Interpreting Ordered Partition Model Parameters from ConQuest

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1 Berkeley Evaluaion Assessmen Research Cener Graduae School of Educaion Universiy of California Berkeley Berkeley CA hp:bearcenerberkeleyedu Technical Paper Series No Inerpreing Ordered Pariion Model Parameers from ConQues Nahaniel J S Brown Ocober 2004

2 Brown Inerpreing OPM parameers Page 2 of Absrac The ordered pariion model OPM described in Wilson 992 is an exension of Masers 982 parial credi model PCM ha allows more han one response caegory o have a paricular score The generalized Rasch modeling sofware ConQues Wu Adams Wilson 998 provides parameer esimaes for he OPM bu using a differen parameerizaion han given in Wilson 992 complicaing inerpreaion of hose esimaes This repor describes and illusraes he conversion of ConQues OPM parameers ino he Andersen level parameers for he OPM and PCM described in Wilson 992 and ino ConQues PCM parameers Inroducion Wilson 992 describes he ordered pariion model OPM as an exension of Masers 982 parial credi model PCM ha breaks he one-o-one correspondence beween response caegories and scores allowing more han one response o have a paricular score The OPs useful for siuaions in which cerain responses represen differen bu equally valued sraegies or ypes of reasoning In equaions and 27 and he accompanying discussion Wilson 992 provides means of inerpreing he parameers of he OPM relaing hem o he relaive probabiliy of achieving paricular response caegories As a submodel of he random coefficiens mulinomial logi RCML model Adams Wilson 995 he OPM can easily be esimaed using he score saemen in ConQues Wu Adams Wilson 998 However while he discussion in Wilson 992 is based upon Andersen level parameers ConQues uses iem and sep parameers prevening he direc applicaion of he inerpreive sraegies discussed in Wilson 992 Moreover because of he iem-sep parameerizaion ConQues OPM parameers canno be direcly compared o ConQues PCM parameers despie he fac ha he wo models are hierarchically relaed

3 Brown Inerpreing OPM parameers Page 3 of In he ineres of faciliaing he use of ConQues in he esimaion of he OPM his repor describes and illusraes he conversion of ConQues OPM parameers ino he Andersen level parameers for he OPM and PCM described in Wilson 992 and ino ConQues PCM parameers Firs all four ses of parameers are defined using a common noaion Then equaions are presened ha effec he sequenial conversion of he ConQues OPM parameers ino heir Andersen OPM Andersen PCM and ConQues PCM parameerizaions as well as a direc conversion from he ConQues OPM o he ConQues PCM parameers Model Parameerizaion Consider I iems i = I Each iem has response caegories k = 0 which are graded ino possible score levels m = 0 For iem i response k is assigned o level m by he scoring funcion k so ha k = m ConQues OPM parameers ConQues parameerizes he OPM using an iem parameer and sep parameers For his parameerizaion he iem response probabiliy model is: PX ni = k = exp " n k # $ i k # exp " n h # $ i h # h= k h where X ni is a random variable ha represens he response of person n wih abiliy θ n o iem i; ξ i is an iem parameer for iem i; ξ ik is a sep parameer for iem i associaed wih reaching caegory k from k ; ξ i0 0; and #j $ 0

4 Brown Inerpreing OPM parameers Page 4 of Andersen OPM parameers Wilson 992 parameerizes he OPM using Andersen level parameers For his parameerizaion he iem response probabiliy model is: PX ni = k = h= 0 exp [ " n k # $ ik ] [ ] exp " n h # $ ih 2 where X ni is a random variable ha represens he response of person n wih abiliy θ n o iem i; η ik is a level parameer for iem i associaed wih caegory k; and η i0 0 The model in 2 is equivalen o 2 in Wilson 992 where he symbol η replaces ξ and he caegory index k begins wih 0 insead of Andersen PCM parameers Wilson 992 parameerizes he PCM using Andersen level parameers For his parameerizaion he iem response probabiliy model is: [ ] PX ni = m = exp m" n # $ im h= 0 [ ] exp h" n # $ ih 3 where X ni is a random variable ha represens he response of person n wih abiliy θ n o iem i; κ im is a level parameer for iem i associaed wih score m; and κ i0 0 The model in 3 is equivalen o 3 in Wilson 992 where he symbol κ replaces η ConQues PCM parameers ConQues parameerizes he PCM using an iem parameer and sep parameers For his parameerizaion he iem response probabiliy model is:

5 PX ni = m = exp m" n # m$ i # exp h" n # h$ i # h= 0 m h Brown Inerpreing OPM parameers Page 5 of 4 where X ni is a random variable ha represens he response of person n wih abiliy θ n o iem i; δ i is an iem parameer for iem i; δ im is a sep parameer for iem i associaed wih reaching level m from m ; δ i0 0; and j # $ 0 Parameer Conversion ConQues OPM parameers o Andersen OPM parameers Comparing and 2 which mus be equivalen reveals: k k k $ 5 where he idenifying consrains ξ i0 0 and #j $ 0 should be kep in mind Example ConQues gave he following oupu for an iem wih 8 response caegories = 7 assigned o 5 score levels = 4 so ha: 0 = 0 = 2 = 3 = 4 = 5 = 6 = 7 = 8 = 2 9 = 0 = = 2 = 3 = 4 = 5 = 3 and 6 = 7 = 4

6 Brown Inerpreing OPM parameers Page 6 of =========================================================================== Ordered Pariion Model TABLES OF RESPONSE MODEL PARAMETER ESTIMATES =========================================================================== TERM : iem VARIABLES UNWGHTED FIT WGHTED FIT iem ESTIMATE ERROR MNSQ T MNSQ T =========================================================================== TERM 2: iemsep VARIABLES UNWGHTED FIT WGHTED FIT iem sep ESTIMATE ERROR MNSQ T MNSQ T Using 5 o calculae he Andersen OPM parameers gives: 0 0 # i0 = = 0 # i 0 # i = $745 = $ # i 0 # i L # i6 = $ = = 0760 where he definiions ξ i0 0 and #j $ 0 have been used 0 $745 L 0895 = 0064

7 Brown Inerpreing OPM parameers Page 7 of These parameers can be used o inerpre he resuls of he OPM analysis following he discussion and equaions and 27 in Wilson 992 Andersen OPM parameers o Andersen PCM parameers Following he discussion and equaion 22 in Wilson 992 he Andersen OPM parameers can be convered ino he equivalen Andersen PCM parameers using: m = ln = m # = m exp #$ i exp #$ i 6 keeping in mind ha κ i0 0 Example Coninuing o use he daa from above 6 gives: = ln = 0 = exp #$ i exp #$ i = ln exp #$ i0 exp #$ i = ln exp 0 exp 555 = #555 where here is only one addend in boh he numeraor and denominaor because here is only one caegory in each of he firs wo score levels Andersen PCM parameers o ConQues PCM parameers The ConQues iem parameer is he average of he Andersen level parameers: = $ # M ih 7 i h= and he ConQues sep parameers are he Andersen level parameers adjused by his average: δ im = κ im δ i 8 ConQues OPM parameers o ConQues PCM parameers Alernaively he ConQues PCM parameers can be deermined direcly from he ConQues OPM parameers using:

8 Brown Inerpreing OPM parameers Page 8 of ln - = 0 = 0 9 and m $ ln - = m$ = m 0 0 which are derived in he Appendix Noe ha 9 implies ha in mos cases he iem parameers for he ConQues OPM and PCM will no be equivalen Only for he special case ha here is a single caegory in boh he lowes m = 0 and highes m = score levels will 9 simplify o: exp $ ln - 0 = # i ln exp 0 M i - exp 0 0 = # i ln 0 resuling in equal iem parameers for he ConQues OPM and PCM Example Coninuing o use he daa from above 9 gives: 0 exp $ 4 ln 6 7 = 090 # 4 ln exp 0 ij - exp exp 0 0 = $0085 where he definiions ξ i0 0 and #j $ 0 have been used The value of δ i deermined by ConQues running he same daa using he PCM was 0086

9 Brown Inerpreing OPM parameers Page 9 of Using 0 gives: exp $ $ ln - 0 = 090 $ $ ln exp 0 - exp = $470 where he definiion ξ i0 0 has been used The value of δ i deermined by ConQues running he same daa using he PCM was 47 In general he ConQues PCM parameers were fully recoverable from he ConQues OPM parameers in his example References Adams R J Wilson M 995 Formulaing he Rasch model as a mixed coefficiens mulinomial logi In G Engelhard M Wilson Eds Objecive measuremen: Theory ino pracice Vol 3 pp Norwood NJ: Ablex Publishing Masers G N 982 A Rasch model for parial credi scoring Psychomerika Wilson M 992 The ordered pariion model: An exension of he parial credi model Applied Psychological Measuremen Wu M L Adams R J Wilson M 998 ACERConQues user guide Hawhorn Ausralia: ACER Press

10 Brown Inerpreing OPM parameers Page 0 of Appendix The derivaion of 9 and 0 follows Subsiuing 5 ino 6 gives: m = ln - = m # = m exp #$ i # exp #$ i # 0 2 which simplifies: m = ln - = m # = m exp # m #$ i m = ln exp #m$ i - m = ln exp # i - m ln - exp # m #$ i # exp #m$ i # 0 exp # = m# exp # = m $ = m = m $ = m = m

11 Brown Inerpreing OPM parameers Page of Plugging 6 ino 7 gives: = exp $ B # M i ln i = h$ i 7 h= - - = h 0 exp $ B ln i = h$ 8 M i h= - = h 0 - ln ln - h= = h$ = 0 = h = Likewise plugging 6 ino 8 gives 0 direcly

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