26GLM - GENERALIZEDLINEARMODELS

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Unformatted text preview: GENERALIZEDLINEARMODELS ConsiderthenormaltheoryGauss-Markovlinearmodel y = X + , N ( , 2 I ) . Doesnothavetobewrittenasfunction+error Couldspecifydistributionandmodel(s)foritsparameters i.e., y i N ( i , 2 ) , where i = X i forall i = 1 ,..., n and y 1 ,..., y n independent. Thisisoneexampleofageneralizedlinearmodel. HereisanotherexampleofaGLM: y i Bernoulli ( i ) ,where i = exp ( X i ) 1 + exp ( X i ) forall i = 1 ,..., n and y 1 ,..., y n independent. c 2011Dept.Statistics(IowaStateUniversity) Stat511section26 1/16 Ineachexample,allresponsesareindependentandeach responseisadrawfromonetypeofdistributionwhose parametersmaydependonexplanatoryvariablesthrougha knownfunctionofalinearpredictor X i . ThenormalandBernouliimodels(andmanyothers)arespecial casesofageneralizedlinearmodel. Thesearemodelswhere: Theparametersarespecifiedfunctionsof X Thedistributionisintheexponentialscalefamily i.e., y i hasdensity(orp.m.f.) exp ( ) T ( y i ) b ( ) a ( ) + c ( y i , ) (1) where () , T () , a () , b () ,and c () areknownfunctionsand isa vectorofunknownparametersdependingon X and iseithera knownorunknownparameter. Exponentialfamily/exponentialclassis(1)withoutthe a ( ) a ( ) includesoverdisperseddistributionsinthefamily c 2011Dept.Statistics(IowaStateUniversity) Stat511section26 2/16 Forexample,thepdfforanormaldistributioncanbewrittenas: exp 1 2 2 y 2 i + 2 2 y i 2 2 2 1 2 log ( 2 2 ) fromwhich: ( )= ( 2 , 1 2 2 ) and T ( y i )= ( y i , y 2 i ) Thisfamilyincludesmanycommondistributions: Distribution ( ) T ( y i ) a ( ) Normal ( 2 , 1 2 2 ) ( y i , y 2 i ) 1 Bernoulii log 1 y i 1 Poisson log y i 1 Overdisp.Poisson log y i Gamma ( 1 , ( k 1 ) ) ( y i , log y i ) 1 c 2011Dept.Statistics(IowaStateUniversity) Stat511section26 3/16 Alotofstattheoryresultsfollowimmediatelyfromtheexponential form e.g. T ( y i ) isthevectorofsufficientstatistics Alotoftheoryismuchnicerwhendistributionparameterizedin termsof ( ) insteadof...
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26GLM - GENERALIZEDLINEARMODELS

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