Latent Trait Measurement Models for Other (not Binary ... · Polytomous Items = Categorical Outcomes •Polytomous = more than 2 options •Polytomous models are not numbered like

Latent Trait Measurement Models for Other (not Binary) Responses

Today’sTopics:Ø Reviewofgeneralizedmodelsforcategoricaloutcomes

Ø OrderedCategoriesà GradedResponse

Ø MaybeOrderedCategoriesà PartialCreditØ UnorderedCategoriesà NominalResponseØ CountOutcomesà PoissonandNegativeBinomial

Ø ToomanyZeros Outcomesà Zero-Inflated/Hurdle/Two-Part

EPSY906 / CLP 948: Lecture 6 1

Our Friend the Logit• Thelogit isalsothebasisformanyothergeneralizedmodelsforpredictingcategoricaloutcomes

• Nextwe’llseehow𝐶 possibleresponsecategoriescanbepredictedusing𝐶 − 1 binary“submodels”thatinvolvecarvingupthecategoriesindifferentways,inwhicheachbinarysubmodelusesalogitlinktopredictitsoutcome

• Typesofcategoricaloutcomes:Ø Definitelyorderedcategories:“cumulativelogit”

Ø Maybeorderedcategories:“adjacentcategorylogit”(notusedmuch)

Ø DefinitelyNOTorderedcategories:“generalizedlogit”


Logit-Based Models for C Ordinal Categories• Knownas“cumulativelogit”or“proportionalodds”modelingeneralizedmodels;willbeknownas“gradedresponsemodel”inIRTØ LINK=CLOGIT,DIST=MULTinSASGLIMMIX

• Modelstheprobabilityoflowervs.higher cumulativecategoriesvia𝐶 − 1submodels(e.g.,if𝐶 = 4 possibleresponsesof𝑐 = 0,1,2,3):

0 vs. 1,2,3 0,1 vs. 2,3 0,1,2 vs. 3

• InSAS,whatthebinarysubmodelspredictdependsonwhetherthemodelispredictingDOWN (𝐲𝐢 = 𝟎,thedefault)orUP(𝐲𝐢 = 𝟏)cumulatively

• ExamplepredictingUPinanemptymodel(subscripts=parm,submodel)

• Submodel1:Logit y5 > 0 = β89 à𝑝 y5 > 0 = exp β89 / 1 + exp β89• Submodel2:Logit y5 > 1 = β8@ à 𝑝 y5 > 1 = exp β8@ / 1 + exp β8@• Submodel3:Logit y5 > 2 = β8Aà𝑝 y5 > 2 = exp β8A / 1 + exp β8A

Submodel3Submodel2Submodel1

I’venamedthesesubmodelsbasedonwhattheypredict,butprogramswillusetheirownnamesintheoutput.


Logit-Based Models for C Ordinal Categories• Modelstheprobabilityoflowervs.higher cumulativecategoriesvia𝐶 − 1submodels(e.g.,if𝐶 = 4 possibleresponsesof𝑐 = 0,1,2,3):

0 vs. 1,2,3 0,1 vs. 2,3 0,1,2 vs. 3

• InSAS,whatthebinarysubmodelspredictdependsonwhetherthemodelispredictingDOWN (𝐲𝐢 = 𝟎,thedefault)orUP(𝐲𝐢 = 𝟏)cumulatively

Ø Eitherway,themodelpredictsthemiddlecategoryresponsesindirectly

• ExampleifpredictingUPwithanemptymodel:

Ø Probabilityof0=1– Prob1Probabilityof1=Prob1– Prob2Probabilityof2=Prob2– Prob3Probabilityof3=Prob3– 0

Submodel3à Prob3

Submodel2à Prob2

Submodel1à Prob1

Thecumulativesubmodelsthatcreatetheseprobabilitiesareeachestimatedusingallthedata(good,especiallyforcategoriesnotchosenoften),butassumeorderindoingso(maybebadorok,dependingonyourresponseformat).

Logit y5 > 2 = β8A

à𝑝 y5 > 2 = BCD EFG9HBCD EFG


Logit-Based Models for C Categories• Usesmultinomialdistributionforresiduals,whosePDFfor𝐶 = 4categoriesof𝑐 = 0,1,2,3,anobserved𝑦J = 𝑐,andindicators𝐼 if𝑐 =𝑦J

𝑓 y5 = c = 𝑝58N[PQR8]𝑝59

N[PQR9]𝑝5@N[PQR@]𝑝5A

N[PQRA]

Ø Maximumlikelihoodisthenusedtofindthemostlikelyparametersinthemodeltopredicttheprobabilityofeachresponsethroughthe(usuallylogit)linkfunction;probabilitiessumto1:∑ 𝑝5UV

UR9 = 1

• Othermodelsforcategoricaldatathatusethemultinomial:Ø Adjacentcategorylogit(partialcredit): Modelstheprobabilityof

eachnexthighestcategoryvia𝐶 − 1 submodels(e.g.,if𝐶 = 4):0 vs. 1 1 vs. 2 2 vs. 3

Ø Baselinecategorylogit(nominal): Modelstheprobabilityofreferencevs.othercategoryvia𝐶 − 1 submodels (e.g.,if𝐶 = 4 and0 = ref):

0 vs. 1 0 vs. 2 0 vs. 3

Only𝑝JY fortheresponse𝑦J = 𝑐 getsused


Innominal models,allparametersareestimatedseparately persubmodel

Polytomous Items = Categorical Outcomes• Polytomous=morethan2options

• Polytomousmodelsarenotnumberedlikebinarymodels,butinsteadgetcalleddifferentnamesØ Mosthavea1-PLvs.2-PLversionthatgobydifferentnames

Ø Withineach,differentconstraintsonwhattodowithmultipleoptions

• Threemainkindsofpolytomousmodels:Ø Responseoptionsareorderedforsureà CumulativeLogit

§ GradedResponseorModifiedGradedResponseModel(IRTandIFA)

Ø Responseoptionsmaybeorderedà AdjacentCategoryLogit§ (Generalized)PartialCreditModelorRatingScaleModel(IRTonly)

Ø Nowayaretheseresponseoptionsorderedà BaselineCategoryLogit§ NominalResponseModel(IRTandIFA)


The Threshold Concept

• Eachcategoricalvariableisreally thechopped-upversionofapretendunderlyingcontinuousvariable(𝐲∗)withmean=0(variance=1.00or3.29)

• PolytomousmodelswilldifferinhowtheymakeuseofmultiplethresholdsperiteminwhichC=#categories,so#thresholdsperitem=k =C−1


y5 = 0 y5 = 2y5 = 1

𝐲𝐢∗ = 𝒕𝒉𝒓𝒆𝒔𝒉𝒐𝒍𝒅 + 𝐫𝐞𝐬𝐭𝐨𝐟𝐲𝐨𝐮𝐫𝐦𝐨𝐝𝐞𝐥 + 𝐞𝐢

Threshold1 Threshold2

Transformedy5 (y5∗)

Graded Response Model for Ordinal Categories• Idealforitemswithclearunderlyingresponsecontinuum(e.g.,Likert)

• #responseoptionsdon’thavetobethesameacrossitems

• GRMisan“indirect”or“difference”modelØ Computedifferencebetweenmodelstogetprobabilityofeachresponse

• Estimate1ai peritemand𝑘 = C − 1 difficulties(4optionsà 3difficulties)

• Modelstheprobabilityoflowervs.higher cumulativecategoriesvia𝑘 submodels(e.g.,if𝐶 = 4 possibleresponsesof𝑐 = 0,1,2,3):

0 vs. 1,2,3 0,1 vs. 2,3 0,1,2 vs. 3

• Eachsubmodelisestimatedusingallthedatacumulatively(assumesorder)

• Aswithbinaryitems,lavaan andMplus estimatestheIFAmodeldirectly(loadingsandthresholds),butunlikebinaryitems,itwon’tdotheIRTconversionintodiscriminationsanddifficulties(butseemyspreadsheetforhelp)


Mpluswillrefertothesethresholdsusing$1,$2,$3aftertheitemnameLavaan |t1,|t2,|t3


Example Graded Response Model (GRM)IRTversionoftheGRM:Notethat𝐚𝐢 isthesameacrosssubmodels…

• 0 vs.1,2,3:Logit y5v > 0 = a5 θv − b95 à 𝑝1 y5v > 0 = BCD yQ z{|}~Q9HBCD yQ z{|}~Q

• 0,1 vs.2,3:Logit y5v > 1 = a5 θv − b@5 à𝑝2 y5v > 1 = BCD yQ z{|}�Q9HBCD yQ z{|}�Q

• 0,1,2 vs.3:Logit y5v > 2 = a5 θv − bA5 à𝑝3 y5v > 2 = BCD yQ z{|}GQ9HBCD yQ z{|}GQ

IFAversionoftheGRM—whatisactuallyestimatedinMplus:

• 0 vs.1,2,3:Logit y5v > 0 = −τ95 + λ5Fv à 𝑝1 y5v > 0 = BCD |�~QH�Q�{9HBCD |�~QH�Q�{

• 0,1 vs.2,3:Logit y5v > 1 = −τ@5 + λ5Fv à𝑝2 y5v > 1 = BCD |��QH�Q�{9HBCD |��QH�Q�{

• 0,1,2 vs.3:Logit y5v > 2 = −τA5 + λ5Fv à𝑝3 y5v > 2 = BCD |�GQH�Q�{9HBCD |�GQH�Q�{

GRMindirectlypredictsprobabilityofeachcategory

Ø 𝑝 y5v = 0 = 1 − 𝑝1

Ø 𝑝 y5v = 1 = 𝑝1 − 𝑝2

Ø 𝑝 y5v = 2 = 𝑝2 − 𝑝3

Ø 𝑝 y5v = 3 = 𝑝3 − 0


IRT𝐛𝐤𝐢 =traitlevelneededfora50%probability(logit=0)ofthehigherbinarycategory

IFA𝛕𝐤𝐢 =logitoftheprobabilityofthelowerbinarycategorywhenFactor=0

Cumulative Item Response Curves:GRM for 4-Category (0123) Item, ai=1

b1 =-2 b3 =2b2 =0

ai =1à allcurveshavesameslope


Cumulative Item Response Curves:GRM for 4-Category (0123) Item, ai=2

b1 =-2 b3 =2b2 =0

ai =2àslopeissteeper


Category Response Curves:GRM for 4-Category (0123) Item, ai=2

GRMindirectlypredictstheprobabilityofeachcategoryresponseacrossTheta:

𝑝 y5v = 0 = 1 − 𝑝1𝑝 y5v = 1 = 𝑝1 − 𝑝2𝑝 y5v = 2 = 𝑝2 − 𝑝3𝑝 y5v = 3 = 𝑝3 − 0

MpluswillmakethistypeofPlotforyou.


Category Response Curves:GRM for 4-Category (0123) Item, ai=.5

This is exactly what you do NOT want to see.

Althoughtheyareordered,themiddlecategoriesareworthless (notdifferentiated).


EPSY906 / CLP 948: Lecture 6

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Figure4:SchoolFormItemParameters

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TestReliability

Trait(M=0,SD=1)

TopPanel:Redbarsdisplayitemdiscriminations.

BottomPanel:Itemlocationsfor4responseoptions:Never,Seldom,Sometimes,andOften

Thelocationofy>sometimes rangesfrom~0to.5acrossitems,creatinglowerreliabilityfortraits>.5

Beingabletoselectahigheroption(e.g.,almostalways)wouldimprovereliabilityforhigh-traitrespondents.

“Modified” (“Rating Scale”)Graded Response Model (MGRM)

• MoreparsimoniousversionofGRMforitemswithsame responseoptions

• InGRM,k*#itemsdifficulties+1discriminationareestimatedperitem• InMGRM,eachitemgetsownslopeandownoverall‘location’parameter,butthedifferencesbetweencategoriesaroundthatlocationareconstrainedtobeequalacrossitems(geta“c”shiftforeachthreshold)Ø So,differentai andbi peritem,butsamec1,c2,andc3 acrossitems(onec=0)Ø NotdirectlyavailablewithinMplus,butcanbedoneusingthresholdconstraints

• 0 vs.1,2,3:Logit y5v > 0 = a5 θv − b5 + c9 à 𝑝1 y5v > 0 = BCD yQ z{|}QHU~9HBCD yQ z{|}QHU~

• 0,1 vs.2,3:Logit y5v > 1 = a5 θv − b5 + c@ à𝑝2 y5v > 1 = BCD yQ z{|}QHU�9HBCD yQ z{|}QHU�

• 0,1,2 vs.3:Logit y5v > 2 = a5 θv − b5 + cA à𝑝3 y5v > 2 = BCD yQ z{|}QHUG9HBCD yQ z{|}QHUG


Modified vs. Regular GRM


ModifiedGRMàk“shift”cterms: Allcategorydistancesaresameacrossitems,andeachitemgetsitsown“bi” overalllocation

OriginalGRMàklocationsperitem:Allcategorydistancesdifferacrossitems

b3

b2

b1c1

c2 c3

b11 b12 b13

b21 b22 b23

b31 b32 b33

Summary of Models forOrdered Categorical Responses

SomeoftheseinMplusvia“CATEGORICALARE”

DifficultyPerItemOnly(categorydistances

equal)

DifficultyPerCategoryPerItem

Equaldiscriminationacrossitems(1-PLish)?

(possible,butnospecialname)

(possible,butnospecialname)

Unequaldiscriminations(2-PLish)?

“ModifiedGRM”or“RatingScaleGRM”(sameresponseoptions)

“GradedResponseModel”

“CumulativeLogit”

• GRMandModifiedGRMarereliablemodelsfororderedcategoricaldataØ Commonlyusedinreal-worldtesting;moststabletouseinpractice

Ø LeastdatademandbecausealldatagetusedinestimatingeachbkiØ Onlymajordeviationsfromthemodelwillendupcausingproblems


(Generalized) Partial Credit Model (PCM)• Whenyouwanttotest anassumptionofanorderedunderlyingcontinuum

• #responseoptionsdoesn’thavetobesameacrossitems,butthereisnoguaranteethateverycategorywillbemostlikelyatsomepoint

• Isa“direct,divide-by-total”model(prob ofeachcategoryisgivendirectly)

• Estimatekdifficulty-like𝛅𝐤𝐢“step”parameters,whicharetheThetavaluesatwhichthenextcategorybecomesmorelikely(notnecessarily50%)

• Modelstheprobabilityofadjacentcategories (“adjacentcategorylogit”)Ø Divideitemintoaseriesofbinaryitems,butwithoutorderconstraintsbeyondadjacent

categoriesbecauseitonlyusesthose2categories(𝑐 = 0,1,2,3):

IRTforPCM(nowinMplus):Is“generalized”if𝐚𝐢 isusedinsteadof𝐚

• If0 or1:Logit y5v = 1 = a5 θv − δ95 à 𝑝1 y5v = 1 = BCD yQ z{|�~Q9HBCD yQ z{|�~Q

• If1 or2:Logit y5v = 2 = a5 θv − δ@5 à𝑝2 y5v = 2 = BCD yQ z{|��Q9HBCD yQ z{|��Q

• If2 or3:Logit y5v = 3 = a5 θv − δA5 à𝑝3 y5v = 3 = BCD yQ z{|�GQ9HBCD yQ z{|�GQ


Category Response Curves:PCM for 4-Category (0123) Item, ai=1

PCMdirectlypredictstheprobabilityofeachcategoryresponseacrossTheta:

𝑝 y5v = 0 = 1 − 𝑝1𝑝 y5v = 1 = 𝑝1𝑝 y5v = 2 = 𝑝2𝑝 y5v = 3 = 𝑝3

These curves look similar to the GRM, but the location parameters are interpreted differently because they are NOT cumulative, they are only adjacent…



δ01δ12

δ23

The𝛅𝐤𝐢 termsarethelocationwherethenextcategoryresponsebecomesmorelikely(notnecessarily50%).



δ01δ12

δ23…a score of 2 instead of 1 requires less Theta than 1 instead of 0 …This is called a ‘reversal’

But here, this likely only happens because of a very low frequency of 1’s


Response Categories0 = green = Time-Out1 = pink = 30 – 45 s2 = blue = 15 – 30 s3 = black = < 15 s

*Misfit (p < .05)

Moreofwhatyoudon’twanttosee…categoryresponsecurvesfromaPCMwherereversalsareaplenty…

…andthemiddlecategoriesarefairlyuseless.


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Latent Trait Score

PCMExample:GeneralIntrusiveThoughts(5options)

WBSI 9: a = .84, b = -.34

WBSI 6: a = 1.04, b = -.30

WBSI 4: a = .94, b = -.10

WBSI 3: a = 1.09, b = -.10

WBSI 5: a = .65, b = .01

WBSI 7: a = .57, b = .17

Notethatthe4thresholdscovera widerange ofthelatenttrait,andwhatthedistributionofThetalookslikeasaresult...

Butthemiddle3categoriesareusedinfrequentlyand/orarenotdifferentiable


-3 -2 -1 0 1 2 3Latent Trait Score

IES 5: a = 1.25, b = .06

IES 11: a = 1.14, b = .10

IES 1: a = 1.10, b = .13

IES 10: a = .94, b = .19

IES 14: a = .96, b = .20

IES 4: a = .76, b = .57

IES 6: a = .88, b = .68

PartialCreditModelExample:Event-SpecificIntrusiveThoughts(4options)

Notethatthe3thresholdsdonot coverawiderange ofthelatenttrait,andwhatthedistributionofthetalookslikeasaresult…


Rating Scale Model (RSM)• MoreparsimoniousversionofPCMforitemswithsame responseformat

• InPCM,k*#itemsstepparameters+1discriminationareestimatedperitem

• InRSM,eachitemgetsownslopeandownoverall‘location’parameter,butthedifferencesbetweencategoriesaroundthatlocationareconstrainedtobeequalacrossitems(geta“c”shiftforeachthreshold;onec=0)Ø So,differentbi (andpossiblyai)peritem,butsamec1,c2,andc3 acrossitems

Ø AlsonotavailablewithinMplus

• If0 or1:Logit y5v = 1 = a θv − δ5 + c9 à 𝑝1 y5v = 1 = BCD y z{|�QHU~9HBCD y z{|�QHU~

• If1 or2:Logit y5v = 2 = a θv − δ5 + c9 à𝑝2 y5v = 2 = BCD y z{|�QHU�9HBCD y z{|�QHU�

• If2 or3:Logit y5v = 3 = a θv − δ5 + c9 à𝑝3 y5v = 3 = BCD y z{|�QHUG9HBCD y z{|�QHUG


Rating Scale vs. Regular PCM


RatingScaleàk“shift”cterms: Allcategorydistancesaresameacrossitems,andeachitemgetsitsown“𝛅i” overalllocation

OriginalPCMàk locationsperitem:Allcategorydistancesdifferacrossitems

𝛅3

𝛅2

𝛅1c1

c2 c3

𝛅11 𝛅12 𝛅13

𝛅21 𝛅22 𝛅23

𝛅31 𝛅32 𝛅33

Summary of Models for Maybe-Ordered Categorical Responses

NotdirectlyavailableinMplus DifficultyPerItemOnly(categorydistances

equal)

DifficultyPerCategoryPerItem

Equaldiscriminationacrossitems(1-PLish)?

“RatingScalePCM” “PartialCreditModel”

Unequaldiscriminations(2-PLish)?

“GeneralizedRatingScalePCM”??

(sameresponseoptions)

“GeneralizedPCM”“AdjacentCategory

Logit”

• PartialCreditModelstesttheassumptionoforderedcategoriesØ Thiscanbeusefulforitemscreening,butperhapsnotforactualanalysis

• ThesemodelshaveadditionaldatademandsrelativetoGRMØ Onlydatafromthatthresholdgetused(i.e.,for1vs.2,0and3don’tcontribute)

Ø Solargersamplesizesareneededtoidentifyallmodelparameters

Ø Sometimescategorieshavetobeconsolidatedtogetthemodeltonotblowup


Nominal Response Model for Unordered Categories

• Idealforitemswithnoorderingofanykind(e.g.,favoritecolor)

• #responseoptionsdon’thavetobethesameacrossitems

• NRMisa“direct”modelà prob ofeachcategorygivendirectly

• Estimate𝑘 ai peritemand𝑘 “intercepts”peritem

• Modelstheprobabilityofeachcategoryrelativetoabaselinecategory(sortoflikedummy-codingtheoutcomevariable,herebaseline=0):

0 vs. 1 0 vs. 2 0 vs. 3

• AvailableinMpluswithNOMINALAREoption(estimatedasanIFAmodel)

• Canbeusefultoexamineutilityofdistractorsinmultiplechoicetests


Mpluswillrefertothese“intercepts”using#1,#2,#3aftertheitemname


Example Nominal Response Model (NRM)IRTversionoftheNRM:Notethat𝐚𝐢 isNOTthesameacrosssubmodels…

• If0or1:Logit y5v = 1 = a95 θv − b95 à 𝑝1 y5v = 1 = BCD y~Q z{H}~Q9HBCD y~Q z{H}~Q

• If0 or2:Logit y5v = 2 = a@5 θv − b@5 à𝑝2 y5v = 2 = BCD y�Q z{H}�Q9HBCD y�Q z{H}�Q

• If0 or3:Logit y5v = 3 = aA5 θv − bA5 à𝑝3 y5v = 3 = BCD yGQ z{H}GQ9HBCD yGQ z{H}GQ

IFAversionoftheNRM—whatisactuallyestimatedinMplus:

• If0or1:Logit y5v = 1 = −τ95 + λ95Fv à 𝑝1 y5v = 1 = BCD �~QH�~Q�{9HBCD �~QH�~Q�{

• If0 or2:Logit y5v = 2 = −τ@5 + λ@5Fv à𝑝2 y5v = 2 = BCD ��QH��Q�{9HBCD ��QH��Q�{

• If0 or3:Logit y5v = 3 = −τA5 + λA5Fv à𝑝3 y5v = 3 = BCD �GQH�GQ�{9HBCD �GQH�GQ�{

NRMdirectlypredictsprobabilityofeachcategory

Ø 𝑝 y5v = 0 = 1 − 𝑝1 + 𝑝2 + 𝑝3

Ø 𝑝 y5v = 1 = 𝑝1

Ø 𝑝 y5v = 2 = 𝑝2

Ø 𝑝 y5v = 3 = 𝑝3


IRT𝐛𝐤𝐢 =traitlevelneededfora50%prob (logit=0)ofthelowerbinarycategoryIFA𝛍𝐤𝐢 =logitoftheprobabilityofthehigherbinarycategorywhenFactor=0

Category Response Curves(NRM for 5-Category Item)

Nominal Response Item Response Function

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ExampleAnalysisofMultipleChoiceDistractors:

PeoplelowinThetaaremostlikelytopickd,butc istheirsecondchoice

PeoplehighinThetaaremostlikelytopicka,butb istheirsecondchoice


Summary: Polytomous ModelsManykindsofpolytomousIRT/IFAmodels…• Someassumeorder ofresponseoptions…

Ø GradedResponseModelFamilyà “cumulativelogitmodel”

§ Modelcumulativechangeincategoriesusingalldataforeach

• Someallowyoutotestorder ofresponseoptions…Ø PartialCreditModelFamilyà “adjacentcategorylogitmodel”

§ Modeladjacentcategorythresholdsonly,sotheyallowyoutoseereversals(empiricalmis-orderingofyourresponseoptionswithrespecttoTheta)

§ PCMusefulforidentifyingseparability andadequacyofcategories

• Someassumenoorder ofresponseoptions…Ø NominalModelà “baselinecategorylogitmodel”

§ Usefultoexamineprobabilityofeachresponseoption

§ Isveryunparsimonious andthuscanbehardtoestimate


Models for Count Outcomes• Counts:non-negativeintegerunboundedresponses

Ø e.g.,howmanycigarettesdidyousmokethisweek?

Ø Traditionallyusesnaturalloglinksothatpredictedoutcomesstay≥0

• 𝐠 ⦁ Log E y5 = Log µ5 = modelà predictsmeanofy5• 𝐠|𝟏 ⦁ E(y5) = exp(model)à toun-logit,useexp(model)

• IFAmodel(noIRTanalog):𝐋𝐨𝐠 𝐲𝐢𝐬 = 𝛍𝐢 + 𝛌𝐢𝐅𝐬 + (𝐞𝐢𝐬)Ø Modelhasinterceptsandloadings,justpredictinglog(yis)instead

Ø InMplus,identifyoutcomesasCOUNT=inVARIABLE:section

• Whataboutanestimatedresidualvariance?Itdepends…


Poisson Conditional Distribution• Poissondistributionhasoneparameter,𝜆,whichisbothitsmeananditsvariance (so𝜆 =mean=varianceinPoisson)

• 𝑓 y5|λ = Prob y5 = y = ��∗BCD |�P!

• PDF:Prob y5 = y|β8, β9, β@ = �Q�∗BCD |�Q

P!

𝑦! isfactorialof𝑦

Thedotsindicatethatonlyintegervaluesareobserved.

Distributionswithasmallexpectedvalue(meanor𝜆)arepredictedtohavealotof0’s.

Once𝜆 > 6 orso,theshapeofthedistributionisclosetoathatofanormaldistribution.𝑦


3 potential problems for Poisson…• ThestandardPoissondistributionisrarelysufficient,though

• Problem#1:Whenmean≠varianceØ Ifvariance<mean,thisleadsto“under-dispersion”(notthatlikely)Ø Ifvariance>mean,thisleadsto“over-dispersion”(happensfrequently)

• Problem#2:Whenthereareno 0valuesØ Some0valuesareexpectedfromcountmodels,butinsomecontextsy5 > 0 always(butsubtracting1won’tfixit;needtoadjustthemodel)

• Problem#3:Whentherearetoomany0valuesØ Some0valuesareexpectedfromthePoissonandNegativeBinomialmodels

already,butmanytimesthereareevenmore0valuesobservedthanthatØ Tofixit,therearetwomainoptions,dependingonwhatyoudotothe0’s

• Eachoftheseproblemsrequiresamodeladjustmenttofixit…


Problem #1: Variance > mean = over-dispersion

• Tofixit,wemustaddaparameterthatallowsthevariancetoexceedthemeanà aNegativeBinomial(NB)distributionØ SaysresidualsareamixtureofPoissonandgammadistributions,

suchthat𝜆 itselfisarandomvariablewithagammadistribution

Ø Soexpectedmeanisstillgivenby𝜆,butthevariancewillbe>Poisson

• IFAModel:𝐋𝐨𝐠 𝐲5v = 𝛍𝐢 + 𝛌5𝐅𝐬 + 𝐞𝐢𝐬𝐆

Ø NBhasa𝑘 dispersionparameter,suchthat: Var y5v = 𝑘(1 + 𝑘α)Ø Poissonisnestedwithinnegativebinomial(cando–2ΔLLtestofα ≠0)

• InMplus,specifywhichresidualdistributionyouwant:Ø COUNT=y1(p)y2(nb);à y1isPoisson;y2isnegativebinomial


Negative Binomial (NB) = “Stretchy” Poisson…

• Becauseits𝑘 dispersionparameterisfixedto0,thePoissonmodelisnestedwithintheNegativeBinomialmodel—totestimprovementinfit:

• Is−2 𝐿𝐿£¤J¥¥¤¦ − 𝐿𝐿§¨©ªJ¦ > 3.84for𝑑𝑓 = 1? Then𝑝 < .05,keepNB

Mean = λDispersion=kVar y5v = 𝑘(1 + 𝑘α)

ANegativeBinomialmodelcanbeusefulforcountresidualswithextraskewness,butotherwisefollowaPoissondistribution.

0.00

0.05

0.10

0.15

0.20

0 5 10 15 20 25 30

Percen

tofO

bservatio

ns

OriginalCountVariable

"Poisson"Mean=5,k=0,Variance=5

"NB"Mean=5,k=0.25,Variance=11.25

"Poisson"Mean=10,k=0,Variance=10

"NB"Mean=10,k=0.25,Variance=35

PoissonandNegativeBinomialDistributionbyMeanandDispersionParameters


Problem #2: There are no 0 values• “Zero-Altered”or“Zero-Truncated”PoissonorNegativeBinomial:ZAP/ZANBorZTP/ZTNB(usedinhurdlemodels)Ø IsusualNBdistribution,justnotallowingany0values

Ø InMplus,COUNT=var1(nbt);à NegativeBinomialTruncated

Ø ItdoesNOTworktojustsubtract1anduseausualcountdistribution

• PoissonPDFwas:Prob y5 = y|µ5 =�Q�∗BCD |�Q

P!

• Zero-TruncatedPoissonPDFis:

Ø Prob y5 = y|µ5,y5 > 0 = �Q�∗BCD |�Q

P! 9|BCD |�Q

Ø Prob y5 = 0 = exp −µ5 ,soProb y5 > 0 = 1 − exp −µ5Ø Dividesbyprobabilityofnon-0outcomessoprobabilitystillsumsto1


Problem #3: Too many 0 values, Option #1• “Zero-Inflated”Poisson(pi)orNegativeBinomial(nbi);availablewithinMplus usingCOUNT=Ø Distinguishestwokindsof0values:expected andinflated (“structural”)

throughamixtureofdistributions(Bernoulli+Poisson/NB)

Ø Createstwosubmodelstopredict“ifextra 0”and“ifnot,howmuch”?§ Doesnotreadilymapontomosthypotheses(inmyopinion)§ ButaZIPexamplewouldlooklikethis…(ZINBwouldaddk dispersion,too)

• Submodel1:Logit y5v = extra0 = −τ59 + λ59Fv9Ø Predictbeinganextra0 usingLink=Logit,Distribution=Bernoulli

Ø Don’thavetospecifyafactormodelforthispart,cansimplyallowathresholdthatsaysyourdatahaveextra0valuesrelativetotheusualcountdistribution

• Submodel2:Log y5v = µ5@ + λ5@Fv@Ø Predictrestofcounts(including0’sexpectedfromcountdistribution)


Example of Zero-Inflated OutcomesZero-inflateddistributionshaveextra“structuralzeros”notexpectedfromPoissonorNB(“stretchedPoisson”)distributions.

Thiscanbetrickytoestimateandinterpretbecausethemodeldistinguishesbetweenkindsofzeros ratherthanzeroornot...

ImageborrowedfromAtkins&Gallop,2007


“Extra”0’srelativetoPoissonorNeg Bin

Problem #3: Too many 0 values, Option #2

• “Hurdle”modelsforPoissonorNegativeBinomialØ PH(notinMplus)orNBH:Explicitlyseparates0fromnon-0valuesthrougha

mixtureofdistributions(Bernoulli+Zero-AlteredPoisson/NB)Ø Createstwosubmodelstopredict“ifany0”and“ifnot0,howmuch”?

§ Easiertothinkaboutintermsofprediction(inmyopinion)

• Submodel1:Logit y5v = 0 = −τ59 + λ59Fv9Ø Predictbeingany0 usingLink=Logit,Distribution=BernoulliØ Don’thavetospecifypredictorsforthispart,cansimplyallowittoexist

• Submodel2: Log y5v|y5v > 0 = µ5@ + λ5@Fv@Ø PredictrestofpositivecountsviaLink=Log,Distribution=ZAP/ZANB

• Thesemodelsarefordiscreteoutcomes,butthereisananalogousversionforcontinuousoutcomes


Two-Part Models for Continuous Outcomes• Atwo-partmodelisananalogtohurdlemodelsforzero-inflatedcountoutcomes(andcouldbeusedwithcountoutcomes,too)Ø Explicitlyseparates0fromnon-0valuesthroughamixtureofdistributions

(Bernoulli+NormalorLogNormal)à usuallymucheasiertoexplain!

• Submodel1:Logit y5 > 0 = −τ59 + λ59Fv9 à predictbeingNOT0

• Submodel2: y5|y5 > 0 = µ5@ + λ5@Fv@ à predictsnon-0usingnormalorlognormalresiduals

Ø Two-partmodelusesMplusDATATWOPART:command§ NAMESAREy1-y4; à listoutcomestobesplitinto2parts§ CUTPOINTIS0; à wheretosplitobservedoutcomes§ BINARYAREb1-b4; à createnamesfor“0ornot”part§ CONTINUOUSAREc1-c4;à createnamesfor“howmuch”part§ TRANSFORMISLOG; à transformationofcontinuouspart§ 0ornot:predictedbylogitofbeingNOT0(“something”isthe1)§ Howmuch:predictedbytransformednormaldistribution(likelog)


Two-Part Measurement Model Example• Two-partmodelforasensation-seekingtask:7drivingtrialsinwhichlightà yellow…

• Twooutcomes:Ø Gothroughlight?

Ø Latencytobreakifnotgoingthrough?

• Createdtwolatentfactors,eachofwhichthenpredictedotheroutcomes


Pile of 0’s Taxonomy


• Whatkindofamount doyouwanttopredict?Ø Discrete:Countà Poisson,StretchyCountà NegativeBinomial

Ø Continuous:Normal,Log-Normal,Gamma(notinMplus)

• WhatkindofIf0 doyouwanttopredict?Ø Discrete:Extra0beyondpredictedbyamount?à Zero-inflatedPoissonorZero-inflatedNegativeBinomial

Ø Discrete:Any0atall?à HurdlePoissonorHurdleNegativeBinomial

Ø Continuous:Any0atall?à Two-PartwithContinuousAmount(seeabove)

Ø Note:Giventhesameamountdistribution,thesealternativewaysofpredicting0willresultinthesameemptymodelfit

Wrapping Up…• Whenfittinglatentfactormodels(orwhenjustpredictingobservedoutcomesfromobservedpredictorsinstead),youhavemanyoptionstofitnon-normaldistributionsØ CFA: Continuousoutcomeswithnormalresiduals,Xà Yislinear

§ IfresidualsmaynotbenormalbutalinearXà Yrelationshipisstillplausible,youcanuseMLRestimationinsteadofMLtocontrolforthat

Ø IRTandIFA: CategoricalorordinaloutcomeswithBernoulli/multinomialresiduals,Xà transformedYislinear;Xà originalYisnonlinear§ FullinformationMMLtraditionallypairedwithIRTversionofmodel;limited

informationWLSMVtraditionallypairedwithIFAversionofmodelinstead

Ø Countfamily: Non-negativeintegeroutcomes,Xà Log(Y)islinear§ ResidualscanbePoisson(wheremean=variance)ornegativebinomial(where

variance>mean);eithercanbezero-inflatedorzero-truncated§ Hurdleortwo-partmaybemoredirectwaytopredict/interpretexcesszeros

(predictzeroornotandhowmuchratherthantwokindsofzeros)


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Latent Trait Measurement Models for Other (not Binary ... · Polytomous Items = Categorical Outcomes •Polytomous = more than 2 options •Polytomous models are not numbered like