{"id":738,"date":"2017-02-09T13:38:06","date_gmt":"2017-02-09T05:38:06","guid":{"rendered":"https:\/\/www.beneplot.com\/?p=738"},"modified":"2020-07-23T15:13:24","modified_gmt":"2020-07-23T07:13:24","slug":"%e5%a4%9a%e7%ba%a7irt%e6%a8%a1%e5%9e%8b-polytomous-irt-models%e4%b8%ad%e5%90%84%e7%b1%bb%e5%8f%82%e6%95%b0%e7%9a%84%e8%ae%a1%e7%ae%97%e6%96%b9%e6%b3%95","status":"publish","type":"post","link":"https:\/\/www.beneplot.com\/?p=738","title":{"rendered":"\u591a\u7ea7IRT\u6a21\u578b (Polytomous IRT Models)\u4e2d\u5404\u7c7b\u53c2\u6570\u7684\u8ba1\u7b97\u65b9\u6cd5"},"content":{"rendered":"


\n\u5728\u300aIRT\u4e2d\u7684\u80fd\u529b\u503c\u548c\u96be\u5ea6\u503c\u662f\u5982\u4f55\u8ba1\u7b97\u7684\uff1f<\/a>\u300b\u4e00\u6587\u4e2d\uff0c\u4ee5Rasch\u6a21\u578b\u4e3a\u4f8b\uff0c\u8bb2\u8ff0\u4e86\u5bf9\u4e24\u7ea7\u8ba1\u5206\uff08\u53730,1\u8ba1\u5206\uff09\u7684\u9898\u76ee\u5982\u4f55\u6765\u8ba1\u7b97IRT\u96be\u5ea6\u503c\uff0c\u5728\u300a\u4e0d\u540c\u591a\u7ea7IRT\u6a21\u578b (Polytomous IRT Models) \u7684\u533a\u522b<\/a><\/span><\/strong>\u300b<\/span><\/strong>\u4e00\u6587\u4e2d\uff0c\u63d0\u5230\u4e86\u591a\u79cd\u7528\u4e8e\u591a\u7ea7\u8ba1\u5206\u9898\u76ee\u7684IRT\u6a21\u578b\u3002\u672c\u6587\u5c06\u4ee5PCM\u4e3a\u4f8b\uff0c\u8bb2\u8ff0\u5176\u5404\u7c7b\u53c2\u6570\u503c\u5982\u4f55\u8ba1\u7b97\u3002<\/span><\/strong><\/p>\n

\u5c06\u300a\u4e0d\u540c\u591a\u7ea7IRT\u6a21\u578b (Polytomous IRT Models) \u7684\u533a\u522b<\/a><\/strong>\u300b\u4e00\u6587\u4e2d\u7684\u7b49\u5f0f\u4e00\u7b80\u5316\u540e\u5f97\uff1a<\/span><\/p>\n

\u516c\u5f0fA\uff1a<\/span><\/strong><\/p>\n

P =(EXP \u03a3<\/strong>(<\/strong>\u03b8n <\/sub>-\u03b4ij<\/sub> )) \/ (\u03a3<\/strong> EXP \u03a3<\/strong>\u00a0(<\/strong>\u03b8n <\/sub>-\u03b4ij<\/sub> ))\uff0c<\/strong><\/span><\/p>\n

\u5728PCM\u4e2d\uff0c\u6d4b\u8bd5\u8005\u4e0e0\u5206\u662f\u6ca1\u6709\u8ddd\u79bb\u7684\uff0c\u5373\u8bbe\u5b9a\uff1a\u03a3<\/strong>\u00a0(<\/strong>\u03b8n <\/sub>-\u03b4i0<\/sub> ) = 0<\/strong>\u3002<\/span><\/p>\n

\u5047\u8bbe\u6709\u4e00\u90530,1,2,3\u8ba1\u5206\u7684\u9898\u76ee\uff0c\u5373\u6839\u636e\u516c\u5f0fA\u53ef\u5f97\uff0c\uff1a<\/span><\/p>\n

Step0\uff1a \u03a3<\/strong> (\u03b8n <\/sub>-\u03b4i0<\/sub> )=0 \u00a0<\/sub>, <\/strong><\/span><\/p>\n

Step1\uff1a \u03a3<\/strong> (\u03b8n <\/sub>-\u03b4i0<\/sub> ) + \u03b8n <\/sub>-\u03b4i1<\/sub> = \u03b8n <\/sub>-\u03b4i1\u00a0 <\/sub>,<\/strong><\/span><\/p>\n

Step2\uff1a \u03b8n <\/sub>-\u03b4i1<\/sub><\/strong> + \u03b8n <\/sub>-\u03b4i2<\/sub> = 2<\/strong>\u03b8n <\/sub>-\u03b4i1<\/sub> -\u03b4i2\u00a0 <\/sub>,<\/strong><\/span><\/p>\n

Step3\uff1a 2\u03b8n <\/sub>-\u03b4i1<\/sub> -\u03b4i2<\/sub><\/strong> +\u03b8n <\/sub>-\u03b4i3<\/sub> = 3<\/strong>\u03b8n <\/sub>-\u03b4i1<\/sub> -\u03b4i2 <\/sub>-\u03b4i3<\/sub><\/strong> \u00a0<\/strong><\/sub>,<\/strong><\/span><\/p>\n

\u7ee7\u7eed\u6839\u636e\u516c\u5f0fA\u8f6c\u5316\uff0c\u53ef\u5f97\uff1a<\/span><\/p>\n

Step0\uff1aS1= EXP (0) = 1 ,<\/strong><\/span><\/p>\n

Step1\uff1aS2= EXP (<\/strong>\u03b8n <\/sub>-\u03b4i1<\/sub>) ,<\/strong><\/span><\/p>\n

Step2\uff1aS3= EXP (2<\/strong>\u03b8n <\/sub>-\u03b4i1<\/sub> -\u03b4i2<\/sub>) ,<\/strong><\/span><\/p>\n

Step3\uff1aS4= EXP (3<\/strong>\u03b8n <\/sub>-\u03b4i1<\/sub> -\u03b4i2 <\/sub>-\u03b4i3<\/sub><\/strong> ) ,<\/strong><\/span><\/p>\n

Denominator(of formula A) = S1+S2+S3+S4 = 1+ (EXP (<\/strong>\u03b8n <\/sub>-\u03b4i1<\/sub>)) +(EXP (2\u03b8n <\/sub>-\u03b4i1<\/sub> -\u03b4i2<\/sub>)) + (EXP (3\u03b8n <\/sub>-\u03b4i1<\/sub> -\u03b4i2 <\/sub>-\u03b4i3<\/sub><\/strong> )) ,<\/strong><\/span><\/p>\n

\u7531\u6b64\uff0c\u6839\u636e\u516c\u5f0fA\uff0c\u80fd\u529b\u4e3a\u03b8n<\/sub>\u7684\u4eba\u5728\u4f5c\u7b54\u8fd9\u9053\u9898\u76ee\u65f6\uff0c<\/span><\/p>\n

\u5f970\u5206\u7684\u6982\u7387\u4e3a\uff1a<\/span><\/p>\n

P0 = S1 \/ Denominator,<\/strong><\/span><\/p>\n

\u5f971\u5206\u7684\u6982\u7387\u4e3a\uff1a<\/span><\/p>\n

P1 = S2 \/ Denominator,<\/strong><\/span><\/p>\n

\u5728\u8fd9\u91cc\u53ef\u4ee5\u53d1\u73b0\uff0cRasch\u6a21\u578b\u662f\u53ea\u67090,1\u4e24\u7c7b\u5206\u503c\u7684PCM\u7684\u7279\u4f8b<\/strong>\uff0c\u5373\uff0c<\/span><\/p>\n

P1 = EXP (\u03b8n <\/sub>-\u03b4i1<\/sub>) \/ 1+ EXP (\u03b8n <\/sub>-\u03b4i1<\/sub>)\u3002<\/span><\/strong><\/p>\n

\u5f972\u5206\u7684\u6982\u7387\u4e3a\uff1a<\/span><\/p>\n

P2 = S3 \/ Denominator,<\/strong><\/span><\/p>\n

\u5f973\u5206\u7684\u6982\u7387\u4e3a\uff1a<\/span><\/p>\n

P3 = S4 \/ Denominator,<\/strong><\/span><\/p>\n

\u4ece\u300aIRT\u4e2d\u7684\u80fd\u529b\u503c\u548c\u96be\u5ea6\u503c\u662f\u5982\u4f55\u8ba1\u7b97\u7684\uff1f<\/a><\/strong><\/span>\u300b<\/span>\u4e00\u6587\u4e2d\u53ef\u77e5\uff0c\u5bf9\u4e8e0,1\u8ba1\u5206\u7684\u9898\u76ee\u6765\u8bf4\uff0c<\/span>\u9898\u76ee\u96be\u5ea6\u5373\u4e3a\u03b4<\/strong>\uff0c<\/span>\u672a\u8fed\u4ee3\u524d\u9898\u76ee\u96be\u5ea6\u503c\u7684\u8ba1\u7b97\u65b9\u6cd5\u4e3a\uff1a<\/strong><\/p>\n

0,1<\/strong>\u8ba1\u5206\u9898\u76ee\u96be\u5ea6\u503c\u03b4= ln((1-p)\/p)<\/strong>\uff0c<\/span><\/p>\n

\u5176\u4e2dP\u4e3a\u9898\u76ee\u7684\u7b54\u5bf9\u7387\uff0cln()\u610f\u4e3a\u6c42\u62ec\u53f7\u5185\u6570\u503c\u7684\u81ea\u7136\u5bf9\u6570\u3002<\/span><\/p>\n

\u591a\u7ea7\u8ba1\u5206\u9898\u76ee\u7684\u8ba1\u7b97\u65b9\u5f0f\u6709\u6240\u4e0d\u540c\u3002<\/strong><\/span><\/p>\n

\u4ee5\u4e00\u90530,1,2,3\u8ba1\u5206\u9898\u76ee\u7684\u5404\u4e2a\u03b4\u503c\u8ba1\u7b97\u4e3a\u4f8b\uff0c\u8be5\u9898\u76ee\u53ef\u6c42\u51fa3\u4e2a\u03b4ik<\/sub>\u503c\uff0c\u5206\u522b\u8bbe\u5b9a\u4e3a\uff1a\u03b4i1<\/sub>\u3001\u03b4i2<\/sub>\u3001\u03b4i3 <\/sub>\uff0c\u4ee5\u53ca\u8be5\u9898\u7684\u6574\u4f53\u96be\u5ea6\u503c\u03b4i<\/sub> = (\u03b4i1 <\/sub>+ \u03b4i2 <\/sub>+ \u03b4i3<\/sub>) \/ 3\u3002<\/strong><\/span><\/p>\n

\u63a5\u4e0b\u6765\u4ee5\u5b9e\u4f8b\u8bb2\u8ff0\u5404\u4e2a\u03b4\u503c\u5982\u4f55\u8ba1\u7b97\u3002<\/span><\/p>\n

\u5047\u5982\u8be5\u9898\u670930\u4e2a\u88ab\u8bd5\u4f5c\u7b54\uff0c\u4f5c\u7b54\u5f97\u5206\u5206\u522b\u662f: c(0,2,0,0,1,2,0,1,3,2,2,1,2,1,3,2,2,2,3,1,1,1,3,2,1,3,1,3,0,3)\uff0c<\/span><\/p>\n

\u9996\u5148\u6c42\u5404\u5206\u503c\u7684\u9891\u7387\uff0c\u5373<\/strong>\uff1a\u5f970\u5206\u7684\u67095\u4eba\uff0c\u5f971\u5206\u7684\u67099\u4eba\uff0c\u5f972\u5206\u7684\u67099\u4eba\uff0c\u5f973\u5206\u7684\u67097\u4eba\uff0c\u90a3\u4e48\uff1a<\/span><\/p>\n

\u03b4i1<\/sub>= ln((1-p1<\/sub>)\/p1<\/sub>)<\/strong> = ln((1-0.643)\/ 0.643) = -0.588<\/strong>\uff0c<\/span><\/p>\n

\u5176\u4e2dp1<\/sub>\u7b49\u4e8e(\u5f971\u5206\u7684\u4eba\u6570)\u9664\u4ee5(\u5f971\u5206\u7684\u4eba\u6570\u52a0\u5f970\u5206\u7684\u4eba\u6570)<\/strong>\uff0c<\/span><\/p>\n

\u03b4i2<\/sub><\/strong>= ln((1-p2<\/sub>)\/p2<\/sub>) = ln((1-0.438)\/ 0.438) = 0<\/strong>\uff0c<\/span><\/p>\n

\u03b4i3<\/sub><\/strong>= ln((1-p3<\/sub>)\/p3<\/sub>) = ln((1-0.438)\/ 0.438) = 0.251<\/strong>\uff0c<\/span><\/p>\n

\u9898\u76ee\u96be\u5ea6\u03b4i<\/sub> = (\u03b4i1 <\/sub>+ \u03b4i2 <\/sub>+ \u03b4i3<\/sub>) \/ 3 <\/strong>= (-0.588 + 0 + 0.251)\/3 = -0.112<\/strong>\u3002<\/span><\/p>\n

\u5047\u8bbe\u67d0\u4f5c\u7b54\u8005\u7684\u80fd\u529b\u503c\u03b8\u4e3a1<\/strong>\uff0c\u90a3\u4e48\uff0c\u4ed6\u4f5c\u7b54\u6b64\u9898<\/span><\/p>\n

\u5f970\u5206\u7684\u6982\u7387\u4e3a\uff1a<\/span><\/strong><\/p>\n

P0 = 1 \/<\/strong> 1+ (EXP (<\/strong>\u03b8n <\/sub>-\u03b4i1<\/sub>)) +(EXP (2\u03b8n <\/sub>-\u03b4i1<\/sub> -\u03b4i2<\/sub>)) + (EXP (3\u03b8n <\/sub>-\u03b4i1<\/sub> -\u03b4i2 <\/sub>-\u03b4i3 <\/sub>)) <\/strong>= 1\/(1+EXP(1-(-0.588))+EXP(2-(-0.588)-0)+EXP(3-(-0.588)-0-(-0.112))) = 0.017 = 1.7% ,<\/strong><\/span><\/p>\n

\u5176\u4e2d\uff0cDenominator(of P0) = 59.64\uff0c\u4e0b\u540c\uff0c<\/strong><\/span><\/p>\n

\u5f971\u5206\u7684\u6982\u7387\u4e3a\uff1a<\/span><\/strong><\/p>\n

P1 = S2 \/ Denominator = EXP (<\/strong>\u03b8n <\/sub>-\u03b4i1<\/sub>) \/ 59.64 = <\/strong>EXP(1-(-0.588)) \/ 59.64 = 4.894 \/ 59.64 = 0.082 = 8.2% ,<\/strong><\/span><\/p>\n

\u5f972\u5206\u7684\u6982\u7387\u4e3a\uff1a<\/span><\/strong><\/p>\n

P2 = S3 \/ Denominator = EXP (2<\/strong>\u03b8n <\/sub>-\u03b4i1<\/sub> -\u03b4i2<\/sub>) \/ 59.64 = <\/strong>EXP(2-(-0.588)-0) \/ 59.64 = 0.223 = 22.3% ,<\/strong><\/span><\/p>\n

\u5f973\u5206\u7684\u6982\u7387\u4e3a\uff1a<\/span><\/strong><\/p>\n

P3 = S4 \/ Denominator = EXP (3<\/strong>\u03b8n <\/sub>-\u03b4i1<\/sub> -\u03b4i2 <\/sub>-\u03b4i3<\/sub><\/strong> ) \/ 59.64 = <\/strong>EXP(3-(-0.588)-0-(-0.112)) \/ 59.64 = 0.678 = 67.8% <\/strong>\u3002<\/span><\/p>\n

\u53ef\u4ee5\u53d1\u73b0\uff1aP0 + P1 + P2 + P3 = 1.7% + 8.2% + 22.3% + 67.8% = 100% \u3002<\/strong><\/span><\/p>\n

\u7531\u300a\u4e0d\u540c\u591a\u7ea7IRT\u6a21\u578b (Polytomous IRT Models) \u7684\u533a\u522b<\/a><\/strong><\/span>\u300b<\/span>\u4e00\u6587\u53ef\u77e5\uff0c\u5728PCM\u4e2d\uff0c\u6709\u4e09\u4e2a\u4e3b\u8981\u7684\u53c2\u6570\uff0c\u5206\u522b\u662f\uff1a<\/span>\ud835\udeff\ud835\udc56\ud835\udc58<\/sub><\/strong> \u88ab\u79f0\u4e3a<\/span>\u4f4d\u7f6e\u53c2\u6570<\/strong>\u6216<\/span>\u7c7b\u522b\u53c2\u6570(Category Parameter)\uff0c<\/strong>\ud835\udeff\ud835\udc56<\/sub><\/strong>\u88ab\u79f0\u4e3a<\/span>\u9898\u76ee\u96be\u5ea6\u503c(Difficulty Parameter)\uff0c<\/strong>\ud835\udf0f<\/strong>\ud835\udc56\ud835\udc58<\/sub><\/strong> \u88ab\u79f0\u4e3a<\/span>\u6b65\u9aa4\u53c2\u6570(Step Parameter)\u3002\u5728\u4e0a\u9762\uff0c<\/strong>\u8bb2\u4e86<\/strong>\u4f4d\u7f6e\u53c2\u6570\u548c\u96be\u5ea6\u53c2\u6570\u7684\u8ba1\u7b97\u65b9\u6cd5\u3002<\/strong>\u5bf90,1,2,3\u8ba1\u5206\u7684\u9898\u76ee\u6765\u8bf4\uff0c\u540c\u6837<\/span>\u6709\u4e09\u4e2a<\/strong>\ud835\udf0f<\/strong>\ud835\udc56\ud835\udc58<\/sub><\/strong>\u503c<\/strong>\u3002\u63a5\u4e0a\u9762\u7684\u4f8b\u5b50\uff1a<\/span><\/p>\n

\u7531\ud835\udf0f\ud835\udc56\ud835\udc58<\/sub> = \ud835\udeff\ud835\udc56<\/sub> \u2212 \ud835\udeff\ud835\udc56\ud835\udc58<\/sub>\u5f97\uff1a<\/span><\/strong><\/p>\n

\ud835\udf0f\ud835\udc561<\/sub><\/strong> = <\/strong>\ud835\udeff\ud835\udc56<\/sub><\/strong> \u2212<\/strong> \ud835\udeff\ud835\udc561<\/sub><\/strong> =<\/strong> -0.112 \u2013 (-0.588) = 0.476,<\/strong><\/span><\/p>\n

\ud835\udf0f\ud835\udc562<\/sub><\/strong> = <\/strong>\ud835\udeff\ud835\udc56<\/sub><\/strong> \u2212<\/strong> \ud835\udeff\ud835\udc562<\/sub><\/strong> =<\/strong> -0.112 \u2013 0 = <\/strong>-0.112<\/strong>,<\/strong><\/span><\/p>\n

\ud835\udf0f\ud835\udc563<\/sub><\/strong> = <\/strong>\ud835\udeff\ud835\udc56<\/sub><\/strong> \u2212<\/strong> \ud835\udeff\ud835\udc563<\/sub><\/strong> =<\/strong> -0.112 \u2013 0.251 = -0.363<\/strong>\u3002<\/strong><\/span><\/p>\n

\u7531<\/span> \u03a3\ud835\udf0f\ud835\udc56\ud835\udc58<\/sub>\u00a0<\/sub> = 0 \u9a8c\u7b97\uff1a<\/strong><\/span><\/p>\n

\ud835\udf0f\ud835\udc561<\/sub><\/strong> + \ud835\udf0f\ud835\udc562<\/sub> + \ud835\udf0f\ud835\udc563<\/sub> = 0.476 – 0.112 – 0.363 \u2248 0<\/strong><\/span><\/p>\n

\u00a0<\/strong><\/span><\/p>\n

END<\/strong><\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

\u5728\u300aIRT\u4e2d\u7684\u80fd\u529b\u503c\u548c\u96be\u5ea6\u503c\u662f\u5982\u4f55\u8ba1\u7b97\u7684\uff1f\u300b\u4e00\u6587\u4e2d\uff0c\u4ee5Rasch\u6a21\u578b\u4e3a\u4f8b\uff0c\u8bb2\u8ff0\u4e86\u5bf9\u4e24\u7ea7\u8ba1\u5206\uff08\u53730,1\u8ba1\u5206\uff09\u7684\u9898 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[10],"tags":[],"yoast_head":"\n\n\n\n\n\n\n\n\n\n\n\n\n\n