Collateral Information for Equating in Small Samples: A Preliminary Investigation

This article describes a preliminary investigation of an empirical Bayes (EB) procedure for using collateral information to improve equating of scores on test forms taken by small numbers of examinees. Resampling studies were done on two different forms of the same test. In each study, EB and non-EB versions of two equating methods—chained linear and chained mean—were applied to repeated small samples drawn from a large data set collected for a common-item equating. The criterion equating was the chained linear equating in the large data set. Equatings of other forms of the same test provided the collateral information. New-form sample size was varied from 10 to 200; reference-form sample size was constant at 200. One of the two new forms did not differ greatly in difficulty from its reference form, as was the case for the equatings used as collateral information. For this form, the EB procedure improved the accuracy of equating with new-form samples of 50 or fewer. The other new form was much more difficult than its reference form; for this form, the EB procedure made the equating less accurate.     

A Comparison of Chained Linear and Poststratification Linear Equating Under Different Testing Conditions

In this study I compared results of chained linear, Tucker, and Levine-observed score equatings under conditions where the new and old forms samples were similar in ability and also when they were different in ability. The length of the anchor test was also varied to examine its effect on the three different equating methods. The three equating methods were compared to a criterion equating to obtain estimates of random equating error, bias, and root mean squared error (RMSE). Results showed that, for most studied conditions, chained linear equating produced fairly good equating results in terms of low bias and RMSE. Levine equating also produced low bias and RMSE in some conditions. Although the Tucker method always produced the lowest random equating error, it produced a larger bias and RMSE than either of the other equating methods. As noted in the literature, these results also suggest that either chained linear or Levine equating be used when new and old form samples differ on ability and/or when the anchor-to-total correlation is not very high. Finally, by testing the missing data assumptions of the three equating methods, this study also shows empirically why an equating method is more or less accurate under certain conditions .     

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.     

Equivalent Pass/Fail Decisions

In competency testing, it is sometimes difficult to properly equate scores of different forms of a test and thereby assure equivalent cutting scores. Under such circumstances, it is possible to set standards separately for each test form and then scale the judgments of the standard setters to achieve equivalent pass/fail decisions. Data from standard setters and examinees for a medical certifying examination were reanalyzed. Cutting score equivalents were derived by applying a linear procedure to the standard-setting results. These were compared against criteria along with the cutting score equivalents derived from typical examination equating procedures. Results indicated that the cutting score equivalents produced by the experts were closer to the criteria than standards derived from examinee performance, especially when the number of examinees used in equating was small. The root mean square error estimate was about 1 item on a 189-item test.     

Comparison of the One‐ and Bi‐Direction Chained Equipercentile Equating

This study investigated differences between two approaches to chained equipercentile (CE) equating (one‐ and bi‐direction CE equating) in nearly equal groups and relatively unequal groups. In one‐direction CE equating, the new form is linked to the anchor in one sample of examinees and the anchor is linked to the reference form in the other sample. In bi‐direction CE equating, the anchor is linked to the new form in one sample of examinees and to the reference form in the other sample. The two approaches were evaluated in comparison to a criterion equating function (i.e., equivalent groups equating) using indexes such as root expected squared difference, bias, standard error of equating, root mean squared error, and number of gaps and bumps. The overall results across the equating situations suggested that the two CE equating approaches produced very similar results, whereas the bi‐direction results were slightly less erratic, smoother (i.e., fewer gaps and bumps), usually closer to the criterion function, and also less variable.     

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.     

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.     

Rater Comparability Scoring and Equating: Does Choice of Target Population Weights Matter in This Context?

When a constructed‐response test form is reused, raw scores from the two administrations of the form may not be comparable. The solution to this problem requires a rescoring, at the current administration, of examinee responses from the previous administration. The scores from this “rescoring” can be used as an anchor for equating. In this equating, the choice of weights for combining the samples to define the target population can be critical. In rescored data, the anchor usually correlates very strongly with the new form but only moderately with the reference form. This difference has a predictable impact: the equating results are most accurate when the target population is the reference form sample, least accurate when the target population is the new form sample, and somewhere in the middle when the new form and reference form samples are equally weighted in forming the target population.     

Random-Groups Equating with Samples of 50 to 400 Test Takers

Five methods for equating in a random groups design were investigated in a series of resampling studies with samples of 400, 200, 100, and 50 test takers. Six operational test forms, each taken by 9,000 or more test takers, were used as item pools to construct pairs of forms to be equated. The criterion equating was the direct equipercentile equating in the group of all test takers. Equating accuracy was indicated by the root-mean-squared deviation, over 1,000 replications, of the sample equatings from the criterion equating. The methods investigated were equipercentile equating of smoothed distributions, linear equating, mean equating, symmetric circle-arc equating, and simplified circle-arc equating. The circle-arc methods produced the most accurate results for all sample sizes investigated, particularly in the upper half of the score distribution. The difference in equating accuracy between the two circle-arc methods was negligible.     

Comparisons among Small Sample Equating Methods in a Common-Item Design

Score equating based on small samples of examinees is often inaccurate for the examinee populations. We conducted a series of resampling studies to investigate the accuracy of five methods of equating in a common-item design. The methods were chained equipercentile equating of smoothed distributions, chained linear equating, chained mean equating, the symmetric circle-arc method, and the simplified circle-arc method. Four operational test forms, each containing at least 110 items, were used for the equating, with new-form samples of 100, 50, 25, and 10 examinees and reference-form samples three times as large. Accuracy was described in terms of the root-mean-squared difference (over 1,000 replications) of the sample equatings from the criterion equating. Overall, chained mean equating produced the most accurate results for low scores, but the two circle-arc methods produced the most accurate results, particularly in the upper half of the score distribution. The difference in equating accuracy between the two circle-arc methods was negligible.     

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.     

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.     

Maintaining Equivalent Cut Scores for Small Sample Test Forms

This study examines the effectiveness of three approaches for maintaining equivalent performance standards across test forms with small samples: (1) common‐item equating, (2) resetting the standard, and (3) rescaling the standard. Rescaling the standard (i.e., applying common‐item equating methodology to standard setting ratings to account for systematic differences between standard setting panels) has received almost no attention in the literature. Identity equating was also examined to provide context. Data from a standard setting form of a large national certification test (N examinees = 4,397; N panelists = 13) were split into content‐equivalent subforms with common items, and resampling methodology was used to investigate the error introduced by each approach. Common‐item equating (circle‐arc and nominal weights mean) was evaluated at samples of size 10, 25, 50, and 100. The standard setting approaches (resetting and rescaling the standard) were evaluated by resampling (N = 8) and by simulating panelists (N = 8, 13, and 20). Results were inconclusive regarding the relative effectiveness of resetting and rescaling the standard. Small‐sample equating, however, consistently produced new form cut scores that were less biased and less prone to random error than new form cut scores based on resetting or rescaling the standard.     

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.     

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.     

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.     

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.     

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.     

A Comparison of IRT Proficiency Estimation Methods Under Adaptive Multistage Testing

This inquiry is an investigation of item response theory (IRT) proficiency estimators’ accuracy under multistage testing (MST). We chose a two‐stage MST design that includes four modules (one at Stage 1, three at Stage 2) and three difficulty paths (low, middle, high). We assembled various two‐stage MST panels (i.e., forms) by manipulating two assembly conditions in each module, such as difficulty level and module length. For each panel, we investigated the accuracy of examinees’ proficiency levels derived from seven IRT proficiency estimators. The choice of Bayesian (prior) versus non‐Bayesian (no prior) estimators was of more practical significance than the choice of number‐correct versus item‐pattern scoring estimators. The Bayesian estimators were slightly more efficient than the non‐Bayesian estimators, resulting in smaller overall error. Possible score changes caused by the use of different proficiency estimators would be nonnegligible, particularly for low‐ and high‐performing examinees.     

Accuracy of Random Groups Equating with Very Small Samples

This study investigated the effectiveness of equating with very small samples using the random groups design. Of particular interest was equating accuracy at specific scores where performance standards might be set. Two sets of simulations were carried out, one in which the two forms were identical and one in which they differed by a tenth of a standard deviation in overall difficulty. These forms were equated using mean equating, linear equating, unsmoothed equipercentile equating, and equipercentile equating using two through six moments of log-linear presmoothing with samples of 25, 50, 75, 100, 150, and 200. The results indicated that identity equating was preferable to any equating method when samples were as small as 25. For samples of 50 and above, the choice of an equating method over identity equating depended on the location of the passing score relative to examinee performance. If passing scores were located below the mean, where data were sparser, mean equating produced the smallest percentage of misclassified examinees. For passing scores near the mean, all methods produced similar results with linear equating being the most accurate. For passing scores above the mean, equipercentile equating with 2- and 3-moment presmoothing were the best equating methods. Higher levels of presmoothing did not improve the results.