5/13/2015 Notation (multinomial logistic regression algorithms) http://www01.ibm.com/support/knowledgecenter/api/content/nl/enus/SSLVMB_21.0.0/com.ibm.spss.statistics.help/alg_nomreg_notation.htm 1/1 Notation (multinomial logistic regression algorithms) The following notation is used throughout this chapter unless otherwise stated: Y The response variable, which takes integer values from 1 to J. J The number of categories of the nominal response. m The number of subpopulations. X A m × p A matrix with vectorelement x A i , the observed values at the i th subpopulation, determined by the independent variables specified in the command. X m × p matrix with vectorelement x i , the observed values of the location model’s independent variables at the i th subpopulation. f ijs The frequency weight for the sth observation which belongs to the cell corresponding to Y=j at subpopulation i . n ij The sum of frequency weights of the observations that belong to the cell corresponding to Y=j at subpopulation i . N The sum of all n ij ’s. π ij The cell probability corresponding to Y=j at subpopulation i . log ( π ij / π ik ) The logit of response category j to response category k. β j = ( β j 1 , ... , β jp ) ´ p ×1 vector of unknown parameters in the jth logit (i.e., logit of response category j to response category J). p Number of parameters in each logit. p≥1. p nr j Number of nonredundant parameters in logit j after maximum likelihood estimation. p ≥ p nr j ≥0. p nr The total number of nonredundant parameters after maximum likelihood estimation. p nr = Σ J −1 j =1 p nr j . B = ( β ´ 1 , ... , β ´ J −1 ) ' ( J −1 ) p ×1 vector of unknown parameters in the model. = ( ´ 1 ,..., ´ J −1 ) ´ The maximum likelihood estimate of B. ij The maximum likelihood estimate of π ij © Copyright IBM Corporation 1989, 2012. B ^ β ^ β ^ π ^

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5/13/2015 Notation(multinomiallogisticregressionalgorithms)

http://www01.ibm.com/support/knowledgecenter/api/content/nl/enus/SSLVMB_21.0.0/com.ibm.spss.statistics.help/alg_nomreg_notation.htm 1/1

Notation(multinomiallogisticregressionalgorithms)Thefollowingnotationisusedthroughoutthischapterunlessotherwisestated:

Y Theresponsevariable,whichtakesintegervaluesfrom1toJ.J Thenumberofcategoriesofthenominalresponse.m Thenumberofsubpopulations.

X Am p Amatrixwithvectorelement xAi ,theobservedvaluesattheithsubpopulation,determinedbytheindependentvariablesspecifiedinthecommand.

Xm pmatrixwithvectorelement x i ,theobservedvaluesofthelocationmodelsindependentvariablesattheithsubpopulation.

f i j s ThefrequencyweightforthesthobservationwhichbelongstothecellcorrespondingtoY=jatsubpopulationi.

n i j ThesumoffrequencyweightsoftheobservationsthatbelongtothecellcorrespondingtoY=jatsubpopulationi.N Thesumofallnijs.

i j ThecellprobabilitycorrespondingtoY=jatsubpopulationi.

log ( i j / i k ) Thelogitofresponsecategoryjtoresponsecategoryk.

j = ( j1 , ... , j p ) p1vectorofunknownparametersinthejthlogit(i.e.,logitofresponsecategoryjtoresponsecategoryJ).p Numberofparametersineachlogit.p1.

p nrjNumberofnonredundantparametersinlogitjaftermaximumlikelihoodestimation. p p nrj 0.

p nrThetotalnumberofnonredundantparametersafter

maximumlikelihoodestimation. p nr= J1j=1 p nrj .B= ( 1 , ... , J1) ' ( J1 ) p1vectorofunknownparametersinthemodel.

= ( 1 ,..., J1) ThemaximumlikelihoodestimateofB. i j Themaximumlikelihoodestimateof i j

CopyrightIBMCorporation1989,2012.

B^ ^ ^

^