Multidimensional CAT Item Selection Methods for Domain Scores and Composite Scores With Item Exposure Control and Content Constraints

The intent of this research was to find an item selection procedure in the multidimensional computer adaptive testing (CAT) framework that yielded higher precision for both the domain and composite abilities, had a higher usage of the item pool, and controlled the exposure rate. Five multidimensional CAT item selection procedures (minimum angle; volume; minimum error variance of the linear combination; minimum error variance of the composite score with optimized weight; and Kullback‐Leibler information) were studied and compared with two methods for item exposure control (the Sympson‐Hetter procedure and the fixed‐rate procedure, the latter simply refers to putting a limit on the item exposure rate) using simulated data. The maximum priority index method was used for the content constraints. Results showed that the Sympson‐Hetter procedure yielded better precision than the fixed‐rate procedure but had much lower item pool usage and took more time. The five item selection procedures performed similarly under Sympson‐Hetter. For the fixed‐rate procedure, there was a trade‐off between the precision of the ability estimates and the item pool usage: the five procedures had different patterns. It was found that (1) Kullback‐Leibler had better precision but lower item pool usage; (2) minimum angle and volume had balanced precision and item pool usage; and (3) the two methods minimizing the error variance had the best item pool usage and comparable overall score recovery but less precision for certain domains. The priority index for content constraints and item exposure was implemented successfully.     

Dual‐Objective Item Selection Criteria in Cognitive Diagnostic Computerized Adaptive Testing

The development of cognitive diagnostic‐computerized adaptive testing (CD‐CAT) has provided a new perspective for gaining information about examinees' mastery on a set of cognitive attributes. This study proposes a new item selection method within the framework of dual‐objective CD‐CAT that simultaneously addresses examinees' attribute mastery status and overall test performance. The new procedure is based on the Jensen‐Shannon (JS) divergence, a symmetrized version of the Kullback‐Leibler divergence. We show that the JS divergence resolves the noncomparability problem of the dual information index and has close relationships with Shannon entropy, mutual information, and Fisher information. The performance of the JS divergence is evaluated in simulation studies in comparison with the methods available in the literature. Results suggest that the JS divergence achieves parallel or more precise recovery of latent trait variables compared to the existing methods and maintains practical advantages in computation and item pool usage.     

Restrictive Stochastic Item Selection Methods in Cognitive Diagnostic Computerized Adaptive Testing

This paper proposes two new item selection methods for cognitive diagnostic computerized adaptive testing: the restrictive progressive method and the restrictive threshold method. They are built upon the posterior weighted Kullback‐Leibler (KL) information index but include additional stochastic components either in the item selection index or in the item selection procedure. Simulation studies show that both methods are successful at simultaneously suppressing overexposed items and increasing the usage of underexposed items. Compared to item selection based upon (1) pure KL information and (2) the Sympson‐Hetter method, the two new methods strike a better balance between item exposure control and measurement accuracy. The two new methods are also compared with Barrada et al.'s (2008) progressive method and proportional method.     

Item Selection in Computerized Adaptive Testing: Should More Discriminating Items be Used First?

During computerized adaptive testing (CAT), items are selected continuously according to the test-taker's estimated ability. The traditional method of attaining the highest efficiency in ability estimation is to select items of maximum Fisher information at the currently estimated ability. Test security has become a problem because high-discrimination items are more likely to be selected and become overexposed. So, there seems to be a tradeoff between high efficiency in ability estimations and balanced usage of items. This series of four studies with simulated data addressed the dilemma by focusing on the notion of whether more or less discriminating items should be used first in CAT. The first study demonstrated that the common maximum information method with Sympson and Hetter (1985) control resulted in the use of more discriminating items first. The remaining studies showed that using items in the reverse order (i.e., less discriminating items first), as described in Chang and Ying's (1999) stratified method had potential advantages: (a) a more balanced item usage and (b) a relatively stable resultant item pool structure with easy and inexpensive management. This stratified method may have ability-estimation efficiency better than or close to that of other methods, particularly for operational item pools when retired items cannot be totally replenished with similar highly discriminating items. It is argued that the judicious selection of items, as in the stratified method, is a more active control of item exposure, which can successfully even out the usage of all items.     

Stratified and Maximum Information Item Selection Procedures in Computer Adaptive Testing

In this study we evaluated and compared three item selection procedures: the maximum Fisher information procedure (F), the a-stratified multistage computer adaptive testing (CAT) (STR), and a refined stratification procedure that allows more items to be selected from the high a strata and fewer items from the low a strata (USTR), along with completely random item selection (RAN). The comparisons were with respect to error variances, reliability of ability estimates and item usage through CATs simulated under nine test conditions of various practical constraints and item selection space. The results showed that F had an apparent precision advantage over STR and USTR under unconstrained item selection, but with very poor item usage. USTR reduced error variances for STR under various conditions, with small compromises in item usage. Compared to F, USTR enhanced item usage while achieving comparable precision in ability estimates; it achieved a precision level similar to F with improved item usage when items were selected under exposure control and with limited item selection space. The results provide implications for choosing an appropriate item selection procedure in applied settings.     

A Conditional Exposure Control Method for Multidimensional Adaptive Testing

In computerized adaptive testing (CAT), ensuring the security of test items is a crucial practical consideration. A common approach to reducing item theft is to define maximum item exposure rates, i.e., to limit the proportion of examinees to whom a given item can be administered. Numerous methods for controlling exposure rates have been proposed for tests employing the unidimensional 3-PL model. The present article explores the issues associated with controlling exposure rates when a multidimensional item response theory (MIRT) model is utilized and exposure rates must be controlled conditional upon ability. This situation is complicated by the exponentially increasing number of possible ability values in multiple dimensions. The article introduces a new procedure, called the generalized Stocking-Lewis method, that controls the exposure rate for students of comparable ability as well as with respect to the overall population. A realistic simulation set compares the new method with three other approaches: Kullback-Leibler information with no exposure control, Kullback-Leibler information with unconditional Sympson-Hetter exposure control, and random item selection.     

Evaluating Comparability in Computerized Adaptive Testing: Issues, Criteria and an Example

When a computerized adaptive testing (CAT) version of a test co-exists with its paper-and-pencil (P&P) version, it is important for scores from the CAT version to be comparable to scores from its P&P version. The CAT version may require multiple item pools for test security reasons, and CAT scores based on alternate pools also need to be comparable to each other. In this paper, we review research literature on CAT comparability issues and synthesize issues specific to these two settings. A framework of criteria for evaluating comparability was developed that contains the following three categories of criteria: validity criterion, psychometric property/reliability criterion, and statistical assumption/test administration condition criterion. Methods for evaluating comparability under these criteria as well as various algorithms for improving comparability are described and discussed. Focusing on the psychometric property/reliability criterion, an example using an item pool of ACT Assessment Mathematics items is provided to demonstrate a process for developing comparable CAT versions and for evaluating comparability. This example illustrates how simulations can be used to improve comparability at the early stages of the development of a CAT. The effects of different specifications of practical constraints, such as content balancing and item exposure rate control, and the effects of using alternate item pools are examined. One interesting finding from this study is that a large part of incomparability may be due to the change from number-correct score-based scoring to IRT ability estimation-based scoring. In addition, changes in components of a CAT, such as exposure rate control, content balancing, test length, and item pool size were found to result in different levels of comparability in test scores.     

A Comparative Study of Item Exposure Control Methods in Computerized Adaptive Testing

This study compared the properties of five methods of item exposure control within the purview of estimating examinees' abilities in a computerized adaptive testing (CAT) context. Each exposure control algorithm was incorporated into the item selection procedure and the adaptive testing progressed based on the CAT design established for this study. The merits and shortcomings of these strategies were considered under different item pool sizes and different desired maximum exposure rates and were evaluated in light of the observed maximum exposure rates, the test overlap rates, and the conditional standard errors of measurement. Each method had its advantages and disadvantages, but no one possessed all of the desired characteristics. There was a clear and logical trade-off between item exposure control and measurement precision. The Stocking and Lewis conditional multinomial procedure and, to a slightly lesser extent, the Davey and Parshall method seemed to be the most promising considering all of the factors that this study addressed.     

A Framework for Measuring the Amount of Adaptation of Rasch‐based Computerized Adaptive Tests

A key consideration when giving any computerized adaptive test (CAT) is how much adaptation is present when the test is used in practice. This study introduces a new framework to measure the amount of adaptation of Rasch‐based CATs based on looking at the differences between the selected item locations (Rasch item difficulty parameters) of the administered items and target item locations determined from provisional ability estimates at the start of each item. Several new indices based on this framework are introduced and compared to previously suggested measures of adaptation using simulated and real test data. Results from the simulation indicate that some previously suggested indices are not as sensitive to changes in item pool size and the use of constraints as the new indices and may not work as well under different item selection rules. The simulation study and real data example also illustrate the utility of using the new indices to measure adaptation at both a group and individual level. Discussion is provided on how one may use several of the indices to measure adaptation of Rasch‐based CATs in practice.     

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.     

Improving Item-Exposure Control in Adaptive Testing

One of the methods of controlling test security in adaptive testing is imposing random item-ineligibility constraints on the selection of the items with probabilities automatically updated to maintain a predetermined upper bound on the exposure rates. Three major improvements of the method are presented. First, a few modifications to improve the initialization of the method and accelerate the impact of its feedback mechanism on the observed item-exposure rates are introduced. Second, the case of conditional item-exposure control given the uncertainty of examinee's ability parameter is addressed. Third, although rare for a well-designed item pool, when applied in combination with the shadow-test approach to adaptive testing the method may meet occasional infeasibility of the shadow-test model. A big M method is proposed that resolves the issue. The practical advantages of the improvements are illustrated using simulated adaptive testing from a real-world item pool under a variety of conditions.     

Increasing the Homogeneity of CAT's Item-Exposure Rates by Minimizing or Maximizing Varied Target Functions While Assembling Shadow Tests

A computerized adaptive testing (CAT) algorithm that has the potential to increase the homogeneity of CAT's item-exposure rates without significantly sacrificing the precision of ability estimates was proposed and assessed in the shadow-test ( van der Linden & Reese, 1998 ) CAT context. This CAT algorithm was formed by a combination of maximizing or minimizing varied target functions while assembling shadow tests. There were four target functions to be separately used in the first, second, third, and fourth quarter test of CAT. The elements to be used in the four functions were associated with (a) a random number assigned to each item, (b) the absolute difference between an examinee's current ability estimate and an item difficulty, (c) the absolute difference between an examinee's current ability estimate and an optimum item difficulty, and (d) item information. The results indicated that this combined CAT fully utilized all the items in the pool, reduced the maximum exposure rates, and achieved more homogeneous exposure rates. Moreover, its precision in recovering ability estimates was similar to that of the maximum item-information method. The combined CAT method resulted in the best overall results compared with the other individual CAT item-selection methods. The findings from the combined CAT are encouraging. Future uses are discussed.     

On the Performance of Semi- and Nonparametric Item Response Functions in Computer Adaptive Tests

Large-scale assessments often use a computer adaptive test (CAT) for selection of items and for scoring respondents. Such tests often assume a parametric form for the relationship between item responses and the underlying construct. Although semi- and nonparametric response functions could be used, there is scant research on their performance in a CAT. In this work, we compare parametric response functions versus those estimated using kernel smoothing and a logistic function of a monotonic polynomial. Monotonic polynomial items can be used with traditional CAT item selection algorithms that use analytical derivatives. We compared these approaches in CAT simulations with a variety of item selection algorithms. Our simulations also varied the features of the calibration and item pool: sample size, the presence of missing data, and the percentage of nonstandard items. In general, the results support the use of semi- and nonparametric item response functions in a CAT.     

A Comparison of Item Exposure Control Methods in Computerized Adaptive Testing

Two new methods for item exposure control were proposed. In the Progressive method, as the test progresses, the influence of a random component on item selection is reduced and the importance of item information is increasingly more prominent. In the Restricted Maximum Information method, no item is allowed to be exposed in more than a predetermined proportion of tests. Both methods were compared with six other item-selection methods (Maximum Information, One Parameter, McBride and Martin, Randomesque, Sympson and Hetter, and Random Item Selection) with regard to test precision and item exposure variables. Results showed that the Restricted method was useful to reduce maximum exposure rates and that the Progressive method reduced the number of unused items. Both did well regarding precision. Thus, a combined Progressive-Restricted method may be useful to control item exposure without a serious decrease in test precision.     

The Relationship Between Item Exposure and Test Overlap in Computerized Adaptive Testing

The purpose of this article is to present an analytical derivation for the mathematical form of an average between-test overlap index as a function of the item exposure index, for fixed-length computerized adaptive tests (CATs). This algebraic relationship is used to investigate the simultaneous control of item exposure at both the item and test levels. The results indicate that, in fixed-length CATs, control of the average between-test overlap is achieved via the mean and variance of the item exposure rates of the items that constitute the CAT item pool. The mean of the item exposure rates is easily manipulated. Control over the variance of the item exposure rates can be achieved via the maximum item exposure rate (r_max). Therefore, item exposure control methods which implement a specification of r_max (e.g., Sympson & Hetter, 1985) provide the most direct control at both the item and test levels.     

Application of IRT Fixed Parameter Calibration to Multiple-Group Test Data

ABSTRACT

In applications of item response theory (IRT), fixed parameter calibration (FPC) has been used to estimate the item parameters of a new test form on the existing ability scale of an item pool. The present paper presents an application of FPC to multiple examinee groups test data that are linked to the item pool via anchor items, and investigates the performance of FPC relative to an alternative approach, namely independent 0–1 calibration and scale linking. Two designs for linking to the pool are proposed that involve multiple groups and test forms, for which multiple-group FPC can be effectively used. A real-data study shows that the multiple-group FPC method performs similarly to the alternative method in estimating ability distributions and new item parameters on the scale of the item pool. In addition, a simulation study shows that the multiple-group FPC method performs nearly equally to or better than the alternative method in recovering the underlying ability distributions and the new item parameters.     

Item Selection and Exposure Control Methods for Computerized Adaptive Testing with Multidimensional Ranking Items

The use of computerized adaptive testing algorithms for ranking items (e.g., college preferences, career choices) involves two major challenges: unacceptably high computation times (selecting from a large item pool with many dimensions) and biased results (enhanced preferences or intensified examinee responses because of repeated statements across items). To address these issues, we introduce subpool partition strategies for item selection and within-person statement exposure control procedures. Simulations showed that the multinomial method reduces computation time while maintaining measurement precision. Both the freeze and revised Sympson-Hetter online (RSHO) methods controlled the statement exposure rate; RSHO sacrificed some measurement precision but increased pool use. Furthermore, preventing a statement's repetition on consecutive items neither hindered the effectiveness of the freeze or RSHO method nor reduced measurement precision.     

Minimizing the Influence of Item Parameter Estimation Errors in Test Development: A Comparison of Three Selection Procedures

In test development, item response theory (IRT) is a method to determine the amount of information that each item (i.e., item information function) and combination of items (i.e., test information function) provide in the estimation of an examinee's ability. Studies investigating the effects of item parameter estimation errors over a range of ability have demonstrated an overestimation of information when the most discriminating items are selected (i.e., item selection based on maximum information). In the present study, the authors examined the influence of item parameter estimation errors across 3 item selection methods—maximum no target, maximum target, and theta maximum—using the 2- and 3-parameter logistic IRT models. Tests created with the maximum no target and maximum target item selection procedures consistently overestimated the test information function. Conversely, tests created using the theta maximum item selection procedure yielded more consistent estimates of the test information function and, at times, underestimated the test information function. Implications for test development are discussed.     

Item Selection and Ability Estimation Procedures for a Mixed-Format Adaptive Test

In this study we compared five item selection procedures using three ability estimation methods in the context of a mixed-format adaptive test based on the generalized partial credit model. The item selection procedures used were maximum posterior weighted information, maximum expected information, maximum posterior weighted Kullback-Leibler information, and maximum expected posterior weighted Kullback-Leibler information procedures. The ability estimation methods investigated were maximum likelihood estimation (MLE), weighted likelihood estimation (WLE), and expected a posteriori (EAP). Results suggested that all item selection procedures, regardless of the information functions on which they were based, performed equally well across ability estimation methods. The principal conclusions drawn about the ability estimation methods are that MLE is a practical choice and WLE should be considered when there is a mismatch between pool information and the population ability distribution. EAP can serve as a viable alternative when an appropriate prior ability distribution is specified. Several implications of the findings for applied measurement are discussed.     

Detection of Test Collusion via Kullback–Leibler Divergence

The development of statistical methods for detecting test collusion is a new research direction in the area of test security. Test collusion may be described as large‐scale sharing of test materials, including answers to test items. Current methods of detecting test collusion are based on statistics also used in answer‐copying detection. Therefore, in computerized adaptive testing (CAT) these methods lose power because the actual test varies across examinees. This article addresses that problem by introducing a new approach that works in two stages: in Stage 1, test centers with an unusual distribution of a person‐fit statistic are identified via Kullback–Leibler divergence; in Stage 2, examinees from identified test centers are analyzed further using the person‐fit statistic, where the critical value is computed without data from the identified test centers. The approach is extremely flexible. One can employ any existing person‐fit statistic. The approach can be applied to all major testing programs: paper‐and‐pencil testing (P&P), computer‐based testing (CBT), multiple‐stage testing (MST), and CAT. Also, the definition of test center is not limited by the geographic location (room, class, college) and can be extended to support various relations between examinees (from the same undergraduate college, from the same test‐prep center, from the same group at a social network). The suggested approach was found to be effective in CAT for detecting groups of examinees with item pre‐knowledge, meaning those with access (possibly unknown to us) to one or more subsets of items prior to the exam.