Local Linear Observed‐Score Equating

Two methods of local linear observed‐score equating for use with anchor‐test and single‐group designs are introduced. In an empirical study, the two methods were compared with the current traditional linear methods for observed‐score equating. As a criterion, the bias in the equated scores relative to true equating based on Lord's (1980) definition of equity was used. The local method for the anchor‐test design yielded minimum bias, even for considerable variation of the relative difficulties of the two test forms and the length of the anchor test. Among the traditional methods, the method of chain equating performed best. The local method for single‐group designs yielded equated scores with bias comparable to the traditional methods. This method, however, appears to be of theoretical interest because it forces us to rethink the relationship between score equating and regression.     

A Comparison Between Linear IRT Observed‐Score Equating and Levine Observed‐Score Equating Under the Generalized Kernel Equating Framework

In this article, linear item response theory (IRT) observed‐score equating is compared under a generalized kernel equating framework with Levine observed‐score equating for nonequivalent groups with anchor test design. Interestingly, these two equating methods are closely related despite being based on different methodologies. Specifically, when using data from IRT models, linear IRT observed‐score equating is virtually identical to Levine observed‐score equating. This leads to the conclusion that poststratification equating based on true anchor scores can be viewed as the curvilinear Levine observed‐score equating.     

Observed Score Equating Using Discrete and Passage-Based Anchor Items

Equating of tests composed of both discrete and passage-based multiple choice items using the nonequivalent groups with anchor test design is popular in practice. In this study, we compared the effect of discrete and passage-based anchor items on observed score equating via simulation. Results suggested that an anchor with a larger proportion of passage-based items, more items in each passage, and/or a larger degree of local dependence among items within one passage produces larger equating errors, especially when the groups taking the new form and the reference form differ in ability. Our findings challenge the common belief that an anchor should be a miniature version of the tests to be equated. Suggestions to practitioners regarding anchor design are also given.     

On the Choice of Anchor Tests in Equating

The choice of anchor tests is crucial in applications of the nonequivalent groups with anchor test design of equating. Sinharay and Holland (2006, 2007) suggested “miditests,” which are anchor tests that are content‐representative and have the same mean item difficulty as the total test but have a smaller spread of item difficulties. Sinharay and Holland (2006, 2007), Cho, Wall, Lee, and Harris (2010), Fitzpatrick and Skorupski (2016), Liu, Sinharay, Holland, Curley, and Feigenbaum (2011a), Liu, Sinharay, Holland, Feigenbaum, and Curley (2011b), and Yi (2009) found the miditests to lead to better equating than minitests, which are representative of the total test with respect to content and difficulty. However, these findings recently came into question as Trierweiler, Lewis, and Smith (2016) concluded, based on a comparison of correlation coefficients of miditests and minitests with the total test, that making an anchor test a miditest does not generally increase the anchor to total score correlation and recommended the continuation of the practice of using minitests over miditests. Their recommendation raises the question, “Should miditests continue to be considered in practice?” This note defends the miditests by citing literature that favors miditests and then by showing that miditests perform as well as the minitests in most realistic situations considered in Trierweiler et al. (2016), which implies that miditests should continue to be seriously considered by equating practitioners.     

How Often Do Subscores Have Added Value? Results from Operational and Simulated Data

Recently, there has been an increasing level of interest in subscores for their potential diagnostic value. Haberman suggested a method based on classical test theory to determine whether subscores have added value over total scores. In this article I first provide a rich collection of results regarding when subscores were found to have added value for several operational data sets. Following that I provide results from a detailed simulation study that examines what properties subscores should possess in order to have added value. The results indicate that subscores have to satisfy strict standards of reliability and correlation to have added value. A weighted average of the subscore and the total score was found to have added value more often.     

On Attempting to Do What Lord Said Was Impossible: Commentary on van der Linden's “Some Conceptual Issues in Observed‐Score Equating”

van der Linden (this issue) uses words differently than Holland and Dorans. This difference in language usage is a source of some confusion in van der Linden's critique of what he calls equipercentile equating. I address these differences in language. van der Linden maintains that there are only two requirements for score equating. I maintain that the requirements he discards have practical utility and are testable. The score equity requirement proposed by Lord suggests that observed score equating was either unnecessary or impossible. Strong equity serves as the fulcrum for van der Linden's thesis. His proposed solution to the equity problem takes inequitable measures and aligns conditional error score distributions, resulting in a family of linking functions, one for each level of θ. In reality, θ is never known. Use of an anchor test as a proxy poses many practical problems, including defensibility.     

Determining the Anchor Composition for a Mixed-Format Test: Evaluation of Subpopulation Invariance of Linking Functions

This study examined the appropriateness of the anchor composition in a mixed-format test, which includes both multiple-choice (MC) and constructed-response (CR) items, using subpopulation invariance indices. Linking functions were derived in the nonequivalent groups with anchor test (NEAT) design using two types of anchor sets: (a) MC only and (b) a mix of MC and CR. In each anchor condition, the linking functions were also derived separately for males and females, and those subpopulation functions were compared to the total group function. In the MC-only condition, the difference between the subpopulation functions and the total group function was not trivial in a score region that included cut scores, leading to inconsistent pass/fail decisions for low-performing examinees in particular. Overall, the mixed anchor was a better choice than the MC-only anchor to achieve subpopulation invariance between males and females. The research reinforces subpopulation invariance indices as a means of determining the adequacy of the anchor.     

Test Score Equating Using a Mini‐Version Anchor and a Midi Anchor: A Case Study Using SAT® Data

This study explores an anchor that is different from the traditional miniature anchor in test score equating. In contrast to a traditional “mini” anchor that has the same spread of item difficulties as the tests to be equated, the studied anchor, referred to as a “midi” anchor (Sinharay & Holland), has a smaller spread of item difficulties than the tests to be equated. Both anchors were administered in an operational SAT administration and the impact of anchor type on equating was evaluated with respect to systematic error or equating bias. Contradicting the popular belief that the mini anchor is best, the results showed that the mini anchor does not always produce more accurate equating functions than the midi anchor; the midi anchor was found to perform as well as or even better than the mini anchor. Because testing programs usually have more middle difficulty items and few very hard or very easy items, midi external anchors are operationally easier to build. Therefore, the results of our study provide evidence in favor of the midi anchor, the use of which will lead to cost saving with no reduction in equating quality.     

Choosing Among Tucker or Chained Linear Equating in Two Testing Situations: Rater Comparability Scoring and Randomly Equivalent Groups With an Anchor

Tucker and chained linear equatings were evaluated in two testing scenarios. In Scenario 1, referred to as rater comparability scoring and equating, the anchor‐to‐total correlation is often very high for the new form but moderate for the reference form. This may adversely affect the results of Tucker equating, especially if the new and reference form samples differ in ability. In Scenario 2, the new and reference form samples are randomly equivalent but the correlation between the anchor and total scores is low. When the correlation between the anchor and total scores is low, Tucker equating assumes that the new and reference form samples are similar in ability (which, with randomly equivalents groups, is the correct assumption). Thus Tucker equating should produce accurate results. Results indicated that in Scenario 1, the Tucker results were less accurate than the chained linear equating results. However, in Scenario 2, the Tucker results were more accurate than the chained linear equating results. Some implications are discussed.     

全测验与锚测验题型分值比对等值误差的影响

为探讨全测验与锚测验不同的客观题与主观题分值比对等值误差造成的影响,本文设计两种全测验与锚测验题型分值比,以等值标准误为因变量,构建2X2的两因素完全随机化设计进行等值误差的方差分析。结果表明,全测验题型分值比与锚测验题型分值比两因素的主效应显著（P〈0．001）,交互作用显著（P〈0．01）,简单效应检验表明两因素在各水平上差异显著（P〈0．01）。全测验题型分值比与锚测验题型分值比对等值误差产生一定的影响,在等值过程中应该考虑这两个影响因素,为了减小等值过程的误差,锚测验题型分值比应该尽量与全测验题型分值比相一致。     

Addressing the Extreme Assumptions of Presumed Linkings

The interpretability of score comparisons depends on the design and execution of a sound data collection plan and the establishment of linkings between these scores. When comparisons are made between scores from two or more assessments that are built to different specifications and are administered to different populations under different conditions, the validity of the comparisons hinges on untestable assumptions. For example, tests administered across different disability groups or tests administered to different language groups produce scores for which implicit linkings are presumed to hold. Presumed linking makes use of extreme assumptions to produce links between scores on tests in the absence of common test material or equivalent groups of test takers. These presumed linkings lead to dubious interpretations. This article suggests an approach that indirectly assesses the validity of these presumed linkings among scores on assessments that contain neither equivalent groups nor common anchor material.     

Local Observed‐Score Kernel Equating

Three local observed‐score kernel equating methods that integrate methods from the local equating and kernel equating frameworks are proposed. The new methods were compared with their earlier counterparts with respect to such measures as bias—as defined by Lord's criterion of equity—and percent relative error. The local kernel item response theory observed‐score equating method, which can be used for any of the common equating designs, had a small amount of bias, a low percent relative error, and a relatively low kernel standard error of equating, even when the accuracy of the test was reduced. The local kernel equating methods for the nonequivalent groups with anchor test generally had low bias and were quite stable against changes in the accuracy or length of the anchor test. Although all proposed methods showed small percent relative errors, the local kernel equating methods for the nonequivalent groups with anchor test design had somewhat larger standard error of equating than their kernel method counterparts.     

Small‐Sample Equating Using a Synthetic Linking Function

This study addressed the sampling error and linking bias that occur with small samples in a nonequivalent groups anchor test design. We proposed a linking method called the synthetic function, which is a weighted average of the identity function and a traditional equating function (in this case, the chained linear equating function). Specifically, we compared the synthetic, identity, and chained linear functions for various‐sized samples from two types of national assessments. One design used a highly reliable test and an external anchor, and the other used a relatively low‐reliability test and an internal anchor. The results from each of these methods were compared to the criterion equating function derived from the total samples with respect to linking bias and error. The study indicated that the synthetic functions might be a better choice than the chained linear equating method when samples are not large and, as a result, unrepresentative.     

Musculoskeletal anatomy knowledge in Australian chiropractors

Anatomy is a key knowledge area in chiropractic and is formally offered in the undergraduate component of chiropractic education. There is the potential for loss of anatomy knowledge before the opportunity to apply it in a clinical setting. This study aimed to determine whether chiropractic clinicians retain a level of anatomy knowledge comparable to that of chiropractic students and to compare chiropractors' self-rating of their anatomical knowledge against an objective knowledge assessment tool. A previously validated multiple-choice test was utilized to measure retention of limb musculoskeletal (MSK) knowledge in Australian chiropractors. One hundred and one registered chiropractors completed the questionnaire and responses were scored, analyzed, and compared to scores attained by undergraduate and postgraduate chiropractic students who had previously completed the same questionnaire. The results indicated that practitioners retained their anatomy knowledge, with a significantly higher total mean score than the undergraduate group [total mean score = 36.5% (±SD 13.6%); P < 0.01] but not significantly different to the postgraduate group [total mean score = 52.2% (±SD 14.1%); P = 0.74]. There was a weak positive correlation between chiropractors' self-rated knowledge and test performance scores indicating the effectiveness of this Australian chiropractic group in self-assessing their anatomy knowledge. This study found that Australian chiropractors' knowledge of MSK anatomy was retained during the transition from university to clinical practice and they accurately evaluated their own test performance.     

Standard Error of Linear Observed‐Score Equating for the NEAT Design With Nonnormally Distributed Data

In the nonequivalent groups with anchor test (NEAT) design, the standard error of linear observed‐score equating is commonly estimated by an estimator derived assuming multivariate normality. However, real data are seldom normally distributed, causing this normal estimator to be inconsistent. A general estimator, which does not rely on the normality assumption, would be preferred, because it is asymptotically accurate regardless of the distribution of the data. In this article, an analytical formula for the standard error of linear observed‐score equating, which characterizes the effect of nonnormality, is obtained under elliptical distributions. Using three large‐scale real data sets as the populations, resampling studies are conducted to empirically evaluate the normal and general estimators of the standard error of linear observed‐score equating. The effect of sample size (50, 100, 250, or 500) and equating method (chained linear, Tucker, or Levine observed‐score equating) are examined. Results suggest that the general estimator has smaller bias than the normal estimator in all 36 conditions; it has larger standard error when the sample size is at least 100; and it has smaller root mean squared error in all but one condition. An R program is also provided to facilitate the use of the general estimator.     

Guidelines for Interpreting and Reporting Subscores

Recent research has proposed a criterion to evaluate the reportability of subscores. This criterion is a value‐added ratio (VAR), where values greater than 1 suggest that the true subscore is better approximated by the observed subscore than by the total score. This research extends the existing literature by quantifying statistical significance and effect size for using VAR to provide practical guidelines for subscore interpretation and reporting. Findings indicate that subscores with VAR ≥ 1.1 are a minimum requirement for a meaningful contribution to a user's score interpretation; subscores with .9 < VAR < 1.1 are redundant with the total score and subscores with VAR ≤ .9 would be misleading to report. Additionally, we discuss what to do when subscores do not add value, yet must be reported, as well as when VAR ≥ 1.1 may be undesirable.     

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。     

Social Pragmatics: Preschoolers Rely on Commonsense Psychology to Resolve Referential Underspecification

Four experiments show that 4- and 5-year-olds (total N = 112) can identify the referent of underdetermined utterances through their Naïve Utility Calculus—an intuitive theory of people’s behavior structured around an assumption that agents maximize utilities. In Experiments 1–2, a puppet asked for help without specifying to whom she was talking (“Can you help me?”). In Experiments 3–4, a puppet asked the child to pass an object without specifying what she wanted (“Can you pass me that one?”). Children’s responses suggest that they considered cost trade-offs between the members in the interaction. These findings add to a body of work showing that reference resolution is informed by commonsense psychology from early in childhood.     

NCME 2008 Presidential Address: The Impact of Anchor Test Configuration on Student Proficiency Rates

Examined in this study were the effects of reducing anchor test length on student proficiency rates for 12 multiple‐choice tests administered in an annual, large‐scale, high‐stakes assessment. The anchor tests contained 15 items, 10 items, or five items. Five content representative samples of items were drawn at each anchor test length from a small universe of items in order to investigate the stability of equating results over anchor test samples. The operational tests were calibrated using the one‐parameter model and equated using the mean b‐value method. The findings indicated that student proficiency rates could display important variability over anchor test samples when 15 anchor items were used. Notable increases in this variability were found for some tests when shorter anchor tests were used. For these tests, some of the anchor items had parameters that changed somewhat in relative difficulty from one year to the next. It is recommended that anchor sets with more than 15 items be used to mitigate the instability in equating results due to anchor item sampling. Also, the optimal allocation method of stratified sampling should be evaluated as one means of improving the stability and precision of equating results.     

Minimum reading fluency necessary for comprehension among second‐grade students

The current study examined the relationship between oral reading fluency (ORF) and reading comprehension for students in second grade. A total of 84 participants were randomly assigned to one of four conditions that involved reading a grade‐appropriate passage with either 0%, 10%, 20%, or 30% scrambled words and answering subsequent comprehension questions. The correlation coefficient between ORF and the number of comprehension questions correctly answered was r = .54. Receiver operating characteristics were then used to empirically derive a minimum ORF score necessary for comprehension, indicating that when these students read 63 words correct per minute they successfully comprehended what they read. Finally, the diagnostic accuracy of the derived criterion of 63 words read correctly per minute was tested and resulted in overall correct classification of .80. © 2010 Wiley Periodicals, Inc.