Observed Score Linear Equating with Covariates

This paper examined observed score linear equating in two different data collection designs, the equivalent groups design and the nonequivalent groups design, when information from covariates (i.e., background variables correlated with the test scores) was included. The main purpose of the study was to examine the effect (i.e., bias, variance, and mean squared error) on the estimators of including this additional information. A model for observed score linear equating with covariates first was suggested. As a second step, the model was used in a simulation study to show that the use of covariates such as gender and education can increase the accuracy of an equating by reducing the mean squared error of the estimators. Finally, data from two administrations of the Swedish Scholastic Assessment Test were used to illustrate the use of the model.     

The Chain and Post-Stratification Methods for Observed-Score Equating: Their Relationship to Population Invariance

The Non-Equivalent-groups Anchor Test (NEAT) design has been in wide use since at least the early 1940s. It involves two populations of test takers, P and Q, and makes use of an anchor test to link them. Two linking methods used for NEAT designs are those (a) based on chain equating and (b) that use the anchor test to post-stratify the distributions of the two operational test scores to a common population (i.e., Tucker equating and frequency estimation). We show that, under different sets of assumptions, both methods are observed score equating methods and we give conditions under which the methods give identical results. In addition, we develop analogues of the Dorans and Holland (2000) RMSD measures of population invariance of equating methods for the NEAT design for both chain and post-stratification equating methods.     

How to Equate Tests With Little or No Data

Standard procedures for equating tests, including those based on item response theory (IRT), require item responses from large numbers of examinees. Such data may not be forthcoming for reasons theoretical, political, or practical. Information about items' operating characteristics may be available from other sources, however, such as content and format specifications, expert opinion, or psychological theories about the skills and strategies required to solve them. This article shows how, in the IRT framework, collateral information about items can be exploited to augment or even replace examinee responses when linking or equating new tests to established scales. The procedures are illustrated with data from the Pre-Professional Skills Test (PPST).     

Evaluating Equating Results: Percent Relative Error for Chained Kernel Equating

This article presents a method for evaluating equating results. Within the kernel equating framework, the percent relative error (PRE) for chained equipercentile equating was computed under the nonequivalent groups with anchor test (NEAT) design. The method was applied to two data sets to obtain the PRE, which can be used to measure equating effectiveness. The study compared the PRE results for chained and poststratification equating. The results indicated that the chained method transformed the new form score distribution to the reference form scale more effectively than the poststratification method. In addition, the study found that in chained equating, the population weight had impact on score distributions over the target population but not on the equating and PRE results.     

An Approach to Evaluating the Missing Data Assumptions of the Chain and Post-stratification Equating Methods for the NEAT Design

Two important types of observed score equating (OSE) methods for the non-equivalent groups with Anchor Test (NEAT) design are chain equating (CE) and post-stratification equating (PSE). CE and PSE reflect two distinctly different ways of using the information provided by the anchor test for computing OSE functions. Both types of methods include linear and nonlinear equating functions. In practical situations, it is known that the PSE and CE methods will give different results when the two groups of examinees differ on the anchor test. However, given that both types of methods are justified as OSE methods by making different assumptions about the missing data in the NEAT design, it is difficult to conclude which, if either, of the two is more correct in a particular situation. This study compares the predictions of the PSE and CE assumptions for the missing data using a special data set for which the usually missing data are available. Our results indicate that in an equating setting where the linking function is decidedly non-linear and CE and PSE ought to be different, both sets of predictions are quite similar but those for CE are slightly more accurate .     

A Comparison of the Kernel Equating Method with Traditional Equating Methods Using SAT® Data

This study applied kernel equating (KE) in two scenarios: equating to a very similar population and equating to a very different population, referred to as a distant population, using SAT^® data. The KE results were compared to the results obtained from analogous traditional equating methods in both scenarios. The results indicate that KE results are comparable to the results of other methods. Further, the results show that when the two populations taking the two tests are similar on the anchor score distributions, different equating methods yield the same or very similar results, even though they have different assumptions.     

Review of Cutscores: A Manual for Setting Standards of Performance on Educational and Occupational Tests

Combinations of five methods of equating test forms and two methods of selecting samples of students for equating were compared for accuracy. The two sampling methods were representative sampling from the population and matching samples on the anchor test score. The equating methods were the Tucker, Levine equally reliable, chained equipercentile, frequency estimation, and item response theory (IRT) 3PL methods. The tests were the Verbal and Mathematical sections of the Scholastic Aptitude Test. The criteria for accuracy were measures of agreement with an equivalent-groups equating based on more than 115,000 students taking each form. Much of the inaccuracy in the equatings could be attributed to overall bias. The results for all equating methods in the matched samples were similar to those for the Tucker and frequency estimation methods in the representative samples; these equatings made too small an adjustment for the difference in the difficulty of the test forms. In the representative samples, the chained equipercentile method showed a much smaller bias. The IRT (3PL) and Levine methods tended to agree with each other and were inconsistent in the direction of their bias.     

Statistical Models and Inference for the True Equating Transformation in the Context of Local Equating

Based on Lord's criterion of equity of equating, van der Linden (this issue) revisits the so‐called local equating method and offers alternative as well as new thoughts on several topics including the types of transformations, symmetry, reliability, and population invariance appropriate for equating. A remarkable aspect is to define equating as a standard statistical inference problem in which the true equating transformation is the parameter of interest that has to be estimated and assessed as any standard evaluation of an estimator of an unknown parameter in statistics. We believe that putting equating methods in a general statistical model framework would be an interesting and useful next step in the area. van der Linden's conceptual article on equating is certainly an important contribution to this task.     

Checking the Statistical Equivalence of Nearly Identical Test Editions

A procedure for checking the score equivalence of nearly identical editions of a test is described. This procedure is used early in the score equating process to help determine whether it is necessary to conduct separate equating analyses (using a variety of equating methods) for the two nearly identical versions of the test. The procedure employs the standard error of equating and utilizes graphical representation of score conversion deviation from the identity function in standard error units. Two illustrations of the procedure involving Scholastic Aptitude Test (SAT) data are presented. Advice about what to do if statistical equivalence does not obtain is given in the discussion section. Alternative strategies for assessing score equivalence are also discussed.     

The Performance of a Method for the Long-term Equating of Mixed-Format Assessment

The goal of this study was the development of a procedure to predict the equating error associated with the long-term equating method of Tate (2003) for mixed-format tests. An expression for the determination of the error of an equating based on multiple links using the error for the component links was derived and illustrated with simulated data. Expressions relating the equating error for single equating links to relevant factors like the equating design and the history of the examinee population ability distribution were determined based on computer simulation. Use of the resulting procedure for the selection of a long-term equating design was illustrated.     

HSK和MHK的等值

HSK是为测试母语为非汉语者(包括外国人和华侨)的汉语水平而设立的国家级标准化考试。MHK是专门测试母语为非汉语的中国少数民族汉语学习者汉语水平的国家级标准化考试。HSK和MHK都是证书考试。如果证书授予标准缺乏稳定性和公平性,如果对使用这一份试卷的人一个标准,对使用另一份试卷的人又一个标准,那么,不仅会大大影响HSK的信度和效度,而且会对有关的决策产生误导,会使考生受到不公平的对待。在HSK和MHK的开发和实施过程中,一直坚持了对考试分数的统计等值处理。在HSK和MHK的等值设计方面,我们综合采用了共同组等值、共同题等值和分半组合的混合设计。在HSK和MHK的等值数据处理方面,我们综合采用了线性等值、等百分位等值和IRT等值。本文介绍了HSK和MHK的等值方法。讨论了各种方法的得失,讨论了今后继续改进的可能性。     

Futility1 of Log‐Linear Smoothing when Equating with Unrepresentative Small Samples

The impact of log‐linear presmoothing on the accuracy of small sample chained equipercentile equating was evaluated under two conditions . In the first condition the small samples differed randomly in ability from the target population. In the second condition the small samples were systematically different from the target population. Results showed that equating with small samples (e.g., N < 25 or 50) using either raw or smoothed score distributions led to considerable large random equating error (although smoothing reduced random equating error). Moreover, when the small samples were not representative of the target population, the amount of equating bias also was quite large. It is concluded that although presmoothing can reduce random equating error, it is not likely to reduce equating bias caused by using an unrepresentative sample. Other alternatives to the small sample equating problem (e.g., the SiGNET design) which focus more on improving data collection are discussed.     

The Effects of Dimensionality on Equating the Law School Admission Test

Using factor analysis, we conducted an assessment of multidimensionality for 6 forms of the Law School Admission Test (LSAT) and found 2 subgroups of items or factors for each of the 6 forms. The main conclusion of the factor analysis component of this study was that the LSAT appears to measure 2 different reasoning abilities: inductive and deductive. The technique of N. J. Dorans & N. M. Kingston (1985) was used to examine the effect of dimensionality on equating. We began by calibrating (with item response theory [IRT] methods) all items on a form to obtain Set I of estimated IRT item parameters. Next, the test was divided into 2 homogeneous subgroups of items, each having been determined to represent a different ability (i.e., inductive or deductive reasoning). The items within these subgroups were then recalibrated separately to obtain item parameter estimates, and then combined into Set II. The estimated item parameters and true-score equating tables for Sets I and II corresponded closely.     

Small-Sample Equating With Log-Linear Smoothing

This study investigated the extent to which log-linear smoothing could improve the accuracy of common-item equating by the chained equipercentile method in small samples of examinees. Examinee response data from a 100-item test were used to create two overlapping forms of 58 items each, with 24 items in common. The criterion equating was a direct equipercentile equating of the two forms in the full population of 93,283 examinees. Anchor equatings were performed in samples of 25, 50, 100, and 200 examinees, with 50 pairs of samples at each size level. Four equatings were performed with each pair of samples: one based on unsmoothed distributions and three based on varying degrees of smoothing. Smoothing reduced, by at least half, the sample size required for a given degree of accuracy. Smoothing that preserved only two moments of the marginal distributions resulted in equatings that failed to capture the curvilinearity in the population equating.     

An NCME Instructional Module on Population Invariance in Linking and Equating

A goal for any linking or equating of two or more tests is that the linking function be invariant to the population used in conducting the linking or equating. Violations of population invariance in linking and equating jeopardize the fairness and validity of test scores, and pose particular problems for test‐based accountability programs that require schools, districts, and states to report annual progress on academic indicators disaggregated by demographic group membership. This instructional module provides a comprehensive overview of population invariance in linking and equating and the relevant methodology developed for evaluating violations of invariance. A numeric example is used to illustrate the comparative properties of available methods, and important considerations for evaluating population invariance in linking and equating are presented.     

Population Invariance and the Equatability of Tests: Basic Theory and The Linear Case

How does the fact that two tests should not be equated manifest itself? This paper addresses this question through the study of the degree to which equating functions fail to exhibit population invariance across subpopulations. Equating fimctions are supposed to be population invariant by definition. But, when two tests are not equatable, it is possible that the linking functions, used to connect the scores of one to the scores of the other, are not invariant across different populations of examinees. While no acceptable equating function is ever completely population invariant, in the situations where equating is usually performed we believe that the dependence of the equating function on the population used to compute it is usually small enough to be ignored. We introduce two root‐mean‐square difference measures of the degree to which the functions used to link two tests computed on different subpopulations differ from the linking function computed for the whole population. We also introduce the system of “parallel‐linear” linking functions for multiple subpopulations and show that, for this system, our measure of population invariance can be computed easily from the standardized mean differences between the scores of the subpopulations on the two tests. For the parallel‐linear case, we develop a correlation‐based upper bound on our measure that holds for all systems of subpopulations. We illustrate these ideas using data from the SAT I and from a concordance study of several combinations of ACT and SAT I scores, In the appendices, we give some theoretical results bearing on the other equating “requirements” of “same construct,”“same reliability” and one aspect of Lord's concept of equity.     

Equating Minimum-Competency Tests: Comparisons of Methods

The 1986 scores from Florida's Statewide Student Assessment Test, Part II (SSAT-II), a minimum-competency test required for high school graduation in Florida, were placed on the scale of the 1984 scores from that test using five different equating procedures. For the highest scoring 84 % of the students, four of the five methods yielded results within 1.5 raw-score points of each other. They would be essentially equally satisfactory in this situation, in which the tests were made parallel item by item in difficulty and content and the groups of examinees were population cohorts separated by only 2 years. Also, the results from six different lengths of anchor items were compared. Anchors of 25, 20, 15, or 10 randomly selected items provided equatings as effective as 30 items using the concurrent IRT equating method, but an anchor of 5 randomly selected items did not     

Section Preequating Under the Equivalent Groups Design Without IRT

In this article, we introduce a section preequating (SPE) method (linear and nonlinear) under the randomly equivalent groups design. In this equating design, sections of Test X (a future new form) and another existing Test Y (an old form already on scale) are administered. The sections of Test X are equated to Test Y, after adjusting for the imperfect correlation between sections of Test X, to obtain the equated score on the complete form of X. Simulations and a real‐data application show that the proposed SPE method is fairly simple and accurate.     

Considerations for Equating Alternate Assessments: Two Case Studies of Alternate Assessments Based on Alternate Achievement Standards

The development of alternate assessments for students with disabilities plays a pivotal role in state and national accountability systems. An important assumption in the use of alternate assessments in these accountability systems is that scores are comparable on different test forms across diverse groups of students over time. The use of test equating is a common way that states attempt to establish score comparability on different test forms. However, equating presents many unique, practical, and technical challenges for alternate assessments. This article provides case studies of equating for two alternate assessments in Michigan and an approach to determine whether or not equating would be preferred to not equating on these assessments. This approach is based on examining equated score and performance-level differences and investigating population invariance across subgroups of students with disabilities. Results suggest that using an equating method with these data appeared to have a minimal impact on proficiency classifications. The population invariance assumption was suspect for some subgroups and equating methods with some large potential differences observed.     

THE EFFECTS OF VIOLATIONS OF UNIDIMENSIONALITY ON THE ESTIMATION OF ITEM AND ABILITY PARAMETERS AND ON ITEM RESPONSE THEORY EQUATING OF THE GRE VERBAL SCALE

One of the major assumptions of item response theory (IRT)models is that performance on a set of items is unidimensional, that is, the probability of successful performance by examinees on a set of items can be modeled by a mathematical model that has only one ability parameter. In practice, this strong assumption is likely to be violated. An important pragmatic question to consider is: What are the consequences of these violations? In this research, evidence is provided of violations of unidimensionality on the verbal scale of the GRE Aptitude Test, and the impact of these violations on IRT equating is examined. Previous factor analytic research on the GRE Aptitude Test suggested that two verbal dimensions, discrete verbal (analogies, antonyms, and sentence completions)and reading comprehension, existed. Consequently, the present research involved two separate calibrations (homogeneous) of discrete verbal items and reading comprehension items as well as a single calibration (heterogeneous) of all verbal item types. Thus, each verbal item was calibrated twice and each examinee obtained three ability estimates: reading comprehension, discrete verbal, and all verbal. The comparability of ability estimates based on homogeneous calibrations (reading comprehension or discrete verbal) to each other and to the all-verbal ability estimates was examined. The effects of homogeneity of item calibration pool on estimates of item discrimination were also examined. Then the comparability of IRT equatings based on homogeneous and heterogeneous calibrations was assessed. The effects of calibration homogeneity on ability parameter estimates and discrimination parameter estimates are consistent with the existence of two highly correlated verbal dimensions. IRT equating results indicate that although violations of unidimensionality may have an impact on equating, the effect may not be substantial.