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.     

The Accuracy and Consistency of a Series of IRT True Score Equatings

This study investigates a sequence of item response theory (IRT) true score equatings based on various scale transformation approaches and evaluates equating accuracy and consistency over time. The results show that the biases and sample variances for the IRT true score equating (both direct and indirect) are quite small (except for the mean/sigma method). The biases and sample variances for the equating functions based on the characteristic curve methods and concurrent calibrations for adjacent forms are smaller than the biases and variances for the equating functions based on the moment methods. In addition, the IRT true score equating is also compared to the chained equipercentile equating, and we observe that the sample variances for the chained equipercentile equating are much smaller than the variances for the IRT true score equating with an exception at the low scores.     

Preequating With Empirical Item Characteristic Curves: An Observed‐Score Preequating Method

Preequating is in demand because it reduces score reporting time. In this article, we evaluated an observed‐score preequating method: the empirical item characteristic curve (EICC) method, which makes preequating without item response theory (IRT) possible. EICC preequating results were compared with a criterion equating and with IRT true‐score preequating conversions. Results suggested that the EICC preequating method worked well under the conditions considered in this study. The difference between the EICC preequating conversion and the criterion equating was smaller than .5 raw‐score points (a practical criterion often used to evaluate equating quality) between the 5th and 95th percentiles of the new form total score distribution. EICC preequating also performed similarly or slightly better than IRT true‐score preequating.     

Stability of Rasch Scales Over Time

Item response theory (IRT) methods are generally used to create score scales for large-scale tests. Research has shown that IRT scales are stable across groups and over time. Most studies have focused on items that are dichotomously scored. Now Rasch and other IRT models are used to create scales for tests that include polytomously scored items. When tests are equated across forms, researchers check for the stability of common items before including them in equating procedures. Stability is usually examined in relation to polytomous items' central “location” on the scale without taking into account the stability of the different item scores (step difficulties). We examined the stability of score scales over a 3–5-year period, considering both stability of location values and stability of step difficulties for common item equating. We also investigated possible changes in the scale measured by the tests and systematic scale drift that might not be evident in year-to-year equating. Results across grades and content areas suggest that equating results are comparable whether or not the stability of step difficulties is taken into account. Results also suggest that there may be systematic scale drift that is not visible using year-to-year common item equating.     

An Extension of IRT-Based Equating to the Dichotomous Testlet Response Theory Model

ABSTRACT

Current procedures for equating number-correct scores using traditional item response theory (IRT) methods assume local independence. However, when tests are constructed using testlets, one concern is the violation of the local item independence assumption. The testlet response theory (TRT) model is one way to accommodate local item dependence. This study proposes methods to extend IRT true score and observed score equating methods to the dichotomous TRT model. We also examine the impact of local item dependence on equating number-correct scores when a traditional IRT model is applied. Results of the study indicate that when local item dependence is at a low level, using the three-parameter logistic model does not substantially affect number-correct equating. However, when local item dependence is at a moderate or high level, using the three-parameter logistic model generates larger equating bias and standard errors of equating compared to the TRT model. However, observed score equating is more robust to the violation of the local item independence assumption than is true score equating.     

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.     

Equating With Miditests Using IRT

The equating performance of two internal anchor test structures—miditests and minitests—is studied for four IRT equating methods using simulated data. Originally proposed by Sinharay and Holland, miditests are anchors that have the same mean difficulty as the overall test but less variance in item difficulties. Four popular IRT equating methods were tested, and both the means and SDs of the true ability of the group to be equated were varied. We evaluate equating accuracy marginally and conditional on true ability. Our results suggest miditests perform about as well as traditional minitests for most conditions. Findings are discussed in terms of comparability to the typical minitest design and the trade‐off between accuracy and flexibility in test construction.     

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).     

IRT Equating Methods

The purpose of this instructional module is to provide the basis for understanding the process of score equating through the use of item response theory (IRT). A context is provided for addressing the merits of IRT equating methods. The mechanics of IRT equating and the need to place parameter estimates from separate calibration runs on the same scale are discussed. Some procedures for placing parameter estimates on a common scale are presented. In addition, IRT true-score equating is discussed in some detail. A discussion of the practical advantages derived from IRT equating is offered at the end of the module.     

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.     

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.     

Situations Where It Is Appropriate to Use Frequency Estimation Equipercentile Equating

In operational equating situations, frequency estimation equipercentile equating is considered only when the old and new groups have similar abilities. The frequency estimation assumptions are investigated in this study under various situations from both the levels of theoretical interest and practical use. It shows that frequency estimation equating can be used under circumstances when it is not normally used. To link theoretical results with practice, statistical methods are proposed for checking frequency estimation assumptions based on available data: observed‐score distributions and item difficulty distributions of the forms. In addition to the conventional use of frequency estimation equating when the group abilities are similar, three situations are identified when the group abilities are dissimilar: (a) when the two forms and the observed conditional score distributions are similar the two forms and the observed conditional score distributions are similar (in this situation, the frequency estimation equating assumptions are likely to hold, and frequency estimation equating is appropriate); (b) when forms are similar but the observed conditional score distributions are not (in this situation, frequency estimation equating is not appropriate); and (c) when forms are not similar but the observed conditional score distributions are (frequency estimation equating is not appropriate). Statistical analysis procedures for comparing distributions are provided. Data from a large‐scale test are used to illustrate the use of frequency estimation equating when the group difference in ability is large.     

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.     

Standard Errors of IRT Parameter Scale Transformation Coefficients: Comparison of Bootstrap Method,Delta Method,and Multiple Imputation Method

The present study evaluated the multiple imputation method, a procedure that is similar to the one suggested by Li and Lissitz (2004), and compared the performance of this method with that of the bootstrap method and the delta method in obtaining the standard errors for the estimates of the parameter scale transformation coefficients in item response theory (IRT) equating in the context of the common‐item nonequivalent groups design. Two different estimation procedures for the variance‐covariance matrix of the IRT item parameter estimates, which were used in both the delta method and the multiple imputation method, were considered: empirical cross‐product (XPD) and supplemented expectation maximization (SEM). The results of the analyses with simulated and real data indicate that the multiple imputation method generally produced very similar results to the bootstrap method and the delta method in most of the conditions. The differences between the estimated standard errors obtained by the methods using the XPD matrices and the SEM matrices were very small when the sample size was reasonably large. When the sample size was small, the methods using the XPD matrices appeared to yield slight upward bias for the standard errors of the IRT parameter scale transformation coefficients.     

A General Linear Method for Equating With Small Samples

Research on equating with small samples has shown that methods with stronger assumptions and fewer statistical estimates can lead to decreased error in the estimated equating function. This article introduces a new approach to linear observed‐score equating, one which provides flexible control over how form difficulty is assumed versus estimated to change across the score scale. A general linear method is presented as an extension of traditional linear methods. The general method is then compared to other linear and nonlinear methods in terms of accuracy in estimating a criterion equating function. Results from two parametric bootstrapping studies based on real data demonstrate the usefulness of the general linear method.     

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.     

The Examination of the Classification of Students into Performance Categories by Two Different Equating Methods

This study investigates the comparability of two item response theory based equating methods: true score equating (TSE), and estimated true equating (ETE). Additionally, six scaling methods were implemented within each equating method: mean-sigma, mean-mean, two versions of fixed common item parameter, Stocking and Lord, and Haebara. Empirical test data were examined to investigate the consistency of scores resulting from the two equating methods, as well as the consistency of the scaling methods both within equating methods and across equating methods. Results indicate that although the degree of correlation among the equated scores was quite high, regardless of equating method/scaling method combination, non-trivial differences in equated scores existed in several cases. These differences would likely accumulate across examinees making group-level differences greater. Systematic differences in the classification of examinees into performance categories were observed across the various conditions: ETE tended to place lower ability examinees into higher performance categories than TSE, while the opposite was observed for high ability examinees. Because the study was based on one set of operational data, the generalizability of the findings is limited and further study is warranted.     

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 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.     

The effects of reducing correlation of external anchors on test equating methods for the equivalent groups and non-equivalent groups designs

Six equating methods were compared: a one-parameter Item Response Theory (IRT) method; two equipercentile methods (direct and by frequency estimation); and three linear methods (Tucker, Levine Equally Reliable and Levine Unequally Reliable) in a situation in which different forms were administered to different groups, thus necessitating the use of an anchor test. The groups were simulated as either equivalent groups or groups of variable ability representing the two types of class groupings that can exist in schools (i.e. parallel or streamed classes). The correlation between the ability measured by an external anchor and the tests to be equated was systematically manipulated. A discrepancy index summarised the discrepancy of each equating method from an IRT criterion, an equipercentile criterion, and from each other. Large discrepancies were interpreted with the aid of graphs and discussed in terms of examinee indifference to the alternative transformations. The direct equipercentile and Levine Unequally Reliable methods were the only methods that consistently increased their level of the discrepancy from criterion following reduction in correlation for the two equatings examined in the equivalent groups design. For the non-equivalent groups design, a reduction in correlation resulted in a systematic effect in favour of those taking an easier form (usually the less able) for all equating methods. What was observed, however, was that for small reductions in correlation, the discrepancy of some of the equating methods from the IRT criterion was reduced. The implications of these findings are discussed and recommendations made for further work.