A Comparison of Linking Methods for Two Groups for the Two-Parameter Logistic Item Response Model in the Presence and Absence of Random Differential Item Functioning

This article investigates the comparison of two groups based on the two-parameter logistic item response model. It is assumed that there is random differential item functioning in item difficulties and item discriminations. The group difference is estimated using separate calibration with subsequent linking, as well as concurrent calibration. The following linking methods are compared: mean-mean linking, log-mean-mean linking, invariance alignment, Haberman linking, asymmetric and symmetric Haebara linking, different recalibration linking methods, anchored item parameters, and concurrent calibration. It is analytically shown that log-mean-mean linking and mean-mean linking provide consistent estimates if random DIF effects have zero means. The performance of the linking methods was evaluated through a simulation study. It turned out that (log-)mean-mean and Haberman linking performed best, followed by symmetric Haebara linking and a newly proposed recalibration linking method. Interestingly, linking methods frequently found in applications (i.e., asymmetric Haebara linking, recalibration linking used in a variant in current large-scale assessment studies, anchored item parameters, concurrent calibration) perform worse in the presence of random differential item functioning. In line with the previous literature, differences between linking methods turned out be negligible in the absence of random differential item functioning. The different linking methods were also applied in an empirical example that performed a linking of PISA 2006 to PISA 2009 for Austrian students. This application showed that estimated trends in the means and standard deviations depended on the chosen linking method and the employed item response model.

Linking of Rasch-Scaled Tests: Consequences of Limited Item Pools and Model Misfit

In the context of item response theory (IRT), linking the scales of two measurement points is a prerequisite to examine a change in competence over time. In educational large-scale assessments, non-identical test forms sharing a number of anchor-items are frequently scaled and linked using two− or three-parametric item response models. However, if item pools are limited and/or sample sizes are small to medium, the sparser Rasch model is a suitable alternative regarding the precision of parameter estimation. As the Rasch model implies stricter assumptions about the response process, a violation of these assumptions may manifest as model misfit in form of item discrimination parameters empirically deviating from their fixed value of one. The present simulation study investigated the performance of four IRT linking methods—fixed parameter calibration, mean/mean linking, weighted mean/mean linking, and concurrent calibration—applied to Rasch-scaled data with a small item pool. Moreover, the number of anchor items required in the absence/presence of moderate model misfit was investigated in small to medium sample sizes. Effects on the link outcome were operationalized as bias, relative bias, and root mean square error of the estimated sample mean and variance of the latent variable. In the light of this limited context, concurrent calibration had substantial convergence issues, while the other methods resulted in an overall satisfying and similar parameter recovery—even in the presence of moderate model misfit. Our findings suggest that in case of model misfit, the share of anchor items should exceed 20% as is currently proposed in the literature. Future studies should further investigate the effects of anchor item composition regarding unbalanced model misfit.

Flexibility at the Price of Volatility: Concurrent Calibration in Multistage Tests in Practice Using a 2PL Model

Multistage test (MST) designs promise efficient student ability estimates, an indispensable asset for individual diagnostics in high-stakes educational assessments. In high-stakes testing, annually changing test forms are required because publicly known test items impair accurate student ability estimation, and items of bad model fit must be continually replaced to guarantee test quality. This requires a large and continually refreshed item pool as the basis for high-stakes MST. In practice, the calibration of newly developed items to feed annually changing tests is highly resource intensive. Piloting based on a representative sample of students is often not feasible, given that, for schools, participation in actual high-stakes assessments already requires considerable organizational effort. Hence, under practical constraints, the calibration of newly developed items may take place on the go in the form of a concurrent calibration in MST designs. Based on a simulation approach this paper focuses on the performance of Rasch vs. 2PL modeling in retrieving item parameters when items are for practical reasons non-optimally placed in multistage tests. Overall, the results suggest that the 2PL model performs worse in retrieving item parameters compared to the Rasch model when there is non-optimal item assembly in the MST; especially in retrieving parameters at the margins. The higher flexibility of 2PL modeling, where item discrimination is allowed to vary, seems to come at the cost of increased volatility in parameter estimation. Although the overall bias may be modest, single items can be affected by severe biases when using a 2PL model for item calibration in the context of non-optimal item placement.

A Comparison of the Separate and Concurrent Calibration Methods for the Full-Information Bifactor model

When calibrating items using multidimensional item response theory (MIRT) models, item response theory (IRT) calibration programs typically set the probability density of latent variables to a multivariate standard normal distribution to handle three types of indeterminacies: (a) the location of the origin, (b) the unit of measurement along each coordinate axis, and (c) the orientation of the coordinate axes. However, by doing so, item parameter estimates obtained from two independent calibration runs on nonequivalent groups are on two different coordinate systems. To handle this issue and place all the item parameter estimates on a common coordinate system, a process called linking is necessary. Although various linking methods have been introduced and studied for the full MIRT model, little research has been conducted on linking methods for the bifactor model. Thus, the purpose of this study was to provide detailed descriptions of two separate calibration methods and the concurrent calibration method for the bifactor model and to compare the three linking methods through simulation. In general, the concurrent calibration method provided more accurate linking results than the two separate calibration methods, demonstrating better recovery of the item parameters, item characteristic surfaces, and expected score distribution.

Evaluating Anchor-Item Designs for Concurrent Calibration With the GGUM

Concurrent calibration using anchor items has proven to be an effective alternative to separate calibration and linking for developing large item banks, which are needed to support continuous testing. In principle, anchor-item designs and estimation methods that have proven effective with dominance item response theory (IRT) models, such as the 3PL model, should also lead to accurate parameter recovery with ideal point IRT models, but surprisingly little research has been devoted to this issue. This study, therefore, had two purposes: (a) to develop software for concurrent calibration with, what is now the most widely used ideal point model, the generalized graded unfolding model (GGUM); (b) to compare the efficacy of different GGUM anchor-item designs and develop empirically based guidelines for practitioners. A Monte Carlo study was conducted to compare the efficacy of three anchor-item designs in vertical and horizontal linking scenarios. The authors found that a block-interlaced design provided the best parameter recovery in nearly all conditions. The implications of these findings for concurrent calibration with the GGUM and practical recommendations for pretest designs involving ideal point computer adaptive testing (CAT) applications are discussed.

Longitudinal Stability of IRT and Equivalent-Groups Linear and Equipercentile Equating

A multiscale criterion-referenced test that featured two presumably equivalent forms (A and B), was administered to 1,667 Head Start children at each of four points over an academic year. Using a randomly equivalent groups design, three equating methods were applied: common-item IRT equating using concurrent calibration, linear transformation, and equipercentile transformation. The methods were compared by examining mean score differences, weighted mean squared difference, and Kolmogorov's D statistics for each subscale. The results indicated that over time the IRT equating method and conventional equating methods exhibited different patterns of discrepancy between the two test forms. IRT equating yielded marginally smaller form-to-form mean score differences and generated slightly f ewer distributional discrepancies between Forms A and B than both linear and equipercentile equating. However, the results were mixed indicating that more studies are needed to provide additional information on the relative merits and weaknesses of each approach.

Effects of Vertical Scaling Methods on Linear Growth Estimation

Vertical scaling is necessary to facilitate comparison of scores from test forms of different difficulty levels. It is widely used to enable the tracking of student growth in academic performance over time. Most previous studies on vertical scaling methods assume relatively long tests and large samples. Little is known about their performance when the sample is small or the test is short, challenges that small testing programs often face. This study examined effects of sample size, test length, and choice of item response theory (IRT) models on the performance of IRT-based scaling methods (concurrent calibration, separate calibration with Stocking–Lord, Haebara, Mean/Mean, and Mean/Sigma transformation) in linear growth estimation when the 2-parameter IRT model was appropriate. Results showed that IRT vertical scales could be used for growth estimation without grossly biasing growth parameter estimates when sample size was not large, as long as the test was not too short (≥20 items), although larger sample sizes would generally increase the stability of the growth parameter estimates. The optimal rate of return in total estimation error reduction as a result of increasing sample size appeared to be around 250. Concurrent calibration produced slightly lower total estimation error than separate calibration in the worst combination of short test length (≤20 items) and small sample size ( n ≤ 100), whereas separate calibration, except in the case of the Mean/Sigma method, produced similar or somewhat lower amounts of total error in other conditions.