高階層試題反應理論及其成效探討

(1)

୯ҥᆵύ௲ػεᏢ௲ػෳᡍ಍ीࣴز܌౛ᏢᅺγፕЎ

ࡰᏤ௲௤Ǻ೾դԽ റγ

ଯ໘ቫ၂ᚒϸᔈ౛ፕϷځԋਏ௖૸

ࣴزғǺ݅٫ᐇ ኗ

(2)

(3)

ᖴᜏ

ӣ२྽߃ٰډ೭္ޑόӼǴӆჹྣӵϞջஒ౥཰ǵԏԋ഻৹ޑךǴ೭΋Ϫ೿ ाགᖴ೚ӭΓჹךޑගឫᆶᔅշǶ२ӃགᖴࡰᏤ௲௤೾դԽറγǴӧԴৣޑಒЈ ௲ᏤΠǴ٬ךளа΋ᑍ௲ػෳᡍሦୱޑు༫Ǵόਔޑ૸ፕ٠ࡰᗺך҅ዴޑБӛǴ ό໻௲௤ᙦ൤ޑޕ᛽Ǵҭ௲ᏤࡑΓೀШޑၰ౛Ǵ٬ךӧ೭ٿԃύᕇ੻ঘభǶӕਔ Ψགᖴα၂ہ঩මࡌሎԴৣǵࡼቼᡕԴৣࡰᏤךޑፕЎǴ๏ϒ೚ӭᝊ຦ޑࡌ᝼Ǵ ٬ҁፕЎૈ୼׳ֹ᏾ԶᝄᙣǶ! ೭ٿԃޑВηǴࣴز࠻္ӅӕޑғࢲᗺᗺᅀᅀǴԖ៿኷ΨԖൿ໾Ǵ೿ஒԋࣁ നऍӳޑӣᏫǴགᖴᏢߏۆॺǵӕᏢǵᏢ׌ۂॺޑϕ࣬ࠀᓰǴգ0یॺޑഉՔᡣ ٿԃޑࣴزғࢲᡂளӭߍӭ࠮Ƕ! གᖴࡹଈǵڂՙǵཫറǵࡏ໋ǵඵࣁǵػໜᏢߏǵች㧌ǵ☰ॳᏢۆॺόჇځ ྠޑࡰрךࣴزύޑલѨǴӧך଎ைਔࣁךှൽǴ੝ձགᖴڂՙᏢߏ຤ЈޑࡰᏤ ᆶႴᓰǴΨགᖴნጩǵሎᇬǵ٫ᑉǵذԹǵҺῑаϷሡӆᏟର΋ԃޑדയǵϘണǵ γരӕᏢॺޑϕ࣬ן࡭ᆶᔅԆǶࣴز࠻ޑቼࣤǵٍػᏢ׌ǵۏ։ǵ܃զᏢۂॺ྽ ฅΨόૈב૶Ǵգ0یॺޑ዗Ј࣬շךሎ૶ӧЈǶ! تܻ϶৓ޱӧङࡕޑᓨᓨЍ࡭׳ࢂך߻຾ޑ୏ΚǴؒԖգޑᡏፊǵх৒Ǵ࣬ ߞ೭ٿԃޑғࢲஒ཮ࡐό΋ኬǶќѦǴाགᖴ΋ՏഉךࡷᐩڹᏯǵεъڹϕ࣬у ݨѺ਻Ъեፓޑӳ϶Ƕ! നࡕǴᙣаԜЎ᝘๏ךኑངޑৎΓǶ! ݅٫ᐇ ύ๮҇୯ΐΜΖԃϤД

(4)

(5)

ᄔा

Ҟ߻୯ሞ΢ၨ๱Ӝϐεࠠ኱ྗϯෳᡍࣣឦܭଯ໘ቫޑຑໆࢎᄬǴҗܭ೭٤ෳ ᡍ܌٬ҔϐෳໆኳԄό΋ǴЪڙज़ܭ౜ԖБݤᆶ೬ᡏϐࡺǴ٬Ҕ໺಍ޑෳໆኳԄ ຾ՉൂቫወӧૈΚޑ՗ीǴԶ҂٬Ҕֹ᏾ኳԄ຾ՉໆЁ՗ीǴӢԜǴຑໆࢎᄬᆶ ෳໆኳԄϐ᏾ӝҭۘԖׯ๓ޑޜ໔Ƕ! ҁࣴزЬाҞޑࣁଛӝၨፄᚇϐຑໆࢎᄬǴගр΋፾Ҕܭଯ໘ቫຑໆࢎᄬϐ ෳໆኳԄǴࡺҁࣴزЬाа PISA ϐຑໆࢎᄬբࣁ୷ᘵǴ೛ीόӕޑଯ໘ቫ၂ᚒ ϸᔈኳԄǴϩձа PISA ܌٬Ҕϐ՗ीБԄ຾Չ՗ीǴаϷҁࣴز܌ගрϐֹ᏾ ኳԄ՗ीБݤ຾Չ՗ीǴᙖа௖૸ӕਔᆶϩ໒՗ीБݤϐ՗ीਏ݀Ǵ٠ှ،໺಍ ෳໆኳԄᆶຑໆࢎᄬό࣬ଛӝϐୢᚒǶ ҁࣴز೸ၸኳᔕჴᡍБԄ௖૸ҁࣴز܌ගрϐֹ᏾ኳԄ՗ीБݤᆶ PISA ύ ܌٬Ҕϐϩ໒՗ीБݤځԋਏৡ౦Ǵ٠٬Ҕ֡БਥᇤৡȐroot mean square error, RMSEȑբࣁຑ՗ࡰ኱ǹҗჴᡍ่݀ёޕǴҁࣴز܌ගрϐֹ᏾ኳԄёаӕਔ՗ ीЬाໆЁϷԛભໆЁǴЪ՗ीᇤৡࣣௗ߈܈ᓬܭ PISA ϐ՗ीБԄǶ

ᜢᗖӷǺεࠠ኱ྗϯෳᡍǵଯ໘ቫ၂ᚒϸᔈ౛ፕǵӭӛࡋ၂ᚒϸᔈ౛ፕǵ PISA

(6)

Abstract

The assessment framework of the many large-scale standardized tests, such as the programme for international student assessment (PISA), is the higher-order assessment framework. The unidimensional item response theories (UIRT) are often used to estimate the overall ability and the multidimensional item response theories (MIRT) are often used to estimate the subscales. This estimation procedure is named as a separated estimation. However, there is no research on the effects of using the full models to estimate the overall ability and subscales concurrently.

The main purpose of this study is to propose the higher-order IRT models being suitable for higher-order assessment frameworks based on the PISA and to estimate the subscales and overall ability concurrently.

By using the simulation data, the performances of the two estimation procedures, the separated estimation used by PISA and the full model estimation proposed by this study, are compared. The root mean square errors (RMSEs) are the indicators of the performances. The results show that the performances of the full models proposed by this study are better than the that used by PISA.

Keyword Κ Large-scale standardized test, Higher-order item response theory, Multidimensional item response theory, PISA

(7)

Ҟᒵ

ᄔा ... I Ҟᒵ ...III ߄Ҟᒵ ... IV კҞᒵ ...V ಃ΋ക! ᆣፕ ...1 ಃ΋࿯! ࣴز୏ᐒᆶҞޑ ...2 ಃΒ࿯! ࡑเୢᚒ ...3 ಃΟ࿯! Ӝຒှញ ...4 ಃΒക! Ў᝘௖૸ ...6 ಃ΋࿯! PISAϐኧᏢຑໆࢎᄬ ...6 ಃΒ࿯! ၂ᚒϸᔈ౛ፕ ...10 ಃΟ࿯! ୖኧ՗ीݤ ...18 ಃΟക! ࣴزБݤ ...24 ಃ΋࿯! ΒӢη HO-IRT ኳԄ...24 ಃΒ࿯! ࣴز೛ी ...25 ಃΟ࿯! ຑ՗ࡰ኱ ...35 ಃѤ࿯! ࣴزπڀ ...36 ಃѤക! ࣴز่݀ ...38 ಃ΋࿯! ୖኧ՗ीᇤৡ่݀ ...38 ಃΒ࿯! ᆕӝКၨ ...50 ಃϖക! ่ፕᆶࡌ᝼ ...54 ಃ΋࿯! ่ፕ ...54 ಃΒ࿯! ࡌ᝼ ...55 ୖԵЎ᝘ ...56 ύЎ೽ҽ ...56 मЎ೽ҽ ...57 ߕᒵ΋! ՗ीኳԄୖኧ೛ۓ ...60

(8)

߄Ҟᒵ!

߄ 2-1-1! PISAӚηӛࡋϐኧᏢຑໆࢎᄬ ...7 ߄ 4-1-1! H1L2-1_EL ϐୖኧ՗ीᇤৡ...39 ߄ 4-1-2! H1L2-1_EH ϐୖኧ՗ीᇤৡ ...39 ߄ 4-1-3! H1L2-1_HL ϐୖኧ՗ीᇤৡ ...40 ߄ 4-1-4! H1L2-2_EL ϐୖኧ՗ीᇤৡ...41 ߄ 4-1-5! H1L2-2_EH ϐୖኧ՗ीᇤৡ ...42 ߄ 4-1-6! H1L2-2_EHL ϐୖኧ՗ीᇤৡ...43 ߄ 4-1-7! H1L4-1_EL ϐୖኧ՗ीᇤৡ...44 ߄ 4-1-8! H1L4-1_EH ϐୖኧ՗ीᇤৡ ...44 ߄ 4-1-9! H1L4-1_EHL ϐୖኧ՗ीᇤৡ...45 ߄ 4-1-10! H1L4-2_EL ϐୖኧ՗ीᇤৡ...46 ߄ 4-1-11! H1L4-2_EH ϐୖኧ՗ीᇤৡ ...47 ߄ 4-1-12! H1L4-2_EHL ϐୖኧ՗ीᇤৡ...48 ߄ 4-1-13! H2L4_EL ϐୖኧ՗ीᇤৡ...49 ߄ 4-1-14! H2L4_EH ϐୖኧ՗ीᇤৡ ...49 ߄ 4-1-15! H2L4_EHL ϐୖኧ՗ीᇤৡ ...50 ߄ 4-2-1! H1L2-1 ϐୖኧ՗ीᇤৡ ...51 ߄ 4-2-2! H1L2-2 ϐୖኧ՗ीᇤৡ ...52 ߄ 4-2-3! H1L4-1 ϐୖኧ՗ीᇤৡ ...52 ߄ 4-2-4! H1L4-2 ϐୖኧ՗ीᇤৡ ...52 ߄ 4-2-5! H2L4 ϐୖኧ՗ीᇤৡ...53 !

(9)

კҞᒵ!

კ 2-1-1! PISA ኧᏢࣽຑໆࢎᄬ...8 კ 2-1-2! ӭӛࡋ IRT ኳԄ ...9 კ 2-1-3! ൂӛࡋ IRT ኳԄ ...9 კ 2-2-1! ᚒ໔ӭӛࡋෳᡍ ...12 კ 2-2-2! ᚒϣӭӛࡋෳᡍ ...12 კ 2-2-3! բ཰ᆶૈΚޑᜢ߯ ...13 კ 2-2-4! HO-IRT ኳԄᔈҔܭ΋ঁD ᆢࡋޑෳᡍ...16 კ 3-1-1! ΒӢη HO-IRT ኳԄ ...25 კ 3-2-1! ࣴزࢬำკ ...26 კ 3-2-2! H1L2 ϐ HO-IRT ኳԄ...29 კ 3-2-3! H1L2_EH ϐ IRT ኳԄ...29 კ 3-2-4! H1L2_EL ϐ MIRT ኳԄ ...30 კ 3-2-5! H1L4 ϐ HO-IRT ኳԄ...30 კ 3-2-6! H1L4_EH ϐ IRT ኳԄ...31 კ 3-2-7! H1L4_EL ϐ MIRT ኳԄ ...31 კ 3-2-8! H2L4 ϐ HO-IRT ኳԄ...32 კ 3-2-9! H2L4_EH ϐ MIRT ኳԄ ...32 კ 3-2-10! H2L4_EL ϐ MIRT ኳԄ ...33 !

(10)

ಃ΋ക! ᆣፕ

୯ϣ໺಍ޑຑໆϷڮᚒБԄε೽ϩமፓϣ৒ޕ᛽ޑᕇڗำࡋǴჹܭଯ໘ޑᏢ ࣽૈΚȐӵኧᏢનᎦȑ೭Бय़ޑຑໆ٠ؒԖϼӭ๱ᏀǴӢځຑໆࢎᄬၨᜤۓကЪ ीϩኳԄᆶीϩೕ߾ၨࣁፄᚇǶ

୯Ѧ೚ӭӃ຾୯ৎޑ௲ػس಍ǴჹܭᏢғ୷ҁૈΚ߄౜೿Ԗ࣬྽ుϪޑᜢᚶ ϷڀᡏܴዴޑᇡޕǴӵNAEPȐThe National Assessment of Educational Progressȑǵ PISAȐThe Programme for International Student AssessmentȑکTIMSSȐThe Trends in International Mathematics and Science Studyȑޑຑໆࢎᄬջගٮךॺؼӳϐጄ ٯǹฅԶǴ೭٤୯ሞ΢ၨޕӜޑεࠠ኱ྗϯෳᡍࢂឦܭଯ໘ቫޑᏢࣽૈΚෳᡍǴ ՠӧෳໆኳԄޑଛӝ΢ࠅϝԖόىϐೀǴٯӵǺNAEPǵTIMSSϝ٬Ҕൂӛࡋ၂ ᚒϸᔈ౛ፕȐunidimensional item response theory, UIRTȑࣁЬाޑෳໆኳԄǴ໻ ૈჹόӕᏢࣽૈΚаൂ΋ૈΚॶ຾ՉඔॊȐLee, Grigg & Dion, 2007; Mullis, Martin, Ruddock, O`Sullivan, Arora, Erberber, 2007ȑǹPISAᗨ٬Ҕӭӛࡋ၂ᚒϸᔈ ౛ፕȐmultidimensional item response theory, MIRTȑύϐӭӛࡋᒿᐒ߯ኧӭ໨logit ኳԄȐmultidimensional random coefficients multinomial logit model, MRCMLMȑǴ ՠ໻ଞჹӚᏢࣽϐԛભໆЁȐsubscaleȑ຾Չ՗ीǴჹܭPISAӚᏢࣽϐЬाໆЁ ϝ٬ҔൂӛࡋIRT຾Չ՗ीǶ

Ҟ ߻ Ԗ ೚ ӭ ೬ ᡏ ё Ҕ ܭ UIRT ᆶ MIRT ϐ ୖ ኧ ՗ ी Ǵ ٯ ӵ Ǻ BILOG-MG ȐZimowski, Muraki, Mislevy, & Bock, 1996ȑǵNOHARMȐFraser, 1988ȑǵ TESTFACTȐWilson, Wood, & Gibbons, 1991ȑǵMAXLOGȐMckinley & Reckase, 1983ȑǵACER ConQuest 2.0ȐWu, Adams, & Wilson, 1998ȑ฻ǴPISA߾٬ҔACER ConQuest 2.0຾Չୖኧ՗ीȐOECD, 2005ȑǴځ՗ीБԄࣣคݤ຾Չଯ໘ቫޑૈ ΚໆЁ՗ीǶҞ߻໻ԖϿኧᏢޣගрཥޑीໆኳԄٰӕਔ՗ीଯ໘ቫૈΚໆЁ Ȑde la Torre & Douglas, 2004; Sheng, 2005; Song, 2007ȑǴځύde la Torre and

(11)

DouglasȐ2004ȑ܌ว৖ޑ໘ቫԄወӧ੝፦ϩ݋ኳԄȐhierarchical latent analysis modelȑᆶShengȐ2005ȑ܌ว৖ޑΒୖኧதᄊު׎໘ቫϩ݋ኳࠠȐtwo-parameter normal ogive analysis modelȑǴ೭ٿᅿኳࠠࢎᄬ೿Ԗჴ୍ᆶཷۺ΢ޑલѨǶ໘ቫ Ԅወӧ੝፦ϩ݋ኳԄӧЬाໆЁ΢ࢂೱុໆЁǴՠӧԛભໆЁ΢߾ࢂ٬Ҕᚆණໆ ЁǹΒୖኧதᄊު׎໘ቫϩ݋ኳࠠӧЬाໆЁᆶԛભໆЁ΢ࣣࣁೱុໆЁǴՠѝ ፾ҔܭΒୖኧதᄊު׎ኳԄ΢ǶSONGȐ2007ȑ܌໒วޑ΋Ӣηଯ໘ቫ၂ᚒϸᔈ ౛ፕኳԄȐHigh-order IRT modelǴᙁᆀHO-IRTȑό໻ӧЬाໆЁᆶԛભໆЁ΢ ࣣࣁೱុໆЁǴҭ፾Ҕܭ1PLǵ2PLϷ3PLኳԄǴׯ຾߻ॊٿኳԄϐલѨ٠ёຎࣁ ଯ໘ቫޑ΋૓ϯኳԄǶӢԜǴ௦ҔѬٰբࣁҁࣴز௖زޑኳԄǶ! SongȐ2007ȑ໒ว΋ӢηHO-IRTǴԜኳԄёଞჹ΋Ӣηଯ໘ቫޑૈΚ຾Չ ௢ፕǴᓬܭMRCMLMѝૈ຾Չൂ΋໘ቫޑૈΚ՗ीǶҗܭଯ໘ቫޑᏢࣽૈΚෳ ᡍǴ۳۳όѝෳໆൂ΋ଯ໘ૈΚǴӢԜǴҁࣴزۯ՜HO-IRTኳԄว৖рΒӢη HO-IRTኳԄǴ٠௖૸όӕޑHO-IRTኳԄჹ՗ीᆒྗࡋޑቹៜǶ ҁകϩࣁΟ࿯Ǵϩձϟಏࣴز୏ᐒᆶҞޑǵࡑเୢᚒϷӜຒှញǴ૟ϩॊӵ ΠǶ

ಃ΋࿯! ࣴز୏ᐒᆶҞޑ

߈ԃٰ೚ӭ୯ሞຑໆȐӵ PISAǵNAEPǵTMISSȑޑ่݀ుڙӚ୯ख़ຎǴ೭ ٤ຑໆ܌ϦѲޑຑໆࢎᄬև౜Ҟ߻ኧᏢ௲ػᏢࣚ܌ख़ຎޑኧᏢનᎦࣁՖǶᖐٯٰ ᇥǴPISA ኧᏢࣽຑໆࢎᄬࣁଯ໘ቫޑຑໆࢎᄬǴځࢎᄬх֖ٿ໘ቫޑኧᏢૈΚǴ ಃ΋ቫޑૈΚໆЁȐԛભໆЁȑх֖ኧໆȐquantityȑǵޜ໔ᆶ׎ᡏȐspace and shapeȑǵׯᡂᆶᜢ߯Ȑchange and relationshipsȑϷόዴۓ܄ȐuncertaintyȑѤঁኧ ᏢૈΚǴಃΒቫޑૈΚໆЁȐЬाໆЁȑࣁኧᏢનᎦǹҁࣴزύۓက೭ᅿх֖ٿ ໘ቫޑຑໆࢎᄬࣁၨֹ᏾ϐଯ໘ቫຑໆࢎᄬǶ൩಍ीϩ݋ԶقǴ܌Ԗ҂ޕୖኧᔈ

(12)

ܭֹ᏾ኳԄΠ΋ଆ՗ीၨ٫Ǵฅ PISA ᗨԖܴዴଛӝޑᏢࣽຑໆࢎᄬǴՠڙज़ܭ ౜ԖБݤᆶ೬ᡏϐࡺǴ٠҂٬Ҕֹ᏾ኳԄ຾ՉໆЁ՗ीǴPISA ஒӚቫભໆЁϩ ໒຾Չ՗ीǴځБݤӵΠǺಃ΋ቫ٬Ҕӭӛࡋ IRT ύϐ MRCMLM ຾ՉኧᏢࣽԛ ભໆЁ՗ीǴಃΒቫ٬Ҕൂӛࡋ IRT ϐ Rasch ኳԄ຾ՉኧᏢࣽЬाໆЁ՗ी ȐOECD, 2005ȑǶԜᅿϩ໒՗ीޑБԄёૈ཮Ӣ۹ౣӚ໘ቫ໔۶Ԝ࣬٩ϐ௃׎Ǵ Ꮴठ՗ीᆒྗࡋफ़եǶ җܭ SongȐ2007ȑኳԄว৖ϐࣴز٠คֹ᏾ǴӧኳԄБय़Ǵѝ೛ۓ΋ঁЬा ໆЁϩኧǹӧୖኧ՗ीБय़ࢂճҔςޕ၂ᚒୖኧ຾ՉૈΚୖኧϐ՗ीǹӧځдᡂ ໨೛ीБय़ǴԛભໆЁϩኧኧҞǵᡂኧ໔࣬ᜢ฻ჹܭኳԄୖኧ՗ीϐቹៜǴ೭٤ ཮٬ࣴزޣคݤుΕΑှࣗԿϒаᔈҔ၀ኳԄܭ΋૓௲ػෳᡍϐ౜൑ǶӢԜǴҁ ࣴزޑЬाҞޑࢂа PISA ϐຑໆࢎᄬբࣁ୷ᘵǴۯ՜ HO-IRT ว৖рΒӢη HO-IRTኳԄǴ٠೛ीόӕޑ HO-IRT ኳԄǴ٬Ҕၨڀቸ܄ϐ՗ी೬ᡏ WinBUGSǴ ٠ගрόӕୖኧ՗ीБԄ຾Չ՗ीǴයఈ೸ၸόӕޑ HO-IRT ኳԄǴྗዴ՗ीр ᏢғϐЬाໆЁϩኧϷ࣬ჹᔈϐԛભໆЁϩኧǴаှ،ଯ໘ቫޑຑໆࢎᄬᆶෳໆ ኳԄό࣬ଛӝޑୢᚒǶ

ಃΒ࿯! ࡑเୢᚒ

ਥᏵ΢ॊޑࣴزҞޑǴҁࣴزஒ૸ፕΠӈୢᚒǺ ΋ǵʳӧϩ໒՗ीЬाໆЁᆶԛભໆЁਔǴаPISA܌٬Ҕϐ՗ीБԄ຾Չ՗ीǴа Ϸҁࣴز܌ගрϐୖኧ՗ीኳԄ຾Չ՗ीჹ՗ीᆒྗࡋޑቹៜࣁՖǻ! Βǵʳֹ᏾՗ीᆶϩ໒՗ीჹୖኧ՗ीᆒྗࡋޑቹៜࣁՖǻ Οǵʳ଑ᘜୖኧ೛ीჹHO-IRTኳԄޑୖኧ՗ीᆒྗࡋޑቹៜࣁՖǻ ѤǵʳԛભໆЁঁኧ೛ीჹHO-IRTኳԄޑୖኧ՗ीᆒྗࡋޑቹៜࣁՖǻ ϖǵʳ΋ӢηᆶΒӢηHO-IRTኳԄჹୖኧ՗ीᆒྗࡋޑቹៜࣁՖǻ

(13)

ಃΟ࿯! Ӝຒှញ

ଞჹҁࣴزதـޑӜຒǴញကӵΠȅ

൘ǵʳԛભໆЁ!

ԛભໆЁࢂෳໆᏢғӧόӕࡰ኱ΠޑૈΚ߄౜ȐᏢಞԋ݀ȑǴ೭٤ࡰ኱ёа ࢂᏢಞҞ኱ǵηෳᡍȐsubtestsȑǵᏢಞೕጄȐlearning standardsȑ฻ǶӵPISAኧ ᏢࣽύǴኧໆǵޜ໔ᆶ׎ᡏǵׯᡂᆶᜢ߯Ϸόዴۓ܄ࣁځ܌ۓကϐԛભໆЁǶ

ມǵʳЬाໆЁ!

ЬाໆЁࢂ᏾ӝԛભໆЁటෳໆϐଯ໘ޑᏢࣽૈΚȐનᎦȑǶӵPISAЬाෳ ໆϐ᎙᠐નᎦǵԾฅનᎦǵኧᏢનᎦࣁځ܌ۓကϐଯ໘ޑᏢࣽૈΚໆЁǴջࣁҁ ࣴز܌ॊϐЬाໆЁǶ

ୖǵʳଯ໘ቫޑຑໆࢎᄬ

ଯ໘ቫޑຑໆࢎᄬЬाх֖ٿ໘ቫ܈а΢ޑᏢࣽૈΚǴಃ΋ቫޑૈΚໆЁࣁ ԛભໆЁǴಃΒቫޑૈΚໆЁࣁЬाໆЁǴҁࣴزύۓက೭ᅿх֖ٿ໘ቫޑຑໆ ࢎᄬࣁଯ໘ቫޑຑໆࢎᄬǶ !

စǵʳଯ໘ቫ၂ᚒϸᔈ౛ፕኳԄ

ଯ໘ቫ၂ᚒϸᔈ౛ፕኳԄȐHigh-order IRT modelǴᙁᆀHO-IRTȑǴᆶଯ໘ ቫޑຑໆࢎᄬ࣬ӕǴх֖ٿ໘ቫޑૈΚໆЁǶ!

(14)

ᅿ՗ीБԄǶ

ഌǵʳୖኧ՗ीᆒྗࡋ

ୖኧ՗ीᆒྗࡋࢂࡰ՗ीᇤৡޑελǴҭջ՗ीᇤৡຫλǴ߾ж߄՗ी่݀ ຫྗዴǴҁࣴز٬Ҕ֡БਥᇤৡȐroot mean square error, RMSEȑբࣁຑ՗ࡰ኱Ƕ!

(15)

ಃΒക! Ў᝘௖૸

ҁࣴزЬा٬Ҕ PISA ϐຑໆࢎᄬ೛ीόӕޑ HO-IRT ኳԄǴϩձа PISA ܌ ٬Ҕϐ՗ीБԄ຾Չ՗ीᆶҁࣴز܌ගрϐֹ᏾՗ीБԄ຾Չ՗ीǶӢԜǴҁക ஒϩձ௖૸ PISA ϐኧᏢຑໆࢎᄬǵ၂ᚒϸᔈ౛ፕϷୖኧ՗ीݤǶ

ಃ΋࿯! PISA ϐኧᏢຑໆࢎᄬ

1999 ԃȨ୯ሞ࿶ᔮӝբᆶว৖ಔᙃȐThe Organization for Economic and Cooperation Development, OECDȑȩว৖ၠ୯ຑໆᏢғޑीฝǴԜीฝջᆀϐࣁȨ୯ ሞ܄ᏢғຑໆीฝǴPISAȩǶPISA ௢୏ኧԃϐࡕǴᙖҗၠ୯܄ࣴزࢎᄬޑЍ࡭Ǵ ӧ୯ሞςڀԖ࣬྽ޑቹៜΚȐ஭႒൤ǵЦШमǵֆችηǵڬЎ๭Ǵ2006ȑǶPISA ၮҔኧᏢનᎦޑཷۺٰඔॊᏢғගрǵှ،ϷှញӚԄӚኬ౐ੋډኧໆǵޜ໔ǵ ᐒ౗܈ࢂځдኧᏢཷۺޑୢᚒ௃ნਔǴૈԖਏ຾Չϩ݋ǵ௢౛аϷྎ೯ޑૈΚȐ݅ ྨ౺ǵቅဃ۸ǵ݅ન༾ǵ׵ཧǴ2008ȑǶPISA ϐຑໆࢎᄬЬाх֖ΟεӛࡋǺ௃ ნکે๎Ȑsituation and context, SCȑǵኧᏢᐕำȐmathematical process, MPȑǵа ϷኧᏢϣ৒Ȑmathematical content, MCȑǴ؂΋ӛࡋӚԖځ܌ឦϐηӛࡋȐ၁ـ߄ 2-1-1ȑǶ

(16)

߄2-1-1! PISAӚηӛࡋϐኧᏢຑໆࢎᄬ ຑໆࢎᄬ ჹᔈϐηӛࡋ ঁΓޑȐpersonalȑ ௲ػޑȐeducationalȑ ᙍ཰ޑȐoccupationalȑ ϦӅޑȐpublicȑ ௃ ნ ک ે ๎ ࣽᏢޑȐscientificȑ ኧໆȐquantityȑ

ޜ໔ᆶ׎ᡏ Ȑspace and shapeȑ

ׯᡂᆶᜢ߯Ȑchange and relationshipsȑ ኧ Ꮲ ϣ ৒ όዴۓ܄Ȑuncertaintyȑ ፄᇙဂಔȐreproduction clusterȑ ೱ่ဂಔȐconnection clusterȑ ૈ Κ ဂ ಔ ϸࡘဂಔȐreflection clusterȑ

ࡘԵϷ௢౛Ȑthinking and reasoningȑ ፕ᛾Ȑargumentationȑ

ྎ೯Ȑcommunicationȑ ࡌኳȐmodellingȑ

ᔕᚒϷှᚒȐproblem posing and solvingȑ ߄ቻȐrepresentationȑ

ၮҔ಄ဦǵ׎ԄϯϷࣽמޑᇟقϷၮᆉȐusing symbolic, formal and technical language and operationsȑ

ኧ Ꮲ ᐕ ำ ኧ Ꮲ ૈ Κ

٬ҔᇶշπڀȐuse of aids and toolsȑ

PISAޑҞޑࢂຑໆᏢғှ،੿ჴୢᚒޑૈΚǴ܌а PISA ۓကຑໆϣ఼฼ౣ ࢂа౜ຝᏢޑڗӛඔॊኧᏢޑཷۺǵ่ᄬ܈གྷݤǶᗨฅ܌఼ᇂޑϣ৒ёૈΨӕਔ р౜ӧځѬኧᏢຑໆ܈୯ሞኧᏢፐำǴՠ၀ڗӛዴߥຑໆޑขᗺکሦୱޑۓကࢂ ΋ठޑǶኧᏢख़ाཷۺёаԖ೚ӭǴଞჹኧᏢનᎦޑۓကԶقǴനख़ाޑԵቾࢂ ाྍܭኧᏢว৖ޑᐕўǵкϩᄆᡉኧᏢख़ाҁ፦ޑుࡋکቶࡋǵ٠ૈӝ౛֖ࡴ౜ ՉኧᏢፐำϐϣ৒Ƕ

(17)

ёޕ PISA ኧᏢࣽຑໆࢎᄬх֖ٿ໘ቫޑኧᏢૈΚǴಃ΋ቫޑૈΚໆЁȐԛભໆ Ёȑх֖ኧໆǵޜ໔ᆶ׎ᡏǵׯᡂᆶᜢ߯ǵόዴۓ܄ѤঁኧᏢૈΚǴಃΒቫޑૈ ΚໆЁȐЬाໆЁȑࣁኧᏢનᎦǴځ՗ीБݤ௦ҔӚ໘ቫϩ໒՗ीޑБݤ຾Չଯ ໘ቫૈΚໆЁϐ՗ीǺಃ΋ቫ٬Ҕ MIRT ϐ MRCMLM ຾ՉኧᏢࣽԛભໆЁ՗ी Ȑӵკ 2-1-2ȑǴಃΒቫ٬Ҕൂӛࡋ IRT ϐ Rasch ኳԄ຾ՉኧᏢࣽЬाໆЁ՗ीȐӵ კ 2-1-3ȑȐOECD, 2005ȑǶ კ2-1-1! PISA ኧᏢࣽຑໆࢎᄬ Item 1-1 Item 1-i Item 2-1 Item 2-j Item 3-1 Item 3-k ኧໆ Item 4-s Item 4-1 ޜ໔ᆶ׎ᡏ _ኧ Ꮲ ન Ꭶ! ׯᡂᆶᜢ߯ όዴۓ܄ ЬाໆЁ ԛભໆЁ

(18)

კ2-1-2! ӭӛࡋ IRT ኳԄ Item 1-1 Item 1-i Item 2-1 Item 2-j Item 3-1 Item 3-k Item 4-s Item 4-1 ኧ Ꮲ ન Ꭶ! Item 1-1 Item 1-i Item 2-1 Item 2-j Item 3-1 Item 3-k ኧໆ Item 4-s Item 4-1 ޜ໔ᆶ׎ᡏ ׯᡂᆶᜢ߯ όዴۓ܄

(19)

ಃΒ࿯! ၂ᚒϸᔈ౛ፕ

൘ǵൂӛࡋ IRT ϐ Rasch ኳԄ

ෳᡍ౛ፕࢂ΋ᅿှញෳᡍၗ਑໔ჴ᛾ᜢ߯ޑ౛ፕᏢᇥȐէ҇ჱǴ1992aǴ 1992bȑǴЬाϩԋٿεᜪǺ΋ࣁђڂෳᡍ౛ፕȐclassical test theory, CTTȑǴࢂа ੿ჴϩኧኳԄࣁࢎᄬǴځኧᏢኳԄᙁൂǴीϩ৒ܰԶቶڙ௦ҔǴՠࢂǴӧෳໆޑ ࠔ፦΢Ǵђڂෳᡍ౛ፕϝԖߚጕ܄ᆶኬҁ٩ᒘǵ၂ᚒ٩ᒘޑલᗺǹќ΋ᅿࣁ྽ж ෳᡍ౛ፕȐmodern test theoryȑǴࢂа၂ᚒϸᔈ౛ፕȐitem response theory, IRTȑ ࣁࢎᄬǴӧෳໆ΢Ԗၨӳޑ܄፦Ǵёၲډጕ܄ᆶ࠼ᢀ܄ޑा؃Ƕ

၂ᚒϸᔈ౛ፕࡌҥӧٿঁ୷ҁཷۺ΢Ǻ(1)ڙ၂ޣӧࢌ΋ෳᡍ၂ᚒ΢ޑ߄౜௃ ׎Ǵёҗ΋ಔӢનٰуаႣෳ܈ှញǴ೭ಔӢનћբወӧ੝፦Ȑlatent traitsȑ܈ ૈΚȐabilitiesȑǹ(2)ڙ၂ޣޑ߄౜௃׎ᆶځૈΚ໔ޑᜢ߯Ǵё೸ၸ΋చೱុ܄ሀ ቚޑڄኧٰуа၍ញǴԜڄኧᆀࣁ၂ᚒ੝ቻԔጕȐitem characteristic curve, ICCȑǶ ҺՖ΋చ၂ᚒ੝ቻԔጕж߄ڙ၂ޣเჹࢌ΋၂ᚒޑᐒ౗ǴࢂҗځૈΚک၂ᚒޑ੝ ܄܌Ӆӕ،ۓȐէ҇ჱǴ1992cȑǶฅԶǴा຾Չෳᡍၗ਑ϩ݋ਔǴIRTኳԄѸ໪ ಄ӝൂӛ܄Ȑunidimensionalityȑǵֽ೽ᐱҥ܄Ȑlocal independenceȑǵߚೲࡋ܄ ȐnonspeednessȑϷȨޕၰ-҅ዴȩଷ೛Ȑ“know-correct” assumptionȑѤ໨୷ҁޑ ଷ೛ȐWeiss & Yoes, 1991ȑǶ

΋ǵൂӛ܄Ǻࢌ΋ෳᡍѝଞჹൂ΋ૈΚ܈ወӧ੝፦຾ՉෳໆǶ Βǵֽ೽ᐱҥ܄Ǻڙ၂ޣӧόӕ၂ᚒ΢ޑբเϸᔈࢂϕ࣬ᐱҥޑǴΨ൩ࢂڙ၂ޣ ӧෳᡍ΢ࢌ΋ᚒޑբเϸᔈǴό཮ڙځѬ၂ᚒޑቹៜǶ Οǵߚೲࡋ܄Ǻࡼෳਔ໔όڙೲࡋޑज़ڋǴΨ൩ࢂڙ၂ޣޑԋ൩߄౜Ǵࢂҗወӧ ੝፦܈ૈΚ܌،ۓǴԶόࢂҗܭਔ໔ޑज़ڋ೷ԋ҂เֹ᏾Ǵቹៜځ߄౜Ƕ Ѥǵޕၰջ҅ዴǺऩڙ၂ޣޕၰࢌ΋၂ᚒޑ҅ዴเਢǴ߾཮เჹ၀၂ᚒǴΨ൩ࢂ ڙ၂ޣเᒱࢌ΋၂ᚒǴ߾߄Ңόޕၰ၀၂ᚒޑเਢǶ

(20)

၂ᚒϸᔈ౛ፕޑीϩБԄёϩࣁΒϡीϩᆶӭᗺीϩǴځኳԄԖൂୖኧჹኧ ኳԄȐone-parameter logistic modelȑǵᚈୖኧჹኧኳԄȐtwo-parameter logistic modelȑϷΟୖኧჹኧኳԄȐthree-parameter logistic modelȑǶаΠ໻ଞჹҁࣴز ܌٬ҔϐൂୖኧჹኧኳԄǴջ Rasch ኳԄ຾ՉϟಏǶ n i e e i i b b i 1,2,3,..., 1 ) ( P ₍ ₎ ) ( T T T Ȑ1ȑ ځύǴP_i(T)ǺૈΚࣁTϐڙ၂ޣǴӧಃiᚒเჹޑᐒ౗ i b Ǻಃiᚒޑ၂ᚒᜤࡋୖኧ nǺෳᡍߏࡋ

ມǵӭӛࡋ IRT ϐ MRCML ኳԄ

ӭӛࡋෳᡍёаϩࣁᚒ໔ӭӛࡋෳᡍȐbetween-item multidimensional testȑᆶ ᚒϣӭӛࡋෳᡍȐwithin-item multidimensional testȑٿᅿȐAdams, Wilson & Wang, 1997ȑǶऩӧෳᡍ္ޑ؂ঁ၂ᚒѝෳໆ΋ᅿૈΚǴջൂӛࡋޑ၂ᚒǴऩ᏾ҽෳᡍ х֖ӭঁෳໆόӕૈΚޑൂӛࡋ၂ᚒǴ߾ᆀԜෳᡍࣁᚒ໔ӭӛࡋෳᡍȐӵკ 2-2-1ȑǹऩӧෳᡍ္ޑ؂ঁ၂ᚒόѝෳໆൂ΋ᅿૈΚǴΨ൩ࢂ၂ᚒϣ൩х֖ӭӛ ࡋǴᆀԜෳᡍࣁᚒϣӭӛࡋෳᡍȐӵკ 2-2-2ȑǶ

(21)

კ2-2-1! ᚒ໔ӭӛࡋෳᡍ

კ2-2-2! ᚒϣӭӛࡋෳᡍ

Ҟ߻தـޑӭӛࡋ၂ᚒϸᔈ౛ፕኳԄεӭࢂൂӛࡋ၂ᚒϸᔈ౛ፕኳԄޑ़ ғኳԄǴMRCMLMջࢂۯ՜RaschኳԄԶԋϐӭӛࡋIRTኳԄȐHoskens, & De BoeckǴ1997ǹWang, Wilson, & ChengǴ2000ǹWilson, & AdamsǴ1995ȑǴځኳ ԄۓကӵΠǺ

¦

i K k ik ik ik ik ik X 1 ) ' ' exp( ) ' exp( ) , ; 1 ( P ȟ a ș b ȟ a' ș b ș | ȟ B A, Ȑ2ȑ Item1 Item2 Item3 Item4 Item5 1 T 2 T Item1 Item2 Item3 Item4 Item5 1 T 2 T

(22)

ځύǴX Ǻڙ၂ޣϐ଺เϸᔈಔࠠ_ik i K Ǻಃi ၂ᚒޑीϩᜪձኧ șǺڙ၂ޣޑૈΚୖኧંତȐӭӛࡋૈΚȑ ȟǺ၂ᚒୖኧӛໆ ik a Ǻಃi ᚒύಃkঁϸᔈᜪձޑ೛ीӛໆȐdesign vectorȑ ik b Ǻಃi ᚒӧಃkঁϸᔈᜪձ΢ޑीϩӛໆȐscoring vectorȑ A Ǻ᏾ҽෳᡍޑ೛ीંତȐdesign matrixȑ B Ǻ᏾ҽෳᡍޑीϩંତȐscoring matrixȑ MRCMLMନΑё௢ፕൂӛࡋӭᗺीϩޑෳᡍၗ਑ǵჹܭෳໆኳԄऩӸӧߚ ᐱҥޑ௃ݩёаᙖҗᚒಔམଛ၀ෳໆኳԄٰ଺ၗ਑ϩ݋ϷֹऍޑኧᏢ܄፦฻ؼ ӳޑឦ܄ѦǴ׳ёаೀ౛ӭӛࡋӭᗺीϩޑෳᡍၗ਑ǴځᔈҔጄൎىа఼ᇂӛ ࡋǵीϩ׎ԄόӕᡂϯޑෳᡍᜪࠠǴՠ MRCMLM ϝԖځज़ڋǴѝёᔈҔܭൂ΋ ໘ቫၗ਑ϐ௢ፕǴӵΠࣁᔈҔ MRCMLM բࣁෳໆኳԄޑጄٯǴឦܭൂ΋໘ቫϐ ၗ਑ǴҗᏢғϐբเϸᔈၗ਑XǴ௢ፕԿಃ΋ቫట௢ፕϐૈΚॶT Ƕ ȜጄٯȝǺ Ꮲғֹԋࢌ໨բ཰Ѹ໪ڀഢٿᅿૈΚT1ǵT2Ǵځբ཰ᆶૈΚޑᜢ߯ӵΠკ 2-2-3܌ҢǺ კ2-2-3! բ཰ᆶૈΚޑᜢ߯ X 1 T 2 T

(23)

җ஑ৎۓကीϩંତ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¹ · ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ © § 2 1 1 1 0 1 2 0 1 0 0 0 B Ǵᆶ೛ीંତ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¹ · ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ © § 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 A Ƕ! ᙖҗ MRCMLM ёаीᆉрΠӈӚᅿளϩᜪࠠр౜ޑᐒ౗ॶǺ

¦

5 0 5 0 2 1 )) (( exp( 1 ) 0 ( k j j )-į ș ș X P

¦

5 0 5 0 2 1 1 2 )) ) (( exp( ) exp( ) 1 ( k j j -į ș ș X P T G

¦

5 0 5 0 2 1 2 2 ) ) ) (( exp( ) 2 exp( ) 2 ( k j j -į ș ș X P T G

¦

5 0 5 0 2 1 3 1 )) ) (( exp( ) exp( ) 3 ( k j j -į ș ș X P T G

¦

5 0 5 0 2 1 4 2 1 )) ) (( exp( ) exp( ) 4 ( k j j -į ș ș X P T T G

¦

5 0 5 0 2 1 5 2 1 )) ) (( exp( ) 2 exp( ) 5 ( k j j -į ș ș X P T T G ඤقϐǴ،ۓीϩંତǵ೛ीંତࡕǴёҗMRCMLM՗ᆉᏢғޑૈΚǶӧ ҁࣴز٬ҔACER ConQuest 2.0೬ᡏٰ຾Չୖኧ՗ीǴځ՗ीБԄ٬Ҕᜐሞനε ཷ՟ݤȐmarginal maximum likelihood estimation, MMLEȑ՗ी၂ᚒୖኧǴ٬Ҕය ఈࡕᡍݤȐexpected a posteriori, EAPȑ՗ीૈΚॶǴஒܭಃΟ࿯ϟಏMMLEݤǵ EAPݤǶ

(24)

ୖǵ΋Ӣη HO-IRT ኳԄ

ᒿ ๱ ෳ ᡍ ׎ Ԅ ޑ ׯ ᡂ Ϸ ሡ ؃ ໆ ޑ ז ೲ ቚ у Ǵ ε ࠠ ෳ ᡍ Ȑ large-scale assessmentsȑޑ᝼ᚒቶݱڙډᢋҞǶҞ߻୯ሞ΢ၨ๱Ӝϐεࠠෳᡍࣣឦܭଯ໘ቫ ෳໆኳԄǴऩ٬Ҕ໺಍ൂӛࡋෳᡍ౛ፕȐconventional unidimensional item response theory, CU-IRTȑǴёૈ཮Ӣၴङځଷ೛Զ٬ଯ໘ቫૈΚ՗ीόྗዴǴ܈྽ԛભໆ Ё܌ჹᔈޑᚒኧၨϿਔǴᏤठ՗ीਏ݀όё᎞ǶԖ᠘ܭԜǴSongȐ2007ȑගр΋ Ӣηଯ໘ቫ IRTȐone-factor higher-order IRTȑኳԄǴԜኳԄӕਔх֖ၨଯ໘ޑૈ ΚȐoverall abilityȑᆶၨե໘ޑૈΚȐdomain abilityȑǴջӕਔх֖ЬाໆЁᆶԛ ભໆЁǶHO-IRT ኳԄ఼ᇂΑ CU-IRTǴ܌а CU-IRT ࢂ HO-IRT ޑ΋ঁ੝ٯǶ٬ Ҕӧ໘ቫنМࢎᄬΠޑ MCMC ݤȐMarkov Chain Monte Carlo methodȑӕਔ՗ी ЬाໆЁǵԛભໆЁϷ࣬ᜢ߯ኧǴਥᏵ SongȐ2007ȑኳᔕࣴزёޕǴ྽ԛભໆЁ ϐ໔ό࣬٩ਔǴHO-IRT ՗ीЬाໆЁޑ่݀཮࣬՟ܭ CU-IRTǹԶ྽ૈΚ໔۶Ԝ ࣬٩ਔǴHO-IRT ՗ीԛભໆЁ཮К CU-IRT ׳ྗዴǶ ΋ǵHO-IRT ኳԄϟಏ HO-IRTኳԄύǴ΋ෳᡍЬाёᢀჸӭঁൂӛࡋޑηෳᡍȐsubtestȑǴջԛભ ໆЁ (d) i T Ǵ (d) i T ߄Ңಃi Տڙ၂ޣӧԛભໆЁ d ޑ߄౜ǴځύǴd 1,2,3,...,DǶ྽ όӕԛભໆЁࣣෳໆ࣬ӕޑૈΚਔǴ߾᏾ҽෳᡍ೏ᇡࣁࢂൂӛࡋޑෳᡍǶऩόӕ ԛભໆЁ໔ԖᜢᖄǴ߾཮ᙖҗ΋ଯ໘ቫૈΚT_iٰೱௗ೭٤ԛભໆЁǴT_iࣁಃi Տ ڙ၂ޣӧЬाໆЁޑ߄౜ǴΨ൩ࢂԛભໆЁࢂૈΚໆЁޑ΋ጕ܄ڄኧǴ , ) ( ) ( id i d d i O T H T ځύǴ_O(d) ࣁ଑ᘜୖኧǴH_idࣁᇤৡ໨Ƕଷ೛H_idܺவதᄊϩଛǴځѳ֡ኧࣁ 0Ǵᡂ ౦ኧࣁ 1 _O(d)2 ǴЪ| ( )| 1 d d O ǴਥᏵ೭٤ଷ೛ёளޕ (d) i T ޑϩଛᆶT_i࣬՟Ǵឦܭ኱

(25)

࣬ᐱҥǶ (_d) O ё߄ҢЬाໆЁᆶԛભໆЁ໔ޑ࣬ᜢǴԶԛભໆЁd ᆶ 'd ໔ޑ࣬ᜢ ߾ࣁ (_d) (_d') O O u Ƕᗨฅ (_d) O ёࣁॄኧǴՠӧ௲ػෳᡍޑᔈҔ΢ǴЬाໆЁϷԛભໆ Ё໔ࣣࣁ҅࣬ᜢǴࡺӧ՗ीਔǴѝԵቾ0dO(d) d1Ƕ კ 2-2-4 ࣁ HO-IRT ޑኳԄკǴಃ΋ቫ߄Ңಃi Տڙ၂ޣӧԛભໆЁ d ύޑಃ th j ၂ᚒϐϸᔈ௃׎ (d) ij X ǴಃΒቫ߄Ңڙ၂ޣޑϸᔈ೸ၸ IRT ኳԄύޑ၂ᚒୖኧ ) (d j E ೱ่ډԛભໆЁǴಃΟቫ߄Ңڙ၂ޣޑԛભໆЁϩኧ೸ၸ଑ᘜୖኧ (d) O ೱ่ ډ࣬ჹᔈϐЬाໆЁT_iǶ კ2-2-4! HO-IRT ኳԄᔈҔܭ΋ঁ D ᆢࡋޑෳᡍ Βǵ՗ीБݤ ᗨฅ HO-IRT ёаׯ๓၂ᚒୖኧϐ՗ीǴӧ SongȐ2007ȑ໻௖૸ڙ၂ޣޑૈ Κ߄౜ǴӢԜǴଷ೛၂ᚒୖኧςޕǶନΑӕਔ՗ीЬाໆЁᆶԛભໆЁѦǴᗋሡ ) I ( O (II) O (D) O Ti ) I ( i T (II) i T (D) i T ) I ( ij X (II) ij X (D) ij X ) I ( j E (II) j E (D) j E ᢀჸᡂ໨а༝୮߄Ңǹ ڰۓᡂ໨аБਣ߄Ңǹ ځдᡂ໨߄Ңࡑࣁ՗ीǶ

(26)

՗ी଑ᘜୖኧ_O(d)_{Ǵӧ໘ቫنМࢎᄬΠǴԜኳԄё߄ҢࣁǺ} , ) 1 , 0 ( N ~ i T , ) 1 , 1 ( U ~ ) ( d O Ъ Ƕ ) 1 , ( N ~ , | ( ) ( ) ( )2 ) ( d i d d i d i T O O T O T җԜёޕǴԛભໆЁޑᜐሞϩଛё߄Ңࣁ኱ྗதᄊϩଛȐT_i(d) ~N(0,1)ȑǶ ऩ଑ᘜୖኧςޕǴ߾ёճҔ΋૓ޑ՗ीݤȐٯӵǺMLEȑ٬ЬाໆЁᆶԛભ ໆЁϐ՗ी׳ᙁܰǶฅԶǴӕਔ՗ी଑ᘜୖኧᆶૈΚॶ཮٬՗ीၸำፄᚇϯǴӢ ԜǴ٬Ҕ MCMC ݤ຾Չୖኧ՗ीǶ ଷ೛ș* {ș(1),ș(2),...,ș(k)}ǴЪP(X|T*)߄Ңཷ՟ڄኧǴӚୖኧޑࡕᡍᐒ౗ϩ Ѳ߄Ңࣁ ) ( P ) | X ( P ) X | ( P ș,ș*,Ȝ v ș,ș*,Ȝ ș,ș*,Ȝ ΰ3α ) )P( , | * )P( * | P(X ș ș ș Ȝ ș,Ȝ ) )P( )P( | * )P( * | P(X ș ș ș,Ȝ ș Ȝ șǵ *ș Ϸ Ȝ ޑֹӄచҹϩଛȐfull conditional distributionȑࣁ

) ( P ) | * ( P ) | X ( P ) , X | ( P ș ș*,Ȝ v ș,ș*,Ȝ ș ș,Ȝ ș|Ȝ Ȑ4ȑ ) )P( | * )P( * | P(X ș ș ș,Ȝ ș,Ȝ ; ) )P( | * P(ș ș,Ȝ ș ) | * ( P ) , * | X ( P ) , X | * ( P ș ș,Ȝ v ș ș,Ȝ ș ș,Ȝ Ȑ5ȑ ; ) | * )P( * | P(X ș ș ș,Ȝ ) ( P ) | * ( P ) * | X ( P ) * , X | ( P Ȝ ș,ș v Ȝ,ș,ș ș ș,Ȝ Ȝ|ș Ȑ6ȑ ) )P( P(ș*|ș,Ȝ Ȝ v ྽

N i i i i*,Ȝ ș ș ș*,Ȝ ș|X, ) P( |X , ) ( P ǴЪჹ؂ঁڙ၂ޣޑֹӄచҹϩଛё߄Ңࣁ

(27)

, c , 1 c ~ ) , X | ( P _i ₁ ₂ _¸¸ ¹ · ¨ ¨ © §

¦

Dd (d) (d) i (d) i i Ȝ ș Ȝ N *,Ȝ ș ș ΰ7α ځύǴ

_¦

D_d _(d) (d) Ȝ Ȝ 1 2 2 1 1 c Ƕᗨฅಃi Տڙ၂ޣϐЬाໆЁϐܜኬёҗ฻ԄȐ7ȑύ ᙖҗதᄊϩଛܜڗрǴՠș_i(d)کȜ(d)คݤᙖҗ฻ԄȐ5ȑǵȐ6ȑޑֹӄచҹϩଛύڗ рǴӢԜǴ೭٤ୖኧࢂ೸ၸ MH ݤȐMetropolis-Hastings algorithmȑٰڗኬǶҗ ܭ MH ݤࢂ΋ԛଞჹ܌Ԗୖኧ຾Չӕਔ՗ीǴऩܜኬၸำύǴԖ΋ୖኧ՗ीၨৡ ਔǴ߾཮ቹៜ܌Ԗୖኧ՗ीԶᜤаӕਔ՗ीрၨӳޑ่݀ǴӢԜǴҁࣴز௦Ҕ Gibbs samplingݤٰ຾Չୖኧ՗ीǴځ՗ीБݤஒܭಃΟ࿯၁ಒϟಏǶ

ಃΟ࿯! ୖኧ՗ीݤ

൘ǵACER ConQuest 2.0 ୖኧ՗ीݤ

ACER ConQuest 2.0٬ҔMMLEݤ՗ी၂ᚒୖኧǴ՗ीૈΚୖኧޑБݤԖന εཷ՟՗ीݤȐMLEȑǵයఈࡕᡍ՗ीݤȐEAPȑǵу៾ཷ՟՗ीݤȐWLEȑǵ ወӧ՗ीݤȐlatentȑѤᅿȐWu, Adams & Wilson, 1998ȑǴҗܭEAPݤޑ֡Бᇤ ৡȐmean square errorȑၨλȐBock & Mislevy, 1982ȑǴࡺҁࣴزӧૈΚ՗ी΢٬ ҔEAPݤǶ ΋ǵᜐሞനεཷ՟ݤ җ฻ԄȐ2ȑёޕ )] A B exp[( ) ( ) ; x ( ȟ|ș ș,ȟ ș ȟ f < Ȑ8ȑ ځύǴ

¦

ȍ z 1 ]} A exp[(B { ) ȥ(ș,ȟ ș ȟ ΰ9α ೯தӧൂӛࡋޑ၂ᚒϸᔈኳԄύǴᏢғܜኬޑ҆ဂ೯தٰԾதᄊϩଛǴѳ֡ ኧࣁ P Ǵᡂ౦ኧࣁ_V2 Ƕᐒ౗ஏࡋڄኧ߄ҢࣁǺ

(28)

» ¼ º « ¬ ª { ₂ 2 2 2 2 ) ( exp 2 1 ) , ; ( ) ; ( V P T SV V P T D T _T T f f Ȑ10ȑ ܈ E P T Ȑ11ȑ ځύǴE~N(0,V2)ǶऩᏢғܜኬޑ҆ဂٰԾӭᡂໆதᄊϩଛǴ߾ᐒ౗ஏࡋڄ ኧ߄ҢࣁǺ »¼ º «¬ ª 6 6 6 ) ȖW ( )' ȖW ( 2 1 exp | | ) 2 ( ) , Ȗ , W ; ( 2 1 1 2 n n n n d n n fT T S T T Ȑ12ȑ ځύǴȖࢂu×dޑ଑ᘜ߯ኧંତǴ6ࢂd×dޑӅᡂ౦ኧંତǴЪW_nࢂu×1ޑڰ ۓᡂኧӛໆǶӵ݀฻ԄȐ12ȑҔٰ྽բ҆ဂϩଛǴٗሶୖኧȖǵ6Ϸ[ ஒ೏՗ीǶ ځ՗ीБݤࢂ٬ҔMLEݤٰ՗ीୖኧȖǵ6Ϸ[ Ǵ่ӝచҹ၂ᚒϸᔈኳԄȐ10ȑ ᆶ҆ဂϩଛȐ12ȑǴёளᜐሞ၂ᚒϸᔈኳԄǺ

³

¦ ¦ T T T J T T [ J [ f x f d x fx( ; , , ) x( ; | ) ( ; , ) Ȑ13ȑ ځཷ՟ڄኧࣁǺ

¦ / N n x n x f 1 ) , , ; ( [ J Ȑ14ȑ ځύǴNࢂᏢғኬҁޑᕴኧǶ ؂ঁୖኧ໔ޑᜢ߯ᆶᜐሞࡕᡍᐒ౗ϐۓကӵΠǺ ) , Ȗ , , W ; x ( ) , Ȗ , W ; ( ) | ; ( ) x | , Ȗ , , W ; ( 6 6 6 [ T T [ [ T T T n n x n n n n x n n n _f f x f h Ȑ15ȑ

¦

³

» ¼ º « ¬ ª 6 N n 1 n _nE n h n n xn n z(z| ) ( ;Y , ,Ȗ, | )d 0 x ' A T T T [ T T Ȑ16ȑ 1 ' W W ' W Ȗˆ ¸ · ¨ § ¸ · ¨ §

¦

N ș N Ȑ17ȑ

(29)

Ъ n n n n n n N n n n d h N _n T T T T [ T T ) x | , Ȗ , , Y ; ( )' ȖW ( ) ȖW ( 1 ˆ 1 6 6

_{¦ ³}

Ȑ18ȑ ځύǴ

¦

: < z n n n z E (z|T ) (T ,[) zexp[z'(bT A[)] Ȑ19ȑ n n n n n n n h T [ T T T T T( ;Y , ,Ȗ, |x )d

³

6 Ȑ20ȑ ฻ԄȐ16ȑǵȐ17ȑکȐ18ȑࢂճҔEMᄽᆉݤٰ؃ှǶ฻ԄȐ16ȑǵȐ17ȑک Ȑ18ȑᑈϩޑ೽ҽࢂ೸ၸᆾӦьᛥݤٰ؃߈՟ॶǶ೭္ۓက4_qࢂDᆢޑӛໆȐᆀ ࣁ࿯ᗺȑǴq 1,...,Qǹଞჹ؂ঁ࿯ᗺۓက΋៾ख़W_q(Ȗ,6)Ǵ߾ᜐሞ၂ᚒϸᔈᐒ౗ Ȑ13ȑޑ߈՟ॶࣁ ; ) , Ȗ ( ) | ; x ( ) , , ; ( 1 6 4 6

_¦

_p _p Q p x x x f W f [ J [ Ȑ21ȑ ЪᜐሞࡕᡍȐ10ȑޑ߈՟ॶࣁ

¦

4 6 6 4 6 4 4 Q p x n p p q q n x n n q f f h 1 ) , Ȗ ( W ) | ; x ( ) , Ȗ ( W ) | ; x ( ) x | , Ȗ , , W ; ( [ [ [ Ȑ22ȑ ځύǴq 1,...,QǶ EMᄽᆉݤޑ؁ᡯӵΠǺ ؁ᡯ΋Ǻа_J(t) ǵ (t) 6 ٰ೛ۓ΋ಔ࿯ᗺᆶ៾ख़Ǵ_J(t) ǵ (t) 6 ࢂಃt ԛॏжJ ǵ 6 ޑ՗ ीॶǶ ؁ᡯΒǺ๏ۓx_nǴճҔ

¦

4 6 6 4 6 4 4 Q p t t p p t n x t t q q t n x n t t t n q f f h 1 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , Ȗ ( W ) | ; x ( ) , Ȗ ( W ) | ; x ( ) x | , Ȗ , , W ; ( [ [ [ Ȑ23ȑ

(30)

[ǵJ Ϸ6 ޑ՗ीॶǶ ؁ᡯΟǺճҔФႥɡऊՕහݤȐNewton-Raphson methodȑှ_[(t1)

¦

4 »¼ º «¬ ª 6 4 4 N n Q r n t t t n r r n E h A 1 1 ) ( ) ( ) ( z(z| ) ( ;W , ,Ȗ , |x ) 0 x ' [ Ȑ24ȑ ؁ᡯѤǺճҔ 1 1 1 ) 1 ( ' W W ' W Ȗ ¸ ¹ · ¨ © § ¸ ¹ · ¨ © § ₄

¦

N n n n N n n n t _Ȑ25ȑ ) x | , Ȗ , , Y ; ( ' ) W Ȗ )( W Ȗ ( 1 () () ( ) 1 1 ) 1 ( ) 1 ( ) 1 ( n t t t n r N n Q r n t r n t r t _h N 4 4 4 6 6 ₄

¦¦

[ Ȑ26ȑ ՗ीJ(t1)ǵ ( 1) 6t ǴځύǴ . ) x | , Ȗ , , W ; ( 1 ) ( ) ( ) (

¦

4 4 4 6 4 Q r n t t t n n r n h [ Ȑ27ȑ ؁ᡯϖǺӣډ؁ᡯ΋Ƕ ख़ፄ΢ॊ؁ᡯǴޔډኧॶԏᔙࣁЗǶ Βǵයఈࡕᡍݤ ᜐሞ၂ᚒϸᔈኳԄȐ12ȑόх֖ወӧૈΚॶT_nǴӢԜคݤ຾ՉወӧૈΚॶT_n ϐ՗ीǴACER ConQuest 2.0 ගٮޑ EAP ૈΚ՗ीݤჹಃn ঁᏢғޑૈΚ՗ीϦ ԄࣁǺ ) | ˆ , ˆ , ˆ , ; ( 1 r n n Q r r EAP n

¦

4 h4 4 W [ J 6 x T Ȑ28ȑ

ມǵWinBUGS ୖኧ՗ीݤ

౜ϞIRTኳԄޑว৖ཇٰཇፄᚇǴځ՗ीኳԄ࣬྽ڙख़ຎǴऩ٬Ҕ໺಍ෳᡍ ౛ፕȐӵђڂෳᡍ౛ፕ܈ൂୖኧ၂ᚒϸᔈ౛ፕȑ຾Չϩ݋Ǵёૈ཮೷ԋෳໆ่݀ ࠼ᢀ܄ϷёКၨ܄όىǶ΋૓ၨፄᚇޑIRTኳԄϐୖኧ՗ीࢂMMLE/EMȐBock &

(31)

Aitkin, 1981ȑǴՠ྽ኳԄཇٰཇፄᚇਔǴEMᄽᆉݤஒᜤаޔௗᔈҔǶMCMCݤࢂ ӧӭᡂໆኳԄύኳᔕᒿᐒܜኬϐБݤǴόӕܭEMᄽᆉݤǴӢMCMCϐीᆉၸำ ό཮౐ੋᑈϩ܈༾ϩǴࡺёᙁൂӦ೏ᔈҔȐPatz & Junker, 1999ȑǶ

MCMCࢂ೸ၸӭԛޑख़ፄሀ଑ܜኬǴࡌᄬрଭёϻ᜘ȐMarkov chainȑǴ຾ Զ؃ள΋ѳᛙϩଛǴջࢂنМࢎᄬΠޑࡕᡍȐposteriorȑϩଛǴᙖҗଭёϻ᜘ύ ޑᒿᐒᡂኧёΑှᡂኧޑ੝፦ǶMCMCӧ಍ी΢ᔈҔޑጄൎߚதቶݱǴࡌᄬଭё ϻ᜘ޑБݤҭԖࡐӭǴаΠ໻ଞჹҁࣴز܌٬ҔϐWinBUGS೬ᡏύ܌٬ҔޑGibbs samplingݤ଺ϟಏǶ ΋ǵଭёϻ᜘ ଷ೛ᒿᐒᡂኧX_n,nt0ࣁ΋ଭёϻ᜘Ǵ೯தX_nޑёૈ໣ӝS ᆀࣁX_nޑރᄊ ޜ໔Ȑstate spaceȑǴځύ؂΋ঁᒿᐒᡂኧ܌วғޑᐒ౗೿ѝک߻΋ঁᒿᐒᡂኧԖ ᜢǴΨ൩ࢂX ࢂவచҹᐒ౗ϩଛn1 P(Xn1|Xn)ύౢғǴځύP(.|.)ᆀࣁଭёϻ᜘ޑ ᙯ࿼ਡȐtransition kernelȑǶԶ܌ᒏѳᛙޑଭёϻ᜘൩ࢂP(X_n₁|X_n)ύޑᙯඤᐒ ౗کރᄊԖᜢǴՠکਔ໔n คᜢǶ ӧIRTኳԄύǴଷ೛ሡा௢ፕTǵb ٿୖኧǴ߾ଭёϻ᜘ޑᙯ࿼ਡࣁǺ )] , ( | ) , ( [ P )] , ( ), , [( 0 b0 1 b1 X ₁ 1 b1 X 0 b0 t T T n T n T Ȑ29ȑ ᒿ๱n ቚуǴP(X_n₁| X_n)཮ԏᔙډѳᛙϩଛS(T,b)ǴӧنМࢎᄬύǴයఈ೸ၸଭ ёϻ᜘ٰۓကࡕᡍϩଛP(T,b|X)ࣁS(T,b)ǴԶѳᛙϩଛࣁǺ ) , ( ) , ( ) , ( )] , ( ), , [( 0 0 1 1 , 0 0 1 1 0 0 _b _b _b _d _b _b t b T T S T T S T T

³

Ȑ30ȑ ځύǴS(T1,b1)ࣁࡕᡍϩଛǶҗԜёޕǴѝाפډଭёϻ᜘ޑࡌᄬБݤǴջё௢ ᆉрයఈ؃ளޑࡕᡍϩଛǶGibbs samplingջࢂࡌᄬଭёϻ᜘ޑБݤϐ΃Ƕ ΒǵGibbs sampling

(32)

ଷ೛S(T,b) P(T,b| X)ǴЪୖኧT ǵb ֹӄచҹᐱҥǴҗԄηȐ30ȑǴёஒ ᙯ࿼ਡϦԄۓကӵΠȐGeman & Geman, 1984ȑǺ

)] , | ( P ) , | P( )] , ( ), , [( 0 b0 1 b1 1 b0 X b1 1 X t T T T T Ȑ31ȑ

ځύǴP(T |b,X)کP(b|T,X)ᆀࣁֹӄచҹϩଛȐfull conditional distributionȑǶ Gibbs samplingޑᄽᆉ؁ᡯӵΠǺ Ȑ΋ȑ๏ۓ܌ԖୖኧଆۈॶǺ

0 0

(33)

ಃΟക! ࣴزБݤ!

ҁࣴزۯ՜ SongȐ2007ȑ٬Ҕϐ΋Ӣη HO-IRT ኳԄว৖рΒӢη HO-IRT ኳԄǴ௖૸όӕޑ HO-IRT ኳԄჹ՗ीᆒྗࡋޑቹៜǶҁകϩࣁѤ࿯Ǵϩձϟಏ ϟಏΒӢη HO-IRT ኳԄǵࣴز೛ीǵຑ՗ࡰ኱ϷࣴزπڀǶ

ಃ΋࿯! ΒӢη HO-IRT ኳԄ

ҁࣴزࣁ௖૸ӧЬाໆЁ΢ӭу΋ଯ໘ૈΚॶჹځ՗ीᆒྗࡋޑቹៜǴࡺว ৖рΒӢη HO-IRT ኳԄȐӵკ 3-1-1ȑǶL ࢂڙ၂ޣӧԛભໆЁ_t t ޑૈΚॶǴ T ,..., 1 t ǴҗόӕޑԛભໆЁёෳໆр 2 ঁၨଯ໘ޑЬाໆЁϐૈΚॶǴЬाໆ ЁᆶԛભໆЁ໔ࣁጕ܄ᜢ߯Ǵ

¦

u 2 1 s st s t t H L O H ځύǴO_stࣁወӧ଑ᘜୖኧǴH_tࣁᇤৡ໨Ǵଷ೛H_tܺவதᄊϩଛǴځѳ֡ኧࣁ 0Ǵ ᡂ౦ኧࣁ

_¦

2 1 2 1 s st O ǴЪ0dO_st d1ǶਥᏵ೭٤ଷ೛ёளޕL_tޑϩଛ࣬՟ܭH_sǴឦ ܭ኱ྗதᄊϩଛ N(0,1)Ƕ

(34)

კ3-1-1! ΒӢη HO-IRT ኳԄ

ಃΒ࿯! ࣴز೛ी

ҁ࿯ϩࣁΟ೽ҽǴ२Ӄϟಏࣴز؁ᡯǴځԛϟಏҁࣴزϐᡂ໨೛ीǴനࡕϟ ಏࣴزำׇǶ

൘ǵࣴز؁ᡯ

ҁࣴز٬ҔPISAϐຑໆࢎᄬٰ೛ीόӕޑHO-IRTኳԄǴ೸ၸኳᔕࣴزБԄ а௖૸ֹ᏾ᆶϩ໒՗ीБݤǴаϷόӕ೬ᡏϐ՗ीਏ݀ǶځύǴ٬ҔACER ConQuest 2.0೬ᡏ຾Չൂ΋໘ቫȐӚ໘ቫϩ໒՗ीȑϐ՗ीǴ٬ҔWinBUGS೬ᡏ ຾Չൂ΋໘ቫᆶٿ໘ቫȐֹ᏾՗ीȑϐ՗ीǴ٠аRMSEբࣁຑ՗ࡰ኱Ƕკ3-2-1 ࣁҁࣴزϐࣴزࢬำკǶ X1 X2 X3 XT L1 L3 L2 LT H1 H2 11 O 12 O T 1 O 13 O 21 O 22 O 23 O T 2 O

(35)

კ 3-2-1! ࣴزࢬำკ

ມǵᡂ໨೛ी

ҁࣴزኳᔕၗ਑ᡂ໨ᆶ՗ी೬ᡏ՗ीБԄǵኳԄ೛ीӵΠǺ ΋ǵʳΓኧ/ᚒኧǺ1000 Γ/40 ᚒ ҁࣴزϐΓኧ೛ۓа΋૓ୖኧ՗ी೬ᡏ܌ሡޑኬҁεऊа1000Γࣁྗ߾Ƕᚒ ኧ೛ۓа΋૓ෳᡍߏࡋ30~40ᚒࣁྗ߾Ǵ٠ଛӝځд೛ीǴࡺᕴᚒኧ೛ۓࣁ40ᚒǶ Βǵʳीϩ/ෳᡍࠠᄊǺΒϡीϩ/ᚒ໔ӭӛࡋ ӭϡຑໆࣣឦܭӭӛࡋෳᡍǴฅӭӛࡋෳᡍЬाϩԋᚒ໔ӭӛࡋෳᡍϷᚒϣ ӭӛࡋෳᡍٿᅿȐAdams, Wilson, & Wang, 1997ȑǴीϩࠠᄊёϩࣁΒϡीϩᆶӭ ᗺ૶ϩǶӧҁࣴزѝଞჹΒϡीϩޑᚒ໔ӭӛࡋٰ຾Չ௖૸Ƕ ΟǵʳHO-IRT ኳԄǺ Ȑ΋ȑӵკ 3-2-2ǴЬाໆЁኧࣁ 1ǴԛભໆЁኧࣁ 2ǴځύǴ଑ᘜୖኧOϩձ೛ ᔕۓࣴزЬᚒ Ў᝘ᇆ໣ᆶ௖૸ ኗቪࣴزൔ֋ ኳᔕڙ၂ޣϐЬाૈΚǵԛाૈΚϷ၂ᚒୖኧǴ٠ኳᔕբเϸᔈ ճҔ ConQuest ຾Չ ୖኧ՗ी ୖኧ՗ीਏ݀ຑ՗ ճҔ WinBugs ຾Չ ୖኧ՗ी

(36)

ीࣁO₁₁=0.8ǵO₁₂=0.9ǴϷO₁₁=0.8ǵO₁₂=0.2ǹӧҁࣴزஒа H1L2-1 Ϸ H1L2-2 ٰ߄Ң೭ٿᅿ೛ीǶ ȐΒȑӵკ 3-2-5ǴЬाໆЁኧࣁ 1ǴԛભໆЁኧࣁ 4ǴځύǴ଑ᘜୖኧOϩձ೛ ीࣁO11=0.9ǵO12=0.8ǵO13=0.7ǵO14=0.6ǴϷO11=0.9ǵO12=0.8ǵO13=0.5ǵ 14 O =0.2ǹӧҁࣴزஒа H1L4-1 Ϸ H1L4-2 ٰ߄Ң೭ٿᅿ೛ीǶ ȐΟȑӵკ 3-2-8ǴЬाໆЁኧࣁ 2ǴԛભໆЁኧࣁ 4ǴځύǴ଑ᘜୖኧOϩձ೛ ीࣁO11=0.9ǵO12=0.8ǵO13=0.5ǵO14=0.2ǵO21=0.5ǵO22=0.8ǹӧҁࣴزஒ а H2L4 ٰ߄ҢԜ೛ीǶ ѤǵʳኳᔕԛኧǺ50 ԛ ҁࣴزϐኳᔕԛኧ೛ۓа΋૓ኳᔕࣴزϐԛኧ೛ۓࣁྗ߾Ƕ ϖǵʳа PISA ܌٬Ҕϐ՗ीБԄ຾Չ՗ीǺ

PISA٬Ҕൂӛࡋ IRT ϐ Rasch ኳԄ຾ՉЬाໆЁȐHȑϐ՗ीǹ٬Ҕ MIRT ϐ MRCMLM ຾ՉԛભໆЁȐLȑϐ՗ीǴ٠٬Ҕ ACER ConQuest 2.0 ೬ᡏ຾Չ ՗ीǴճҔനεཷ՟ݤ(maximum likelihood method, MLE/EM)՗ीᜤࡋୖኧǴճ ҔයఈࡕᡍݤȐexpected a posteriori, EAPȑ՗ीൂ΋໘ቫૈΚॶǶҗܭ ACER ConQuest 2.0೬ᡏѝૈ຾Չൂ΋໘ቫϐૈΚ՗ीǴࡺҁࣴز٬ҔԜ೬ᡏਔǴஒϩ ԋٿᅿБԄ຾Չ՗ीǺ Ȑ΋ȑѝ՗ीЬाໆЁȐӵკ 3-2-3ǵ3-2-6ǵ3-2-9ȑǴа EH ߄ҢϐǶ ȐΒȑѝ՗ीԛભໆЁȐӵკ 3-2-4ǵ3-2-7ǵ3-2-10ȑǴа EL ߄ҢϐǶ Ϥǵʳаҁࣴز܌ගрϐୖኧ՗ीኳԄ຾Չ՗ीǺ ҁࣴز٬Ҕ MCMC ࢎᄬΠϐ Gibbs sampling ՗ीБݤǴֹ᏾՗ीЬाໆЁ ȐHȑǵԛભໆЁȐLȑϷ၂ᚒୖኧȐbȑǴ٠٬Ҕ WinBUGS ೬ᡏ຾Չ՗ीǶҗܭ WinBUGS ೬ᡏё຾Չൂ΋໘ቫǵଯ໘ቫϐૈΚ՗ीǴࡺҁࣴز٬ҔԜ೬ᡏਔǴ

(37)

ۓǴԵቾ٬Ҕ΋૓ϯȨόࡰۓૈΚॶǵ၂ᚒᜤࡋӃᡍϩଛϐຬୖኧȐfull bayesian modelȑȩБԄǴҗܭԜБԄёૈ཮Ӣ՗ीୖኧၸӭԶᏤठ՗ीᆒྗࡋफ़եǴࡺҁ ࣴزϩձаѤᅿ՗ीኳԄ຾Չ՗ीǴ٠௖૸όӕ՗ीኳԄჹ՗ीᆒྗࡋޑቹៜǶ Ȑ΋ȑ຾ՉȨֹ᏾՗ीЬाໆЁᆶԛભໆЁȩǵȨѝ՗ीЬाໆЁȩϷȨѝ՗ीԛ ભໆЁȩǴ٠ȨࡰۓځૈΚໆЁǵ၂ᚒᜤࡋӃᡍϩଛϐຬୖኧȩǴϩձа EHL-1ǵEH-1 Ϸ EL-1 ߄ҢϐǶ ȐΒȑ຾ՉȨֹ᏾՗ीЬाໆЁᆶԛભໆЁȩǵȨѝ՗ीЬाໆЁȩϷȨѝ՗ीԛ ભໆЁȩǴ٠ȨࡰۓૈΚॶӃᡍϩଛϐຬୖኧȩǴϩձа EHL-2ǵEH-2 Ϸ EL-2߄ҢϐǶ ȐΟȑ຾ՉȨֹ᏾՗ीЬाໆЁᆶԛભໆЁȩǵȨѝ՗ीЬाໆЁȩϷȨѝ՗ीԛ ભໆЁȩǴ٠Ȩࡰۓ၂ᚒᜤࡋӃᡍϩଛϐຬୖኧȩǴϩձа EHL-3ǵEH-3 Ϸ EL-3 ߄ҢϐǶ ȐѤȑ຾ՉȨֹ᏾՗ीЬाໆЁᆶԛભໆЁȩǵȨѝ՗ीЬाໆЁȩϷȨѝ՗ीԛ ભໆЁȩǴȨόࡰۓૈΚॶǵ၂ᚒᜤࡋӃᡍϩଛϐຬୖኧȩǴϩձа EHL-4ǵ EH-4Ϸ EL-4 ߄ҢϐǶ

(38)

კ 3-2-2! H1L2 ϐ HO-IRT ኳԄ X01 X10 X11 X20 X21 X30 L1 L2 X40 X31 H1 11 O 12 O ԛભໆЁ! ЬाໆЁ X01 X10 X11 X20 X21 X30 X40 X31 H1 ЬाໆЁ

(39)

კ3-2-4! H1L2_EL ϐ MIRT ኳԄ 3-2-5! H1L4 ϐ HO-IRT ኳԄ X01 X10 X11 X20 X21 X30 L1 L3 X40 X31 L2 L4 H1 11 O 12 O 13 O 14 O ԛભໆЁ! ЬाໆЁ X01 X10 X11 X20 X21 X30 L1 L2 X40 X31 ԛભໆЁ!

(40)

კ3-2-6! H1L4_EH ϐ IRT ኳԄ X01 X10 X11 X20 X21 X30 L1 L3 X40 X31 L2 L4 ԛભໆЁ! X01 X10 X11 X20 X21 X30 X40 X31 H1 ЬाໆЁ

(41)

კ 3-2-8! H2L4 ϐ HO-IRT ኳԄ X01 X10 X11 X20 X21 X30 L1 L3 X40 X31 L2 L4 H1 H2 11 O 12 O 13 O 14 O 21 O 22 O ԛભໆЁ! ЬाໆЁ X01 X10 X11 X20 X21 X30 X40 X31 H1 H2 ЬाໆЁ

(42)

კ3-2-10! H2L4_EL ϐ MIRT ኳԄ

ୖǵࣴزำׇ

΋ǵኳࠠϟಏ ҁࣴزஒ HO-IRT ኳԄϩԋٿεᜪǴኳԄ΋ࣁЬाໆЁѝԖൂ΋ૈΚӛࡋޑ ΋Ӣη HO-IRT ኳԄǴኳԄΒࣁЬाໆЁԖٿঁૈΚӛࡋޑΒӢη HO-IRT ኳԄǶ Βǵౢғኳᔕၗ਑ (΋) ኳࠠ΋ ճҔ኱ྗதᄊϩଛᒿᐒౢғЬाໆЁȐHȑϐୖኧǴH_s ~ N(0,1)Ǵ s 1Ǵ٠ ਥᏵࣴز೛ीύ଑ᘜୖኧOޑ೛ۓǴౢғᆶЬाໆЁϕࣁጕ܄ᜢ߯ϐԛભໆЁ ȐLȑϐୖኧǴL_t O_st uH_s H_tǴH_t ~ N(0,1O_st2)Ǵ s 1Ǵ t 1,2 ܈ t 1,2,3,4ǹ ќѦǴᒿᐒౢғ኱ྗதᄊϩଛϐ 40 ᚒ၂ᚒᜤࡋୖኧǴb_j ~N(0,1)Ǵj 1,2,...,40Ƕ X01 X10 X11 X20 X21 X30 L1 L3 X40 X31 L2 L4 ԛભໆЁ!

(43)

ճҔ MRCMLM ౢғڙ၂ޣӧ؂ᚒޑเჹᐒ౗Ǵӆ೸ၸᒿᐒౢғϐ֡Ϭϩଛ U(0,1) ղۓڙ၂ޣܭ၀ᚒϐเჹᆶցǶ (Β) ኳࠠΒ ճҔ኱ྗதᄊϩଛᒿᐒౢғЬाໆЁȐHȑϐୖኧǴH_s ~ N(0,1)Ǵ s 1,2Ǵ ٠ਥᏵࣴز೛ीύ଑ᘜୖኧOޑ೛ۓǴౢғᆶЬाໆЁϕࣁጕ܄ᜢ߯ϐԛભໆЁ ȐLȑϐୖኧǴ °¯ ° ® u u

¦

2

¦

1 2 1 2 2 4 , 3 if , ) 1 , 0 ( ~ , H L 2 , 1 if , ) 1 , 0 ( ~ , H L s st s t t s st t st t t s st t t N t N O H H O O H H O ǹ ќѦǴᒿᐒౢғ኱ྗதᄊϩଛϐ 40 ᚒ၂ᚒᜤࡋୖኧǴb_j ~N(0,1)Ǵ j 1,2,...,40Ƕ ճҔ MRCMLM ౢғڙ၂ޣӧ؂ᚒޑเჹᐒ౗Ǵӆ೸ၸᒿᐒౢғϐ֡Ϭϩଛ U(0,1) ղۓڙ၂ޣܭ၀ᚒϐเჹᆶցǶ Οǵֹ᏾ኳԄୖኧ՗ीБݤ (΋) ኳࠠ΋ ځ՗ीኳԄϐ܌Ԗୖኧ೛ۓӵΠȐаȨόࡰۓૈΚॶǵ၂ᚒᜤࡋӃᡍϩଛϐ ຬୖኧȩࣁٯǴځᎩ೛ۓ၁ـߕᒵ΋ȑǺ s H ~ N( , 2 s s H H V P ) s H P ~ N(0,1) 2 s H V ~ * (100 , 0.01) st O ~ U(0,1) t H ~ N(0 , 1-O_st2 2 s H V u ) t L ₌O₁_* H_s₊H₁ b ~ N(P_b,V_b2) b P _{~ N(0,1)} V_b2_{~ * (100 , 0.01)} ځύǴ s 1Ǵ t 1,2 ܈ t 1,2,3,4Ƕ

(44)

(Β) ኳࠠΒ ځ՗ीኳԄϐ܌Ԗୖኧ೛ۓӵΠȐаȨόࡰۓૈΚॶǵ၂ᚒᜤࡋӃᡍϩଛϐ ຬୖኧȩࣁٯǴځᎩ೛ۓ၁ـߕᒵ΋ȑǺ s H ~ N( , 2 s s H H V P ) s H P ~ N(0,1) 2 s H V ~ * (100 , 0.01) st O ~ U(0,1) °¯ ° ® u u u u

¦

2

¦

1 2 1 2 2 2 2 4 , 3 if , ) 1 , 0 ( ~ , H L 2 , 1 if , ) 1 , 0 ( ~ , H L s st s t t s st H t H st t t s st t t N t N s s V O H H O V O H H O b ~ N( 2 , _b b V P ₎ b P _{~ N(0,1)} 2 b V _{~ * (100 , 0.01)} ځύǴ s 1,2Ǵ t 1,2,3,4Ƕ

ಃΟ࿯! ຑ՗ࡰ኱

ҁࣴزаኳᔕࣴزౢғၗ਑Ǵ٠٬Ҕ RMSE բࣁຑ՗ࡰ኱ǴᙖаᕇளኳԄ՗ ीϐᆒྗࡋǶRMSE ϐीᆉБݤӵΠǺ

൘ǵʳЬाໆЁ

¦

N j Hj Hj N H 1 2 ) ˆ ( 1 ) ( RMSE ځύ H Ǻಃ_j jՏЬाໆЁϐ੿ॶ j Hˆ Ǻಃ jՏЬाໆЁϐ՗ीॶ N Ǻڙ၂ޣΓኧ

(45)

ມǵʳԛભໆЁ

¦

N j j j L L N L 1 2 ) ˆ ( 1 ) ( RMSE ځύ L Ǻಃ_j jՏԛભໆЁϐ੿ॶ j Lˆ Ǻಃ jՏԛભໆЁϐ՗ीॶ N Ǻڙ၂ޣΓኧ

ୖǵʳ၂ᚒୖኧ

¦

n i bi bi n b 1 2 ) ˆ ( 1 ) ( RMSE ځύ b Ǻಃ_i iᚒ၂ᚒୖኧ੿ॶ i bˆ Ǻಃiᚒ၂ᚒୖኧ՗ीॶ n Ǻ၂ᚒᚒኧ

ಃѤ࿯! ࣴزπڀ

ҁࣴز٬ҔޑπڀԖMATLAB೬ᡏǵACER ConQuest 2.0೬ᡏϷWinBUGS೬ ᡏǴ૟ϩॊӵΠǶ

൘ǵMATLAB 7

ҁࣴز٬Ҕ Matlab 7 ำԄౢғڙ၂ޣϐЬाૈΚǵԛाૈΚϷ၂ᚒୖኧǴ຾ ԶኳᔕբเϸᔈǴ٠ीᆉୖኧ՗ीᇤৡǶ

(46)

ኳԄǶҁࣴز٬ҔACER ConQuest 2.0೬ᡏ຾Չൂ΋໘ቫૈΚȐӚ໘ቫϩ໒՗ीȑ ᆶ၂ᚒୖኧ՗ीǴճҔMMLݤ՗ी၂ᚒୖኧǴճҔEAPݤ՗ीൂ΋໘ቫૈΚॶǶ

ୖǵWinBUGS

WinBUGSࣁ೸ၸMCMCޑБݤೀ౛نМኳԄࢎᄬ܌ࣴวޑ಍ी೬ᡏǴ٬Ҕ Бݤࡐቸ܄ǴҞ߻ހҁȐWinBUGS1.4ȑԖำԄኗቪکკ׎ᏹբٿᅿȐቅ޲྆Ǵ 2006ȑǶWinBUGS೬ᡏёᔈҔޑኳԄ࣬྽ޑቶݱǴхࡴ΋૓ጕ܄کߚጕ܄ኳԄǵ ೀ౛ೱុکߚೱុ܄ၗ਑ϷӭᡂໆኳԄȐCowles, 2004; Qiu, Song, & Tan, 2002; Sturtz, Ligges, & Gelman, 2005ȑǶҁࣴز٬ҔWinBUGS೬ᡏ຾Չൂ΋໘ቫૈΚ՗ ीᆶֹ᏾՗ीǴճҔMCMCࢎᄬΠϐGibbs samplingݤ՗ीᜤࡋୖኧᆶൂ΋໘ ቫǵଯ໘ቫૈΚॶǶ

(47)

ಃѤക! ࣴز่݀

ҁകӅϩԋٿ࿯Ǵಃ΋࿯ࣁӚ௃ნΠୖኧ՗ीᇤৡ่݀ϐКၨǴಃΒ࿯ࣁᆕ ӝКၨǶϩॊӵΠǺ

ಃ΋࿯! ୖኧ՗ीᇤৡ่݀

ҁࣴزϐኳᔕࣴزӅϩࣁϖঁჴᡍǴϩॊӵΠǺ

൘ǵჴᡍ΋ȐH1L2-1ȑ

ԜჴᡍϐЬाໆЁኧࣁ 1ǴԛભໆЁኧࣁ 2ǴځύǴ଑ᘜୖኧOϩձ೛ीࣁ 11 O =0.8ǵO12=0.9Ƕ٬Ҕόӕ೬ᡏǵόӕ՗ीБݤϷόӕ՗ीኳԄ຾Չ՗ीǴځ่

݀ܭߕᒵΒև౜ǶځύǴH1L2-1_CQ_EH ߄ҢӧԜჴᡍΠǴճҔ ACER ConQuest 2.0 ೬ᡏ຾Չϩ໒՗ीɡѝ՗ीЬाໆЁϐ՗ी่݀ǹH1L2-1_CQ_EL ߄ҢӧԜ ჴᡍΠǴճҔ ACER ConQuest 2.0 ೬ᡏ຾Չϩ໒՗ीɡѝ՗ीԛભໆЁϐ՗ी่ ݀ǹH1L2-1_WB_EHL-1 ߄ҢӧԜჴᡍΠǴճҔ WinBUGS ೬ᡏ຾Չֹ᏾՗ीٿ ໘ቫૈΚໆЁϐ՗ी่݀Ǵځ՗ीኳԄ٬ҔȨࡰۓځૈΚໆЁǵ၂ᚒᜤࡋӃᡍϩ ଛϐຬୖኧȩǴH1L2-1_WB_EHL-2 ߾߄Ң՗ीኳԄ٬ҔȨࡰۓૈΚॶӃᡍϩଛϐ ຬୖኧȩǴH1L2-1_WB_EHL-3 ߾߄Ң՗ीኳԄ٬ҔȨࡰۓ၂ᚒᜤࡋӃᡍϩଛϐຬ ୖኧȩǴH1L2-1_WB_EHL-4 ߾߄Ң՗ीኳԄ٬ҔȨόࡰۓૈΚॶǵ၂ᚒᜤࡋӃᡍ ϩଛϐຬୖኧȩǶ ΋ǵϩ໒՗ीܭόӕ՗ीኳԄϐԋਏ௖૸ ߄ 4-1-1 ࣁ٬Ҕϩ໒՗ीБԄ຾Չಃ΋ቫȐԛભໆЁȑϐ՗ी่݀Ǵҗ߄ύ ёޕӧ H1L2-1_WB_EL-3 ᆶ H1L2-1_WB_EL-4 ௃ნΠޑୖኧ՗ीᇤৡࣣᆶ H1L2-1_CQ_EL คܴᡉৡ౦Ƕ߄ 4-1-2 ࣁ٬Ҕϩ໒՗ीБԄ຾ՉಃΒቫȐЬाໆ Ёȑϐ՗ी่݀Ǵҗ߄ύёޕӧ H1L2-1_WB_EH-1 ᆶ H1L2-1_WB_EH-2 ௃ნΠ

(48)

ޑୖኧ՗ीᇤৡࣣᆶ H1L2-1_CQ_EH คܴᡉৡ౦Ƕ ᆕӝ΢ॊКၨёޕǴа PISA ܌٬Ҕϐ՗ीБԄ຾Չ՗ीᆶҁࣴز܌ගрϐ ୖኧ՗ीኳԄ຾Չ՗ीǴКၨࡕёว౜ӧࢌ٤௃ნΠᆶคܴᡉৡ౦Ǵ߄Ңҁࣴز ܌ගрϐୖኧ՗ीኳԄҔܭϩ໒՗ीࢂёߞǵёՉޑǴ٠ё຾Չֹ᏾՗ीϐԋਏ ௖૸Ƕ ߄ 4-1-1! H1L2-1_EL ϐୖኧ՗ीᇤৡ жዸ ՗ीୖኧ ኳԄ_೬ᡏ_՗ीБԄ-՗ीኳԄ L1 L2 B RMSE 0.3818 0.3907 0.0771 H1L2-1_CQ_EL STD 0.0089 0.0092 0.0086 RMSE 0.4083 0.4101 0.0783 H1L2-1_WB_EL-1 STD 0.0104 0.0111 0.0093 RMSE 0.4082 0.4102 0.0791 H1L2-1_WB_EL-2 STD 0.0104 0.0109 0.0094 RMSE 0.3830 0.3918 0.0765 H1L2-1_WB_EL-3 STD 0.0089 0.0094 0.0083 RMSE 0.3833 0.3917 0.0767 H1L2-1_WB_EL-4 STD 0.0091 0.0092 0.0083 ߄ 4-1-2! H1L2-1_EH ϐୖኧ՗ीᇤৡ жዸ ՗ीୖኧ ኳԄ_೬ᡏ_՗ीБԄ-՗ीኳԄ H1 B RMSE 0.4519 0.0764 H1L2-1_CQ_EH STD 0.0102 0.0084 RMSE 0.4456 0.0771 H1L2-1_WB_EH-1 STD 0.0098 0.0087 RMSE 0.4454 0.0767 H1L2-1_WB_EH-2 STD 0.0100 0.0086 RMSE 0.4989 0.2021 H1L2-1_WB_EH-3 STD 0.0613 0.1217

(49)

Βǵֹ᏾՗ीᆶϩ໒՗ीܭόӕ՗ीኳԄϐԋਏ௖૸ җܭ ACER ConQuest 2.0 ѝૈ٬Ҕϩ໒՗ीБԄ຾Չ՗ीǴࡺ߄ 4-1-3 ࣁ ConQuest٬Ҕϩ໒՗ीБԄ຾Չಃ΋ቫǵಃΒቫϐ՗ी่݀ǴϷ WinBUGS ٬Ҕ ֹ᏾՗ीܭόӕ՗ीኳԄϐ՗ी่݀Ǵҗ߄ύёޕǴӧ H1L2-1_WB_EHL-1ǵ H1L2-1_WB_EHL-2 ௃ ნ Π ޑ ୖ ኧ ՗ ी ᇤ ৡ ࣣ ϩ ձ ᆶ H1L2-1_CQ_EL ǵ H1L2-1_CQ_EH คܴᡉৡ౦Ƕஒࡷᒧрҁࣴز܌ගрϐୖኧ՗ीኳԄԋਏၨ٫ ޣǴ຾ՉಃΒ࿯ϐᆕӝКၨǶ ߄ 4-1-3! H1L2-1_HL ϐୖኧ՗ीᇤৡ жዸ ՗ीୖኧ ኳԄ_೬ᡏ_՗ीБԄ-՗ीኳԄ H1 L1 L2 B RMSE 0.4519 NA NA 0.0764 H1L2-1_CQ_EH STD 0.0102 NA NA 0.0084 RMSE NA 0.3818 0.3907 0.0771 H1L2-1_CQ_EL STD NA 0.0089 0.0092 0.0086 RMSE 0.4477 0.3855 0.3911 0.0768 H1L2-1_WB_EHL-1 STD 0.0129 0.0091 0.0094 0.0089 RMSE 0.4475 0.3856 0.3909 0.0769 H1L2-1_WB_EHL-2 STD 0.0141 0.0093 0.0094 0.0089 RMSE 0.4845 0.4221 0.4270 0.1681 H1L2-1_WB_EHL-3 STD 0.0510 0.0427 0.0445 0.0884 RMSE 0.5867 0.5177 0.5235 0.3186 H1L2-1_WB_EHL-4 STD 0.1507 0.1408 0.1404 0.2054

ມǵჴᡍΒȐH1L2-2ȑ

ԜჴᡍϐЬाໆЁኧࣁ 1ǴԛભໆЁኧࣁ 2ǴځύǴ଑ᘜୖኧOϩձ೛ीࣁ 11 O =0.8ǵO₁₂=0.2Ƕϩձ٬Ҕόӕ೬ᡏǵόӕ՗ीБݤϷόӕ՗ीኳԄ຾Չ՗ीǴ ่݀ܭߕᒵΒև౜Ƕځжዸ߄Ңݤᆶჴᡍ΋ӕǶ

(50)

΋ǵϩ໒՗ीܭόӕ՗ीኳԄϐԋਏ௖૸

߄ 4-1-4 ࣁ٬Ҕϩ໒՗ीБԄ຾Չಃ΋ቫȐԛભໆЁȑϐ՗ी่݀Ǵҗ߄ύ ё ޕ Ǵ ӧ H1L2-2_WB_EL-1 ǵ H1L2-2_WB_EL-2 ǵ H1L2-2_WB_EL-3 Ϸ H1L2-2_WB_EL-4 ௃ნΠޑୖኧ՗ीᇤৡࣣᆶ H1L2-2_CQ_EL คܴᡉৡ౦Ƕ߄ 4-1-5 ࣁ٬Ҕϩ໒՗ीБԄ຾ՉಃΒቫȐЬाໆЁȑϐ՗ी่݀Ǵҗ߄ύёޕӧ H1L2-2_WB_EH-1 ᆶ H1L2-2_WB_EH-2 ௃ ნ Π ޑ ୖ ኧ ՗ ी ᇤ ৡ ࣣ ᆶ H1L2-2_CQ_EHคܴᡉৡ౦Ƕ ᆕӝ΢ॊКၨёޕǴа PISA ܌٬Ҕϐ՗ीБԄ຾Չ՗ीᆶҁࣴز܌ගрϐ ୖኧ՗ीኳԄ຾Չ՗ीǴКၨࡕёว౜ӧࢌ٤௃ნΠᆶคܴᡉৡ౦Ǵ߄Ңҁࣴز ܌ගрϐୖኧ՗ीኳԄҔܭϩ໒՗ीࢂёߞǵёՉޑǴ٠ё຾Չֹ᏾՗ीϐԋਏ ௖૸Ƕ ߄ 4-1-4! H1L2-2_EL ϐୖኧ՗ीᇤৡ жዸ ՗ीୖኧ ኳԄ_೬ᡏ_՗ीБԄ-՗ीኳԄ L1 L2 B RMSE 0.4473 0.4638 0.0787 H1L2-2_CQ_EL STD 0.0116 0.0110 0.0092 RMSE 0.4516 0.4635 0.0791 H1L2-2_WB_EL-1 STD 0.0120 0.0112 0.0095 RMSE 0.4516 0.4638 0.0792 H1L2-2_WB_EL-2 STD 0.0121 0.0113 0.0094 RMSE 0.4471 0.4634 0.0786 H1L2-2_WB_EL-3 STD 0.0114 0.0112 0.0092 RMSE 0.4472 0.4634 0.0784 H1L2-2_WB_EL-4 STD 0.0114 0.0112 0.0094

(51)

߄ 4-1-5! H1L2-2_EH ϐୖኧ՗ीᇤৡ жዸ ՗ीୖኧ ኳԄ_೬ᡏ_՗ीБԄ-՗ीኳԄ H1 B RMSE 0.7796 0.1000 H1L2-2_CQ_EH STD 0.0147 0.0112 RMSE 0.7808 0.0915 H1L2-2_WB_EH-1 STD 0.0164 0.0106 RMSE 0.7805 0.0908 H1L2-2_WB_EH-2 STD 0.0162 0.0105 RMSE 0.8004 0.1968 H1L2-2_WB_EH-3 STD 0.0295 0.0933 RMSE 0.9001 0.4102 H1L2-2_WB_EH-4 STD 0.1360 0.2562 Βǵֹ᏾՗ीᆶϩ໒՗ीܭόӕ՗ीኳԄϐԋਏ௖૸ ߄ 4-1-6 ࣁ ACER ConQuest 2.0 ٬Ҕϩ໒՗ीБԄ຾Չಃ΋ቫǵಃΒቫϐ՗ ी่݀ǴϷ WinBUGS ٬Ҕֹ᏾՗ीܭόӕ՗ीኳԄϐ՗ी่݀Ǵҗ߄ύёޕǴ ӧ H1L2-2_WB_EHL-1ǵH1L2-2_WB_EHL-2 ௃ნΠޑୖኧ՗ीᇤৡࣣϩձᆶ H1L2-2_CQ_ELǵH1L2-2_CQ_EH คܴᡉৡ౦Ǵՠӧ H1L2-2_WB_EHL-1 ௃ნΠ ܌՗ीޑ၂ᚒᜤࡋȐBȑ߾ܴᡉᓬܭ H1L2-2_CQ_EHǶஒࡷᒧрҁࣴز܌ගрϐ ୖኧ՗ीኳԄԋਏၨ٫ޣǴ຾ՉಃΒ࿯ϐᆕӝКၨǶ

(52)

߄ 4-1-6! H1L2-2_EHL ϐୖኧ՗ीᇤৡ жዸ ՗ीୖኧ ኳԄ_೬ᡏ_՗ीБԄ-՗ीኳԄ H1 L1 L2 B RMSE 0.7796 NA NA 0.1000 H1L2-2_CQ_EH STD 0.0147 NA NA 0.0112 RMSE NA 0.4473 0.4638 0.0787 H1L2-2_CQ_EL STD NA 0.0116 0.0110 0.0092 RMSE 0.7656 0.4511 0.4635 0.0791 H1L2-2_WB_EHL-1 STD 0.0963 0.0121 0.0112 0.0099 RMSE 0.7856 0.4511 0.4632 0.0788 H1L2-2_WB_EHL-2 STD 0.1035 0.0119 0.0112 0.0095 RMSE 0.8889 0.4930 0.5095 0.2014 H1L2-2_WB_EHL-3 STD 0.1832 0.0579 0.0682 0.1062 RMSE 0.9294 0.5289 0.5076 0.2304 H1L2-2_WB_EHL-4 STD 0.1947 0.1206 0.0596 0.1450

ୖǵჴᡍΟȐH1L4-1ȑ

ԜჴᡍϐЬाໆЁኧࣁ 1ǴԛભໆЁኧࣁ 4ǴځύǴ଑ᘜୖኧOϩձ೛ीࣁ 11 O =0.9ǵO12=0.8ǵO13=0.7ǵO14=0.6Ƕϩձ٬Ҕόӕ೬ᡏǵόӕ՗ीБݤϷόӕ ՗ीኳԄ຾Չ՗ीǴ่݀ܭߕᒵΒև౜Ƕځжዸ߄Ңݤᆶჴᡍ΋ӕǶ ΋ǵϩ໒՗ीܭόӕ՗ीኳԄϐԋਏ௖૸ ߄ 4-1-7 ࣁ٬Ҕϩ໒՗ीБԄ຾Չಃ΋ቫȐԛભໆЁȑϐ՗ी่݀Ǵҗ߄ё ޕǴӧ H1L4-1_WB_EL-3 ᆶ H1L4-1_WB_EL-4 ௃ნΠޑୖኧ՗ीᇤৡࣣᆶ H1L4-1_CQ_EL คܴᡉৡ౦Ƕ߄ 4-1-8 ࣁ٬Ҕϩ໒՗ीБԄ຾ՉಃΒቫȐЬाໆ Ёȑϐ՗ी่݀Ǵҗ߄ύёޕӧ H1L4-1_WB_EH-1 ᆶ H1L4-1_WB_EH-2 ௃ნΠ ޑୖኧ՗ीᇤৡࣣК H1L4-1_CQ_EH եȐ՗ीਏ݀ၨᆒྗȑǶ ᆕӝ΢ॊКၨёޕǴа PISA ܌٬Ҕϐ՗ीБԄ຾Չ՗ीᆶҁࣴز܌ගрϐ ୖኧ՗ीኳԄ຾Չ՗ीǴКၨࡕёว౜ӧࢌ٤௃ნΠᆶคܴᡉৡ౦ǴࣗԿԖၨӳ

(53)

ޑ՗ीԋਏǴ߄Ңҁࣴز܌ගрϐୖኧ՗ीኳԄҔܭϩ໒՗ीࢂёߞǵёՉޑǴ ٠ё຾Չֹ᏾՗ीϐԋਏ௖૸Ƕ ߄4-1-7! H1L4-1_ELϐୖኧ՗ीᇤৡ жዸ ՗ीୖኧ ኳԄ_೬ᡏ_՗ीБԄ-՗ीኳԄ L1 L2 L3 L4 B RMSE 0.4478 0.4500 0.4796 0.5150 0.0764 H1L4-1_CQ_EL STD 0.0108 0.0108 0.0115 0.0131 0.0080 RMSE 0.4891 0.4952 0.5110 0.5344 0.0812 H1L4-1_WB_EL-1 STD 0.0123 0.0121 0.0134 0.0138 0.0084 RMSE 0.4888 0.4954 0.5109 0.5346 0.0815 H1L4-1_WB_EL-2 STD 0.0124 0.0117 0.0132 0.0140 0.0086 RMSE 0.4484 0.4512 0.4793 0.5150 0.0764 H1L4-1_WB_EL-3 STD 0.0103 0.0100 0.0111 0.0129 0.0078 RMSE 0.4483 0.4513 0.4796 0.5147 0.0763 H1L4-1_WB_EL-4 STD 0.0102 0.0103 0.0114 0.0129 0.0080 ߄4-1-8! H1L4-1_EHϐୖኧ՗ीᇤৡ жዸ ՗ीୖኧ ኳԄ_೬ᡏ_՗ीБԄ-՗ीኳԄ H1 B RMSE 0.5334 0.0814 H1L4-1_CQ_EH STD 0.0094 0.0077 RMSE 0.5129 0.0788 H1L4-1_WB_EH-1 STD 0.0084 0.0070 RMSE 0.5129 0.0780 H1L4-1_WB_EH-2 STD 0.0085 0.0072 RMSE 0.5627 0.1920 H1L4-1_WB_EH-3 STD 0.0454 0.1069 RMSE 0.6200 0.2846 H1L4-1_WB_EH-4 STD 0.1484 0.2339

(54)

Βǵֹ᏾՗ीᆶϩ໒՗ीܭόӕ՗ीኳԄϐԋਏ௖૸ ߄ 4-1-9 ࣁ ACER ConQuest 2.0 ٬Ҕϩ໒՗ीБԄ຾Չಃ΋ቫǵಃΒቫϐ՗ ी่݀ǴϷ WinBUGS ٬Ҕֹ᏾՗ीܭόӕ՗ीኳԄϐ՗ी่݀Ǵҗ߄ύёޕǴ ӧ H1L4-1_WB_EHL-1 ᆶ H1L4-1_WB_EHL-2 ௃ნΠჹ L1ǵL2ǵL3ǵL4ǵB ޑ ՗ीᆶ H1L4-1_CQ_EL คܴᡉৡ౦Ǵՠ H1 ޑ՗ी߾ᓬܭ H1L4-1_CQ_EHǶஒࡷ ᒧрҁࣴز܌ගрϐୖኧ՗ीኳԄԋਏၨ٫ޣǴ຾ՉಃΒ࿯ϐᆕӝКၨǶ ߄4-1-9! H1L4-1_EHLϐୖኧ՗ीᇤৡ жዸ ՗ीୖኧ ኳԄ_೬ᡏ_՗ीБԄ-՗ीኳԄ H1 L1 L2 L3 L4 B RMSE 0.5334 NA NA NA NA 0.0814 H1L4-1_CQ_EH STD 0.0094 NA NA NA NA 0.0077 RMSE NA 0.4478 0.4500 0.4796 0.5150 0.0764 H1L4-1_CQ_EL STD NA 0.0108 0.0108 0.0115 0.0131 0.0080 RMSE 0.4835 0.4468 0.4529 0.4833 0.5178 0.0772 H1L4-1_WB_EHL-1 STD 0.0088 0.0102 0.0112 0.0117 0.0127 0.0075 RMSE 0.4837 0.4469 0.4528 0.4833 0.5178 0.0774 H1L4-1_WB_EHL-2 STD 0.0088 0.0101 0.0112 0.0120 0.0126 0.0077 RMSE 0.5315 0.4932 0.4935 0.5170 0.5406 0.1806 H1L4-1_WB_EHL-3 STD 0.0570 0.0543 0.0477 0.0403 0.0281 0.0967 RMSE 0.8923 0.7910 0.7768 0.7456 0.7192 0.5580 H1L4-1_WB_EHL-4 STD 0.3130 0.2698 0.2512 0.2126 0.1713 0.3073

စǵჴᡍѤȐH1L4-2ȑ

ԜჴᡍϐЬाໆЁኧࣁ 1ǴԛભໆЁኧࣁ 4ǴځύǴ଑ᘜୖኧOϩձ೛ीࣁ 11 O =0.9ǵO12=0.8ǵO =0.5ǵ13 O14=0.2Ƕϩձ٬Ҕόӕ೬ᡏǵόӕ՗ीБݤϷόӕ՗ ीኳԄ຾Չ՗ीǴ่݀ܭߕᒵΒև౜Ƕځжዸ߄Ңݤᆶჴᡍ΋ӕǶ

(55)

߄ 4-1-10 ࣁ٬Ҕϩ໒՗ीБԄ຾Չಃ΋ቫȐԛભໆЁȑϐ՗ी่݀Ǵҗ߄ύ ёޕǴӧ H1L4-2_WB_EL-3 ᆶ H1L4-2_WB_EL-4 ௃ნΠޑୖኧ՗ीᇤৡࣣᆶ H1L4-2_CQ_ELคܴᡉৡ౦Ƕ߄ 4-1-11 ࣁ٬Ҕϩ໒՗ीБԄ຾ՉಃΒቫȐЬाໆ Ёȑϐ՗ी่݀Ǵҗ߄ύёޕӧ H1L4-2_WB_EH-1 ᆶ H1L4-2_WB_EH-2 ௃ნΠ ޑୖኧ՗ीࣣᓬܭ H1L4-2_CQ_EHǶ ᆕӝ΢ॊКၨёޕǴа PISA ܌٬Ҕϐ՗ीБԄ຾Չ՗ीᆶҁࣴز܌ගрϐ ୖኧ՗ीኳԄ຾Չ՗ीǴКၨࡕёว౜ӧࢌ٤௃ნΠᆶคܴᡉৡ౦ǴࣗԿԖၨӳ ޑ՗ीԋਏǴ߄Ңҁࣴز܌ගрϐୖኧ՗ीኳԄҔܭϩ໒՗ीࢂёߞǵёՉޑǴ ٠ё຾Չֹ᏾՗ीϐԋਏ௖૸Ƕ ߄4-1-10! H1L4-2_ELϐୖኧ՗ीᇤৡ жዸ ՗ीୖኧ ኳԄ_೬ᡏ_՗ीБԄ-՗ीኳԄ L1 L2 L3 L4 B RMSE 0.4841 0.4728 0.5431 0.5949 0.0782 H1L4-2_CQ_EL STD 0.0107 0.0104 0.0153 0.0148 0.0075 RMSE 0.5146 0.5109 0.5513 0.5939 0.0805 H1L4-2_WB_EL-1 STD 0.0114 0.0111 0.0162 0.0145 0.0075 RMSE 0.5147 0.5110 0.5516 0.5940 0.0807 H1L4-2_WB_EL-2 STD 0.0115 0.0111 0.0163 0.0145 0.0071 RMSE 0.4843 0.4734 0.5426 0.5945 0.0778 H1L4-2_WB_EL-3 STD 0.0105 0.0101 0.0154 0.0146 0.0075 RMSE 0.4843 0.4736 0.5427 0.5942 0.0778 H1L4-2_WB_EL-4 STD 0.0109 0.0105 0.0154 0.0149 0.0072

(56)

߄4-1-11! H1L4-2_EHϐୖኧ՗ीᇤৡ жዸ ՗ीୖኧ ኳԄ_೬ᡏ_՗ीБԄ-՗ीኳԄ H1 B RMSE 0.6696 0.1011 H1L4-2_CQ_EH STD 0.0119 0.0108 RMSE 0.6407 0.0912 H1L4-2_WB_EH-1 STD 0.0119 0.0096 RMSE 0.6407 0.0903 H1L4-2_WB_EH-2 STD 0.0119 0.0093 RMSE 0.6924 0.2069 H1L4-2_WB_EH-3 STD 0.0513 0.1279 RMSE 0.7493 0.3303 H1L4-2_WB_EH-4 STD 0.1085 0.2003 Βǵֹ᏾՗ीᆶϩ໒՗ीܭόӕ՗ीኳԄϐԋਏ௖૸ ࡺ߄ 4-1-12 ࣁ ACER ConQuest 2.0 ٬Ҕϩ໒՗ीБԄ຾Չಃ΋ቫǵಃΒቫϐ ՗ी่݀ǴϷ WinBUGS ٬Ҕֹ᏾՗ीܭόӕ՗ीኳԄϐ՗ी่݀Ǵҗ߄ύёޕǴ ӧ H1L4-2_WB_EHL-1 ᆶ H1L4-2_WB_EHL-2 ௃ნΠჹ L1ǵL2ǵL3ǵL4ǵB ޑ ՗ ी ᆶ H1L4-2_CQ_EL ค ܴ ᡉ ৡ ౦ ǹ ӧ H1L4-2_WB_EHL-1 ᆶ H1L4-2_WB_EHL-2 ௃ნΠჹ H1 ޑ՗ीࣣᓬܭ H1L4-2_CQ_EHǶஒࡷᒧрҁࣴ ز܌ගрϐୖኧ՗ीኳԄԋਏၨ٫ޣǴ຾ՉಃΒ࿯ϐᆕӝКၨǶ

(57)

߄4-1-12! H1L4-2_EHLϐୖኧ՗ीᇤৡ жዸ ՗ीୖኧ ኳԄ_೬ᡏ_՗ीБԄ-՗ीኳԄ H1 L1 L2 L3 L4 B RMSE 0.6696 NA NA NA NA 0.1011 H1L4-2_CQ_EH STD 0.0119 NA NA NA NA 0.0108 RMSE NA 0.4841 0.4728 0.5431 0.5949 0.0782 H1L4-2_CQ_EL STD NA 0.0107 0.0104 0.0153 0.0148 0.0075 RMSE 0.5310 0.4832 0.4763 0.5443 0.5934 0.0783 H1L4-2_WB_EHL-1 STD 0.0103 0.0107 0.0100 0.0154 0.0142 0.0070 RMSE 0.5312 0.4833 0.4766 0.5444 0.5935 0.0786 H1L4-2_WB_EHL-2 STD 0.0105 0.0106 0.0097 0.0157 0.0145 0.0070 RMSE 0.5819 0.5292 0.5229 0.5596 0.5955 0.1649 H1L4-2_WB_EHL-3 STD 0.0690 0.0602 0.0603 0.0255 0.0144 0.0897 RMSE 0.6865 0.6071 0.6015 0.5922 0.6016 0.2581 H1L4-2_WB_EHL-4 STD 0.1940 0.1572 0.1573 0.0643 0.0181 0.1861

ҴǵჴᡍϖȐH2L4ȑ

ԜჴᡍϐЬाໆЁኧࣁ 2ǴԛભໆЁኧࣁ 4ǴځύǴ଑ᘜୖኧOϩձ೛ीࣁ 11 O =0.9ǵO12=0.8ǵO13=0.5ǵO14=0.2ǵO21=0.5ǵO22=0.8Ƕϩձ٬Ҕόӕ೬ᡏǵό ӕ՗ीБݤϷόӕ՗ीኳԄ຾Չ՗ीǴ่݀ܭߕᒵΒև౜Ƕځжዸ߄Ңݤᆶჴᡍ ΋ӕǶ ΋ǵϩ໒՗ीܭόӕ՗ीኳԄϐԋਏ௖૸ ߄ 4-1-13 ࣁ٬Ҕϩ໒՗ीБԄ຾Չಃ΋ቫȐԛભໆЁȑϐ՗ी่݀Ǵҗ߄ύ ё ޕ ӧ H2L4_WB_EL-3 ᆶ H2L4_WB_EL-4 ௃ ნ Π ޑ ୖ ኧ ՗ ी ᇤ ৡ ࣣ ᆶ H2L4_CQ_ELคܴᡉৡ౦Ƕ߄ 4-1-14 ࣁ٬Ҕϩ໒՗ीБԄ຾ՉಃΒቫȐЬाໆЁȑ ϐ՗ी่݀Ǵҗ߄ύёޕӧ H2L4_WB_EH-1 ᆶ H2L4_WB_EH-2 ௃ნΠჹ H1 ޑ ՗ीᇤৡᆶ H2L4_CQ_EH คܴᡉৡ౦Ǵՠჹ H2ǵB ޑ՗ीࣣᓬܭ H2L4_CQ_EHǶ ᆕӝ΢ॊКၨёޕǴа PISA ܌٬Ҕϐ՗ीБԄ຾Չ՗ीᆶҁࣴز܌ගрϐ

(58)

ୖኧ՗ीኳԄ຾Չ՗ीǴКၨࡕёว౜ӧࢌ٤௃ნΠᆶคܴᡉৡ౦ǴࣗԿԖၨӳ ޑ՗ीԋਏǴ߄Ңҁࣴز܌ගрϐୖኧ՗ीኳԄҔܭϩ໒՗ीࢂёߞǵёՉޑǴ ٠ё຾Չֹ᏾՗ीϐԋਏ௖૸Ƕ ߄4-1-13! H2L4_ELϐୖኧ՗ीᇤৡ жዸ ՗ीୖኧ ኳԄ_೬ᡏ_՗ीБԄ-՗ीኳԄ L1 L2 L3 L4 B RMSE 0.4839 0.4714 0.4926 0.5335 0.0769 H2L4_CQ_EL STD 0.0120 0.0115 0.0127 0.0148 0.0100 RMSE 0.5147 0.5115 0.5204 0.5474 0.0814 H2L4_WB_EL-1 STD 0.0114 0.0147 0.0138 0.0156 0.0108 RMSE 0.5144 0.5112 0.5204 0.5476 0.0815 H2L4_WB_EL-2 STD 0.0116 0.0148 0.0135 0.0156 0.0108 RMSE 0.4841 0.4724 0.4921 0.5333 0.0765 H2L4_WB_EL-3 STD 0.0118 0.0119 0.0122 0.0144 0.0095 RMSE 0.4841 0.4723 0.4921 0.5334 0.0765 H2L4_WB_EL-4 STD 0.0110 0.0116 0.0123 0.0147 0.0098 ߄4-1-14! H2L4_EHϐୖኧ՗ीᇤৡ жዸ ՗ीୖኧ ኳԄ_೬ᡏ_՗ीБԄ-՗ीኳԄ H1 H2 B RMSE 0.5462 0.8649 0.3605 H2L4_CQ_EH STD 0.0126 0.0228 0.0484 RMSE 0.5494 0.7754 0.0793 H2L4_WB_EH-1 STD 0.0117 0.0154 0.0112 RMSE 0.5493 0.7751 0.0792 H2L4_WB_EH-2 STD 0.0116 0.0153 0.0116 RMSE 0.6307 0.8413 0.2711 H2L4_WB_EH-3 STD 0.0829 0.1156 0.1419 RMSE 0.9713 0.8248 0.7887 H2L4_WB_EH-4 STD 0.2980 0.0991 0.3497

(59)

Βǵֹ᏾՗ीᆶϩ໒՗ीܭόӕ՗ीኳԄϐԋਏ௖૸ ߄ 4-1-15 ࣁ ACER ConQuest 2.0 ٬Ҕϩ໒՗ीБԄ຾Չಃ΋ቫǵಃΒቫϐ՗ ी่݀ǴϷ WinBUGS ٬Ҕֹ᏾՗ीܭόӕ՗ीኳԄϐ՗ी่݀Ǵҗ߄ύёޕǴ ӧ H2L4_WB_EHL-1 ᆶ H2L4_WB_EHL-2 ௃ნΠჹ L1ǵL2ǵL3ǵL4ǵB ޑ՗ी ᆶ H2L4-2_CQ_EL คܴᡉৡ౦Ǵՠჹ H1ǵH2 ޑ՗ीࣣϩձᓬܭ H2L4-2_CQ_EHǶ ஒࡷᒧрҁࣴز܌ගрϐୖኧ՗ीኳԄԋਏၨ٫ޣǴ຾ՉಃΒ࿯ϐᆕӝКၨǶ ߄4-1-15! H2L4_EHLϐୖኧ՗ीᇤৡ жዸ ՗ीୖኧ ኳԄ_೬ᡏ_՗ीБԄ-՗ीኳԄ H1 H2 L1 L2 L3 L4 B RMSE 0.5462 0.8649 NA NA NA NA 0.3605 H2L4_CQ_EH STD 0.0126 0.0228 NA NA NA NA 0.0484 RMSE NA NA 0.4839 0.4714 0.4926 0.5335 0.0769 H2L4_CQ_EL STD NA NA 0.0120 0.0115 0.0127 0.0148 0.0100 RMSE 0.5337 0.6927 0.4835 0.4756 0.4962 0.5359 0.0782 H2L4_WB_EHL-1 STD 0.0120 0.0229 0.0116 0.0115 0.0123 0.0148 0.0104 RMSE 0.5329 0.7065 0.4831 0.4751 0.4962 0.5358 0.0782 H2L4_WB_EHL-2 STD 0.0118 0.0333 0.0118 0.0113 0.0124 0.0149 0.0103 RMSE 0.6234 0.8194 0.5399 0.5348 0.5680 0.6162 0.2613 H2L4_WB_EHL-3 STD 0.0916 0.1281 0.0643 0.0681 0.0789 0.0798 0.1101 RMSE 0.8062 0.8700 0.7089 0.7012 0.7689 0.7216 0.5039 H2L4_WB_EHL-4 STD 0.3168 0.1855 0.2761 0.2780 0.2642 0.1814 0.3059

ಃΒ࿯! ᆕӝКၨ

΋ǵ଑ᘜୖኧϐ೛ۓჹ՗ीᆒྗࡋϐቹៜ Кၨ H1L2-1Ȑ߄ 4-2-1ȑᆶ H1L2-2Ȑ߄ 4-2-2ȑϐ՗ीਏ݀ǴH1L2-1 ϐ՗ी ᆒྗࡋࣣК H1L2-2 ଯǹКၨ H1L4-1Ȑ߄ 4-2-3ȑᆶ H1L4-2Ȑ߄ 4-2-4ȑϐ՗ीਏ ݀ǴH1L4-1 ϐ՗ीᆒྗࡋࣣК H1L4-2 ଯǶҗԜёޕǴ଑ᘜୖኧཇଯǴځ՗ीᆒ

(60)

ΒǵԛભໆЁኧϐ೛ۓჹ՗ीᆒྗࡋϐቹៜ Кၨ H1L2-1_HL ᆶ H1L2-1_HȐ߄ 4-2-1ȑаϷ H1L2-2_HL ᆶ H1L2-2_EH Ȑ߄ 4-2-2ȑϐЬाໆЁ՗ीᆒྗࡋǴځᆒྗࡋคၨεৡ౦ǹКၨ H1L4-1_HL ᆶ H1L4_HȐ߄ 4-2-3ȑаϷ H1L4-2_HL ᆶ H1L4-2_HȐ߄ 4-2-4ȑϐЬाໆЁ՗ी ᆒྗࡋǴځᆒྗࡋԖܴᡉৡ౦ǶҗԜёޕǴԛભໆЁኧቚуǴ߾ WinBUGS ֹ᏾ ՗ीϐЬाໆЁ཮К ACER ConQuest 2.0 ѝ՗ीЬाໆЁਔǴܴᡉᕇளၨଯޑᆒ ྗࡋǶ ΟǵЬाໆЁኧϐ೛ۓჹ՗ीᆒྗࡋϐቹៜ ऩܭ H1L4-2Ȑკ 3-2-5ȑΠቚу΋ঁЬाໆЁ׎ԋ H2L4Ȑკ 3-2-8ȑǴ٠Кၨ H1L4-2_WB_EHL-1Ȑ߄ 4-2-4ȑᆶ H2L4_WB_EHL-2Ȑ߄ 4-2-5ȑǴ߾ӧԛભໆЁ L3ǵL4 ύёளډၨଯϐ՗ीᆒྗࡋǶҗԜёޕǴቚуЬाໆЁኧǴځ࣬ჹᔈԛ ભໆЁϐ՗ीᆒྗࡋКൂ΋ЬाໆЁଯǶ ߄ 4-2-1! H1L2-1 ϐୖኧ՗ीᇤৡ жዸ ՗ीୖኧ ኳԄ_೬ᡏ_՗ीБԄ-՗ीኳԄ H1 L1 L2 B RMSE 0.4519 NA NA 0.0764 H1L2-1_CQ_EH STD 0.0102 NA NA 0.0084 RMSE NA 0.3818 0.3907 0.0771 H1L2-1_CQ_EL STD NA 0.0089 0.0092 0.0086 RMSE 0.4475 0.3856 0.3909 0.0769 H1L2-1_WB_EHL-2 STD 0.0141 0.0093 0.0094 0.0089

(61)

߄ 4-2-2! H1L2-2 ϐୖኧ՗ीᇤৡ жዸ ՗ीୖኧ ኳԄ_೬ᡏ_՗ीБԄ-՗ीኳԄ H1 L1 L2 B RMSE 0.7796 NA NA 0.1000 H1L2-2_CQ_EH STD 0.0147 NA NA 0.0112 RMSE NA 0.4473 0.4638 0.0787 H1L2-2_CQ_EL STD NA 0.0116 0.0110 0.0092 RMSE 0.7656 0.4511 0.4635 0.0791 H1L2-2_WB_EHL-1 STD 0.0963 0.0121 0.0112 0.0099 ߄4-2-3! H1L4-1ϐୖኧ՗ीᇤৡ жዸ ՗ीୖኧ ኳԄ_೬ᡏ_՗ीБԄ-՗ीኳԄ H1 L1 L2 L3 L4 B RMSE 0.5334 NA NA NA NA 0.0814 H1L4-1_CQ_EH STD 0.0094 NA NA NA NA 0.0077 RMSE NA 0.4478 0.4500 0.4796 0.5150 0.0764 H1L4-1_CQ_EL STD NA 0.0108 0.0108 0.0115 0.0131 0.0080 RMSE 0.4835 0.4468 0.4529 0.4833 0.5178 0.0772 H1L4-1_WB_EHL-1 STD 0.0088 0.0102 0.0112 0.0117 0.0127 0.0075 ߄4-2-4! H1L4-2ϐୖኧ՗ीᇤৡ жዸ ՗ीୖኧ ኳԄ_೬ᡏ_՗ीБԄ-՗ीኳԄ H1 L1 L2 L3 L4 B RMSE 0.6696 NA NA NA NA 0.1011 H1L4-2_CQ_EH STD 0.0119 NA NA NA NA 0.0108 RMSE NA 0.4841 0.4728 0.5431 0.5949 0.0782 H1L4-2_CQ_EL STD NA 0.0107 0.0104 0.0153 0.0148 0.0075 RMSE 0.5310 0.4832 0.4763 0.5443 0.5934 0.0783 H1L4-2_WB_EHL-1 STD 0.0103 0.0107 0.0100 0.0154 0.0142 0.0070

(62)

߄4-2-5! H2L4ϐୖኧ՗ीᇤৡ жዸ ՗ीୖኧ ኳԄ_೬ᡏ_՗ीБԄ-՗ीኳԄ H1 H2 L1 L2 L3 L4 B RMSE 0.5462 0.8649 NA NA NA NA 0.3605 H2L4_CQ_EH STD 0.0126 0.0228 NA NA NA NA 0.0484 RMSE NA NA 0.4839 0.4714 0.4926 0.5335 0.0769 H2L4_CQ_EL STD NA NA 0.0120 0.0115 0.0127 0.0148 0.0100 RMSE 0.5329 0.7065 0.4831 0.4751 0.4962 0.5358 0.0782 H2L4_WB_EHL-2 STD 0.0118 0.0333 0.0118 0.0113 0.0124 0.0149 0.0103

(63)

ಃϖക! ่ፕᆶࡌ᝼

ҁകϩࣁΒ࿯Ǵಃ΋࿯ࣁҁࣴزϐ่ፕǴಃΒ࿯൩ҁࣴز҂ᅰֹഢϐೀǴග р΋٤ࣴزࡌ᝼ǴٮࡕុࣴزޣୖԵǶ૟ϩॊӵΠǺ

ಃ΋࿯! ่ፕ

΋ǵϩ໒՗ी җჴᡍ่݀ёޕǴҁࣴز܌ගрϐୖኧ՗ीኳԄӧϩ໒՗ीਔǴ՗ीᇤৡࣣ ௗ߈܈ᓬܭ PISA ϐ՗ीБԄǶҁࣴز܌ගрϐୖኧ՗ीኳԄᗨߚֹӄឦܭ΢ॊ ރݩǴՠࣣԖ΋ঁа΢ϐ௃ნ಄ӝ΢ॊރݩǶ Βǵֹ᏾՗ीᆶϩ໒՗ी җჴᡍ่݀ёޕǴֹ᏾՗ीϐᆒྗࡋࣣᓬܭϩ໒՗ीǴӢԜǴऩၗ਑ឦܭଯ ໘ቫޑຑໆࢎᄬǴՠ՗ीਔࠅஒЬाໆЁϷԛભໆЁϩ໒՗ीǴ߾཮Ӣᒪᅅ΋٤ ૻ৲Զଯ՗ૈΚୖኧЪ೷ԋ၂ᚒୖኧ՗ीϐୃৡǴᏤठ՗ीᆒྗࡋफ़եǶӢԜǴ ᒧҔӝ፾ޑෳໆኳԄࢂ࣬྽ख़ाޑǴ྽ຑໆࢎᄬឦܭଯ໘ቫਔǴ௦Ҕ HO-IRT ኳ Ԅ຾Չ՗ी཮ளډၨଯޑ՗ीᆒྗࡋǹऩຑໆࢎᄬࣁൂ΋໘ቫਔǴҗܭ௦Ҕ໺಍ MIRT ᆶ HO-IRT ኳԄ՗ीϐਏ݀คܴᡉৡ౦Ǵӧ՗ीਔ໔ޑԵໆ΢Ǵё௦Ҕ໺ ಍ MIRT ኳԄ՗ी཮ၨ࣪ਔǶ Οǵ଑ᘜୖኧϐ೛ۓ җჴᡍ่݀ёޕǴ଑ᘜୖኧཇଯǴځ՗ीᆒྗࡋཇଯǶ଑ᘜୖኧॶଯǴ߄Ң ЬाໆЁᆶԛભໆЁ໔ޑ࣬ᜢଯǴҗԜёޕǴ೛ीຑໆࢎᄬਔǴሡԵቾЬाໆЁ ᆶԛભໆЁ໔ޑ࣬ᜢำࡋǴځ࣬ᜢำࡋཇଯǴ՗ीᆒྗࡋཇଯǶ ѤǵԛભໆЁኧϐ೛ۓ җჴᡍ่݀ёޕǴԛભໆЁኧቚуǴ߾ֹ᏾՗ीϐЬाໆЁ཮Кϩ໒՗ीϐ ѝ՗ीЬाໆЁਔǴܴᡉᕇளၨଯޑᆒྗࡋǶऩຑໆࢎᄬϐԛભໆЁኧၨϿਔǴ

(64)

ֹ᏾՗ीᆶϩ໒՗ीϐ՗ीਏ݀ৡ౦όεǴӧёௗڙᇤৡጄൎΠǴё௦Ҕϩ໒՗ ीǴځ՗ीਔ໔ၨזǹऩຑໆࢎᄬϐԛભໆЁኧၨӭਔǴࡌ᝼٬Ҕֹ᏾՗ीኳԄ ຾Չ՗ीǴځ՗ीਏ݀Кϩ໒՗ीᆒྗǶ ϖǵЬाໆЁኧϐ೛ۓ җჴᡍ่݀ёޕǴቚуЬाໆЁኧǴځ࣬ჹᔈԛભໆЁϐ՗ीᆒྗࡋКൂ΋ ЬाໆЁଯǶӢԜǴ೛ीຑໆࢎᄬਔǴጓӈӭঁଯ໘ޑቹៜӢηǴό໻ёளډ׳ ӭޑૻ৲Ǵҭёගϲୖኧ՗ीϐᆒྗࡋǶ

ಃΒ࿯! ࡌ᝼

ҁ࿯൩ҁࣴز҂ᅰֹഢϐೀǴගр΋٤ࣴزࡌ᝼ǴٮࡕុࣴزޣୖԵǶ ΋ǵҗܭҁࣴزϐΓኧϷᚒኧޑ೛ी೿ࢂڰۓޑǴ҂௖૸೭ٿঁᡂ໨ϐ೛ीჹ HO-IRTኳԄޑ՗ीԋਏǴࡕុࣴزޣёаଞჹԜᗺ຾Չ௖૸Ƕ ΒǵҗܭҁࣴزϐीϩࠠᄊឦܭΒϡीϩǴ܌аࡕុࣴزޣёۯ՜ࣴزԿӭᗺी ϩ܈Βϡीϩᆶӭᗺीϩషӝޑ௃׎Ƕ Οǵҗܭҁࣴزϐෳᡍࠠᄊឦܭᚒ໔ӭӛࡋǴࡕុࣴزޣёۯ՜ࣴزԿᚒϣӭӛ ࡋǴа௖૸ᚒϣӭӛࡋჹ HO-IRT ኳԄޑ՗ीԋਏǶ Ѥǵҗܭҁࣴز܌ගрϐୖኧ՗ीኳԄǴ҂ૈӧ܌Ԗ೛ीΠၲډ΋ठᓬܭ PISA ܌٬Ҕϐ՗ीБԄǴࡕុࣴزޣёаଞჹୖኧ՗ीБԄ຾ՉׯؼǶ ϖǵҗܭҁࣴز܌٬Ҕϐ՗ी೬ᡏ WinBUGSǴӧ՗ीਔ໔΢ሡၨΦޑਔ໔Ǵࡕ ុࣴزޣёԾՉኗቪำԄǴаׯ๓՗ीޑਔ໔Ƕ

Ϥǵҗܭҁࣴز҂௖૸ႝတϯ፾܄ෳᡍǵ฻ϯǵDIFȐDifferential Item Functioningȑ ޑᔠۓǾ฻᝼ᚒǴࡕុࣴزޣёаଞჹ೭٤᝼ᚒ຾௖૸Ƕ

(65)

ୖԵЎ᝘

ύЎ೽ҽ!

Цᝊ❲Ȑ1995ȑǶ౜౜жෳᡍ౛ፕǶᆵчѱǺЈ౛рހޗǶ է҇ჱȐ1992aȑǶ၂ᚒϸᔈ౛ፕޑϟಏȐ΋ȑ–ෳᡍ౛ፕޑว৖ᖿ༈Ƕࣴࣴಞၗૻ 8(6)Ǵ13-18Ƕ է҇ჱȐ1992bȑǶ၂ᚒϸᔈ౛ፕޑϟಏȐΒȑ–ෳᡍ౛ፕޑว৖ᖿ༈Ƕࣴࣴಞၗૻ 9(1)Ǵ5-9Ƕ է҇ჱȐ1992cȑǶ၂ᚒϸᔈ౛ፕޑϟಏ(Ο)-၂ᚒϸᔈኳԄϷځ੝܄Ƕࣴࣴಞၗૻ 9(2)Ǵ6-10Ƕ ݅ྨ౺ǵቅဃ۸ǵ݅ન༾ǵ׵ཧȐ2008ȑǶѠѠ᡼ୖу PISA 2006 ԋ݀ൔ֋Ƕ޸ጪ ѱǺ୯ҥ޸ጪ௲ػεᏢǶ ஭႒൤ǵЦШमǵֆችηǵڬЎ๭Ȑ2006ȑǶ୷ҁૈΚຑໆၠ୯ว৖࿶ᡍϐКၨǶ ௲ػၗ਑ᆶࣴزǴ68Ǵ81-99Ƕ ഋ࢙ᐫȐ2006ȑǶૈΚ՗ीБݤჹӭӛࡋႝတϯ፾܄ෳᡍෳໆᆒྗࡋޑቹៜǶ௲௲ ػЈ౛ᏢൔǴ38(2)Ǵ93-210Ƕ ቅ޲྆Ȑ2006ȑǶ၂၂ᚒϸᔈ౛ፕӭ࣬ኳԄϐۯ՜ᆶᔈҔǶ୯ҥύ҅εᏢЈ౛Ꮲࣴ ز܌ᅺγፕЎǶ჏ကᑜǶ

!