Evaluating the Effects of Multidimensionality on IRT True-Score Equating

The performance of the item response theory (IRT) true-score equating method is examined under conditions of test multidimensionality. It is argued that a primary concern in applying unidimensional equating methods when multidimensionality is present is the potential decrease in equity (Lord, 1980) attributable to the fact that examinees of different ability are expected to obtain the same test scores. In contrast to equating studies based on real test data, the use of simulation in equating research not only permits assessment of these effects but also enables investigation of hypothetical equating conditions in which multidimensionality can be suspected to be especially problematic for test equating. In this article, I investigate whether the IRT true-score equating method, which explicitly assumes the item response matrix is unidimensional, is more adversely affected by the presence of multidimensionality than 2 conventional equating methods-linear and equipercentile equating-using several recently proposed equity-based criteria (Thomasson, 1993). Results from 2 simulation studies suggest that the IRT method performs at least as well as the conventional methods when the correlation between dimensions is high (³ 0.7) and may be only slightly inferior to the equipercentile method when the correlation is moderate to low (£ 0.5).     

考生能力分布与被试量对IRT等值的影响

基于项目反应理论中的LOGISTIC双参数模型研究共同题非等组设计下,考生能力分布与被试量对等值的影响。等值方法采用分别校准下的项目特征曲线法、Stocking-Lord法、Haebara法。等值结果采用等值分数标准误、等值系数标准误、共同题参数稳定性三种方法进行评价。研究结果表明,考生能力分布越接近,被试量越大,等值误差越小;且Stocking-Lord法较Haebara法的等值结果更稳定。     

A Comparison of IRT Equating and Beta 4 Equating

Four equating methods (3PL true score equating, 3PL observed score equating, beta 4 true score equating, and beta 4 observed score equating) were compared using four equating criteria: first-order equity (FOE), second-order equity (SOE), conditional-mean-squared-error (CMSE) difference, and the equipercentile equating property. True score equating more closely achieved estimated FOE than observed score equating when the true score distribution was estimated using the psychometric model that was used in the equating. Observed score equating more closely achieved estimated SOE, estimated CMSE difference, and the equipercentile equating property than true score equating. Among the four equating methods, 3PL observed score equating most closely achieved estimated SOE and had the smallest estimated CMSE difference, and beta 4 observed score equating was the method that most closely met the equipercentile equating property.     

考生样本量对项目反应理论(IRT)等值稳定性的影响

测验等值使得不同形式的考试能进行比较,从而保证了测验之间的相对稳定性。基于IRT的分数等值是在估计出参数的基础上进行的参数转换,等值结果的稳定性与考生样本量密不可分。本研究针对汉语水平考试(HSK)阅读分测验,采用真实数据模拟共同组锚测验设计,确定等值的参照标准,考察考生样本量的变化对IRT分数等值稳定性的影响。结果表明,考生样本量为2000左右时各种方案的等值结果均比较稳定。考生样本量进一步增大时,等值误差不降反增。     

A Comparison of Two Procedures for Computing IRT Equating Coefficients

In order to equate tests under Item Response Theory (IRT), one must obtain the slope and intercept coefficients of the appropriate linear transformation. This article compares two methods for computing such equating coefficients–Loyd and Hoover (1980) and Stocking and Lord (1983). The former is based upon summary statistics of the test calibrations; the latter is based upon matching test characteristic curves by minimizing a quadratic loss function. Three types of equating situations: horizontal, vertical, and that inherent in IRT parameter recovery studies–were investigated. The results showed that the two computing procedures generally yielded similar equating coefficients in all three situations. In addition, two sets of SAT data were equated via the two procedures, and little difference in the obtained results was observed. Overall, the results suggest that the Loyd and Hoover procedure usually yields acceptable equating coefficients. The Stocking and Lord procedure improves upon the Loyd and Hoover values and appears to be less sensitive to atypical test characteristics. When the user has reason to suspect that the test calibrations may be associated with data sets that are typically troublesome to calibrate, the Stocking and Lord procedure is to be preferred.     

The Effect of Repeaters on Equating

Test equating might be affected by including in the equating analyses examinees who have taken the test previously. This study evaluated the effect of including such repeaters on Medical College Admission Test (MCAT) equating using a population invariance approach. Three-parameter logistic (3-PL) item response theory (IRT) true score and traditional equipercentile equating methods were used under the random groups equating design. This study also examined whether or not population sensitivity of equating by repeater status varies depending on other background variables (gender and ethnicity). The results indicated that there was some evidence of repeaters' effect on equating with varying amounts of such effect by gender.     

IRT下题量与被试量对参数估计模拟返真性能的影响

在项目反应理论下的题库建设时,进行纸笔测验测试时需要多少被试量、题量,试题的参数估计能够达到较为精确估计?本文使用蒙特卡洛模拟方法模拟测验情境,对此问题进行探讨。分析题量的变化和被试量的变化对a、b参数估计的模拟返真性能的影响。1)从被试量角度来看,在两级、多级记分试题模拟测验情境下,随着被试量逐渐增大,项目参数估计值模拟返真指标均方误差逐渐减小。2)从题量角度来看,在两级记分试题模拟情境下,均方误差曲线在题量为25题左右时有一个拐点,即当题量小于25题时,随着题量增加时RMSE减小幅度较大,而当题量大于25题时,这时再增加题量,RMSE减小幅度很小。在多级记分试题模拟情境下,均方误差曲线在题量为15题左右时有一个拐点,即当题量小于15题时,随着题量增加, RMSE逐渐减小,当题量大于15题时,随着题量增加,RMSE逐渐增大。     

IRT理论不同模型下同时校准等值方法的跨样本研究

本研究基于IRT理论中最常用的LOGISTIC三种模型来探讨等值的跨样本一致性,研究对象为某一汉语类别的测验,等值方法采用同时校准法。研究结果表明,双参数模型下同时校准法等值跨样本一致性最好,最为稳定。     

大规模测评中IRT等值的影响因素研究

通过模拟和实证研究探讨样本量、题本量以及锚题题型对大尺度测评中项目参数等值精度的影响,模拟研究和实证研究的结果均表明:(1)0/1计分项目参数的等值精度在大多数条件下均好于多级计分项目,相对而言,实证研究的差异不如模拟研究明显;(2)相对而言,样本容量的增加对于提高项目参数等值精度有着重要的作用,而增加题本数量的作用甚微;(3)无论是区分度参数还是难度参数,均表现为3个题本和2 000人的搭配已经可以达到较好的等值精度,如果进一步提高等值精度,只需将每一题本的样本容量增加到3 000人即可;在多级计分时,当选用5个题本时,每一个题本2 000人是最适宜的组合。     

应用项目反应理论等值含有多种题型考试的一个实例

本文以美国一个州的高中统考为例介绍应用项目反应理论来对含有多种题型的考试进行等值处理的具体做法,同时也对考试的其他技术环节进行了一些探讨。     

Asymptotic Standard Errors of Observed‐Score Equating With Polytomous IRT Models

In observed‐score equipercentile equating, the goal is to make scores on two scales or tests measuring the same construct comparable by matching the percentiles of the respective score distributions. If the tests consist of different items with multiple categories for each item, a suitable model for the responses is a polytomous item response theory (IRT) model. The parameters from such a model can be utilized to derive the score probabilities for the tests and these score probabilities may then be used in observed‐score equating. In this study, the asymptotic standard errors of observed‐score equating using score probability vectors from polytomous IRT models are derived using the delta method. The results are applied to the equivalent groups design and the nonequivalent groups design with either chain equating or poststratification equating within the framework of kernel equating. The derivations are presented in a general form and specific formulas for the graded response model and the generalized partial credit model are provided. The asymptotic standard errors are accurate under several simulation conditions relating to sample size, distributional misspecification and, for the nonequivalent groups design, anchor test length.     

Comparison of Item Preequating and Random Groups Equating Using IRT and Equipercentile Methods

An item-preequating design and a random groups design were used to equate forms of the American College Testing (ACT) Assessment Mathematics Test. Equipercentile and 3-parameter logistic model item-response theory (IRT) procedures were used for both designs. Both pretest methods produced inadequate equating results, and the IRT item preequating method resulted in more equating error than had no equating been conducted. Although neither of the item preequating methods performed well, the results from the equipercentile preequating method were more consistent with those from the random groups method than were the results from the IRT item pretest method. Item context and position effects were likely responsible, at least in part, for the inadequate results for item preequating. Such effects need to be either controlled or modeled, and the design further researched before the item preequating design can be recommended for operational use.     

The Effect of Differential Motivation on IRT Linking

The purpose of this study was to investigate whether simulated differential motivation between the stakes for operational tests and anchor items produces an invalid linking result if the Rasch model is used to link the operational tests. This was done for an external anchor design and a variation of a pretest design. The study also investigated whether a constrained mixture Rasch model could identify latent classes in such a way that one latent class represented high‐stakes responding while the other represented low‐stakes responding. The results indicated that for an external anchor design, the Rasch linking result was only biased when the motivation level differed between the subpopulations to which the anchor items were administered. However, the mixture Rasch model did not identify the classes representing low‐stakes and high‐stakes responding. When a pretest design was used to link the operational tests by means of a Rasch model, the linking result was found to be biased in each condition. Bias increased as percentage of students showing low‐stakes responding to the anchor items increased. The mixture Rasch model only identified the classes representing low‐stakes and high‐stakes responding under a limited number of conditions.     

A Comparison of Angoff's Design I and Design II for Vertical Equating Using Traditional and IRT Methodology

Practical considerations in conducting an equating study often require a trade-off between testing time and sample size. A counterbalanced design (Angoff's Design II) is often selected because, as each examinee is administered both test forms and therefore the errors are correlated, sample sizes can be dramatically reduced over those required by a spiraling design (Angoff's Design I), where each examinee is administered only one test form. However, the counterbalanced design may be subject to fatigue, practice, or context effects. This article investigated these two data collection designs (for a given sample size) with equipercentile and IRT equating methodology in the vertical equating of two mathematics achievement tests. Both designs and both methodologies were judged to adequately meet an equivalent expected score criterion; Design II was found to exhibit more stability over different samples.     

The Effect of Changing Content on IRT Scaling Methods

Equating test forms is an essential activity in standardized testing, with increased importance with the accountability systems in existence through the mandate of Adequate Yearly Progress. It is through equating that scores from different test forms become comparable, which allows for the tracking of changes in the performance of students from one year to the next. This study compares three different item response theory scaling methods (fixed common item parameter, Stocking & Lord, and Concurrent Calibration) with respect to examinee classification into performance categories, and estimation of the ability parameter, when the content of the test form changes slightly from year to year, and the examinee ability distribution changes. The results indicate that calibration methods, especially concurrent calibration, produced more stable results than the transformation method.     

The Philosophical Aspects of IRT Equating: Modeling Drift to Evaluate Cohort Growth in Large‐Scale Assessments

Calibration and equating is the quintessential necessity for most large‐scale educational assessments. However, there are instances when no consideration is given to the equating process in terms of context and substantive realization, and the methods used in its execution. In the view of the authors, equating is not merely an exhibit of the statistical methodology, but it is also a reflection of the thought process undertaken in its execution. For example, there is hardly any discussion in literature of the ideological differences in the selection of an equating method. Furthermore, there is little evidence of modeling cohort growth through an identification and use of construct‐relevant linking items’ drift, using the common item nonequivalent group equating design. In this article, the authors philosophically justify the use of Huynh's statistical method for the identification of construct‐relevant outliers in the linking pool. The article also dispels the perception of scale instability associated with the inclusion of construct‐relevant outliers in the linking item pool and concludes that an appreciation of the rationale used in the selection of the equating method, together with the use of linking items in modeling cohort growth, can be beneficial to the practitioners.     

The Extent of Mismeasurement for Aberrant Examinees

The person-fit literature assumes that aberrant response patterns could be a sign of person mismeasurement, but this assumption has rarely, if ever, been empirically investigated before. We explore the validity of test responses and measures of 10-year-old examinees whose response patterns on a commercial standardized paper-and-pencil mathematics test were flagged as aberrant. Validity evidence was collected through postexamination reflective interviews with 31 of the 80 pupils flagged as aberrant and their teachers, and teacher assessment (TA) judgments for the whole examination cohort of 674 examinees. Analysis suggested that interview-adjusted scores were significantly better fitting than expected by chance, but only some adjustments suggest serious mismeasurement. In addition, disagreement between TA and test scores was significantly greater for aberrant examinees, and partially predicted the interview adjustments. We conclude that person misfit statistics when combined with TA might be a useful antidote to mismeasurement, and we discuss the implications for assessment research and practice.     

论科举考试的社会经济限制

相较于前科举时代的选才方式,科举考试已具有相当程度的开放性,在形式上也具有平等之精神。尽管如此,它并没有也不可能向所有社会成员提供平等的机会。在科举考试竞争的调节中,财富、权力和家庭背景等这些考试之外的诸因素仍起着重要作用。由于科举考试的制度设计和客观条件的限制,尽管普通平民子弟有着相当广泛的读书和较低层次的应试机会,但往往止步于乡试,想要问鼎举业,实非易事。科举考试并非实行自由竞争主义,而是一种定额制度下的有限竞争,须具备相当程度的经济基础。科举考试成功者往往具备一定的经济基础与闲暇时间。实际上,科举考试有着相当高的经济门槛,并非想象中的那样亲民。     

锚测验难度参数方差特征对测验等值的影响研究

本文使用R-2.15.2软件模拟研究锚测验难度参数方差特征对测验等值误差的影响,采用三种等值方法(链百分位等值法、Levine等值法和Tucker等值法)对锚测验不同类型的难度方差进行比较研究。结果显示,当锚测验难度方差小于全测验难度方差时,其等值的随机误差和系统误差与锚测验难度方差和全测验难度方差一致时(即锚测验为全测验的平行缩减版minitest时)的表现基本相同。因此,对锚测验而言,要求其与全测验具有相同的统计规格可能过于严格。     

The Circle-Arc Method for Equating in Small Samples

This article suggests a method for estimating a test-score equating relationship from small samples of test takers. The method does not require the estimated equating transformation to be linear. Instead, it constrains the estimated equating curve to pass through two pre-specified end points and a middle point determined from the data. In a resampling study with two test forms that differed substantially in difficulty, the proposed method compared favorably with other equating methods, especially for equating scores below the 10th percentile and above the 90th percentile.