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”
EPSY906 / CLP 948: Lecture 6 2
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.
EPSY906 / CLP 948: Lecture 6 3
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
EPSY906 / CLP 948: Lecture 6 4
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
EPSY906 / CLP 948: Lecture 6 5
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)
EPSY906 / CLP 948: Lecture 6 6
The Threshold Concept
• Eachcategoricalvariableisreally thechopped-upversionofapretendunderlyingcontinuousvariable(𝐲∗)withmean=0(variance=1.00or3.29)
• PolytomousmodelswilldifferinhowtheymakeuseofmultiplethresholdsperiteminwhichC=#categories,so#thresholdsperitem=k =C−1
EPSY906 / CLP 948: Lecture 6 7
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)
Submodel3Submodel2Submodel1
Mpluswillrefertothesethresholdsusing$1,$2,$3aftertheitemnameLavaan |t1,|t2,|t3
EPSY906 / CLP 948: Lecture 6 8
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
EPSY906 / CLP 948: Lecture 6 9
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
EPSY906 / CLP 948: Lecture 6 10
Cumulative Item Response Curves:GRM for 4-Category (0123) Item, ai=2
b1 =-2 b3 =2b2 =0
ai =2àslopeissteeper
EPSY906 / CLP 948: Lecture 6 11
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.
EPSY906 / CLP 948: Lecture 6 12
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 13
EPSY906 / CLP 948: Lecture 6
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Figure4:SchoolFormItemParameters
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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
EPSY906 / CLP 948: Lecture 6 15
Modified vs. Regular GRM
EPSY906 / CLP 948: Lecture 6 16
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
EPSY906 / CLP 948: Lecture 6 17
(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
EPSY906 / CLP 948: Lecture 6 18
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…
EPSY906 / CLP 948: Lecture 6 19
Category Response Curves:PCM for 4-Category (0123) Item, ai=1
δ01δ12
δ23
The𝛅𝐤𝐢 termsarethelocationwherethenextcategoryresponsebecomesmorelikely(notnecessarily50%).
EPSY906 / CLP 948: Lecture 6 20
Category Response Curves:PCM for 4-Category (0123) Item, ai=1
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
EPSY906 / CLP 948: Lecture 6 21
Response Categories0 = green = Time-Out1 = pink = 30 – 45 s2 = blue = 15 – 30 s3 = black = < 15 s
*Misfit (p < .05)
Moreofwhatyoudon’twanttosee…categoryresponsecurvesfromaPCMwherereversalsareaplenty…
…andthemiddlecategoriesarefairlyuseless.
EPSY906 / CLP 948: Lecture 6 22
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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
EPSY906 / CLP 948: Lecture 6 23
-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…
EPSY906 / CLP 948: Lecture 6 24
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
EPSY906 / CLP 948: Lecture 6 25
Rating Scale vs. Regular PCM
EPSY906 / CLP 948: Lecture 6 26
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
EPSY906 / CLP 948: Lecture 6 27
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
Submodel3Submodel2Submodel1
Mpluswillrefertothese“intercepts”using#1,#2,#3aftertheitemname
EPSY906 / CLP 948: Lecture 6 28
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
EPSY906 / CLP 948: Lecture 6 29
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
EPSY906 / CLP 948: Lecture 6 30
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
EPSY906 / CLP 948: Lecture 6 31
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…
EPSY906 / CLP 948: Lecture 6 32
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
EPSY906 / CLP 948: Lecture 6 33
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…
EPSY906 / CLP 948: Lecture 6 34
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
EPSY906 / CLP 948: Lecture 6 35
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.
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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
EPSY906 / CLP 948: Lecture 6 36
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
EPSY906 / CLP 948: Lecture 6 37
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)
EPSY906 / CLP 948: Lecture 6 38
Example of Zero-Inflated OutcomesZero-inflateddistributionshaveextra“structuralzeros”notexpectedfromPoissonorNB(“stretchedPoisson”)distributions.
Thiscanbetrickytoestimateandinterpretbecausethemodeldistinguishesbetweenkindsofzeros ratherthanzeroornot...
ImageborrowedfromAtkins&Gallop,2007
EPSY906 / CLP 948: Lecture 6 39
“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
EPSY906 / CLP 948: Lecture 6 40
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)
EPSY906 / CLP 948: Lecture 6 41
Two-Part Measurement Model Example• Two-partmodelforasensation-seekingtask:7drivingtrialsinwhichlightà yellow…
• Twooutcomes:Ø Gothroughlight?
Ø Latencytobreakifnotgoingthrough?
• Createdtwolatentfactors,eachofwhichthenpredictedotheroutcomes
EPSY906 / CLP 948: Lecture 6 42
Pile of 0’s Taxonomy
EPSY906 / CLP 948: Lecture 6 43
• 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)
EPSY906 / CLP 948: Lecture 6 44