistics. You can also
makestatisticalinferenceabouttheparametersbyusingconfidenceintervalsandhypothesistests. However,
specificinferenceproceduresareusuallybasedonasymptoticconsiderations,sinceexactdistributiontheory
isnotavailableorisnotpracticalforallgeneralizedlinearmodels.
Examples of Generalized Linear Models
Youconstructageneralizedlinearmodelbydecidingonresponseandexplanatoryvariablesforyourdataand
choosinganappropriatelinkfunctionandresponseprobabilitydistribution. Someexamplesofgeneralized
linearmodelsfollow. Explanatoryvariablescanbeanycombinationofcontinuousvariables,classification
variables,andinteractions.
Traditional Linear Model
(cid:15) responsevariable: acontinuousvariable
(cid:15) distribution: normal
(cid:15) linkfunction: identity,g.(cid:22)/ D (cid:22)
Logistic Regression
(cid:15) responsevariable: aproportion
(cid:15) distribution: binomial
(cid:18) (cid:19)
(cid:22)
(cid:15) linkfunction: logit,g.(cid:22)/ D log
1(cid:0)(cid:22)
(cid:70)
3066 Chapter45:TheGENMODProcedure
Poisson Regression in Log-Linear Model
(cid:15) responsevariable: acount
(cid:15) distribution: Poisson
(cid:15) linkfunction: log,g.(cid:22)/ D log.(cid:22)/
Gamma Model with Log Link
(cid:15) responsevariable: apositive,continuousvariable
(cid:15) distribution: gamma
(cid:15) linkfunction: log,g.(cid:22)/ D log.(cid:22)/
The GENMOD Procedure
TheGENMODprocedurefitsageneralizedlinearmodeltothedatabymaximumlikelihoodestimationofthe
parametervectorˇ. Thereis,ingeneral,noclosedformsolutionforthemaximumlikelihoodestimatesofthe
parameters. TheGENMODprocedureestimatestheparametersofthemodelnumericallythroughaniterative
fittingprocess. Thedispersionparameter(cid:30) isalsoestimatedbymaximumlikelihoodor,optionally,bythe
residualdevianceorbyPearson’schi-squaredividedbythedegreesoffreedom. Covariances,standarderrors,
andp-valuesarecomputedfortheestimatedparametersbasedontheasymptoticnormalityofmaximum
likelihoodestimators. Anumberofpopularlinkfunctionsandprobabilitydistributionsareavailableinthe
GENMODprocedure. Thebuilt-inlinkfunctionsareasfollows:
(cid:15) identity: g.(cid:22)/ D (cid:22)
(cid:15) logit: g.(cid:22)/ D log.(cid:22)=.1(cid:0)(cid:22)//
(cid:15) probit: g.(cid:22)/ D ˆ(cid:0)1.(cid:22)/,whereˆisthestandardnormalcumulativedistributionfunction
(cid:26) (cid:22)(cid:21) if(cid:21) ¤ 0
(cid:15) power: g.(cid:22)/ D
log.(cid:22)/ if(cid:21) D 0
(cid:15) log: g.(cid:22)/ D log.(cid:22)/
(cid:15) complementarylog-log: g.(cid:22)/ D log.(cid:0)log.1(cid:0)(cid:22)//
Theavailabledistributionsandassociatedvariancefunctionsareasfollows:
(cid:15) normal: V.(cid:22)/ D 1
(cid:15) binomial(proportion): V.(cid:22)/ D (cid:22).1(cid:0)(cid:22)/
(cid:15) Poisson: V.(cid:22)/ D (cid:22)
(cid:15) gamma: V.(cid:22)/ D (cid:22)2
(cid:70)
TheGENMODProcedure 3067
(cid:15) inverseGaussian: V.(cid:22)/ D (cid:22)3
(cid:15) negativebinomial: V.(cid:22)/ D (cid:22)Ck(cid:22)2
(cid:15) geometric: V.(cid:22)/ D (cid:22)C(cid:22)2
(cid:15) multinomial
(cid:15) zero-inflatedPoisson
(cid:15) zero-inflatednegativebinomial
Thenegativebinomialandzero-inflatednegativebinomialaredistributionswithanadditionalparameterkin
thevariancefunction. PROCGENMODestimateskbymaximumlikelihood,oryoucanoptionallysetitto
aconstantvalue. Fordiscussionsofthenegativebinomialdistribution,seeMcCullaghandNelder(1989);
Hilbe(1994,2007);Long(1997);CameronandTrivedi(1998);Lawless(1987).
The multinomial distribution is sometimes used to model a response that can take values from a number
of categories. The binomial is a special case of the multinomial with two categories. See the section
“MultinomialModels”onpage3166andMcCullaghandNelder(1989,Chapter5)foradescriptionofthe
multinomialdistribution.
Thezero-inflatedPoissonandzero-inflatednegativebinomialareincludedinPROCGENMODeventhough
they are not generalized linear models. They are useful extensions of generalized linear models. See the
section“Zero-InflatedModels”onpage3166forinformationaboutthezero-inflateddistributions. Models
fordatawithcorrelatedresponsesfitbytheGEEmethodarenotavailableforzero-inflateddistributions.
Inaddition,youcaneasilydefineyourownlinkfunctionsordistributionsthroughDATAstepprogramming
statementsusedwithintheprocedure.
Animportantaspectofgeneralizedlinearmodelingistheselectionofexplanatoryvariablesinthemodel.
Changes in goodness-of-fit statistics are often used to evaluate the contribution of subsets of explanatory
variables to a particular model. The deviance, defined to be twice the difference between the maximum
attainableloglikelihoodandtheloglikelihoodofthemodelunderconsideration,isoftenusedasameasure
ofgoodnessoffit. Themaximumattainableloglikelihoodisachievedwithamodelthathasaparameterfor
everyobservation. Seethesection“GoodnessofFit”onpage3154forformulasforthedeviance.
Onestrategyforvariableselectionistofitasequenceofmodels,beginningwithasimplemodelwithonlyan
interceptterm,andthentoincludeoneadditionalexplanatoryvariableineachsuccessivemodel. Youcan
measuretheimportanceoftheadditionalexplanatoryvariablebythedifferenceindeviancesorfittedlog
likelihoodsbetweensuccessivemodels. AsymptotictestscomputedbytheGENMODprocedureenableyou
toassessthestatisticalsignificanceoftheadditionalterm.
TheGENMODprocedureenablesyoutofitasequenceofmodels,upthroughamaximumnumberofterms
specifiedinaMODELstatement. Atablesummarizestwicethedifferenceinloglikelihoodsbetweeneach
successivepairofmodels. ThisiscalledaType1analysisintheGENMODprocedure,becauseitisanalogous
to Type I (sequential) sums of squares in the GLM procedure. As with the PROC GLM Type I sums of
squares,theresultsfromthisprocessdependontheorderinwhichthemodeltermsarefit.
TheGENMODprocedurealsogeneratesaType3analysisanalogoustoTypeIIIsumsofsquaresintheGLM
procedure. AType3analysisdoesnotdependontheorderinwhichthetermsforthemodelarespecified. A
GENMODprocedureType3analysisconsistsofspecifyingamodelandcomputinglikelihoodratiostatistics
forTypeIIIcontrastsforeachterminthemodel. Thecontrastsaredefinedinthesamewayastheyareinthe
(cid:70)
3068 Chapter45:TheGENMODProcedure
GLMprocedure. TheGENMODprocedureoptionallycomputesWaldstatisticsforTypeIIIcontrasts. This
iscomputationallylessexpensivethanlikelihoodratiostatistics,butitisthoughttobelessaccuratebecause
thespecifiedsignificancelevelofhypothesistestsbasedontheWaldstatisticmightnotbeasclosetothe
actualsignificancelevelasitisforlikelihoodratiotests.
AType3analysisgeneralizestheuseofTypeIIIestimablefunctionsinlinearmodels. Briefly,aTypeIII
estimable function (contrast) for an effect is a linear function of the model parameters that involves the
parameters of the effect and any interactions with that effect. A test of the hypothesis that the Type III
contrastforamaineffectisequalto0isintendedtotestthesignificanceofthemaineffectinthepresence
of interactions. See Chapter 47, “The GLM Procedure,” and Chapter 15, “The Four Types of Estimable
Functions,”formoreinformationaboutTypeIIIestimablefunctions. AlsoseeLittell,Freund,andSpector
(1991).
AdditionalfeaturesoftheGENMODprocedureincludethefollowing:
(cid:15) likelihood ratio statistics for user-defined contrasts—that is, linear functions of the parameters and
p-valuesbasedontheirasymptoticchi-squaredistributions
(cid:15) estimatedvalues,standarderrors,andconfidencelimitsforuser-definedcontrastsandleastsquares
means
(cid:15) abilitytocreateaSASdatasetcorrespondingtomosttablesdisplayedbytheprocedure(seeTable45.12
andTable45.13)
(cid:15) confidenceintervalsformodelparametersbasedoneithertheprofilelikelihoodfunctionorasymptotic
normality
(cid:15) syntaxsimilartothatofPROCGLMforthespecificationoftheresponseandmodeleffects,including
interactiontermsandautomaticcodingofclassificationvariables
(cid:15) abilitytofitGEEmodelsforclusteredresponsedata
(cid:15) abilitytoperformBayesiananalysisbyGibbssampling
Getting Started: GENMOD Procedure
Poisson Regression
YoucanusetheGENMODproceduretofitavarietyofstatisticalmodels. AtypicaluseofPROCGENMOD
istoperformPoissonregression.
You can use the Poisson distribution to model the distribution of cell counts in a multiway contingency
table. Aitkin et al. (1989) have used this method to model insurance claims data. Suppose the following
hypotheticalinsuranceclaimsdataareclassifiedbytwofactors: agegroup(withtwolevels)andcartype
(withthreelevels).