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Test Equating Zhang Zhonghua Chinese University of Hong Kong

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Question ？ Two sets of Standardized Test which measure the same trait: A and B. A and B were administrated separately to two groups of students (Group 1 and Group 2). Group 1 students only took Test A, and Group 2 students only took Test B. The mean score on Test A for Group 1 is 84. And the mean score on Test B for Group2 is 80. t-test result indicated that there was a statistically significant difference between the mean score for Group 1 and Group 2 (p<0.05). Then, should the conclusion that the Group 1 students were better than the Group 2 students on the trait that the two tests measured be gotten?

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Why Equate ? To compare test scores of different forms of tests (Strictly speaking, Parallel tests) which measure the same latent trait To construct the item bank/pool Computerized Adaptive Testing (CAT)

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What’s Equating ? “Equating is a statistical process that is used to adjust scores on test forms so that scores on the forms can be used interchangeably. Equating adjusts for differences in difficulty among forms that are built to be similar in difficulty and content” (Kolen & Brennan, 2004). The two alternate test forms for equating: Same content and statistical specification Equity Symmetric Group Invariance

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Lord’s Equity Property Examinees with a given true score would have identical observed score means, standard deviations, and distributional shapes of converted scores on Form X and scores on Form Y. First-order Equity Property Examinees with a given true score have the same means converted score on Form X as they have on Form Y.

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Form Y RawForm X 1 RawForm X 2 Raw 124 235...... 131416 141517 151618 161719 171820 181921 ………

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Equating Design Single Group Random Groups Single Group with Counterbalance Anchored/Common-item Nonequivalent Group Preequating

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Single Group SampleForm XForm Y G1√√

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Single Group with Counterbalancing SampleTime 1Time 2 G1 Form XForm Y G2Form YForm X

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Random Groups SampleForm XForm Y G1√ G2√

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Common-item Nonequivalent Groups SampleForm XForm YCommon Items V G1√√ G2√√

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Preequating Precalibrated IRT Parameter Item Bank Items form Bank (Operational items) New Items (Non-Operational Items)

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Equating Methods Based on Classical Testing Theory (CTT) Based on Item Response Theory (IRT)

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Downloadable Equating Procedures Equating/Linking Programs http://www.education.uiowa.edu/casma/Equ atingLinkingPrograms.htm IRT Scale Transformation Programs http://www.education.uiowa.edu/casma/IRT Programs.htm

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Equating Methods Based on CTT Mean Equating Linear Equating Equipercentiel equating

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CTT-Mean Equating In mean equating, Form X is considered to differ in difficulty from Form Y by the difference of the mean scores between the two forms. Example: M X =70, M Y =75. Let Form X as the base Form, Form Y as the target Form. For the score 80 on Form Y, the Equated Score on the scale of Form X is 80-(75-70)=75.

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CTT-Linear Equating In Linear Equating, scores that are an equal distance from their means in standard deviation units are set equal.

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CTT-Equipercentile For a given Form X score, find the percentage of examinees earning scores at or below that Form X score. Find the Form Y score that has the same percentage of examinees at or below it. The Form X and Form Y score are considered to be equivalent. Example: 70% of the examinees got a score 75 or below on Form X. 70% of the examinees got a score 80 or below on Form Y. Then a Form X score of 75 would be considered to represent the same level of achievement as a Form Y score of 80.

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Equating Methods Based on IRT IRT Parameters Equating IRT Observed Score and IRT Truce Score Equating

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Item Response Theory Take IRT Three-Parameter Model as an example, Item parameters: Item Discrimination, Item Difficulty, Guessing

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Probability Item 1Item 2 Scale Score Difficulty Item 1 Item 2

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Probability Item 1Item 2 Scale Score Difficulty Item 1 Item 2

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Probability Item 1Item 2 Scale Score Difficulty Item 1 Item 2

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Item Parameter Equating Linking Separate Calibration (Mean/Mean Method, Mean/Sigma Method, Stocking-Lord Method, Haebara Method) Concurrent Calibration Fixed Common-Precalibrated Item Parameter Method

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IRT-Linking Separate Calibration

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IRT-Moment Methods Mean/Mean Method Mean/Sigma Method

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IRT-Characteristic Curve Method Stocking-Lord method: Haebara method:

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Example Take Form Y as the base test, Form X as the target Test Item 1 on Form X: Item Difficulty is 1.0; Item Discrimination is 1.896; Guessing is 0.18 Equated item parameters for Item 1 on Form X onto the scale of Base Form Y can be computed as follows, Stocking-LordHaebaraMean/MeanMean/Sigma B-0.057-0.063-0.0870.028 A0.9480.9420.9430.770

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IRT- Concurrent Calibration Concurrent calibration method involves estimating item and ability parameters simultaneously on a single computer run. In the procedure, the items that are not taken by one group of subjects are taken as not reached or missing data and the item parameters for all items on the two test forms are simultaneously estimated. This one estimation run makes the item parameters for all items from the two test forms put on the same scale (Kim & Hanson, 2002; Kim & Cohen, 1998). Example

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Concurrent Calibration for Replication 16 >COMMENTS Horizontal Equating Concurrent Calibration for Replication 16 >GLOBAL NPARM=3,DFNAME='D:\RESEARCH\REP16\CONH-16\CONH- 16.DAT',SAVE; >SAVE PARM='D:\RESEARCH\REP16\CONH-16\CONH-16.PAR'; >LENGTH NITEMS=140; >INPUT NTOTAL=80,SAMPLE=2000,NALT=4,NIDCH=4,FORMS=2; (4X,4A1,6X,I1,1X,80A1) >FORM1 LENGTH=80,ITEMS=(1(1)80); >FORM2 LENGTH=80,ITEMS=(1(1)20,81(1)140); >TEST ITEMS=(1(1)140), LINK=(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0); >CALIB CYCLES=20; >SCORE;

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IRT-Fixed Common-Item Parameters This procedure combines the features of concurrent calibration and linking separate calibration methods. In the method, the item parameters for the two test forms are estimated separately. What differs from linking separate calibration is that the common item parameters from the target test will be fixed at the estimated values from the base test. Example

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Fixed Common Item Parameters for Replication 16 >COMMENTS FCIP for Replication 16 Target Test Form B with N (0,1) >GLOBAL NPARM=3,DFNAME='D:\RESEARCH\REP16\FIXV-16\B11-16.DAT',SAVE; >SAVE PARM='D:\RESEARCH\REP16\FIXV-16\FIXV-16.PAR'; >LENGTH NITEMS=(80); >INPUT NTOTAL=80,SAMPLE=1000,NALT=4,NIDCH=4; (4A1,1X,80A1) >TEST ITEMS=(1(1)80); >CALIB TPRIOR,SPRIOR,GPRIOR,READPRI,CYCLES=20; >PRIORS TMU=(-0.639,1.041,1.701,0.482,-1.144,-0.023,0.616,1.133,0.668,0.577, -0.257,0.029,0.904,0.232,1.602,1.642,0.537,-0.228,1.439,0.517,0.0(0)60), TSIGMA=(0.001,0.001,0.001,0.001,0.001,0.001,0.001,0.001,0.001,0.001, 0.001,0.001,0.001,0.001,0.001,0.001,0.001,0.001,0.001,0.001,2.0(0)60), SMU=(-0.688,0.011,-0.810,0.614,-0.811,-0.445,-0.142,-0.387,0.292,-0.449, 0.040,-0.522,0.080,0.660,0.301,0.408,-0.689,-0.079,0.294,-0.174,0.0(0)60), SSIGMA=(0.001,0.001,0.001,0.001,0.001,0.001,0.001,0.001,0.001,0.001, 0.001,0.001,0.001,0.001,0.001,0.001,0.001,0.001,0.001,0.001,0.5(0)60), ALPHA=(2882.990,1329.080,3540.010,4092.470,2694.080,2652.900,2314.660, 2532.500,2336.870,3725.870,2364.700,2545.460,2358.110,2307.760, 3583.990,3117.190,2569.460,1817.030,1057.210,2544.350,6(0)60), BETA=(7119.010,8672.920,6461.990,5909.530,7307.920,7349.100,7687.340,7469.500, 7665.130,6276.130,7637.300,7456.540,7643.890,7694.240,6418.010,6884.810, 7432.540,8184.970,8944.790,7457.650,16(0)60); >SCORE;

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Comparison of Different Equating Methods No agreements have been gotten Methods based on CTT can be used to equate tests. Methods based on IRT are essential to construct item bank/pool. Among the methods based on IRT, some researches indicated that Concurrent Calibration Method could produce more accurate equating results than that of Linking Separate Calibration Method and FCIP method.

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Thank You Very Much!

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