多向度試題反應理論用於次級量尺分數估計之模擬研究

୯ҥᆵύ௲ػεᏢ௲ػෳᡍ಍ीࣴز܌౛ᏢᅺγፕЎ

ࡰᏤ௲௤Ǻ೾դԽ റγ

ӭӛࡋ၂ᚒϸᔈ౛ፕҔܭԛભໆЁ

ϩኧ՗ीϐኳᔕࣴز

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ѐԃޑ೭ঁਔংǴ࣮๱નሺᏢۊעךቪ຾ӴޑፕЎᖴᜏϐύǴ٠֋ນךǺȨᖴ ᜏǴ۳۳ࢂ΋ጇፕЎ္ന੿ЈǵനԖག௃ޑ΋೽ҽȩǴ྽ਔޑך൩ӧЈ္֋ນԾ ρǺܴԃޑ೭ঁਔংǴךΨाӳӳӦҔЈቪΠךޑᖴᜏǶӵϞǴךಖܭΨوډΑ ೭΋څǴёаӧᖴᜏ္੿௃ࢬ៛ӦᇥрךޑགᖴǴࣗࢂഒ៌Ǽ ኗቪፕЎය໔Ǵတੇ္ਔத؅ዬ௢ᄽ๱೚ӭόӕޑᖴᜏހҁǴ೭ኬޑགྷႽࢂ Ѝኖךዖၸ਋שǵֹԋፕЎޑ୏Κϐ΋Ǵᕴӧளډࢌ΋ঁΓޑᔅշਔǴѕᓲԾρ ձבΑགᖴдȐӴȑǶচаࣁȨᖴᜏȩ཮ࢂךӧ೭ҁፕЎ္നளЈᔈЋǵජ฽ҥ ൩ޑ೽ϩǴଁ਑Ԝڅ֤ӧႝတᑻჿ߻ޑךǴЈ္ޑགڙӵԜፄᚇǴԭགҬ໣ǾǴ ך၂๱ᄟѺᗖዬע܌Ԗӧ೭ࢤය໔ഉՔךǵڐշךޑாॺ΋΋ቪΠǴభอޑЎӷ းၩ๱٫ᑉ૱ЈӦགᖴǼ གᖴኑངךޑР҆Ǵாॺගٮ๏ך΋ঁྕཪکᒋޑৎǴᡣךёаӄЈӄΚӦ ֹԋᏢ཰Ƕ྽ךӧࣴزၸำύၶ΢਋שԶݪ഼όςਔǴாॺޑႝ၉ኃୢϷངЈߡ ྽೿஥๏ךคКޑΚໆᆶྕཪǴᡣךԖ߿਻ᝩុय़ჹ֚ᜤǴँઇख़ൎǶᖴᖴݿݿ ༰༰๏ךᅈྈޑངǴؒԖாॺޑЍ࡭ᆶ΋ၡޑ࣬ՔǴஒؒԖ౜ӧޑךǴᖴᖴாॺǼ གᖴ׌׌Ͽ఼ӧᅺΒ΢Ꮲයਔ؂ϺௗଌךǴᡣךӧ؂ঁڹఁ္೿ૈ٦ڙ೏ៈଌǵ ೏ྣ៝ޑྕཪǴգࢂךനё᎞ޑៈ޸٬ޣǶ྽ฅǴךޕၰ೭΋ၡ΢գΨᕴࢂᓨᓨ ӦЍ࡭๱ךǴᖴᖴգǼ གᖴךޑࡰᏤ௲௤೾դԽԴৣǴᖴᖴாவךεᏢਔය൩όᘐႴᓰךǵගܘ ךǴ๏ך೚ӭୖᆶᏢೌࣴزޑᐒ཮ǴᖴᖴாЇሦך፯຾ࣴزϐߐǴ׳གᖴாӧ೭ ٿԃٰคፕࢂғࢲ΢ޑྣៈ܈ࢂࣴز΢ޑࡰᏤǴா೿ගٮך೚ӭၗྍǴᡣךёа ໩ճӦֹԋᏢ཰ǶӢࣁாޑගឫᆶ΋ၡ΢ޑྣ៝ǴᡣךளаֹԋᅺγᏢՏȋ΋໨ ךම࿶གྷႽόډ཮ӧՖਔωૈၲԋޑҞ኱ǴᖴᖴாǼགᖴךޑٿՏα၂ہ঩මࡌ ሎԴৣᆶࡼቼᡕԴৣǴᖴᖴමԴৣӧፐ୸΢ҔЈӦ௲௤ǴᡣךᏢډࡐӭෳᡍሦୱ ޑ࣬ᜢޕ᛽ǹᖴᖴࡼԴৣ௲Ꮴךॺ଺ᏢୢޑᄊࡋǴᡣךᙣ཈Ӧय़ჹךޑࣴزǹ྽ ฅǴ׳ाགᖴΟՏԴৣॺӧፕЎα၂ਔ๏ךޑࡌ᝼ϷගᒬǴᡣךޑፕЎ׳ᖿֹഢǶ ӧך຾ՉኳᔕჴᡍࣴزޑၸำύǴགᖴཫറᏢߏόჇځྠӦᆶךӅӕ૸ፕǴ ךޕၰ྽ਔংޑך΋ۓᡣգ໾೸တโǴՠգᗋࢂऐЈӦ๏ϒך೚ӭୖԵࡌ᝼ǹΨ གᖴۗ൛Ꮲۊ๏ךޑࡰᗺᆶεΚޑڐշǴӳ൳ԛၶ΢ୢᚒӛی؃௱ǴیᕴࢂճҔ πբϐᎩᔅךᔠࢗำԄǵᔅךନᒱǴಒЈӦᔅךפрୢᚒ܌ӧǹᖴᖴች㧌Ꮲۊᆶ ☰ॳᏢۊӧךᢀۺኳጋਔǴ๏ϒך೚ӭගᒬǴᔅךᙶమᢀۺǴԜѦǴΨᖴᖴች㧌 Ꮲۊӧך߃ௗҺीฝࣴزঋҺշ౛ਔǴ๏ϒך೚ӭჴ୍΢ޑࡰᏤᆶڐշǴᡣךૈ ໩ճയҺ೭ঁᙍ୍ǹᗋԖགᖴࣴز࠻္ޑڂՙǵࡏ໋ǵඵࣁǵػໜǵ໋๔ǵЎߪ ฻ፏՏᏢߏǴӧךᅺ੤೭ٿԃ္ම࿶๏ϒךޑᔅշǶ ؂ϺරΐఁΐǴӧࣴز࠻္࣬ೀ᏾᏾ٿԃޑӕᏢॺǴךΨाᖴᖴգॺǼ२Ӄ ࢂךޑӳұՔذԹǴᖴᖴیӧᅺΒ೭΋ԃ္ᔅךӭ܍ᏼΑόϿᏢߏҬᒤޑ٣໨Ǵ

ᡣךό཮ΞӢࣁঋҺշ౛πբԶϩيЮೌǴᅺΒΠᏢය஑Ј׫ΕፕЎޑ೭ࢤਔ ໔Ǵᖴᖴیᕴࢂ᠋ךΟόϖਔޑܤ࡜Ǵ྽ךӧำԄ΢࿘ᏛਔǴᖴᖴیΨᕴࢂගٮ ך೚ӭ௱ජǴ੿ޑࡐଯᑫԖیᆶך΋ଆ٠ުբᏯǹᖴᖴ٫ᐇکҺῑޑഉՔǴԖی ॺ΋ଆոΚǴᡣךॺόමགډېൂǴӕਔᖴᖴ٫ᐇᔅךҙፎஎްߐ࿣ьǴࢌ൳ঁ ڹఁǴ൩ഭΠךॺॵӧࣴز࠻္ࡷᐩڹᏯǴٗࢂᜤבޑӣᏫǹᖴᖴნጩǵሎᇬǵ דയǵϘണǵγരǴࣴز࠻္ԖգॺቚబΑόϿ៿኷ǴΨӢࣁԖգॺǴᡣךӧႝ တ΢ԖୢᚒਔǴᕴૈளډനזޑڐշᆶှ،ǴᖴᖴգॺǼ ᗋԖࣴز࠻္ޑᏢ׌ۂॺǴ܃զǵች࣑ǵۏ։ǵٍػǵቼࣤǴᖴᖴգॺϩᏼ Α΋٤ᚇ୍πբǴᡣךॺૈ୼׳Јค਒ᜰӦֹԋፕЎǴЀځǴ੝ձाགᖴ܃զᏢ ۂǴӧךኗቪፕЎޑനࡕ໘ࢤ္Ǵਔத੝Ӧ੮ӧࣴز࠻ഉՔךǴ฻ך΋ӕӣஎްǴ ӳ൳ԛӢࣁፕЎᓸΚԶᓨᓨࢬఽǴ೿ࢂیӧMSN΢๏ךذذ๏ךӼኃǴᖴᖴیǼ ӧࣴز܌ғఱύǴନΑٰԾৎΓޑᜢངڛៈǵৣߏॺޑࡰᏤǵᏢߏۊޑྣ៝ аϷࣴز࠻္౲ӭұՔॺޑഉՔϐѦǴᗋԖ೚ӭόӧךيᜐࠅుుЍ࡭๱ךޑܻ ϶ॺ஥๏ך೚ӭ߻຾ޑΚໆǴךΨགྷा΋΋གᖴգȐیȑॺǶᖴᖴ؃੿ஏ࠻K958 ޑұՔનሺᏢۊǴᅺ΋ΠᏢයᆶیӧK958 Ӆ٣ޑᗺᗺᅀᅀࢂך΋ࢤᜤבޑӣᏫǴ ᖴᖴᏢۊ๏ךࡐӭΓғᄊࡋޑ໒ᏤǴᖴᖴᏢۊ΋ޔوӧך߻य़ᡣךԖঁոΚޑҞ ኱ǴȨ୲࡭ǵኾΚǵፂፂፂǼȩךಖܭΨوၸΑξࢰ္ޑ໵སਔයǴᖴᖴیӧ೭ ࢤਔ໔္ޑࠀᓰǹᖴᖴךᇡ᛽ΑΜԃа΢ޑӳܻ϶࠮ފᆶҥ൛ǴیॺٿՏ೿ࢂך ࡐख़ाǵࡐࣔெޑܻ϶Ǵჹךޑख़ाำࡋόϩଈၫǴԵቾؼΦǴ،ۓӃགᖴவ୯ ύ൩࣬᛽ޑ࠮ފᇥଆǴӳ༏ǻ࠮ފǴᖴᖴی೭΋ၡаٰޑЍ࡭ᆶഉՔǴךޕၰЍ ࡭ޑΚໆᆶྕཪჹךॺ۶Ԝ೿ࢂख़ाޑЍࢊǴᖴᖴیջ٬ᇻӧऍ୯ϝᙑ஥๏ךؒ ԖਔৡޑӼۓΚໆǴ፾ਔޑ΋೯ႝ၉ǵ΋ࢤ੮قǵ΋࠾ແҹǴیޑ΋ѡᜢЈୢং ᕴࢂૈ஥๏ךྕཪǵᡣךԖᝩុ߻຾ޑΚໆǴךޕၰی΋ۓܴқךޑགᖴǴჹ༏ǻ ҥ൛ǴךനᒃངޑΤηǴᖴᖴیΨᕴࢂӧङࡕЍ࡭๱ךǴ྽ךၶ΢਋שਔǴӣགྷ ྽߃ޑیΨම࿶߿ඪӦوၸ೭΋ࢤၡǴך൩ૈࠀᓰԾρႴଆ߿਻य़ჹᜤᜢǴᖴᖴ یӧԆ࿛ޑπբϐᎩᜤளӣٰᙦচ΋፩൩཮ډѠύٰפךǴᗎऊ္ޑፋ၉хᛥ࿤ ຝǴ۳۳ӧଌیډًઠམًࡕǴךΞૈкᅈରדǴᖴᖴیǼ ᗋԖךޑ΋ဂεᏢӕᏢॺǴ໡ൟǵ٫ॣǵΞϓǵݒ๩ǵ໡൛Ǵךޕၰیॺӧ ठΚܭԾρޑπբ஢Տ΢ਔϝฅኘр΋ᗺᗺޑЈࡘᜢЈ๱ךǴόਔ๏ךႴᓰǵ๏ ךѺ਻ǴԖیॺޑуݨǴךಖܭ΋؁΋؁ӦֹԋΑךޑፕЎǴᖴᖴیॺǼ ӧԜǴᗋा੝ձགᖴᏢᇶύЈޑ٫໥ԴৣǴᖴᖴாӧך௃ᆣၶ΢֚ᘋǵ܍ڙ ᓸΚਔ๏ϒךന፾Ϫޑ໒Ꮴᆶಡ᠋ޑഉՔǴᖴᖴாഉךوၸ೭΋ࢤၡǶ നࡕǴךགྷஒ೭ҽԋ݀᝘๏ךལངޑѦϦǴᖴᖴாᕴࢂჹךԖ๱ుϪޑය ఈǴᡣך஥๱೭ҽᚎ฼ޑΚໆوӧ؃Ꮲޑၡ΢Ǵ׳у୲ۓǼ ᖴ٫ᑉ ᙣᇞ 2009/06/24 ܭࣴز࠻

ᄔ ा

ҁࣴزаኳᔕჴᡍБԄ௖૸όӕԛભໆЁϩኧ՗ीБݤܭൂ΋ෳᡍ೛ीϷ ฻ϯෳᡍ೛ीύǴӧӚᅿჴᡍ௃ნΠϐ՗ीਏ݀Ƕൂ΋ෳᡍ೛ीύԵቾѤᅿӢ નǺᚒҁԛભໆЁঁኧǵԛભໆЁෳᡍߏࡋǵԛભໆЁ࣬ᜢำࡋϷࡼෳΓኧǹ฻ ϯෳᡍ೛ीϐ฻ϯ೛ी٬Ҕۓᗕό฻ಔ೛ीȐnon-equivalent groups with anchor test design, NEATȑᆶѳᑽόֹӄ୔༧೛ीȐbalanced incomplete block, BIBȑǴځ ύԵቾΟᅿӢનǺᚒҁԛભໆЁКٯǵԛભໆЁ࣬ᜢำࡋϷࡼෳΓኧǶ ҁࣴز่݀ว౜Ǻ 1. ԛભໆЁϩኧ՗ीᇤৡᒿ๱ԛભໆЁ࣬ᜢำࡋቚуԶ෧ϿǴԶࡼෳΓኧ߾ό ቹៜ՗ीǹ 2. ൂ΋ෳᡍ೛ीύǴԛભໆЁϩኧ՗ीᇤৡᒿ๱ᚒҁԛભໆЁঁኧቚуԶᡂ εǵᒿ๱ԛભໆЁෳᡍߏࡋቚуԶ෧Ͽǹ 3. ฻ϯෳᡍ೛ीύǴԛભໆЁϩኧ՗ीᇤৡᒿ๱ᚒҁԛભໆЁКٯᝌਸำࡋቚ уԶᡂεǴЪ NEAT ᆶ BIB ٿᅿ฻ϯ೛ीΠǴԛભໆЁϩኧ՗ीคܴᡉৡ౦Ƕ ᜢᗖӷǺ฻ϯǵӭӛࡋ၂ᚒϸᔈ౛ፕǵԛભໆЁǵۓᗕό฻ಔ೛ीǵѳᑽόֹӄ ୔༧೛ी

Abstract

The purpose of this paper is to evaluate the performances of the different subscale scores estimation methods by using the simulation data in two testing design situations, the single test design and the equated test design with non-equivalent groups with anchor test design (NEAT) and balanced incomplete block (BIB). In the single test design, factors taken into consideration include the following: a number of the subscales, the test lengths of the subscales, the correlation coefficients between the subscales, and the sample sizes. In the equated test design, factors taken into consideration include the following: the ratios of the subscales, the correlation coefficients between the subscales, and the sample sizes.

The major findings of this study are summarized as follows:

1. The estimation errors decrease as the correlation coefficients between the subscales increases; however, the estimation errors are not impacted by the sample sizes.

2. In the single test design, the estimation errors increases as a number of the subscales increase and the estimation errors decrease as the test lengths decrease. 3. In the equated test design, the estimation errors increase as the ratios of the

subscales increase and the estimation errors with NEAT and BIB are almost the same.

Ҟᒵ

ᄔा ... I Ҟᒵ ...III ߄Ҟᒵ ... IV კҞᒵ ...V ಃ΋ക ᆣፕ ...1 ಃ΋࿯ ࣴزङඳᆶ୏ᐒ ...1 ಃΒ࿯ ࣴزҞޑ ...3 ಃΟ࿯ ࡑเୢᚒ ...4 ಃѤ࿯ Ӝຒញက ...4 ಃΒക Ў᝘௖૸ ...6 ಃ΋࿯ ၂ᚒϸᔈ౛ፕ ...6 ಃΒ࿯ ԛભໆЁϩኧ՗ीБݤ ...14 ಃΟ࿯ ෳᡍ฻ϯޑཀကᆶ฻ϯ೛ी ...26 ಃΟക ࣴزБݤ ...30 ಃ΋࿯ ࣴزࢬำ ...30 ಃΒ࿯ ࣴزᡂ໨೛ۓ ...31 ಃΟ࿯ ჴᡍ೛ी ...35 ಃѤ࿯ ՗ीᆒྗࡋ ...39 ಃϖ࿯ ࣴزπڀ ...39 ಃѤക ࣴز่݀ ...41 ಃ΋࿯ ൂ΋ෳᡍ೛ीϐ՗ी่݀ ...41 ಃΒ࿯ ฻ϯෳᡍ೛ीϐ՗ी่݀ ...45 ಃϖക ่ፕᆶ҂ٰࣴزࡌ᝼...50 ಃ΋࿯ ่ፕ ...50 ಃΒ࿯ ҂ٰࣴزࡌ᝼ ...54 ୖԵЎ᝘ ...55 ύЎ೽ϩ...55 मЎ೽ϩ...55 ߕᒵ΋ ൂ΋ෳᡍ೛ीϐᇤৡRMSE ...62 ߕᒵΒ ฻ϯෳᡍ೛ीϐᇤৡRMSE ...68 ߕᒵΟ REGPБݤܭ฻ϯෳᡍ೛ीϐᇤৡRMSE...70

߄Ҟᒵ

߄ 2-1! NEAT೛ी ...27 ߄ 3-1 ൂ΋ෳᡍ೛ीϐӅӕᡂ໨೛ۓ ...32 ߄ 3-2 ฻ϯෳᡍ೛ीϐӅӕᡂ໨೛ۓ ...33 ߄ 3-3 NEATᚒҁଛ࿼߄ ...36 ߄ 3-4 BIBᚒҁଛ࿼߄...36

კҞᒵ

კ 2-1 ᚒ໔ӭӛࡋෳᡍኳԄ...9 კ 2-2 ᚒϣӭӛࡋෳᡍኳԄ...10 კ 3-1 ࣴزࢬำკ...31 კ 4-1 ൂ΋ෳᡍ೛ीΠόӕᚒҁԛભໆЁঁኧϐRMSE...41 კ 4-2 ൂ΋ෳᡍ೛ीΠόӕԛભໆЁෳᡍߏࡋϐRMSE...42 კ 4-3 ൂ΋ෳᡍ೛ीΠόӕԛભໆЁ࣬ᜢำࡋϐRMSE...43 კ 4-4 ൂ΋ෳᡍ೛ीΠόӕࡼෳΓኧϐRMSE...44 კ 4-5 ฻ϯෳᡍ೛ीΠόӕᚒҁԛભໆЁКٯϐRMSE...46 კ 4-6 ฻ϯෳᡍ೛ीΠόӕԛભໆЁ࣬ᜢำࡋϐRMSE...47 კ 4-7 ฻ϯෳᡍ೛ीΠόӕࡼෳΓኧϐRMSE...48

ಃ΋കʳ ᆣፕ

ҁࣴزਥᏵ၂ᚒϸᔈ౛ፕȐitem response theory, IRTȑύൂୖኧ Rasch ኳԄ Ȑone-parameter logistic model, 1PLȑᆶӭӛࡋ၂ᚒϸᔈ౛ፕȐmultidimensional item response theory, MIRTȑύӭӛࡋᒿᐒ߯ኧӭ໨ logit ኳԄȐmultidimensional random coefficients multinomial logit model, MRCMLMȑǴаኳᔕჴᡍБԄ௖૸ όӕԛભໆЁϩኧ՗ीБݤӧൂ΋ෳᡍ೛ीᆶ฻ϯෳᡍ೛ीύǴԛભໆЁϩኧ ՗ीϐਏ݀Ƕҁകஒଞჹࣴزङඳᆶ୏ᐒǵࣴزҞޑǵࡑเୢᚒᆶӜຒញက຾ ՉឍॊǶ

ಃ΋࿯ʳ ࣴزङඳᆶ୏ᐒ

ᒿ๱ෳᡍࠠᄊϐᡂᎂᆶሡ؃Ǵ୯ϣѦᏢޣ೴ᅌख़ຎεࠠෳᡍȐlarge-scale assessmentsȑޑ᝼ᚒǶεࠠෳᡍ٩ᏵόӕޑෳᡍфૈǴεठёϩࣁٿᅿᜪࠠǴ΋ ࣁڀԖᑔᒧфૈϐεࠠෳᡍǴҞޑӧܭෳໆᏢғޑᏢࣽૈΚǴගٮᏢғଯύΕᏢ ܈εᏢΕᏢϐୖԵ܈٩ᏵǴٯӵ୯ϣϐȨ୯ύ୷ҁᏢΚෳᡍȩǵȨεᏢᏢࣽૈΚ ෳᡍȩǴऍ୯εᏢΕᏢෳᡍȐAmerican College Test, ACTȑ฻Ƕќ΋ᅿࢂҔаࡌ ҥ௲ػၗ਑৤ϐεࠠෳᡍǴҞޑӧܭࡌ࿼΋঺࠼ᢀЪֹ๓ޑᏢғᏢಞԋ൩ၗ਑ ৤ǴᙖҗٯՉ܄Ӧᇆ໣Ꮲғܭෳᡍύϐ߄౜Ǵ٠уа಍ीǴଓᙫᏢғޑᏢಞԋ݀ Ϸϩ݋ځᏢಞᡂᎂϐᖿ༈Ǵ຾Զᔠຎ୯ৎ௲ػࡹ฼ჴࡼࢂցֹ๓ǴٯӵȨѠ᡼௲ ػߏයଓᙫၗ਑৤ȩȐTaiwan Education Panel Survey, TEPSȑȩǵȨᆵ᡼ᏢғᏢ ಞԋ൩ຑໆၗ਑৤ȐTaiwan Assessment of Student Achievement, TASAȑȩǵȨ୯ ሞኧ౛ᖿ༈ࣴزȐThe Trends in International Mathematics and Science Study, TIMSSȑȩǵȨ୯ৎ௲ػ຾৖ຑໆȐNational Assessment of Educational Progress, NAEPȑȩϷȨ୯ሞᏢғຑໆȐProgram for International Student Assessment, PISAȑȩ ฻ǴࣣࣁԜᜪࠠϐεࠠෳᡍǶεࠠෳᡍӧჴࡼਔத཮ၶډ೚ӭୢᚒǴٯӵǺᚒ৤

Ȑitem bankȑࡌҥǵᚒҁ೛ीȐbooklet designȑǵၗ਑ԏ໣೛ीȐdata collection designȑǵኬҁޑ೛ीȐsample designȑǵ೯ၸ኱ྗ೛ۓȐpassing criteriaȑǵୖኧ ՗ीȐparameter estimationȑǵໆЁϯำׇȐscaling proceduresȑǵϩኧໆЁȐscore scaleȑϐीᆉǵԛભໆЁϩኧȐsubscale scoreȑϐൔ֋฻Ƕ೭٤ୢᚒε೽ϩς࿶ Ԗ೚ӭ࣬ᜢЪֹ᏾ޑࣴزൔ֋Ϸჴࡼำׇޑ௖૸ǴٯӵǺTEPSЈ౛ीໆൔ֋Ȑླྀ ۏ᜽ǵ᛿நᄪǵ໳௵໢Ǵ2003ȑǵThe NAEP 1998 Technical ReportȐNance, John, & Terry, 2001ȑǵTIMSS 2003 Technical ReportȐMartin, Mullis, & Chrostowski, 2004ȑǵNational Indian Education Study 2007 Part IȐMoran, Rampey, Dion, & Donahue, 2008ȑ฻ǴฅԶεӭኧޑεࠠෳᡍמೌൔ֋ύǴࠅᗲϿԖჹܭԛભໆЁ ϩኧϐ૸ፕǶ

ෳᡍޑ᏾ᡏϩኧ೯தҔٰຑᘐঁΓ฻ભǴෳᡍޑԛભໆЁϩኧ࿶தԖշܭ௲ ৣຑᘐᏢғޑ੝ਸ஑ߏϷ১ᗺȐWainer, Vevea, Camacho, Reeve, Rosa, Nelson, Swygert, & Thissen, 2000ǹYen, 1987ȑǶऩाޕၰᏢғӧӚय़ӛϐ߄౜Ǵӵૈޔ ௗෳໆډᏢғӧӚय़ӛޑૈΚǴஒКவᏢғ᏾ᡏԋᕮٰႣෳځ߄౜Ԗ׳ӳޑਏ݀ ȐBock, Thissen, & Zimowski, 1997ȑǶӢԜǴऩૈᆒྗޑ՗ीԛભໆЁϩኧǴߡૈ Ԗਏගٮڙ၂ޣ׳ӭૻ৲Ǵ܌аԛભໆЁϩኧޑൔ֋ҭࣁ೚ӭεࠠෳᡍགᑫ፪ޑ ୢᚒȐKahraman & Kamata, 2004ȑǶᖐٯٰᇥǴ PISA 2006ኧᏢૈΚෳᡍ ȐMathematical Literacy in PISA 2006ȑǴෳᡍϣ৒х֖ኧໆȐquantityȑǵޜ໔ᆶ ׎ރȐspace and shapeȑǵ௢ፕȐreasoningȑϷόዴۓ܄Ȑuncertaintyȑ฻य़ӛȐPISA 2006ȑǶᆕᢀ΢ॊёޕǴᙖҗෳᡍޑ᏾ᡏϩኧૈΑှᏢғޑ᏾ᡏ߄౜ǴԶᙖҗෳ ᡍޑԛભໆЁϩኧൔ֋߾ૈև౜Ꮲғӧኧໆǵޜ໔ᆶ׎ރǵ௢ፕϷόዴۓ܄฻य़ ӛޑᓬ༈ᆶӍ༈ȐаPISA 2006ࣁٯȑǴό໻ԖշܭঁձϯޑᏢಞࡰᏤǴ׳ૈඓ ඝᏢғӚय़ӛޑ߄౜НྗǶ

Ў᝘ύǴGessaroliȐ2004ȑǵTateȐ2004ȑϷ Yao ᆶ BoughtonȐ2007ȑࣣа ӭӛࡋ၂ᚒϸᔈ౛ፕ՗ीԛભϩኧȐsubscoreȑǹ஭ۗ൛Ȑ2008ȑම௖૸ԛભໆ

Ёϩኧ՗ीᔈҔܭ௲ػෳᡍ௃ნϐࣴزǶࡺҁࣴزᔕуΕӭӛࡋ၂ᚒϸᔈ౛ፕ БݤȐMIRT methodȑٰ՗ीԛભໆЁϩኧǶ

Ԝ Ѧ Ǵ ೚ ӭ ε ࠠ ෳ ᡍ ϐ ෳ ᡍ ᚒ ҁ ೱ ่ ೛ ी ௦ Ҕ ۓ ᗕ ό ฻ ಔ ೛ ी Ȑnonequivalent groups with anchor test design, NEATȑϷѳᑽόֹӄ୔༧೛ी Ȑ balanced incomplete block, BIB ȑ ٿ ᅿ ฻ ϯ ೛ ी Ƕ ٯ ӵ Ǻ ഞ Ԁ ᆕ ӝ ෳ ᡍ ȐMassachusetts comprehensive assessment system, MCASȑջ௦ҔNEAT೛ीǴ Զ಻ើޑPPONȐPeriodiek Peilingsonderzoek van het Onderwijsȑǵऍ୯୯ৎ௲ ػ຾৖ຑໆȐNational Assessment of Educational Progress, NAEPȑаϷѠ᡼Ꮲғ Ꮲಞԋ൩ຑໆၗ਑৤ȐTaiwan Assessment of Student Achievement, TASAȑϐࡌ ࿼ीฝࣣ௦ҔBIB೛ीȐЦཫറǴ2006ȑǹࡺҁࣴزᔕаNEATᆶBIBٿᅿ฻ϯ ೛ीբࣁෳᡍᚒҁϐೱ่೛ीǶ

ಃΒ࿯ʳ ࣴزҞޑ

஭ۗ൛ǵڬ໡അǵ೚ϺᆢᆶࡼలীȐ2008ȑම௖૸ԛભໆЁϩኧ՗ीᔈҔ ܭ௲ػෳᡍ௃ნϐࣴزǹЦཫറǵᖴ٫ᑉǵֆҺῑᆶ೚ϺᆢȐ2008ȑҭ௖૸ό ӕԛભໆЁϩኧ՗ीБݤᔈҔӧεࠠෳᡍϐ฻ϯਏ݀Ǵࣴزύ௖૸ϐ฻ϯਏ݀ ӧܭКၨόӕ฻ϯБݤΠǴόӕԛભໆЁϩኧ՗ीБݤϐ՗ीਏ݀ǴЪ໻૸ፕ ൂ΋ᅿ฻ϯ೛ीȐNEATȑϐ฻ϯਏ݀Ƕ୷ܭ΢ॊࣴزϐԋ݀Ǵҁࣴزаኳᔕჴ ᡍБԄ௖૸όӕ฻ϯ೛ीϷόӕᚒҁԛભໆЁКٯჹܭԛભໆЁϩኧ՗ीϐቹ ៜǶЪӧԛભໆЁϩኧ՗ीБݤ΢Ǵቚуӭӛࡋ၂ᚒϸᔈ౛ፕБݤǴ٠૸ፕӚ ՗ीБݤϐਏ݀Ƕ ᆕӝ΢ॊǴ૟ஒҁࣴزीฝҞޑᔕۓӵΠǺ ΋ǵʳ ௖૸ൂ΋ෳᡍ೛ीύǴόӕԛભໆЁϩኧ՗ीБݤܭόӕᚒҁԛભໆЁ ঁኧǵԛભໆЁෳᡍߏࡋǵԛભໆЁ໔࣬ᜢำࡋϷࡼෳΓኧϐ՗ीਏ݀Ƕ

Βǵʳ ௖૸฻ϯෳᡍ೛ीύǴӧόӕ฻ϯ೛ीΠǴόӕԛભໆЁϩኧ՗ीБݤ ܭόӕᚒҁԛભໆЁКٯǵԛભໆЁ໔࣬ᜢำࡋǵࡼෳΓኧϐ՗ीਏ݀Ƕ

ಃΟ࿯ʳ ࡑเୢᚒ

٩Ᏽ΢ॊϐࣴزҞޑǴᔕܭൂ΋ෳᡍ೛ीᆶ฻ϯෳᡍ೛ीϩձගрΠӈ൳ ໨ୢᚒǺ

൘ǵʳൂ΋ෳᡍ೛ी

΋ǵʳ όӕᚒҁԛભໆЁঁኧࢂցቹៜԛભໆЁϩኧ՗ीϐਏ݀ǻ Βǵʳ όӕԛભໆЁෳᡍߏࡋࢂցቹៜԛભໆЁϩኧ՗ीϐਏ݀ǻ Οǵʳ όӕԛભໆЁ໔࣬ᜢำࡋࢂցቹៜԛભໆЁϩኧ՗ीϐਏ݀ǻ Ѥǵʳ όӕࡼෳΓኧࢂցቹៜԛભໆЁϩኧ՗ीϐਏ݀ǻ

ມǵʳ฻ϯෳᡍ೛ी

΋ǵʳ όӕ฻ϯ೛ीࢂցቹៜԛભໆЁϩኧ՗ीϐਏ݀ǻ Βǵʳ όӕᚒҁԛભໆЁКٯࢂցቹៜԛભໆЁϩኧ՗ीϐਏ݀ǻ Οǵʳ όӕԛભໆЁ໔࣬ᜢำࡋࢂցቹៜԛભໆЁϩኧ՗ीϐਏ݀ǻ Ѥǵʳ όӕࡼෳΓኧࢂցቹៜԛભໆЁϩኧ՗ीϐਏ݀ǻ

ಃѤ࿯ʳӜຒញက

൘ǵʳԛભໆЁϩኧ

ԛભໆЁϩኧ߯ࡰ၂ᚒη໣Ȑitem subsetsȑޑϩኧǴҔٰ߄ҢᏢғӧᏢಞ Ҟ኱Ȑlearning objectivesȑǵηෳᡍȐsubsetsȑ܈Ꮲಞ኱ྗȐlearning standardsȑ ϐ߄౜ȐMeyers, Shin, & Nichols, 2008ȑǶӵኧᏢࣽԋ൩ෳᡍх֖ኧᆶໆǵжኧǵ

൳ՖϷ಍ीᆶᐒ౗฻य़ӛǴ೭٤य़ӛࣣࣁԛભໆЁȐsubscaleȑǴӧ၀ԛભໆЁ ϐளϩջࣁԛભໆЁϩኧǶ

ມǵʳൂ΋ෳᡍ೛ी

ൂ΋ෳᡍ೛ीԖձܭ฻ϯෳᡍ೛ीϐෳᡍᚒࠠǴջෳᡍύ໻Ԗൂ΋ᚒҁǴ ܌Ԗڙ၂ޣࣣբเ܌Ԗ၂ᚒǴࡺёᇆ໣ڙ၂ޣӧ܌Ԗ၂ᚒϐբเϸᔈǶҁࣴز ϐൂ΋ෳᡍ೛ी߯җᒧ᏷ᚒȐmultiple choice items, MC itemsȑಔԋϐൂ΋ᚒҁ ෳᡍǶ

ୖǵʳ฻ϯෳᡍ೛ी

ҁࣴزϐ฻ϯෳᡍ೛ीࣁҗΎঁ၂ᚒ୔༧Ȑblockȑ܌ಔԋϐ NEAT ᆶ BIB ฻ϯ೛ीǴNEAT ೛ीύх֖ΟঁෳᡍᚒҁȐbookletȑǴBIB ೛ीύх֖Ύঁ ෳᡍᚒҁǶ

စǵʳൂӛࡋ IRT

ൂӛࡋ IRT ջࣁൂӛࡋ၂ᚒϸᔈ౛ፕȐunidimensional item response theory, UIRTȑǴҁࣴزܭЎύஒа UIRT ᙁᆀϐǶ

Ҵǵʳӭӛࡋ IRT

ӭӛࡋ IRT ջࣁӭӛࡋ၂ᚒϸᔈ౛ፕȐmultidimensional item response theory, MIRTȑǴҁࣴزܭЎύஒа MIRT ᙁᆀϐǶ

ಃΒകʳ Ў᝘௖૸

ҁࣴزҞޑӧ௖૸όӕԛભໆЁϩኧ՗ीБݤҔܭൂ΋ෳᡍ೛ीᆶ฻ϯෳ ᡍ೛ी௃ნΠǴჹෳᡍϩኧ՗ीϐਏ݀ǶӢԜǴҁകஒଞჹԛભໆЁ՗ीБݤǵ ෳᡍ฻ϯޑཀကǵෳᡍ฻ϯ೛ी฻࣬ᜢࣴز຾Չϩ݋᏾౛ǶҁകӅϩࣁΟ࿯Ǵ ಃ΋࿯ࣁ၂ᚒϸᔈ౛ፕǴϩձϟಏ UIRT ᆶ MIRTǹಃΒ࿯ࣁԛભໆЁϩኧ՗ी БݤǴϩձϟಏҁࣴز܌٬ҔϐΎᅿԛભໆЁϩኧ՗ीБݤǹಃΟ࿯ࣁෳᡍ฻ ϯޑཀကᆶ฻ϯ೛ीǹ၁ॊӵΠǶ

ಃ΋࿯ʳ ၂ᚒϸᔈ౛ፕ

၂ᚒϸᔈ౛ፕਥᏵம༈ଷ೛Ȑstrong assumptionȑԶٰǴᇡࣁڙ၂ޣჹ၂ᚒ ϸᔈޑ҅ዴ܄ϐයఈॶёҔΠԄ߄ҢǺ ) , ( ) (X f I A [ Ȑ2-1ȑ ځύǴ X ࣁ၂ᚒϸᔈޑ҅ዴ܄ǹ I ࣁ၂ᚒୖኧӛໆǹ AࣁૈΚୖኧӛໆǶ җԄηȐ2-1ȑёޕǴ X ޑයఈॶࢂҗ၂ᚒୖኧکૈΚୖኧ܌ԋϐڄኧ܌، ۓޑǶฅԶǴ٬Ҕୖኧࠠ၂ᚒϸᔈ౛ፕ຾Չෳᡍၗ਑ϐϩ݋ਔǴIRT ኳԄѸ໪ ಄ӝѤ໨୷ҁଷ೛ȐWeiss & Yoes, 1991ȑǴҭջൂӛ܄Ȑunidimensionalityȑǵֽ ೽ᐱҥȐlocal independenceȑǵߚೲࡋ܄ȐnonspeednessȑϷȨޕၰ-҅ዴȩଷ೛ Ȑ“know-correct” assumptionȑǶ ܌ᒏൂӛ܄ଷ೛ࢂࡰ΋ҽෳᡍѝෳໆڙ၂ޣ΋ᅿૈΚ܈ወӧ੝፦ǹֽ೽ᐱ ҥଷ೛ࢂࡰڙ၂ޣӧෳᡍόӕ၂ᚒਔǴځբเ௃׎۶Ԝ໔ନΑڙ၂ޣҁيޑૈ ΚϐѦǴόڙځдӢનቹៜǹߚೲࡋ܄ଷ೛ࢂࡰڙ၂ޣޑෳᡍளϩࢂڙҁيૈ Κଯե܌ቹៜǴόӢෳᡍਔ໔ߏอቹៜځளϩǹȨޕၰ-҅ዴȩଷ೛߾ࡰڙ၂ޣ ӧޕၰ၂ᚒޑ҅ዴเਢϐΠǴѸૈเჹ၀၂ᚒǴคΓࣁӢનޑᒱᇤ༤เ௃׎Ƕ ୷ܭ၂ᚒϸᔈ౛ፕޑൂӛ܄ଷ೛Ǵ΋૓٬Ҕϐ၂ᚒϸᔈ౛ፕࣁ UIRTǶฅ

ԶǴӧჴሞᔈҔ΢Ǵ೚ӭෳᡍ௃ნ࿶தх֖ӭঁϩໆ߄܈ϩෳᡍǴόൂѝԖෳ ໆൂ΋ૈΚǴࣁΑᗉխڙ၂ޣӢᚒҞၸӭԶౢғޑੲമ౜ຝǴ೭٤ෳᡍ܌х֖ ޑӭঁϩෳᡍ೯தคݤܫΕϼӭ၂ᚒǴऩа UIRT ঁձ຾ՉӚϩෳᡍ܈ϩໆ߄ ޑϩ݋Ǵ߾ϩෳᡍ܈ϩໆ߄ޑߞࡋ೿όଯǹࡺჴሞᔈҔ΢Ǵൂӛ܄ଷ೛٠ό৒ ܰၲԋǴӢԶᏢޣॺ೴ᅌගр MIRTȐAdams, Wilson & Wang, 1997; Bock & Aitkin, 1981; Fraser, 1988; McDonald, 1967; Mckinley & Reckase, 1983; Sympson, 1978; Whitely, 1980ȑǴаှ،ෳᡍჴሞᔈҔ΢ޑୢᚒȐෳᡍ஑཰πբ֝Ǵ2006ȑǶ ӢԜǴҁ࿯ஒϩձଞჹൂӛࡋ၂ᚒϸᔈ౛ፕϷӭӛࡋ၂ᚒϸᔈ౛ፕ຾Չኳ ԄϐϟಏǶ

൘ǵʳ ൂӛࡋ၂ᚒϸᔈ౛ፕ

தҔޑ UIRT ኳԄԖΟᅿǴ٩ኳԄ܌௦ҔޑୖኧঁኧٰڮӜǴϩձࣁൂୖ ኧჹኧኳԄȐone-parameter logistic model, 1PLȑǵΒୖኧჹኧኳԄȐtwo-parameter logistic model, 2PLȑϷΟୖኧჹኧኳԄȐthree-parameter logistic model, 3PLȑǴ ૟ϩॊӵΠǶ ΋ǵʳ ൂୖኧჹኧኳԄ ൂୖኧჹኧኳԄΞԖ Rasch ኳԄϐᆀǴӧ IRT ޑ 1PL ኳԄΠǴଷ೛ڙ၂ޣ jϐૈΚࣁTjǴځբเ၂ᚒi ೯ၸޑᐒ౗ӵΠȐRasch, 1960ȑǺ )] ( exp[ 1 1 ) , | 1 ( i j i j ij b b X P  T   T Ȑ2-2ȑ ځύǴXijࣁڙ၂ޣ jӧ၂ᚒ ޑբเϸᔈǴเჹ૶ࣁ 1Ǵเᒱ૶ࣁ 0ǹ ࣁ၂ᚒ

ϐ၂ᚒᜤࡋୖኧȐitem difficulty parameterȑǴ

i b_i

i fbi fǶ Βǵʳ ΒୖኧჹኧኳԄ

౗ӵΠȐBirnbaum, 1968ȑǺ )] ( exp[ 1 1 ) , , | 1 ( i j i i i j ij b a a b X P  T   T Ȑ2-3ȑ ځύǴXijࣁڙ၂ޣ jӧ၂ᚒi ޑբเϸᔈǴเჹ૶ࣁ 1Ǵเᒱ૶ࣁ 0ǹ ࣁ၂ᚒ

ϐ၂ᚒ᠘ձࡋୖኧȐitem discrimination parameterȑǴ

i a i 0a_iǹ ࣁ၂ᚒ ϐ၂ᚒ ᜤࡋୖኧǴ i b i f   f  b_i Ƕ Οǵʳ ΟୖኧჹኧኳԄ ӧ IRT ޑ 3PL ኳԄΠǴଷۓෳᡍ཮วғ౒ᚒϐ౜ຝǴࡺଷ೛ڙ၂ޣ j ϐૈ ΚࣁTjǴځբเ၂ᚒ ೯ၸޑᐒ౗ӵΠȐBirnbaum, 1968ǹLord, 1980ȑǺi )] ( exp[ 1 ) 1 ( ) , , , | 1 ( i j i i i i i i j ij b a c c c a b X P  T     T Ȑ2-4ȑ ځύǴXijࣁڙ၂ޣ jӧ၂ᚒi ޑբเϸᔈǴเჹ૶ࣁ 1Ǵเᒱ૶ࣁ 0ǹ ࣁ၂ᚒ ϐ၂ᚒ᠘ձࡋୖኧǴ ǹ ࣁ၂ᚒ ϐ၂ᚒᜤࡋୖኧǴ i a i 0a_i b_i i fbi fǹ ࣁ၂

ᚒ ϐ၂ᚒ౒ෳࡋୖኧȐitem guessing parameterȑǴ

i c i 0dci 1Ƕ

ມǵʳ ӭӛࡋ၂ᚒϸᔈ౛ፕ

΋ǵʳ ӭӛࡋෳᡍޑᅿᜪ MIRT ёҔаϩ݋ӭӛࡋෳᡍ္ޑϩෳᡍ܈ϩໆ߄Ǵӭӛࡋෳᡍёаϩࣁ ٿᅿȐAdams, Wilson & Wang, 1997; Wang, Wilson & Adams, 1997ȑǶ΋ᅿࢂᚒ໔ ӭӛࡋෳᡍȐbetween-item multidimensional testȑǴ೭ᅿෳᡍ္ޑ؂ঁᚒҞѝෳ ൂ΋ᅿૈΚǴջൂӛࡋ၂ᚒǴԶ᏾ҽෳᡍх֖೚ӭൂӛࡋޑ၂ᚒǶᒌ୘Ј౛Ꮲ ৎ࿶த٬ҔޑΓ਱ໆ߄ջឦܭᚒ໔ӭӛࡋෳᡍޑ΋ᅿǴ؂ঁ၂ᚒѝෳໆൂ΋ᅿ ૈΚȐӵঁΓ፾ᔈǵޗ཮፾ᔈ܈௃ᆣ֚ᘋȑǴԶ᏾ҽໆ߄߾х֖Α೭٤ൂӛࡋ၂

ᚒǶΞӵ୯ύ୷ҁᏢΚෳᡍύޑޗ཮ࣽǴෳໆΑх֖ᐕўǵӦ౛ǵϦ҇฻Ꮲࣽ ૈΚǴԾฅࣽෳໆΑх֖ނ౛ǵϯᏢǵғނǵӦౚࣽᏢ฻ᏢࣽૈΚǶ೭ᜪࠠޑ ෳᡍࣁᆕӝૈΚෳᡍǴځ၂ᚒෳໆឦ܄࣬՟ޑૈΚǶᚒ໔ӭӛࡋෳᡍӵკ 2-1 ܌ҢǶ

კ2-1 ᚒ໔ӭӛࡋෳᡍኳԄ

ќ΋ᅿࢂᚒϣӭӛࡋෳᡍȐwithin-item multidimensional testȑǴ೭ᅿෳᡍ္ ޑ؂ঁᚒҞ೿ෳໆόѝ΋ᅿૈΚǴࡺൂ΋ᚒ္൩х֖ӭঁӛࡋǶኧᏢෳᡍ္ޑ ᔈҔᚒᚒࠠջࣁᚒϣӭӛࡋෳᡍǴᔈҔᚒҔЎӷ௶ॊኧᏢୢᚒǴڙ၂ޣሡӃᕕ ှᚒཀǴၮҔ߄ቻૈΚȐrepresentationȑஒୢᚒ௃ნҔᆉԄٰ߄ҢǴωૈ຾Չ ኧᏢीᆉǴ܌а΋ᚒᔈҔୢᚒෳໆΑόѝԖኧᏢीᆉૈΚᗋԖୢᚒ߄ቻૈΚǶ ೭ᜪࠠޑෳᡍӵკ 2-2 ܌ҢǶ ၂ᚒ 1 ၂ᚒ 2 ၂ᚒ 3 ၂ᚒ 4 ၂ᚒ 5 ၂ᚒ 6 ၂ᚒ 7 ၂ᚒ 8 ঁΓ፾ᔈ ޗ཮፾ᔈ ௃ᆣ֚ᘋ

კ 2-2 ᚒϣӭӛࡋෳᡍኳԄ

Βǵ

UIRT ኳԄޑ़ғኳԄǶаΠஒϟಏ൳ᅿத

Ȑ΋ȑ

Reckase, 1983ǹ Reckase & Mckinley, 1991ȑǴځኳԄӵϦԄȐ2-5ȑ܌ҢǺ ၂ᚒ 1 ၂ᚒ 2 ၂ᚒ 3 ीᆉૈΚ! ၂ᚒ 4 ୢᚒ߄ቻૈΚ ʳ ӭӛࡋ၂ᚒϸᔈ౛ፕኳԄ Ҟ߻தـޑ MIRT ኳԄεӭࢂ ـޑӭӛࡋ၂ᚒϸᔈ౛ፕኳԄǶ ʳ ӭӛࡋΒୖኧኳԄ

ӭӛࡋΒୖኧኳԄȐmultidimensional two parameters model, M2PLȑࣁΒୖ ኧ logistic ኳԄȐtwo-parameter logistic model, 2PLȑ܌़ғޑኳԄȐMckinley &

)] ( exp[ 1 1 ) , , | 1 ( i j i a ij i ᚒ i j j i i ӭঁӛࡋޑ᠘ձࡋǴӵԜӭঁӛࡋޑ᠘ձࡋคݤֹ᏾߄౜рൂ΋၂ᚒޑ᠘ձ ࡋǴӢԜ Reckase & McKinleyȐ1991ȑۓကрٿঁதҔޑӭӛࡋࡰ኱Ǵ΋ঁࢂ

j i i ij i d d x P  c   ș ș a Ȑ2-5ȑ ځύ ࣁڙ၂ޣϸᔈࠠᄊǴ1 ߄Ңเჹ၀၂ᚒǴ0 ߄Ңเᒱ၀၂ᚒǶ ࣁ၂ ᠘ձࡋӛໆǴ ࣁ၂ᚒᜤࡋǴ ࣁڙ၂ޣૈΚӛໆǶ೭ঁኳԄᆶচҁΒୖኧ IRT ޑৡձࢂஒচҁޑڙ၂ޣૈΚॶ ᆶ၂ᚒ᠘ձࡋ ᘉ৖ࣁӛໆ Ϸ Ǵ೸ၸӛ ໆٰ߄ҢǴаஒӭӛࡋޑૈΚӕਔх֖ӧኳԄύǶҗܭ၂ᚒ᠘ձࡋӛໆ х֖

ಃ ᚒ ޑ ӭ ӛ ࡋ ᠘ ձ ࡋ ୖ ኧ Ȑ multidimensional discrimination parameter, ȑǺ x a d ș ș a ș a a i i MDISC

¦

m k ik i a MDISC 1 2 1 2 ) ( Ȑ2-6ȑ ځύǴ m ࣁૈΚӛࡋኧҞǶ

ќ΋ঁࢂಃ ᚒޑӭӛࡋᜤࡋୖኧȐmultidimensional difficulty parameter, ȑǺ i i MDIFF i i i MDISC d MDIFF  Ȑ2-7ȑ ќѦǴࣁΑૈڀᡏᢀჸ၂ᚒޑӛࡋ่ᄬǴаᡉҢঁձӛࡋ᠘ձࡋ ᆶӭӛ ࡋ᠘ձࡋୖኧ ϐ໔ޑᜢ߯ǴAckermanȐ1996ȑۓက၂ᚒ܌ाෳໆޑૈ ΚБӛᆶӚૈΚӛࡋ໔ޑ֨فӵΠǺ ik a i MDISC i ik ik MDISC a D cos Ǵk 1,...,m Ȑ2-8ȑ ȐΒȑʳ ӭӛࡋΟୖኧኳԄ

ӭӛࡋΟୖኧኳԄȐmultidimensional three parameters model, M3PLȑࣁΟ ୖኧ logistic ኳԄȐthree-parameter logistic model, 3PLȑׯؼԶளǴஒኳԄύޑ ૈΚୖኧᆶ᠘ձࡋୖኧׯԋӛໆޑࠠԄȐHattie, 1981; Sympson, 1978ȑǴځኳԄ ӵϦԄȐ2-9ȑ܌ҢǺ )] ( exp[ 1 1 ) , , , | 1 ( 1 ș a ș a b c c c b U P j i i i j i i i i i  c     Ȑ2-9ȑ ځύǴ ࣁಃiᚒϸᔈࠠᄊǹ ࣁڙ၂ޣૈΚӛໆǹ ࣁ၂ᚒޑ౒ෳୖኧǹ ࣁ ၂ᚒ᠘ձࡋӛໆǹԶࣁΑ٬၂ᚒޑᜤࡋԋࣁӛໆҔаᆶૈΚӛໆ࣬෧Ǵࡺஒᜤ ࡋୖኧ b ᆶӛໆ 1 ࣬४Ƕ i U ș_j ci ai ӭӛࡋΟୖኧޑኳԄԖځд߄ҢݤǴҗ ReckaseȐ1997ȑගрޑӭӛࡋΟୖ ኧࢶ୷ኳԄȐmultidimensional three-parameter logistic model, M-3PLȑӵϦԄ Ȑ2-10ȑ܌ҢǶૈΚࣁși & ޑڙ၂ޣǴӧΒϡीϩ၂ᚒ ޑเჹᐒ౗ࣁǺj ) ( 3 3 1 1 2 1 1 ) , | 1 ( j T i j ȕ ȕ j j j i ij ij e ȕ ȕ ȕ ș x P P  T 4     & & & & Ȑ2-10ȑ

ځύǴxij ࣁڙ၂ޣ ӧ၂ᚒ ޑբเϸᔈǴเჹਔi j xij 1Ǵเᒱਔ ǹ ࣁ 0 ij x ) ,..., ( 2 1 2 2j ȕ j ȕ jD ȕ & Dঁӛࡋޑ၂ᚒ᠘ձࡋୖኧӛໆǹ ࣁ၂ᚒᜤࡋୖኧǹ ࣁ၂ᚒ౒ෳୖኧǹ ǹಃ j ȕ1 ȕ3j

¦

4 _lD jl il T i j ș ȕ ș ȕ2 ₁ 2 & & jᚒޑ၂ᚒୖኧࣁȕj (ȕ2j,ȕ1j,ȕ3j) & & Ƕ ȐΟȑʳӭӛࡋᒿᐒ߯ኧӭ໨ࢶ୷ኳԄ

ӭ ӛ ࡋ ᒿ ᐒ ߯ ኧ ӭ ໨ ࢶ ୷ ኳ Ԅ Ȱ multidimensional random coefficients multinomial logit model, MRCMLMȱࢂҗ AdamsǵWilson ᆶ WangȰ1997ȱ฻Γ ܌ගрǴPISA ኧᏢૈΚϐෳໆኳԄջࢂ٬Ҕ MRCMLM ኳԄǶMRCMLM ࣁ Rasch ኳԄޑ़ғኳԄǴࢂ΋ঁషӝޑ co-efficients ኳࠠȐmixed co-efficients modelȑǴ၂ᚒୖኧࢂҗ҂ޕޑୖኧ ܌ඔॊǴԶڙ၂ޣޑወӧᡂኧ Ǵࢂ΋ঁᒿ ᐒᡂ໨Ƕ ȟ ș ଷ೛Ԗ I ঁ၂ᚒǴ኱Ңࣁi 1,...,Iǹ؂ঁ၂ᚒԖK_i  ঁϸᔈᜪձǴ኱Ңࣁ1 ǶҔӛໆ߄Ңᒿᐒᡂኧ ޑॶࣁ ǴځύǴ i K k 0,1,... X_i T Ki X X X , ,..., ) ( Xi i1 i2 i ¯ ® ­ ځд ঁϸᔈᜪձ բเಃ ӵ݀၂ᚒ ! -! 0 , 1 ij j i X Ҕаࡰр၂ᚒ ޑi K_i 1ঁёૈޑϸᔈǶ i ႟ӛໆ߄Ңբเϸᔈࣁᜪձ 0Ǵ೭ঁ 0 ᜪձࢂ΋ঁୖྣᜪձǴჹኳԄ᠘ۓ ࢂѸाచҹǶҔ೭ঁ྽բୖྣᜪձࢂᒿЈ܌టޑǴԶЪόቹៜኳԄޑЬा೽ϩǶ Ψёаԏ໣ԋ΋ঁൂӛໆ ǴᆀࣁϸᔈӛໆȐ܈ϸᔈಔ ࠠȑǶ i X T K X X X , ,..., ) ( X_{i i1 i2 i} pঁୖኧޑ၂ᚒҗӛໆ ඔॊϐǶ೭٤ጕ܄ಔӝ೏Ҕӧϸᔈ ᐒ౗ኳԄύඔॊ؂΋ঁ၂ᚒޑϸᔈᜪձޑ࿶ᡍ΢ޑ੝ቻǶ ) ȟ ,..., ȟ , ȟ ( ȟ 1 2 p T Dঁӛࡋϐ೛ीӛໆ Ȑdesign vectorȑaijǴځύǴi 1,...,I;j 1,...,KiǴ؂ঁӛໆߏࡋࣁ p ǴѬॺёа

ԏ໣ԋ΋ঁ೛ीંତȐdesign matrixȑ ٰۓ ကѬॺޑጕ܄ಔӝǶ ) ,..., ,..., , ,..., , ( 2 1 21 2 1 12 11 K K IKI T a a a a a a A ኳԄޑӭӛࡋࠠԄଷۓঁձޑϸᔈϐΠԖ D ঁ੝܄Ǵ೭ D ঁወӧ੝܄ۓက ԋ΋ঁ D -ӛࡋޑወӧޜ໔Ƕӛໆș (ș₁,ș₂,...,ș_D)c߄Ң΋ঁӧ D -ӛࡋޑወӧޜ ໔ύޑՏ࿼Ƕ೭ঁኳԄΨ௦ҔीϩБำԄǴҔаᇥܴ؂΋ঁ၂ᚒޑϩኧ܈؂΋ ঁ၂ᚒޑ؂΋ঁёૈϸᔈᜪձޑֹԋቫભǶऩӧ၂ᚒ i ǵӛࡋ D ϐϸᔈࣁᜪձ Ǵ ߾ ځ ϸ ᔈ ϩ ኧ ࣁ Ƕ ၠ ຫ k b_ikd D ঁ ӛ ࡋ ޑ ϩ ኧ ё а ԏ ໣ ԋ ΋ ঁ Չ ӛ ໆ Ǵ Զ ࡕ ჹ ၂ ᚒ i ӆ ԏ ໣ ԋ ΋ ঁ ी ϩ η ં ତ Ȑ scoring submatrixȑ Ǵനࡕჹ᏾ҽෳᡍӆ຾ډ΋ঁीϩંତȐscoring matrixȑ Ƕऩϸᔈӧ 0 ᜪձޑϩኧࣁ 0 ϩǴՠࢂځдϸᔈΨ Ԗёૈीࣁ 0 ϩǶ T ikD ik ik ik (b 1,b 2,...,b ) b T iD i i i (b1,b2,...,b ) B T ) ,..., , (B₁T BT₂ BT_I B ӢԜǴ၂ᚒ i ϸᔈӧᜪձ j ᐒ౗ޑኳԄӵԄηȐ2-11ȑ܌ҢǺ

¦

c  c c  c i 1 ) exp( ) exp( ) ; 1 ( _K k ik ik ik ik ik | P ȟ a ș b ȟ a ș b ș ȟ X Ȑ2-11ȑ ځύ ࣁڙ၂ޣϸᔈࠠᄊǴ ࣁಃi ᚒޑϸᔈᜪձኧǴ ࣁಃiᚒӧಃkঁ ϸᔈᜪձ΢ޑीϩӛໆǹș ࣁڙ၂ޣૈΚӛໆǹ ik X K_i bc_ik ik ac ࣁಃiᚒύಃ ঁϸᔈᜪձޑ ೛ीӛໆǹ ȟ ࣁ၂ᚒୖኧӛໆǶ k PISA 2003ջа MRCMLM ϩ݋ȐPISA 2003ȑǴࣁ٬ҁࣴز܌௖૸ϐԛભໆ Ёϩኧ՗ीᔈҔܭεࠠෳᡍύǴࡺҁࣴزύϐ MIRT Бݤջ௦Ҕ MRCML ኳԄ ٰ՗ीǶ ԜѦǴYao ᆶ SchwarzȐ2006ȑΨගрӭӛࡋ೽ϩ๏ϩኳԄȐmultidimensional version of the partial credit model, M-2PPCȑǶૈΚࣁși

&

ޑڙ၂ޣǴӧӭϡीϩ၂ ᚒ ӣเޑ܌ឦᜪձࣁj k1Ǵځᐒ౗ӵϦԄȐ2-12ȑ܌ҢǺ

¦

 4   4  ¦ ¦ G G  j m t tj T i j k t tj T i j K m ȕ ș ȕ m ȕ ș ȕ k j i ij ijk e e ȕ ș k x P P 1 ) 1 (( ) 1 ( 1 2 1 2 ) , | 1 ( & & & & & & Ȑ2-12ȑ ځύǴ ࣁڙ၂ޣxij iӧ၂ᚒ ޑբเϸᔈj xij 0,...,Kj 1ǹȕ 2j (ȕ2j1,...,ȕ2jD) & ࣁ ঁӛࡋޑ၂ᚒ᠘ձࡋୖኧӛໆǹ D j k G E ࣁ⸣ॶȐthresholdȑǴk 1,2,...,KjЪEG1j 0ǹ ࢂಃ ᚒϸᔈᜪձޑኧໆǹಃ ᚒޑ၂ᚒୖኧࣁ j K j j ( , ,..., ) 2 2j j j j ȕ ȕ ȕ K_j ȕ& & _{G G} Ƕ

ಃΒ࿯ʳ ԛભໆЁϩኧ՗ीБݤ

೚ӭᏢޣӧ΋٤ԛભໆЁϷෳᡍϩኧϐ࣬ᜢࣴزύǴගрૈྗዴ՗ीᢀჸ ϩኧȐobserved scoreȑЪёߞᒘϐ՗ीБݤȐBock, Thissen, & Zimowski, 1997ǹ Gessaroli, 2004ǹKahraman & Kamata, 2004ǹPommerich, Nicewander, & Hanson, 1999ǹShin, 2006ǹShin, Ansley, Tsai, & Mao, 2005ǹTate, 2004ǹWainer, Vevea, Camacho, Reeve, Rosa, Nelson, Swygert, & Thissen, 2000ǹYen, 1987ǹYen, Sykes, Ito, & Julian, 1997ȑǴ೭٤Бݤ೸ၸෳᡍၗ਑ӧόӕԛભໆЁ໔ޑߕឦૻ৲ౢғ ԛભໆЁϩኧ՗ीॶǶ

ҁࣴز٬ҔΎᅿԛભໆЁϩኧ՗ीБݤǴх֖ӭӛࡋ၂ᚒϸᔈ౛ፕБݤ ȐMIRT methodȑǵBock БݤȐBock methodȑǵҞ኱߄౜ࡰ኱БݤȐobjective performance index method, OPI methodȑǵ଑ᘜϩኧБݤȐregressed score method, REG methodȑǵ҅ዴ౗ϩኧБݤȐproportion-correct method, PC methodȑǵW-Bock БݤȐW-Bock methodȑϷ REGP БݤȐREGP methodȑǴҁ࿯ஒଞჹ೭٤Бݤ ଺ϟಏǴ၁ॊӵΠǶ

൘ǵʳMIRT Бݤ

MIRT Бݤࢂаӭӛࡋ၂ᚒϸᔈ౛ፕٰ՗ीԛભໆЁϩኧǴճҔڙ၂ޣܭ ෳᡍύϐ MIRT ໆЁϩኧᙯඤԋԛભໆЁϩኧǶҁࣴز܌௦Ҕϐ MIRT Бݤࢂ

ਥᏵ MIRT ύޑ MRCMLM ٰ՗ीԛભໆЁϩኧǴЪࣴزޣஒ MIRT Бݤԛભ ໆЁ ϐϩኧۓကࣁ Ǵځ՗ीӵԄηȐ2-13ȑ܌ҢȐAdams, Wilson, & Wang, 1997ȑǺ j MIRT T_j

¦

c  c c  c i 1 ) exp( ) exp( K k ik ik ik ik j T MIRT ȟ a ș b ȟ a ș b Ȑ2-13ȑ ځύǴKiࣁಃ i ᚒޑϸᔈᜪձኧǹbc ࣁಃik iᚒӧಃkঁϸᔈᜪձ΢ޑीϩӛໆǹ ࣁڙ၂ޣૈΚӛໆǹa ࣁಃ ᚒύಃ ঁϸᔈᜪձޑ೛ीӛໆǹ ࣁ၂ᚒୖኧ ӛໆǶ ș _ikc i k ȟ

ມǵʳBOCK Бݤ

BOCK Бݤࢂа၂ᚒϸᔈ౛ፕٰ՗ीԛભໆЁϩኧǴճҔڙ၂ޣܭෳᡍύ ϐ IRT ໆЁϩኧᙯඤԋԛભໆЁϩኧǶࣴزޣஒ BOCK БݤԛભໆЁ ϐϩኧ ۓကࣁ Ǵځ՗ीӵԄηȐ2-14ȑ܌ҢȐBock, Thissen, & Zimowski, 1997ȑǺ

j j T IRT

¦

j I i ij j j n T IRT 1 ) ˆ ( 1 T H Ȑ2-14ȑ ځύǴi ࣁ၂ᚒǹ j ࣁԛભໆЁǹ ࣁԛભໆЁI_j jύޑ၂ᚒኧǹ ࣁԛભໆЁn_j jύ നεёૈϩኧǴЪ Ǵ ࣁ၂ᚒ ϐ܌ԖᜪձኧǹT ࣁڙ၂ޣૈΚ ՗ीॶǹ ࣁڙ၂ޣૈΚ՗ीॶࣁT ਔǴԛભໆЁ

¦

 i I i i j m n 1 ) 1 ( m_i i ˆ ) ˆ (T Hij ˆ jӧ၂ᚒi ϐเჹ౗Ƕ ऩෳᡍ၂ᚒࣁᒧ᏷ᚒޑ၂ᚒǴ߾ԛભໆЁ j ӧ၂ᚒ ϐเჹ౗ ջࣁ IRT ύ၀ᚒϐ೯ၸᐒ౗ǴӵΠԄ߄ၲϐǺ i Hij(Tˆ) ) ˆ ( ) ˆ (T ij T ij P İ Ȑ2-15ȑ а၂ᚒϸᔈ౛ፕ 1PL ՗ीਔǴ߾

)] ( exp[ 1 1 ) ˆ ( ) ˆ ( i j ij ij b P İ  T   T T Ȑ2-16ȑ а၂ᚒϸᔈ౛ፕ 2PL ՗ीਔǴ߾ )] ( exp[ 1 1 ) ˆ ( ) ˆ ( i j i ij ij b a P İ  T   T T Ȑ2-17ȑ а၂ᚒϸᔈ౛ፕ 3PL ՗ीਔǴ߾ )] ( exp[ 1 ) 1 ( ) ˆ ( ) ˆ ( i j i i i ij ij b a c c P İ  T     T T Ȑ2-18ȑ! ځύǴaijࣁ၂ᚒ᠘ձࡋୖኧǹ ij b ࣁ၂ᚒᜤࡋୖኧǹ ij c ࣁ၂ᚒ౒ෳࡋୖኧǶ

ୖǵʳҞ኱߄౜ࡰ኱Бݤ

Ҟ኱߄౜ࡰ኱БݤȐobjective performance index, OPI methodȑࢂ΋ᅿ՗ी ؂ঁԛભໆЁ၂ᚒϐ੿ჴϩኧȐtrue scoreȑޑБݤǶࣴزޣஒҞ኱߄౜ࡰ኱Б ݤԛભໆЁ ϐϩኧۓကࣁj OPIT_jǴаΠϟಏ OPI БݤȐYen, 1987ȑǶ

аᒧ᏷ᚒ၂ᚒࣁٯٰᇥܴǴଷ೛΋ҽෳᡍԖ ᚒ၂ᚒЪх֖n J ঁԛભໆ ЁǴӧԛભໆЁ j ύԖ ᚒ၂ᚒǴԶЪ΋ᚒ၂ᚒനӭឦܭ΋ঁԛભໆЁǴёૈ Ԗ٤၂ᚒόឦܭҺ΋ԛભໆЁǶз ࣁԛભໆЁ j n j X jϐᢀჸเჹ၂ᚒϩኧ

Ȑobserved number-correct scoreȑǴЪT_j {E(X_j /n_j)Ƕଷ೛ӧԛભໆЁϐѦёᕇ

ளڙ၂ޣޑᚐѦၗૻǴٯӵ ϐӃᡍϩթȐprior distributionȑǴԜᚐѦޑૻ৲܈ ࢂӃᡍૻ৲Ȑprior informationȑёૈࢂڙ၂ޣϐӧਠԋᕮ܈ࢂځӧځдෳᡍϐ

j

߄౜ǶOPI БݤϐำׇӵΠ܌ॊǺ ΋ǵʳ ଆۈ؁ᡯ ଷ೛ӧ๏ۓڙ၂ޣϐ௃׎ΠǴT_jϐӃᡍϩթࣁȕ(r_j,s_j)ǴջǺ )! 1 ( )! 1 ( ) 1 ( )! 1 ( ) ( 1 1        j j s j r j j j j s r T T s r T g j j for 0dTj d1ǹ rj,sj !0 Ȑ2-19ȑ ٠ଷ೛ӧ๏ۓT_jਔǴX_jܺவΒ໨ϩѲȐbinomial distributionȑǴӵΠ܌ҢǺ j j j n x j x j j j j j j T T x n T x X p _¸¸   ¹ · ¨ ¨ © § ) 1 ( ) | ( for 0dxj dnj;0dTj d1 Ȑ2-20ȑ ӧԄηȐ2-19ȑᆶȐ2-20ȑϐଷۓϐΠǴ๏ۓ ਔǴ ϐࡕᡍϩѲȐposterior distributionȑࣁǺ j x T_j ) ( ) (T_j|X_j x_j ȕ p_j,q_j g Ȑ2-21ȑ ځύǴ j x _{j j j} j j j r x q s n p  Ъ  Ȑ2-22ȑ ߾ۓက OPI ࣁTjࡕᡍϩթϐѳ֡ኧǴӵΠԄ܌ҢǺ j j j j j q p p T T OPI  ~ Ȑ2-23ȑ Βǵʳ ՗ीӃᡍϩթ а n ᚒᒧ᏷ᚒ၂ᚒޑෳᡍԶقǴځ၂ᚒୖኧࢂа IRT ϐ 3PL ຾Չӕਔ՗ीǴ Ъଷ೛Ԗى୼ӭޑኬҁኧ՗ी၂ᚒୖኧǶаԄηȐ2-15ȑϷȐ2-18ȑёޕǴ ࣁڙ၂ޣૈΚ՗ीॶࣁT ਔǴԛભໆЁ ) ˆ (T Hij ˆ

_j

_{ӧ၂ᚒi ϐเჹ౗Ǵз}

¦

nj i ij j j n T 1 ( ) 1 T H Ȑ2-24ȑ

߾๏ۓڙ၂ޣૈΚ՗ीॶࣁ ϐԛભໆЁșˆ

j

ޑ҅ዴ౗ϩኧȐproportion-correct scoreȑ՗ीॶࣁǺ

¦

nj i ij j j n T 1 ) ˆ ( 1 ˆ _{H T Ȑ2-25ȑ} ଷ೛๏ۓڙ၂ޣૈΚ՗ीॶࣁ Ǵѳ֡ኧࣁ ǵᡂ౦ኧࣁ Ǵ ߾ڙ၂ޣϩኧϐӃᡍϩѲࣁ Ƕǹ೭ঁϩթࢂҔٰ՗ी ϐࡕᡍϩթǶ җԄηȐ2-19ȑǴଷ೛ ܺவ Ǵ߾Ԝ beta ϩթϐѳ֡ኧᆶᡂ౦ኧё߄ၲ ࣁǺȐNovick & Jackson, 1974, p.113ȑ

șˆ P(Tˆ_j |ș) 2( ˆ | ) ș T_j V ) | ˆ (T ș g _j Tj j T ȕ(r_j,s_j) j j j j s r r T  ) | ˆ ( T P Ȑ2-26ȑ ) 1 ( ) ( ) | ˆ ( ₂ 2    j j j j j j j s r s r s r T T V Ȑ2-27ȑ ਥᏵԄηȐ2-26ȑᆶȐ2-27ȑёளǺ * ) | ˆ ( _{j j} j T n r P T Ȑ2-28ȑ * j | )] Tˆ ( -1 [ _j j n s P T Ȑ2-29ȑ ځύǴ 1 -) | ˆ ( ] ) | Tˆ ( -1 )[ | ˆ ( 2 j * T V T P T P j j j T T n Ȑ2-30ȑ ٬Ҕ IRT ኳԄǴ_V2(ˆ |_T)ёа၂ᚒୖኧޑᢀᗺ߄ၲȐLord, 1983ȑǴ j T

¦

| j n i ij j j n T 1 ) ( 1 ) | ˆ ( T H T P Ȑ2-31ȑ җܭTjࣁT ޑൂፓᙯϯȐmonotonic transformationȑǴࡺǺ

) | ˆ ( ) | ˆ ( 2 2 j j j T T T T V V Ȑ2-32ȑ ਥᏵ LoadȐ1980, p.71ȑёޕǴ 1 2 ) ˆ , ( ) | ˆ (Tj Tj |I Tj Tj  V Ȑ2-33ȑ ځύǴ ࣁ ගٮᜢܭ ϐૻ৲ໆǴ٬ҔԄηȐ2-32ȑǵȐ2-33ȑϷ Load Ȑ1980, p.85ȑёޕǴ ) ˆ , (T_j T_j I Tˆ_j Tj 2 ] / [ ) ˆ , ( ) ˆ , ( T T w w j j j j T T I T T I Ȑ2-34ȑ ș ș P n T j n i ij j j w » » ¼ º « « ¬ ª w T w w

¦

1 ( ) 1 !

¦

j w_w n i ij j ș ș P n 1 ) ( 1

¦

H T j n i ij j n 1 ' ) ( 1 Ȑ2-35ȑ ځύǴ ) 1 ( ] ) ( )][ ( 1 [ 7 . 1 ) ( ' ij ij ij ij ij ij c c P P a   T T  T H Ȑ2-36ȑ ਥᏵ LoadȐ1980, p.79ȑǴ ) ˆ , ( ) ˆ , (T T I T T I _j | Ȑ2-37ȑ ऩਥᏵڙ၂ޣϐ၂ᚒϸᔈࠠᄊǴҔനεཷ՟Ȑmaximum likelihoodȑำׇ՗ीTǴ ߾ਥᏵ LoadȐ1980, p.74ȑёளǺ

¦ ¦

J c _ j n i _{ij ij} ij j I 1 1 2 )] ( 1 )][ ( [ )] ( [ ) ˆ , ( T H T H T H T T Ȑ2-38ȑ ऩਥᏵڙ၂ޣӧෳᡍϐเჹ၂ᚒϩኧȐnumber-correct scoreȑǴҔനεཷ՟ำׇ

՗ीT Ǵ߾Ǻ

¦ ¦

¦ ¦

c  c c J j ij n i ij J j n i ij j j I 1 1 2 1 1 )] ( 1 [ )] ( [ ] ) ( [ ) ˆ , ( T H T H T H T T Ȑ2-39ȑ ԄηȐ2-38ȑϷȐ2-39ȑᙁϯӃ߻؂ঁ၂ᚒ೿ឦܭ΋ঁԛભໆЁޑଷ೛ǹऩԖ ٤၂ᚒόឦܭҺՖԛભໆЁǴՠࠅԖҔٰ՗ीT Ǵ೭٤၂ᚒගٮϐૻ৲ໆሡу ΕԿԄηȐ2-38ȑϷȐ2-39ȑǶ ) ˆ , ( ) ( 1 ) | ˆ ( 2 1 2 T T T H T V I n T j n i ij j j » » ¼ º « « ¬ ª c |

¦

Ȑ2-40ȑ

ӢԜǴਥᏵԄηȐ2-28ȑԿȐ2-30ȑǴ ϐ beta ӃᡍϩթȐprior beta distributionȑ ޑୖኧё٬ҔԄηȐ2-25ȑѐ՗ीԄηȐ2-31ȑϷ٬ҔԄηȐ2-36ȑǵȐ2-38ȑǵȐ2-39ȑ ϷȐ2-40ȑϐ IRT 3PL ኳԄୖኧޑᢀᗺٰ߄ҢǶࡺҗԄηȐ2-22ȑёޕ ϐ beta ࡕᡍϩթȐposterior beta distributionȑޑୖኧૈҗ IRT ୖኧޑᢀᗺٰ߄ҢӵΠǺ

j T j T j j j j T n x p _ˆ *  Ȑ2-41ȑ j j j j j T n n x q  *   ] ˆ 1 [ Ȑ2-42ȑ ӢԜǴ j j j j j j j j j j n n x n T q p p T T OPI    * * ˆ ~ Ȑ2-43ȑ ऩаӃᡍϩѲ Ϸᢀჸเჹ౗ϩኧTˆj x /j nj࣬ჹଅ᝘ޑᢀᗺǴ߾ OPI ёаቪԋǺ j j j j j j j n x w T w T T OPI ~ ˆ (1 ) Ȑ2-44ȑ ځύǴw_jࣁ๏ۓӃᡍϩѲϐ࣬ჹ៾ख़ǴӵΠԄǺ

j j j j n n n w  * * Ȑ2-45ȑ Ѹ໪ݙཀޑࢂǴӃᡍ՗ीޑ኱ྗᇤǴջԄηȐ2-40ȑޑ໒ਥဦǴᖿ߈ܭ 0 ਔǴ ൳Я཮΋ठǹϸϐǴऩ Ǵ߾ό๏ϒӃᡍ՗ीϐ៾ख़Ƕ j w 0 * j n Οǵʳ ᔠᡍ΋ठ܄! ऩ ૈҔٰඔॊ၂ᚒϸᔈǴջ٬ IRT ኳԄૈᆒዴӦඔॊڙ၂ޣӧ၂ᚒ΢ ޑ߄౜Ǵڙ၂ޣӧԛભໆЁϐ၂ᚒϸᔈёૈࢂӭӛࡋޑȐmultidimensionalȑǶᖐ ٯٰᇥǴ΋ঁ੝ਸޑڙ၂ޣёૈเჹ֚ᜤޑᚒҞǴՠࢂࠅเᒱᙁൂޑᚒҞǹӧ ೭ঁٯηύǴаӃᡍ՗ी Ϸ ߄Ңϐ٠ό፾྽Ƕӧ OPI ޑीᆉำׇύǴ ёճҔΠԄٰղᘐڙ၂ޣӧӚԛભໆЁύϐӃᡍϩѲࢂց಄ӝႣයȐYen, Sykes, Ito, & Julian, 1997ȑǶ

) (T c ij İ j Tˆ x /j nj

¦

_  J j _{j j} j j j j T T T n x n Q 1 2 ) ˆ 1 ( ˆ ) ˆ ( Ȑ2-46ȑ ऩ ǴࡰؒԖပΕܔ๊୔Ǵ߄Ң ᆶ ࢂ፾ଛޑǴ߾ճҔԄη Ȑ2-41ȑԿԄηȐ2-43ȑٰीᆉ OPIǹϸϐǴऩ Ǵ߄Ң ᆶ ࢂ ό፾ଛޑǴӢԜǴзԄηȐ2-41ȑԿԄηȐ2-43ȑϐ ٰीᆉ OPIǶ ) 10 . , ( 2 J QdF Tˆj x /j nj ) 10 . , ( 2 J Q!F Tˆj x /j nj 0 * j n

စǵʳ଑ᘜϩኧБݤ

଑ᘜϩኧ೯தࢂ٬Ҕচۈϩኧٰ՗ी੿ჴϩኧǴKelley ଑ᘜϩኧȐKelley, 1927Ǵ1947ȑǴ߄ҢӵΠԄǺ ) ( ) 1 ( ˆ U U P P U P W x   x Ȑ2-47ȑ ځύǴWˆࣁڙ၂ޣ੿ჴϩኧǹ U ࣁဂᡏڙ၂ޣޑෳᡍߞࡋǹxࣁڙ၂ޣޑᢀჸ

ϩኧǹ P ࣁဂᡏڙ၂ޣޑѳ֡ϩኧǶ Զ Kelley’s ଑ᘜϩኧӧ՗ी੿ჴϩኧਔǴёஒԄηȐ2-47ȑޑUڗжࣁ r ǵ Pڗжࣁ x.Ǵ߄ҢӵΠԄǺ .) ( . ˆ x r xx W Ȑ2-48ȑ ࣴزޣஒ଑ᘜϩኧБݤԛભໆЁ ϐϩኧۓကࣁ Ǵ߾ஒԄηȐ2-48ȑ аӛໆ׎Ԅ߄ၲǴё߄ҢӵΠԄȐShin, 2006ΙShin, Ansley, Tsai, & Mao, 2005Ι Wainer et al., 2000ȑǺ j REGT_j .) ( . ˆ x B xx W j T REG Ȑ2-49ȑ ځύǴ ࣁԛભໆЁޑෳᡍᢀჸϩኧǹ ࣁဂᡏڙ၂ޣޑѳ֡ᢀჸϩኧǹ ࣁ Ҕٰ՗ीෳᡍߞࡋϐӭᡂໆંତǶ x x. B ёаஒંତ ຎࣁ΋ᅿ៾ख़Ǵх่֖ӝ੿ჴϩኧB

W

ᆶᢀჸϩኧxϐᜢ߯Ǵऩ Ǵ߾ж߄ᢀჸϩኧࢂֹӄёߞޑǴջᢀჸϩኧ ࣁ੿ჴϩኧޑ՗ीǹऩ Ǵ߾ж߄܌Ԗ੿ჴϩኧ֡ёҔѳ֡ϩኧ ߄ҢϐǶਥᏵԄηȐ2-49ȑёޕǴ ऩట؃੿ჴϩኧޑ՗ीॶǴ໪Ӄڗள ॶǴаΠஒᇥܴ ॶڗளϐБݤǶ I B x 0 B x. B B

ۓက ࣁόӕԛભໆЁᢀჸϩኧޑӅᡂ౦ંତȐthe observed covariance

matrixȑǴځჹفϡનࣁӚԛભໆЁᢀჸϩኧޑᡂ౦ኧǹ ࣁόӕԛભໆЁ੿ ჴϩኧޑӅᡂ౦ંତǶ obs S true S true S ϐߚჹفϡનࣁόӕԛભໆЁԋჹ੿ჴϩኧޑӅᡂ౦ኧǴҗܭᇤৡک ੿ჴϩኧคᜢǴ߾ёޕ

V

W_jvW_jvc

V

x_jvx_jvcǶ ϐჹفϡનࣁ੿ჴϩኧޑᡂ౦ኧǴ ջ ǹ ჹ ف ϡ ન ࣁ ᢀ ჸ ϩ ኧ ޑ ᡂ ౦ ኧ Ǵ ջ Ƕ Ӣ Ԝ Ǵ ё ޕ ǴځύǴ ࣁԛભໆЁޑߞࡋǶਥᏵԜᜢ߯Ǵૈ୼ ՗ी੿ϩኧޑӅᡂ౦ંତ ǶӵΠԄǺ true S 2 W V obs S V_x2 ) / ( 2 _x2 obs true S S u V_W V V_W2/V_x2 true S

v v for s s obs vv true vv' ' z c Ȑ2-50ȑ v v for s s_vvtrue' U_{v vv}obs c Ȑ2-51ȑ ځύǴv ᆶ ࣁંତϐϡનǹvc UࣁԛભໆЁϐߞࡋǶҁࣴز٬Ҕ Cronbach's D߯ ኧȐCronbach's coefficient alphaȑٰीᆉԛભໆЁޑߞࡋǶीᆉԄηӵΠ܌Ң ȐWainer et al., 2000ȑǺ D V V U » » » » ¼ º « « « « ¬ ª   t

¦

c 2 1 2 1 1 _x n i y x x i n n Ȑ2-52ȑ ԄηȐ2-52ȑύǴଷ೛ x ෳᡍх֖ n ᚒ၂ᚒy₁,y₂,,y_nǹ ෳᡍࣁ' x xෳᡍϐፄҁ ෳᡍȐalternate formȑǶ ӢԜǴаંତ׎Ԅ߄ၲਔǴਥᏵԄηȐ2-50ȑᆶȐ2-51ȑёޕ ᆶ ޑ ᜢ߯ࣁǺ ǶځύǴ ࣁჹفંତǴჹفϡનࣁᇤৡᡂ౦ኧϐ՗ी ॶǴࡺёளǺ true S _Sobs D S Strue obs  D ) (1 ߞࡋ ᢀჸϩኧᡂ౦ኧ ᇤৡᡂ౦ኧ u  Ǵ߾ ߄ҢӵΠǺD » » » » ¼ º « « « « ¬ ª U  U  U  obs vv v obs obs s ) 1 ( ... 0 0 ... ... ... ... 0 ... s ) 1 ( 0 0 ... 0 s ) 1 ( 22 2 11 1 D Ȑ2-53ȑ ௗ๱ǴଷۓԛભໆЁࣁதᄊϩѲǴаΠᖐٯٰᇥܴ՗ीำׇǶऩଷ೛Ԗٿ ᡂኧ Ϸ ǴܺவӭᡂໆதᄊϩѲȐmultivariate normal distributionȑǴ߾ёޕ ȐJohnson & Wichern, 2007ȑǺ

1 y y2 ¸ ¸ ¹ · ¨ ¨ © § » ¼ º « ¬ ª » ¼ º « ¬ ª » ¼ º « ¬ ª

¦

¦

¦

¦

22 21 12 11 2 1 2 1 , ~ P P N y y Ȑ2-54ȑ ӢԜǴӧ๏ۓy2ΠǴᡂኧy1ёа߄ҢӵΠԄǺ

) , ) ( ( ~ ) ( 1 2 1

¦ ¦

_{12 22}1 2 2

¦

₁₁

¦ ¦ ¦

_{12 22}1 ₂₁      P P y y y Ȑ2-55ȑ ࡺёޕǴ๏ۓҺཀॶy2Ǵᡂኧ ϐයఈॶᆶచҹӅᡂ౦ંତǴӵΠԄǺy1

¦ ¦ ¦

¦

¦

¦ ¦

     _{12 22}1 2 2 _{( ) 11 12 22}1 ₂₁ 1 2 1 ) ( ) ₂ ( _y|y 1 y y y| ȝ ȝ ᆶ Ǽ Ȑ2-56ȑ ਥᏵ΢ॊ኱ྗϯϐ่݀ǴჹܭԛભໆЁ՗ीୢᚒǴଷ೛੿ჴϩኧ

W

کᢀჸ ϩኧ ܺவӭᡂໆதᄊϩѲǴѳ֡ኧࣁx PǴёޕǴ ¸ ¸ ¹ · ¨ ¨ © § » » ¼ º « « ¬ ª » ¼ º « ¬ ª » ¼ º « ¬ ª

¦

¦

¦

¦

obs true true true N IJ , ~ P P x Ȑ2-57ȑ җԄηȐ2-56ȑޕǴ๏ۓᢀჸϩኧ ޑచҹΠǴڙ၂ޣ ӧԛભໆЁϐ੿ ჴϩኧ j x j j W ࣁ (W | ) P

_¦

(

_¦

)1( j P)Ƕ߾੿ჴϩኧ՗ीޑϦԄӵΠԄǺ obs true j j E x x .) x ( B . x ) . x ( ) S ( S . x ˆ true 1     W  j j obs j T REG x x Ȑ2-58ȑ ځύǴаԛભໆЁϐѳ֡ϩኧ жඹx. Pǹ_Strueᆶ_Sobsжඹ

¦

true ᆶ ǹЪ Ƕ

¦

obs 1 ) ( obs  true S S B

Ҵǵʳ҅ዴ౗ϩኧБݤ

҅ዴ౗ϩኧБݤȐproportion-correct method, PC methodȑࢂаڙ၂ޣӧෳᡍ ύϐ܌ԖբเϸᔈΠเჹԛኧޑКٯբࣁԛભໆЁϩኧȐGummerman, 1972Ι Shin, 2006Ι஭ۗ൛ǵڬ໡അǵ೚ϺᆢǵࡼలীǴ2008ȑǶࣴزޣஒ҅ዴ౗ϩኧ БݤԛભໆЁ ϐϩኧۓကࣁj PCT_jǴځीᆉԄηӵΠ܌ҢǺ j j j n x T PC Ȑ2-59ȑ ځύǴ j ࣁԛભໆЁǹ ࣁԛભໆЁn_j jޑനεёૈϩኧǹ ࣁԛભໆЁx_j jޑᕴϩ Ȑcomposite scoreȑǶ

ഌǵʳREGP Бݤ

ਥᏵ Wainer et al.(2000)ගрϐа IRT ໆЁϩኧࣁ୷ᘵϐ࿶ᡍنМӣᘜ՗ी Ȑempirical Bayes regressed estimates based on IRT scale scoresȑǴࣴزޣஒԜෳᡍ ϩኧ՗ीБݤڮӜࣁREGPTǴԛભໆЁෳᡍϩኧ߄ҢӵΠԄȐ஭ۗ൛Ǵ2008ȑǺ x.) -B(x x. ) x. x ( ) S ( S x. ˆ true obs 1 _{j j}    W  T REGP Ȑ2-60ȑ ځύǺxjࣁڙ၂ޣ jӧԛભໆЁϐ IRT ՗ीϩኧǹ ࣁԛભໆЁϐѳ֡ϩኧǶx. ӢԜǴӵӕԄηȐ2-47ȑԿԄηȐ2-49ȑёޕ ࣁԛભໆЁ ϐ IRT ໆЁϩ ኧߞࡋǴЪ v ȡ v ]] [ [ ] [ ] [ 2 v v v v SE Average Variance Variance ȡ T  T T Ȑ2-61ȑ ٠ଷ೛ v j * j x x ȡ | Ǵӵӕ REG БݤϐᢀჸϩኧǴЪӵӕԄηȐ2-50ȑᆶȐ2-51ȑǴ ࣁ ϐӅᡂໆંତǹ ϐંତჹفጕϡન཮฻ܭ ४΢ IRT ໆЁϩኧߞࡋ Ǵߚჹفጕϡન฻ܭ ϐߚჹفጕϡનǶ obs S * j x true S obs S v ȡ obs S

ࢠǵʳW-BOCK Бݤ

W-BOCK Бݤࢂа Bock Бݤࣁ୷ᘵǴЇΕȨ៾ख़ȩཷۺϐ՗ीБݤǴҭ ջஒᚒҁ v ϐ IRT ໆЁϩኧߞࡋ ϐКख़ຎࣁ៾ख़Ƕࣴزޣஒ W-BOCK Бݤԛ ભໆЁ v ȡ jϐϩኧۓကࣁWIRTT_jǴځ՗ीӵΠԄ܌ҢȐ஭ۗ൛Ǵ2008ȑǺ j j v j v j n x ȡ T ȡ T WIRT ˆ (1 ) Ȑ2-62ȑ ځύǴ j ࣁԛભໆЁǹ ࣁԛભໆЁnj jύനεёૈϩኧǹ ࣁเჹ౗ϩኧǴҗԄTˆj

ηȐ2-60ȑёளǹȡ_vࣁᚒҁ v ϐߞࡋǶ ߞࡋޑཷۺࢂҗ GreenǵBockǵHumphreyǵLinn ᆶ Reckase(1984)ගрǴी ᆉϦԄӵԄηȐ2-61ȑ܌Ң 2 2 2 ) ( ) ( ș e ș ı ș ı ı ȡ T  Ȑ2-63ȑ ځύǴ ࣁૈΚ՗ीॶޑᡂ౦ኧǴ ࣁૈΚॶෳໆᇤৡᡂ౦ኧȐmeasurement error varianceȑǶ 2 ș ı 2( ) ș ıe

ಃΟ࿯ʳ ෳᡍ฻ϯޑཀကᆶ฻ϯ೛ी

ӧ೚ӭεࠠෳᡍύǴӵ୯ύ୷ҁᏢΚෳᡍǵNAEPǵTIMMS Ϸ PISA ฻Ǵ ࿶த೸ၸෳᡍ฻ϯٰКၨࡼෳόӕෳᡍᚒҁϐڙ၂ޣޑૈΚ੝፦ǶฅԶǴЎ᝘ ϐύۘคଞჹԛભໆЁ଺฻ϯϐ௖૸ǴЪ᠘ܭӭᅿෳᡍ฻ϯ೛ीϐύǴЀа BIB Ϸ NEAT ٿᅿ೛ीനத٬Ҕܭ೭٤εࠠෳᡍϐύǴࡺҁࣴزஒ௖૸ BIB ᆶ NEAT ฻ϯ೛ीҔܭԛભໆЁ฻ϯჹԛભໆЁϩኧ՗ीਏ݀ϐКၨǶҁ࿯ஒჹෳᡍ฻ ϯޑཀကаϷҁࣴز܌٬Ҕϐ BIB ᆶ NEAT ฻ϯ೛ी଺ᇥܴǶϩॊӵΠǶ

൘ǵʳෳᡍ฻ϯޑཀက

΋૓ٰᇥǴӧٿҽόӕޑෳᡍύǴځϩኧ໔คݤޔௗ଺КၨǴѸ໪ճҔ಍ ीБݤǴஒڙ၂ޣӧԜ΋ෳᡍϐϩኧᙯඤԿќ΋ෳᡍϐϩኧໆЁ΢ǴωૈКၨ ٿෳᡍϩኧ໔ޑᜢ߯ǴԶԜ΋ϩኧໆЁᙯඤϐၸำջࣁෳᡍ฻ϯȐtest equatingȑ ȐKolen & Brennan, 1995ȑǶట฻ϯϐෳᡍǴځෳᡍϣ৒Ϸᜤࡋ೿ཱུࣁ࣬՟ǴҔ аෳໆ࣬ӕޑૈΚ܈੝፦ǴԶ฻ϯϐҞޑΏࢂࣁΑਠྗෳᡍ၂ᚒᜤࡋϐৡ౦Զ ߚෳᡍϣ৒ϐৡ౦ǴЪ฻ϯϐ่݀ό཮Ӣਔ໔کΓޑӢનԶׯᡂځཀကȐKolen & Brennan, 2004ȑǴҭόڙ၂ᚒϣ৒Ϸڙ၂ޣૈΚϩթቹៜǴฅԶǴ฻ϯϐ຾Չ

Ѹ໪ᅈىаΠచҹȐHambleton & Swaminathan, 1985ǹLord, 1980ȑǺ ΋ǵʳ ჹᆀ܄Ȑsymmetry propertyȑǺෳᡍϩኧϐ฻ϯѸ໪ࣁё଍ޑǴҭջǴค ፕҗ X ෳᡍ฻ϯԿ Y ෳᡍǴ܈ࢂҗ Y ෳᡍ฻ϯԿ X ෳᡍǴځٿᅿ฻ϯ ่݀Ѹ໪࣬ӕǶ Βǵʳ ࣬฻܄Ȑequity propertiesȑǺऩԖٿෳᡍȐX ෳᡍᆶ Y ෳᡍȑట຾Չ฻ϯǴ ߾όፕڙ၂ޣڙෳ X ෳᡍ܈ࢂڙෳ Y ෳᡍǴځ฻ϯ่݀࣬ӕǶ

Οǵʳ იᡏόᡂ܄Ȑgroup invariance propertyȑǺ฻ϯၸำύǴόڙڙ၂ޣဂᡏޑ ቹៜǴջόፕڙ၂ޣࣁՖǴځ฻ϯ่݀٠คৡ౦Ƕ

Ѥǵʳ ൂӛࡋȐunidimensionality of the testsȑǺٿෳᡍऩట຾Չ฻ϯǴځෳᡍϣ ৒Ѹ໪ࣁෳໆ࣬ӕϐૈΚ੝፦Ƕ

ມǵʳෳᡍ฻ϯ೛ी

ෳᡍ฻ϯ೛ीࡰࡼෳޣԏ໣฻ϯၗ਑ϐБԄǶځ೛ीБݤԖࡐӭᅿǴ૟ϟ ಏҁࣴز܌٬Ҕϐٿᅿ฻ϯ೛ीǺ ΋ǵʳ ۓᗕό฻ಔ೛ी NEAT೛ीஒा฻ϯޑٿෳᡍȐX ෳᡍᆶ Y ෳᡍȑϩձ๏ϒٿಔڙ၂ኬҁ ȐP ک QȑࡼෳǴЪٿಔڙ၂ኬҁሡќѦௗڙ΋ҽӅӕෳᡍ AǴෳᡍ A ջࣁۓ ᗕෳᡍǶۓᗕ၂ᚒӧٿኬҁޑෳᡍ໩ׇࢂ΋ኬޑǴаᗉխ໩ׇӢનޑቹៜǹЪ ۓᗕෳᡍޑෳᡍϣ৒ᆶᜤࡋѸ໪ᆶ X ෳᡍǵY ෳᡍ࣬՟ǶNEAT ೛ीӵ߄ 2-1 ܌ҢȐKolen & Brennan, 1995ǹvon Davier, Holland & Thayer, 2004ȑǶ

߄2-1! NEAT೛ी ڙ၂ኬҁ Xෳᡍ Yෳᡍ ۓᗕෳᡍ A P V V Q V V “V”ࣁڙ၂ޣௗڙϐෳᡍ NEAT ೛ीࣁதـϐෳᡍ฻ϯ೛ीǴѝሡाଷ೛ڙ၂ဂᡏࢂᒿᐒܜڗǴό

Ҕଷ೛ٿڙ၂ޣဂԖ࣬ӕޑૈΚॶǶԜѦǴNEAT ೛ीޑۓᗕෳᡍϣ৒ाᅰё ૈ࣬՟Ъ၂ᚒᜤࡋा࣬߈ǴӢࣁۓᗕ၂ᚒࢂҔٰፓ᏾ٿঁόӕૈΚϐဂᡏ܌೷ ԋޑό฻ȐЦཫറǴ2006ǹPetersen, Kolen & Hoover, 1993ȑǶ

Βǵʳ ѳᑽόֹӄ୔༧೛ी BIB ೛ीࢂஒ၂ᚒϩԋऩυ၂ᚒ୔༧Ǵ୔༧໔ᆶ୔༧ϣޑ၂ᚒࣣόख़ፄǶ ஒڙ၂ޣϩࣁኧဂǴ೛ी൳ঁᚒҁȐbookletȑ൩ϩࣁ൳ဂǴ؂ဂڙ၂ޣѝሡௗ ڙऩυ၂ᚒ୔༧ޑ၂ᚒǴόӕڙ၂ޣёૈௗڙ೽ϩ࣬ӕǵֹӄ࣬ӕǵ܈ֹӄό ӕޑ၂ᚒ୔༧ǶനࡕǴஒ܌Ԗڙ၂ޣޑբเϸᔈၗ਑୴᠄຾Չ฻ϯϩ݋Ǵаၲ ډૈΚ՗ीޑҞޑǶ BIB ೛ीϐᓬᗺࣁ၂ᚒ୔༧ᆶᚒҁϐଛ࿼БԄ௦Ҕᖥ௽ȐspiralȑԄ௨ӈБ ԄǴځё٬؂΋ঁ၂ᚒ୔༧ޑࡼෳԛኧ࣬ӕȐЦཫറǴ2006ǹNemhauser & Wolsey, 1999ǹvan der Linden, Veldkamp & Carlson, 2004ȑǶԜ೛ीӧคբเਔ໔Ȑresponse timeȑϐज़ڋ௃׎ΠǴѸ໪ᅈىаΠज़ڋԄǺ S s k x t i is , 1,..., 1

¦

Ȑ2-64ȑ t i r x S s is , 1,... 1 d

¦

Ȑ2-65ȑ t j i z S s ijs , 1,..., 1  t

¦

O Ȑ2-66ȑ S s t j i z x x_is  _js t2 _ijs,  1,..., , 1,..., Ȑ2-67ȑ ځύǺ t ࡰ၂ᚒ୔༧ኧǹ sࡰᚒҁжဦǴs 1,...,Sǹ kࡰ؂ঁᚒҁଛ࿼ޑ၂ᚒ୔༧ኧǹ rࡰ၂ᚒ୔༧ӧ܌Ԗᚒҁύр౜ޑԛኧǹ iࡰᚒ৤ύঁձ୔༧жဦǴi 1,...tǹ jࡰᚒ৤ύԋჹ୔༧ύಃΒঁ୔༧жဦǴ j 1,...,tǹ

Oࡰԋჹ၂ᚒ୔༧ӧ܌Ԗᚒҁύр౜ޑԛኧǹ is x ࡰ၂ᚒ୔༧ᆶᚒҁޑଛ࿼ಔࠠǴx_is 

^ `

0,1,i 1,...,t,s 1,...,Sǹ ijs z ࡰԋჹ၂ᚒ୔༧ᆶᚒҁޑଛ࿼ಔࠠǴz_ijs 

^ `

0,1,i  j 1,...,t,s 1,...,SǶ ԄηȐ2-64ȑж߄؂΋ঁᚒҁଛ࿼ޑ၂ᚒ୔༧ኧҞǹԄηȐ2-65ȑж߄؂΋ ঁ၂ᚒ୔༧ӧ܌Ԗᚒҁύр౜ޑԛኧǹԄηȐ2-66ȑж߄ԋჹ၂ᚒ୔༧ӧ܌Ԗᚒ ҁύр౜ޑԛኧǹԄηȐ2-67ȑж߄ԋჹ၂ᚒ୔༧ᆶಔࠠޑ΋ठ܄ǶBIB ೛ी໪ ಄ӝԄηȐ2-64ȑԿȐ2-67ȑޑा؃Ǵ؃р಄ӝޑന٫ှǶ ԜѦǴBIB ೛ीԖΟ໨୷ҁज़ڋǺ Ȑ΋ȑ ؂΋ঁᚒҁϣޑ၂ᚒ୔༧ኧा࣬ӕǹ ȐΒȑ ၂ᚒ୔༧բ่ӝа؃рനλᚒҁኧǹ ȐΟȑ ؂΋ঁ၂ᚒ୔༧ӧ܌Ԗᚒҁύр౜ޑԛኧा࣬ӕǶ ฅԶǴ೭ѝࢂ BIB ೛ीѸ໪಄ӝޑΟ໨୷ҁज़ڋǴՠӧჴሞ೛ीਔǴᗋ ሡԵቾ၂ᚒޑϣ৒ǵ׎ԄϷբเਔ໔Ƕ

ಃΟകʳ ࣴزБݤ

ҁകӅϩࣁϖ࿯Ǵಃ΋࿯ࣁࣴزࢬำǹಃΒ࿯ࣁࣴزᡂ໨೛ۓǴᇥܴࣴز ύޑӅӕᡂ໨೛ۓᆶୖኧ೛ۓǹಃΟ࿯ࣁჴᡍ೛ीǴᇥܴࣴزύൂ΋ෳᡍᆶ฻ ϯෳᡍϐ೛ीǹಃѤ࿯ࣁ՗ीᆒྗࡋǴᇥܴҁࣴزҔٰКၨόӕБݤ՗ीᇤৡ ϐࡰ኱ǹಃϖ࿯ϟಏࣴزπڀǹ၁ॊӵΠǶ! !

ಃ΋࿯ʳ ࣴزࢬำ

ҁࣴزа၂ᚒϸᔈ౛ፕࣁ୷ᘵǴ่ӝӭӛࡋ၂ᚒϸᔈ౛ፕǴట௖૸όӕԛ ભໆЁϩኧ՗ीБݤӧൂ΋ෳᡍ೛ीᆶ฻ϯෳᡍ೛ी௃ნΠǴჹෳᡍϩኧ՗ी ϐਏ݀Ƕӧൂ΋ෳᡍ௃ნύǴ௖૸ BOCK БݤǵOPI БݤǵREG БݤǵPC Б ݤǵREGP БݤǵW-BOCK БݤϷ MIRT Бݤ฻ΎᅿБݤჹԛભໆЁϩኧीᆉ ϐᆒྗࡋǹӧ฻ϯෳᡍ௃ნύǴҗܭ REG БݤϷ PC Бݤа CTT ࣁ୷ᘵǴࡺ໻ ௖૸ BOCK БݤǵOPI БݤǵREGP БݤǵW-BOCK БݤϷ MIRT Бݤ฻ϖᅿ БݤჹԛભໆЁϩኧीᆉϐᆒྗࡋǶҁࣴزϐࣴزࢬำӵკ 3-1 ܌ҢǶ२Ӄࢂ ೛ۓࣴزЬᚒǴዴۓЬᚒࡕǴ຾ՉᆶࣴزЬᚒ࣬ᜢϐЎ᝘ᇆ໣ᆶ௖૸Ǵ຾Զ೛ ۓჴᡍ௃ნǵ೛ीෳᡍᚒҁǴх֖ൂ΋ෳᡍ௃ნᆶ฻ϯෳᡍ௃ნϐ೛ۓǴ٩Ᏽ ࣴزޣ܌೛ۓϐࣴز௃ნౢғኳᔕၗ਑ࡕǴջа Acer ConQuest 2.0 ೬ᡏ຾Չ՗ ीǴௗ๱٬Ҕόӕ՗ीБݤीᆉԛભໆЁϩኧǴ٠؃рόӕ՗ीБݤϐ՗ीᆒ ྗࡋǴനࡕኗቪࣴزൔ֋Ƕ!

೛ۓࣴزЬᚒ! ౢғኳᔕၗ਑! ෳᡍᚒҁ೛ी! ൂ΋ෳᡍ೛ी! ฻ϯෳᡍ೛ी! Ў᝘ᇆ໣ᆶ௖૸! ೛ۓჴᡍ௃ნ! ௦Ҕ Acer ConQuest 2.0 ೬ᡏ຾Չୖኧ՗ी! ٬Ҕόӕ՗ीБݤ! ीᆉԛભໆЁϩኧ! ीᆉόӕ՗ीБݤϐ՗ीᆒྗࡋ ኗቪࣴزൔ֋! კ3-1 ࣴزࢬำკ

ಃΒ࿯ʳ ࣴزᡂ໨೛ۓ

ҁࣴزట௖૸όӕԛભໆЁϩኧीᆉБݤӧൂ΋ෳᡍ೛ीᆶ฻ϯෳᡍ೛ी ௃ნΠǴჹෳᡍϩኧ՗ीϐᆒྗࡋǶҁ࿯ஒϩձ൩ൂ΋ෳᡍ೛ीᆶ฻ϯෳᡍ೛ ीύϐӅӕᡂ໨೛ۓᆶࣴزύϐୖኧ೛ۓ଺ᇥܴǶ၁ॊӵΠǶ

൘ǵʳ Ӆӕᡂ໨೛ۓ

΋ǵʳ ൂ΋ෳᡍ೛ी ߄ 3-1 ൂ΋ෳᡍ೛ीϐӅӕᡂ໨೛ۓ ࣴزᡂ໨ ᡂ໨೛ۓ ෳᡍᚒҁߏࡋ 24ᚒǵ36 ᚒϷ 72 ᚒ ᚒҁԛભໆЁঁኧ 2ঁǵ4 ঁ܈ 6 ঁ ԛભໆЁෳᡍߏࡋ 6ᚒǵ12 ᚒ܈ 18 ᚒ ԛભໆЁ࣬ᜢำࡋ 0.2ǵ0.5ǵ0.8 ڙ၂Γኧ 500ǵ1000 Ϸ 3000 Γ

ԛભໆЁ՗ीБݤ MIRTݤǵBOCK ݤǵOPI ݤǵREG ݤǵPC ݤǵ REGPݤǵW-BOCK ݤ ؂΋ᅿ௃׎ኳᔕԛኧ 100ԛ ਥᏵࣴزҞޑǴҁࣴزኳᔕόӕ௃ნϐෳᡍၗ਑Ƕൂ΋ෳᡍ೛ी೛ۓϖᅿ όӕᡂ໨Ǵӵ߄ 3-1 ܌ҢǶ٩ྣӚ௃ნᡂ໨ౢғኳᔕၗ਑ǴϤᅿࣴزᡂ໨௶ॊ ӵΠǺ 1. ӧෳᡍᚒҁߏࡋޑ೛ۓǴኳᔕ 24 ᚒǵ36 ᚒϷ 72 ᚒΟᅿᚒҁߏࡋǶ 2. ӧᚒҁԛભໆЁঁኧޑ೛ۓǴኳᔕ 2 ঁǵ4 ঁ܈ 6 ঁΟᅿᚒҁԛભໆЁঁኧǶ 3. ӧԛભໆЁෳᡍߏࡋޑ೛ۓǴኳᔕ 6 ᚒǵ12 ᚒ܈ 18 ᚒΟᅿԛભໆЁෳᡍߏ ࡋǶ ᡂ໨೛ۓύǴԖ΋Ҟޑట௖૸όӕޑԛભໆЁෳᡍߏࡋϷόӕޑᚒҁԛભ ໆЁঁኧჹܭԛભໆЁϩኧ՗ीޑቹៜǴࡺҁࣴزڰۓෳᡍᚒҁߏࡋȐ24 ᚒǵ 36ᚒϷ 72 ᚒȑǴ೛ۓᚒҁԛભໆЁޑঁኧǴ٬ԛભໆЁෳᡍߏࡋᒿϐᡂ୏Ƕᖐ ٯٰᇥǴӧෳᡍߏࡋࣁ 24 ᚒޑᚒҁύǴ௖૸ 2 ঁϷ 4 ঁᚒҁԛભໆЁঁኧޑଛ ࿼௃׎Ǵ߾ԛભໆЁෳᡍߏࡋӚࣁ 12 ᚒȐ24y2 12ȑϷ 6 ᚒȐ ȑǹ ӧෳᡍߏࡋࣁ 36 ᚒޑᚒҁύǴ௖૸ 2 ঁϷ 6 ঁᚒҁԛભໆЁঁኧޑଛ࿼௃׎Ǵ ߾ԛભໆЁෳᡍߏࡋӚࣁ 18 ᚒȐ 6 4 24y 8 1 2 36y ȑϷ 6 ᚒȐ36y6 6ȑǹӧෳᡍߏࡋ ࣁ 72 ᚒޑᚒҁύǴ௖૸ 4 ঁϷ 6 ঁᚒҁԛભໆЁঁኧޑଛ࿼௃׎Ǵ߾ԛભໆЁ

ෳᡍߏࡋӚࣁ 18 ᚒȐ72y4 18ȑϷ 12 ᚒȐ72y6 12ȑǶӢԜǴӧ΢ॊΟᅿᡂ ໨ύǴᕴӅԖ3u2 6ᅿଛ࿼௃׎Ƕ 4. ӧԛભໆЁ࣬ᜢำࡋޑ೛ۓǴҁࣴزట௖૸ԛભໆЁ໔ޑ࣬ᜢำࡋჹܭԛભ ໆЁϩኧ՗ीޑቹៜǴࡺ೛ۓڙ၂ޣԛભໆЁૈΚॶ T ϐ໔ޑ࣬ᜢ߯ኧ Ȑcorrelation coefficientsȑӧե࣬ᜢࣁ 0.2ǵӧύ฻࣬ᜢࣁ 0.5ǵӧଯ࣬ᜢࣁ 0.8ǴӢԶኳᔕڙ၂ޣϐૈΚॶ T ܺவ኱ྗӭᡂໆதᄊϩթȐstandardized multivariate normal distributionȑǴ٠ଷ೛ૈΚॶϐ໔ޑ࣬ᜢ߯ኧࣁ 0.2ǵ0.5 Ϸ 0.8 Οᅿ࣬ᜢำࡋǶ

5. ӧڙ၂Γኧޑ೛ۓǴኳᔕࡼෳΓኧࣁ 500 Γǵ1000 ΓϷ 3000 ΓΟᅿ௃׎Ƕ 6. ӧԛભໆЁ՗ीБݤޑ೛ۓǴКၨ MIRT БݤǵBOCK БݤǵOPI БݤǵREG БݤǵPC БݤǵREGP БݤǵW-BOCK Бݤ฻Ύᅿ՗ीБݤޑ՗ीਏ݀Ƕ ӢԜǴਥᏵኳᔕჴᡍϐӚᡂ໨೛ۓǴҁࣴزӧൂ΋ෳᡍ೛ीύǴӅ௖૸ ᅿଛ࿼௃׎Ƕ 54 3 3 6u u Βǵʳ ฻ϯෳᡍ೛ी ߄ 3-2 ฻ϯෳᡍ೛ीϐӅӕᡂ໨೛ۓ ࣴزᡂ໨ ᡂ໨೛ۓ ෳᡍᚒҁߏࡋ 60ᚒ ᚒҁԛભໆЁঁኧ 4ঁ ฻ϯ೛ी NEATǵBIB ᚒҁԛભໆЁКٯ 30%ǵ30%ǵ20%ǵ20%Ϸ 40%ǵ40%ǵ10%ǵ10% ԛભໆЁ࣬ᜢำࡋ 0.2ǵ0.5ǵ0.8 ڙ၂Γኧ 3570ǵ7560 Γ

ԛભໆЁ՗ीБݤ BOCKݤǵOPI ݤǵREGP ݤǵW-BOCK ݤǵMIRT ݤ ؂΋ᅿ௃׎ኳᔕԛኧ 100ԛ

ӧ฻ϯෳᡍ೛ीύ೛ۓϤᅿόӕᡂ໨Ǵӵ߄ 3-2 ܌ҢǶ٩ྣӚ௃ნᡂ໨ౢғ ኳᔕၗ਑ǴϤᅿࣴزᡂ໨௶ॊӵΠǺ

2. ӧᚒҁԛભໆЁঁኧޑ೛ۓǴኳᔕᚒҁԛભໆЁঁኧࣁ 4 ঁǶ 3. ӧ฻ϯ೛ीޑ೛ۓǴҁࣴز௖૸ۓᗕό฻ಔ೛ीϷѳᑽόֹӄ୔༧೛ीٿᅿ ฻ϯ೛ीБݤǶ 4. ӧᚒҁԛભໆЁКٯޑ೛ۓǴҁࣴزట௖૸ӧ؂ঁ၂ᚒ୔༧ϣǴԛભໆЁᚒ ኧޑКٯჹܭԛભໆЁϩኧ՗ीޑቹៜǴࡺኳᔕٿᅿᚒҁԛભໆЁКٯǴѤ ঁԛભໆЁ໔ᚒኧޑКٯϩձࣁ 30%ǵ30%ǵ20%ǵ20%Ϸ 40%ǵ40%ǵ10%ǵ 10%ٿᅿКٯǶځύ 30%ǵ30%ǵ20%ǵ20%ϐКٯࣁࣴزޣୖԵ೚ӭ୯ϣ Ѧεࠠෳᡍמೌൔ֋ύǴኧᏢࣽෳᡍϐԛભໆЁКٯǴӵ PISA 2003 ኧᏢࣽ Ϸ TIMSS 2007 ΖԃભኧᏢࣽ܌х֖Ѥঁϣ৒ሦୱϐКٯջࣁ 30%ǵ30%ǵ 20%ǵ20%ȐPISA 2003ǹTIMSS 2007ȑǹќ΋Кٯ߾ࣁࣴزޣࣁ௖૸ԛભໆ ЁКٯᝌਸၨεਔϐԛભໆЁϩኧ՗ीᆒྗࡋǴԶۓΠ 40%ǵ40%ǵ10%ǵ 10%ϐКٯǶ 5. ӵӕൂ΋ෳᡍ೛ीǴӧౢғኳᔕၗ਑ਔǴҭԵቾΑԛભໆЁ࣬ᜢำࡋϷڙ၂ ޣΓኧޑ೛ۓǶԛભໆЁ࣬ᜢำࡋԖ 0.2ǵ0.5 Ϸ 0.8 Οᅿ࣬ᜢǹڙ၂ޣΓኧ Ԗ 3570 ΓϷ 7560 ΓٿᅿΓኧǶࣁΑ࣬ᔈܭεࠠෳᡍϐࡼෳΓኧ࿶தࣁεኬ ҁǴЪଛӝ฻ϯ೛ीϐᚒҁڙ၂Γኧሡ؃Ǵࡺҁࣴز೛ۓεኬҁڙ၂ޣΓኧ ࣁ 7560 ΓǹԜѦǴࣁΑКၨࡼෳΓኧჹԛભໆЁ՗ीϐਏ݀Ǵࡺҁࣴز೛ ۓλኬҁڙ၂ޣΓኧࣁ 3570 ΓǴа٬ӚᚒҁϐࡼෳΓኧૈӧ 500 Γа΢Ǵ ቚу՗ीϐᛙۓ܄Ƕ 6. ԛભໆЁ՗ीБݤύǴҗܭ REG БݤϷ PC Бݤࢂа CTT ࣁ୷ᘵǴࡺӧҁ ࣴزϐ฻ϯෳᡍ೛ीύǴ໻КၨځᎩϖᅿ՗ीБݤȐBOCK БݤǵOPI Бݤǵ REGPБݤǵW-BOCK БݤǵMIRT Бݤȑޑ՗ीਏ݀Ƕ

ӢԜǴਥᏵኳᔕჴᡍϐӚᡂ໨೛ۓǴҁࣴزӧ฻ϯෳᡍ೛ीύǴӅ௖૸ ᅿଛ࿼௃׎Ƕ 24 2 3 2 2u u u ਥᏵ΢ॊϐࣴزᡂ໨೛ۓǴϩձౢғൂ΋ෳᡍᆶ฻ϯෳᡍϐኳᔕၗ਑Ǵӧ

ҁࣴزύǴჹܭൂ΋ෳᡍ೛ीᆶ฻ϯෳᡍ೛ीޑ؂΋ঁόӕࣴزᡂ໨֡ख़ፄ຾ Չ 100 ԛޑၗ਑ኳᔕǴҔа՗ीԛભໆЁϩኧϐᆒྗࡋǴ՗ीᆒྗࡋаԛભໆ Ёϩኧϐਥ֡БৡȐroot mean square error, RMSEȑբࣁຑ՗ྗ߾Ƕ

ມǵʳ ୖኧ೛ۓ

΋ǵʳ ڙ၂ޣૈΚୖኧ೛ۓ ኳᔕόӕԛભໆЁϐڙ၂ޣૈΚϩѲǴࣁ኱ྗӭᡂໆதᄊϩѲǶଷ೛ ܺவӭᡂໆதᄊϩѲǴ૶ࣁ ) ,..., (T Tj T 1 MN(P,6)ǴځύǴ ϩձࣁᄒ׀ தᄊϩѲǴջ Ǵѳ֡ኧࣁ 0Ǵ኱ྗৡࣁ 1Ǵጄൎࣚۓܭ j ,...,T T1 ) 1 , 0 ( ~ ),..., 1 , 0 ( ~ 1 N Tj N T 3 ~ 3  Ǵ࣬ᜢऊࣁ 0.8ǵ0.5 ᆶ 0.2Ƕ Βǵʳ ၂ᚒᜤࡋୖኧ೛ۓ ኳᔕᜤࡋୖኧϩѲࣁᄒ׀தᄊϩѲN(0,1)Ǵጄൎ3~ 3Ƕ

ಃΟ࿯ʳ ჴᡍ೛ी

ҁࣴزϐኳᔕჴᡍ೛ۓΑൂ΋ෳᡍ೛ीᆶ฻ϯෳᡍ೛ीǴ૟ஒٿᅿჴᡍ೛ ीϷኳᔕჴᡍ؁ᡯϩॊӵΠǺ

൘ǵʳൂ΋ෳᡍ೛ी

ҁࣴزӧൂ΋ෳᡍ௃ნύǴኗቪำԄኳᔕౢғ 72 ᚒΒϡीϩᒧ᏷ᚒ၂ᚒϐ നεᚒ৤ǴаϷኳᔕౢғڙ၂ޣΓኧ 3000 ΓǶ௖૸όӕෳᡍᚒҁߏࡋϷόӕڙ ၂ޣΓኧϐԛભໆЁϩኧᆒྗࡋਔǴӆᒿᐒܜڗᚒ৤ϣϐ၂ᚒᆶܜڗڙ၂ޣΓ ኧǶᖐٯٰᇥǴऩᚒҁх֖ 2 ঁԛભໆЁЪ؂ঁԛભໆЁෳᡍߏࡋࣁ 18 ᚒǴ߾ ሡӧᚒ৤ύܜڗ 36 ᚒ၂ᚒǶऩ௖૸௃ნϐڙ၂ޣΓኧࣁ 1000 ΓਔǴ߾வኳᔕ ϐ 3000 Γύܜڗڙ၂Γኧ 1000 ΓǶ

ມǵʳ฻ϯෳᡍ೛ी

ҁࣴزӧ฻ϯෳᡍ௃ნύǴКၨ NEAT ᆶ BIB ٿᅿ฻ϯ೛ीჹܭԛભໆЁ ϩኧ՗ीᆒྗࡋϐቹៜǶӧЦཫറȐ2006ȑࣴزύࡰрǴܭ࣬ӕڙ၂ޣΓኧϐ ΠǴ၂ᚒ୔༧ኧຫӭǴ߾ୖኧ՗ीᇤৡຫεǴࡺҁࣴزϐ BIB ฻ϯ೛ी߯٩Ᏽ මҏฑǵЦཫറǵ೾դԽᆶ೚ϺᆢȐ2006ȑ܌೛ीϐ BIB1 ೛ीǴջ໻Ύঁ၂ ᚒ୔༧ǵΎঁෳᡍᚒҁϐ೛ीǶࣁΑКၨ NEAT ᆶ BIB ٿᅿ฻ϯ೛ीΠǴԛભ ໆЁϩኧ՗ीϐ่݀Ǵҁࣴزϐ NEAT ฻ϯ೛ीҭ೛ۓࣁΎঁ၂ᚒ୔༧ǴӢԶ ԖΟঁෳᡍᚒҁǶԜѦǴҁࣴز೛ۓ؂ঁ၂ᚒ୔༧ϣࣣ֖ԖѤঁԛભໆЁޑ၂ ᚒǶԜѦǴӧ฻ϯෳᡍ೛ीύҭ௖૸ᚒҁԛભໆЁКٯჹܭԛભໆЁϩኧ՗ी ᆒྗࡋϐቹៜǴࡺ೛ۓѤঁԛભໆЁޑКٯϩձࣁ 30%ǵ30%ǵ20%ǵ20%Ϸ 40%ǵ40%ǵ10%ǵ10%ٿᅿǶNEAT ೛ीᆶ BIB ೛ीӵ߄ 3-3 Ϸ߄ 3-4 ܌ҢǶ ߄ 3-3 NEAT ᚒҁଛ࿼߄ ᚒҁׇဦ ୔༧Ȑk1ȑ ୔༧Ȑk2ȑ ୔༧Ȑk3ȑ S1 M1 M2 M3 S2 M1 M4 M5 S3 M1 M6 M7 ӧ NEAT ೛ीύǴх֖ΟঁෳᡍᚒҁǵΎঁ၂ᚒ୔༧Ǵ؂ঁᚒҁԖΟঁ၂ ᚒ୔༧Ƕ ߄ 3-4 BIB ᚒҁଛ࿼߄ ᚒҁׇဦ ୔༧Ȑk1ȑ ୔༧Ȑk2ȑ ୔༧Ȑk3ȑ S1 M1 M2 M4 S2 M2 M3 M5 S3 M3 M4 M6 S4 M4 M5 M7 S5 M5 M6 M1 S6 M6 M7 M2 S7 M7 M1 M3 ӧ BIB ೛ीύǴх֖ΎঁෳᡍᚒҁǵΎঁ၂ᚒ୔༧Ǵ؂ঁᚒҁԖΟঁ၂ᚒ ୔༧Ƕ

ٿᅿ฻ϯ೛ीޑᚒҁෳᡍߏࡋࣁ 60 ᚒǴ؂ঁ၂ᚒ୔༧Ԗ 20 ᚒǴӧᚒҁԛ ભໆЁКٯࣁ 30%ǵ30%ǵ20%ǵ20%ޑ೛ीύǴ؂ঁ၂ᚒ୔༧ϣϐѤঁԛભໆ Ёޑᚒኧϩձࣁ 6 ᚒǵ6 ᚒǵ4 ᚒǵ4 ᚒǹӧᚒҁԛભໆЁКٯࣁ 40%ǵ40%ǵ 10%ǵ10%ޑ೛ीύǴ؂ঁ၂ᚒ୔༧ϣϐѤঁԛભໆЁޑᚒኧϩձࣁ 8 ᚒǵ8 ᚒǵ 2ᚒǵ2 ᚒǶӢԜǴӧ฻ϯෳᡍ೛ीύǴኳᔕౢғ 140 ᚒȐ ȑMC ၂ ᚒϐᚒ৤ǴаϷኳᔕౢғڙ၂ޣΓኧ 7560 ΓǶ௖૸όӕΓኧϐ฻ϯࡕԛભໆЁ ϩኧᆒྗࡋਔǴӆᒿᐒܜڗ܌ሡϐΓኧǶ 140 7 20u

ୖǵʳ ኳᔕჴᡍ؁ᡯ

ҁࣴزϐኳᔕჴᡍኳᔕΑൂ΋ෳᡍ೛ीᆶ฻ϯෳᡍ೛ीٿᅿჴᡍ௃ნǴ૟ ϩձ൩ٿᅿჴᡍ௃ნϐჴᡍ؁ᡯϩॊӵΠǺ ΋ǵʳ ൂ΋ෳᡍ೛ीϐኳᔕჴᡍ؁ᡯ Ȑ΋ȑኳᔕ၂ᚒᜤࡋୖኧܺவᄒ׀தᄊϩթǴࡌҥᚒ৤Ǵ٠வᚒ৤ύࡷᒧ၂ᚒ ԿӚԛભໆЁಔԋᚒҁǹ ȐΒȑኳᔕӚԛભໆЁϐڙ၂ޣૈΚܺவ኱ྗӭᡂໆதᄊϩѲǴ٠ଷ೛ԛભໆ Ё໔ޑ࣬ᜢऊࣁ 0.8ǵ0.5 ᆶ 0.2ǹ ȐΟȑҔ IRT ൂୖኧ Rasch ኳԄीᆉӚԛભໆЁϐPij(T)ǴځύǴiࣁ၂ᚒǵ ࣁ ԛભໆЁǹ j ȐѤȑ٬Ҕ؁ᡯΟϐPij(T)ीᆉ؂ঁԛભໆЁޑ੿ჴϩኧǶаෳᡍᚒҁߏࡋ 24 ᚒǴᚒҁԛભໆЁঁኧ 4 ঁϐ௃ნࣁٯٰᇥǴ؂ঁԛભໆЁෳᡍߏࡋࣁ 6 ᚒǴ߾ѤঁԛભໆЁޑ੿ჴϩኧϩձࣁ၂ᚒ 1 ډ၂ᚒ 6 ϐPij(T)ޑᕴ کǵ၂ᚒ 7 ډ၂ᚒ 12 ϐPij(T)ޑᕴکǵ၂ᚒ 13 ډ၂ᚒ 18 ϐPij(T)ޑᕴک Ϸ၂ᚒ 19 ډ၂ᚒ 24 ϐPij(T)ޑᕴکǶࣴزύଷ೛Ԝࣁ੿ჴϩኧǴҔٰբ

ࣁКၨόӕԛભໆЁϩኧीᆉБݤϐ୷ྗǹ Ȑϖȑ٬Ҕ؁ᡯȐΟȑϐPij(T)ౢғբเϸᔈȐresponseȑXijǹ ȐϤȑ٬Ҕ؁ᡯȐϖȑϐբเϸᔈXijϷ Acer ConQuest 2.0 ೬ᡏ຾Չୖኧ՗ीǹ ȐΎȑϩձҔ BOCKǵOPIǵREGǵPCǵREGPǵW-BOCK Ϸ MIRT ฻ΎᅿБݤ ՗ीԛભໆЁϩኧǹ ȐΖȑஒ΢ॊϐ؁ᡯȐ΋ȑډ؁ᡯȐΎȑख़ፄ຾Չ 100 ԛǴКၨόӕБݤϐԛ ભໆЁϩኧޑ RMSEǶ Βǵʳ ฻ϯෳᡍ೛ीϐኳᔕჴᡍ؁ᡯ Ȑ΋ȑኳᔕ၂ᚒᜤࡋୖኧܺவᄒ׀தᄊϩթǴࡌҥᚒ৤Ǵ٠٩ྣᚒҁԛભໆЁ КٯǴࡷᒧ၂ᚒԿӚԛભໆЁಔԋᚒҁǹ ȐΒȑኳᔕӚԛભໆЁϐڙ၂ޣૈΚܺவ኱ྗӭᡂໆதᄊϩѲǴ٠ଷ೛ԛભໆ Ё໔ޑ࣬ᜢऊࣁ 0.8ǵ0.5 ᆶ 0.2ǹ ȐΟȑҔ IRT ൂୖኧ Rasch ኳԄीᆉӚԛભໆЁϐPij(T)ǴځύǴ ࣁ၂ᚒǵ ࣁ ԛભໆЁǹ i j ȐѤȑ٬Ҕ؁ᡯΟϐPij(T)ीᆉ؂ঁԛભໆЁϐ੿ჴϩኧǶаᚒҁԛભໆЁКٯ ࣁ 30%ǵ30%ǵ20%ǵ20%ϐ௃ნࣁٯٰᇥǴಃ΋ঁԛભໆЁԖ 42 ᚒ Ȑ ȑǴԜԛભໆЁϐ੿ჴϩኧࣁ၂ᚒ 1 ډ၂ᚒ 6ǵ ၂ᚒ 21 ډ၂ᚒ 26ǵ၂ᚒ 41 ډ၂ᚒ 46ǵ၂ᚒ 61 ډ၂ᚒ 66ǵ၂ᚒ 81 ډ ၂ᚒ 86ǵ၂ᚒ 101 ډ၂ᚒ 106 Ϸ၂ᚒ 121 ډ၂ᚒ 126 ϐ ᚒ ঁ၂ᚒ୔༧ ᚒ 7 42 6 u ) (T ij P ޑᕴکǴ ಃΒǵΟǵѤঁԛભໆЁϐ੿ჴϩኧ٩ԜीᆉБݤ٩Ԝᜪ௢Ƕࣴزύଷ ೛Ԝࣁ੿ჴϩኧǴҔٰբࣁКၨόӕԛભໆЁϩኧीᆉБݤϐ୷ྗǹ Ȑϖȑ٬Ҕ؁ᡯȐΟȑϐPij(T)ౢғբเϸᔈȐresponseȑXijǹ

ȐϤȑ٬Ҕ؁ᡯȐϖȑϐբเϸᔈXijϷ Acer ConQuest 2.0 ೬ᡏ຾Չୖኧ՗ीǹ ȐΎȑϩձҔ BOCKǵOPIǵREGPǵW-BOCK Ϸ MIRT ฻ϖᅿБݤ՗ीԛભໆ Ёϩኧǹ ȐΖȑஒ΢ॊϐ؁ᡯȐ΋ȑډ؁ᡯȐΎȑख़ፄ຾Չ 100 ԛǴКၨόӕБݤϐԛ ભໆЁϩኧޑ RMSEǶ

ಃѤ࿯ʳ ՗ीᆒྗࡋ

՗ीᆒྗࡋࢂࡰ՗ीᇤৡޑελǴ՗ीᇤৡຫλǴ߾ж߄՗ीຫྗዴǶҁࣴ ز٬ҔԛભໆЁϩኧϐRMSEբࣁ՗ीԛભໆЁϩኧޑྗዴࡰኧǴीᆉԄηӵΠǺ N RMSE N i ij ij j j

¦

[ [ [ [ 1 2 ) ˆ ( ) ˆ , ( Ȑ3-1ȑ ځύǴ j ࣁಃ j ঁԛભໆЁǹ Nж߄ڙ၂ޣΓኧǹ ) ,..., , , ([1j [2j [3j [Nj [ ࣁԛભໆЁ j ϐ੿ჴϩኧǹ ࣁԛભໆЁ ) ˆ ,..., ˆ , ˆ , ˆ ( ˆ 3 2 1j [ j [ j [Nj [ [ jϐ՗ीϩኧǶ

ಃϖ࿯ʳ ࣴزπڀ

൘ǵʳ MATLAB

MATLABᔈҔ೬ᡏ่ӝΑኧॶϩ݋ǵંତၮᆉϷᛤკ฻фૈǴᇟݤᙁൂǵ ᏹբϟय़ᙁܰǴᏱԖфૈமεޑڄኧ৤ǴЪගٮֹ᏾ޑંତၮᆉࡰзǴЬाޑ Ҕ೼ࢂբંତԄޑኧᏢၮᆉǶMATLAB த೏ᔈҔܭࣽᏢᆶπำሦୱޑኧॶၮ ᆉǵϩ݋ᆶኳᔕǶӢԜǴҁࣴز٬ҔԜ೬ᡏٰౢғኳᔕၗ਑ϷኗቪԛભໆЁϩ ኧीᆉϐำԄǴ٠ҔаीᆉԛભໆЁ՗ीБݤϐᆒྗࡋǶ

ມǵʳ Acer ConQuest

Acer ConQuest 2.0ȐWu, Adams, & Wilson, 1998ȑࢂ΋ঁ፾ଛܭ၂ᚒϸᔈኳ ԄȐitem response modelȑکወӧӣᘜኳԄȐlatent regression modelȑޑႝတ೬ᡏǴ ගٮቶݱޑ၂ᚒϸᔈኳԄϩ݋ǴҔനεཷ՟ݤȐmaximum likelihood method, MLȑ ՗ीᜤࡋୖኧǵයఈࡕᡍݤȐexpected a posteriori, EAPȑ՗ीૈΚୖኧǴёҔ ӧӭӛࡋ IRT ϐ՗ीǶӢԜǴҁࣴز٬ҔԜ೬ᡏٰ՗ी၂ᚒᜤࡋୖኧϷڙ၂ޣ ૈΚୖኧǶ