Parameter Recovery and Classification Accuracy Under Conditions of Testlet Dependency: A Comparison of the Traditional 2PL,Testlet, and Bi-Factor Models

ABSTRACT

Testlets, or groups of related items, are commonly included in educational assessments due to their many logistical and conceptual advantages. Despite their advantages, testlets introduce complications into the theory and practice of educational measurement. Responses to items within a testlet tend to be correlated even after controlling for latent ability, which violates the assumption of conditional independence made by traditional item response theory models. The present study used Monte Carlo simulation methods to evaluate the effects of testlet dependency on item and person parameter recovery and classification accuracy. Three calibration models were examined, including the traditional 2PL model with marginal maximum likelihood estimation, a testlet model with Bayesian estimation, and a bi-factor model with limited-information weighted least squares mean and variance adjusted estimation. Across testlet conditions, parameter types, and outcome criteria, the Bayesian testlet model outperformed, or performed equivalently to, the other approaches.     

A Comparison of Methods of Estimating Conditional Standard Errors of Measurement for Testlet-Based Test Scores Using Simulation Techniques

The primary purpose of this study was to investigate the appropriateness and implication of incorporating a testlet definition into the estimation of procedures of the conditional standard error of measurement (SEM) for tests composed of testlets. Another purpose was to investigate the bias in estimates of the conditional SEM when using item-based methods instead of testlet-based methods. Several item-based and testlet-based estimation methods were proposed and compared. In general, item-based estimation methods underestimated the conditional SEM for tests composed for testlets, and the magnitude of this negative bias increased as the degree of conditional dependence among items within testlets increased. However, an item-based method using a generalizability theory model provided good estimates of the conditional SEM under mild violation of the assumptions for measurement modeling. Under moderate or somewhat severe violation, testlet-based methods with item response models provided good estimates.     

Using a New Statistical Model for Testlets to Score TOEFL

Standard item response theory (IRT) models fit to examination responses ignore the fact that sets of items (testlets) often are matched with a single common stimulus (e.g., a reading comprehension passage). In this setting, all items given to an examinee are unlikely to be conditionally independent (given examinee proficiency). Models that assume conditional independence will overestimate the precision with which examinee proficiency is measured. Overstatement of precision may lead to inaccurate inferences as well as prematurely ended examinations in which the stopping rule is based on the estimated standard error of examinee proficiency (e.g., an adaptive test). The standard three parameter IRT model was modified to include an additional random effect for items nested within the same testlet (Wainer, Bradlow, & Du, 2000). This parameter, γ characterizes the amount of local dependence in a testlet.
We fit 86 TOEFL testlets (50 reading comprehension and 36 listening comprehension) with the new model, and obtained a value for the variance of γ for each testlet. We compared the standard parameters (discrimination (a), difficulty (b) and guessing (c)) with what is obtained through traditional modeling. We found that difficulties were well estimated either way, but estimates of both a and c were biased if conditional independence is incorrectly assumed. Of greater import, we found that test information was substantially over-estimated when conditional independence was incorrectly assumed.     

A Mixture Group Bifactor Model for Binary Responses

The presence of nuisance dimensionality is a potential threat to the accuracy of results for tests calibrated using a measurement model such as a factor analytic model or an item response theory model. This article describes a mixture group bifactor model to account for the nuisance dimensionality due to a testlet structure as well as the dimensionality due to differences in patterns of responses. The model can be used for testing whether or not an item functions differently across latent groups in addition to investigating the differential effect of local dependency among items within a testlet. An example is presented comparing test speededness results from a conventional factor mixture model, which ignores the testlet structure, with results from the mixture group bifactor model. Results suggested the 2 models treated the data somewhat differently. Analysis of the item response patterns indicated that the 2-class mixture bifactor model tended to categorize omissions as indicating speededness. With the mixture group bifactor model, more local dependency was present in the speeded than in the nonspeeded class. Evidence from a simulation study indicated the Bayesian estimation method used in this study for the mixture group bifactor model can successfully recover generated model parameters for 1- to 3-group models for tests containing testlets.     

Building Algebra Testlets: A Comparison of Hierarchical and Linear Structures

Earlier (Wainer & Lewis, 1990), we reported the initial development of a testlet-based algebra test. In this account, we provide the details of this excursion into the use of testlets. A pretest of two 15–item algebra tests was carried out in which examinees' performance on a 4-item subset of each test (a 4–item testlet) was used to predict performance on the entire test. Two models for constructing the testlets were considered: hierarchical (adaptive) and linear (fixed format). These models are compared with each other. It was found on cross–validation that, although an adaptive testlet is superior to a fixed format testlet, this superiority is modest, whereas the potential cost of that superiority is considerable. It was concluded that in circumstances similar to those we report a fixed format testlet that uses the best items in a pool can do almost as well as the optimal adaptive testlet of equal length from that same pool.     

Formal Relations and an Empirical Comparison among the Bi-Factor,the Testlet,and a Second-Order Multidimensional IRT Model

Testlet effects can be taken into account by incorporating specific dimensions in addition to the general dimension into the item response theory model. Three such multidimensional models are described: the bi-factor model, the testlet model, and a second-order model. It is shown how the second-order model is formally equivalent to the testlet model. In turn, both models are constrained bi-factor models. Therefore, the efficient full maximum likelihood estimation method that has been established for the bi-factor model can be modified to estimate the parameters of the two other models. An application on a testlet-based international English assessment indicated that the bi-factor model was the preferred model for this particular data set.     

Trace Lines for Testlets: A Use of Multiple-Categorical-Response Models

It is not always convenient or appropriate to construct tests in which individual items are fungible. There are situations in which small clusters of items (testlets) are the units that are assembled to create a test. Using data from a test of reading comprehension constructed of four passages with several questions following each passage, we show that local independence fails at the level of the individual questions. The questions following each passage, however, constitute a testlet. We discuss the application to testlet scoring of some multiple-category models originally developed for individual items, In the example examined, the concurrent validity of the testlet scoring equaled or exceeded that of individual-item-level scoring     

Use of Restricted Item Response Models for Examining Item Difficulty Ordering and Slope Uniformity

This article demonstrates the utility of restricted item response models for examining item difficulty ordering and slope uniformity for an item set that reflects varying cognitive processes. Twelve sets of paired algebra word problems were developed to systematically reflect various types of cognitive processes required for successful performance. This resulted in a total of 24 items. They reflected distance-rate–time (DRT), interest, and area problems. Hypotheses concerning difficulty ordering and slope uniformity for the items were tested by constraining item difficulty and discrimination parameters in hierarchical item response models. The first set of model comparisons tested the equality of the discrimination and difficulty parameters for each set of paired items. The second set of model comparisons examined slope uniformity within the complex DRT problems. The third set of model comparisons examined whether the familiarity of the story context affected item difficulty for two types of complex DRT problems. The last set of model comparisons tested the hypothesized difficulty ordering of the items.     

A Comparison of Different Psychometric Approaches to Modeling Testlet Structures: An Example with C‐Tests

C‐tests are a specific variant of cloze tests that are considered time‐efficient, valid indicators of general language proficiency. They are commonly analyzed with models of item response theory assuming local item independence. In this article we estimated local interdependencies for 12 C‐tests and compared the changes in item difficulties, reliability estimates, and person parameter estimates for different modeling approaches: (a) Rasch, (b) testlet, (c) partial credit, and (d) copula models. The results are complemented with findings of a simulation study in which sample size, number of testlets, and strength of residual correlations between items were systematically manipulated. Results are discussed with regard to the pivotal question whether residual dependencies between items are an artifact or part of the construct.     

Small-Sample Equating Using a Single-Group Nearly Equivalent Test (SiGNET) Design

A single-group (SG) equating with nearly equivalent test forms (SiGNET) design was developed by Grant to equate small-volume tests. Under this design, the scored items for the operational form are divided into testlets or mini tests. An additional testlet is created but not scored for the first form. If the scored testlets are testlets 1–6 and the unscored testlet is testlet 7, then the first form is composed of testlets 1–6 and the second form is composed of testlets 2–7. The seven testlets are administered as a single administered form, and when a sufficient number of examinees have taken the administered form, the second form (testlets 2–7) is equated to the first form (testlets 1–6) using an SG equating design. As evident, this design facilitates the use of an SG equating and allows for the accumulation of data, both of which may reduce equating error. This study compared equatings under the SiGNET and common-item equating designs and found lower equating error for the SiGNET design in very small sample size conditions (e.g., N = 10).     

Exploring differential bundle functioning in mathematics by gender: the effect of hierarchical modelling

Researchers interested in exploring substantive group differences are increasingly attending to bundles of items (or testlets): the aim is to understand how gender differences, for instance, are explained by differential performances on different types or bundles of items, hence differential bundle functioning (DBF). Some previous work has modelled hierarchies in data in this context or considered item responses within persons, but here we model the bundles themselves as explanatory variables at the item level potentially explaining significant intra-class correlation due to gender differences in item difficulty, and thus explaining variation at the second item level. In this study, we analyse DBF using single- and two-level models (the latter modelling random item effects, which models responses at Level 1 and items at Level 2) in a high-stakes National Mathematics test. The models show comparable regression coefficients but the statistical significances of the two-level models are smaller due to the larger values of the estimated standard errors. We discuss the contrasting relevance of this effect for test developers and gender researchers.     

Test fairness: Examining differential functioning of the reading comprehension section of the GSEEE in China

This study investigated differential item functioning (DIF), differential bundle functioning (DBF), and differential test functioning (DTF) across gender of the reading comprehension section of the Graduate School Entrance English Exam in China. The datasets included 10,000 test-takers’ item-level responses to 6 five-item testlets. Both DIF and DBF were examined by using poly-simultaneous item bias test and item-response-theory-likelihood-ratio test, and DTF was investigated with multi-group confirmatory factor analyses (MG-CFA). The results indicated that although none of the 30 items exhibited statistically and practically significant DIF across gender at the item level, 2 testlets were consistently identified as having significant DBF at the testlet level by the two procedures. Nonetheless, DBF does not manifest itself at the overall test score level to produce DTF based on MG-CFA. This suggests that the relationship between item-level DIF and test-level DTF is a complicated issue with the mediating effect of testlets in testlet-based language assessment.     

A Multilevel Testlet Model for Dual Local Dependence

The applications of item response theory (IRT) models assume local item independence and that examinees are independent of each other. When a representative sample for psychometric analysis is selected using a cluster sampling method in a testlet‐based assessment, both local item dependence and local person dependence are likely to be induced. This study proposed a four‐level IRT model to simultaneously account for dual local dependence due to item clustering and person clustering. Model parameter estimation was explored using the Markov Chain Monte Carlo method. Model parameter recovery was evaluated in a simulation study in comparison with three other related models: the Rasch model, the Rasch testlet model, and the three‐level Rasch model for person clustering. In general, the proposed model recovered the item difficulty and person ability parameters with the least total error. The bias in both item and person parameter estimation was not affected but the standard error (SE) was affected. In some simulation conditions, the difference in classification accuracy between models could go up to 11%. The illustration using the real data generally supported model performance observed in the simulation study.     

An investigation of the nature of the influences of item stem and option representation on student responses to a mathematics test

In teaching, representations are used as ways to illustrate the concepts underlying a specific topic. For example, use symbols (e.g., 1?+?2?=?3) to express the concept of addition. To compare students’ abilities to interpret different representations in mathematics, the symbolic representation (SR) test and the pictorial representation (PR) test were designed, and then administered to 681 sixth graders in Taipei, Taiwan. This study adopts two different modeling perspectives, the testlet perspective and the multi-ability perspective, to analyze this SR and PR test data in the context of item response theory. The main results show that:

1. Students scored on average significantly higher on the SR test than the PR test.
2. The effects of the item stem testlets could be large, but they are statistically non-significant; however, the influence of the number of items in the testlet should also be considered.
3. The nature of the option representations, SR and PR, represents two different mathematics abilities.
4. The main factor that influences students’ item responses is students’ abilities to interpret SR and PR, and the testlet effects generated from the shared item stem can be ignored.
5. Regarding the parameter estimates of the best-fitting model: (a) the person ability variance estimates show that the ability distributions on the SR and PR dimension may not be the same, (b) the correlation estimate between the SR and PR dimension indicates that these two abilities are moderately correlated, and (c) the item difficulty estimates for different models are similar.

Suggestions for teaching practice and future studies are provided in the Conclusion.     

A Comparison of the Performance of Simulated Hierarchical and Linear Testlets

A series of computer simulations were run to measure the relationship between testlet validity and the factors of item pool size and testlet length for both adaptive and linearly constructed testlets. We confirmed the generality of earlier empirical findings (Wainer, Lewis, Kaplan, & Braswell, 1991) that making a testlet adaptive yields only modest increases in aggregate validity because of the peakedness of the typical proficiency distribution.     

The Effect of Linguistic Simplification of Science Test Items on Score Comparability

The use of accommodations has been widely proposed as a means of including English language learners (ELLs) or limited English proficient (LEP) students in state and districtwide assessments. However, very little experimental research has been done on specific accommodations to determine whether these pose a threat to score comparability. This study examined the effects of linguistic simplification of 4th- and 6th-grade science test items on a state assessment. At each grade level, 4 experimental 10-item testlets were included on operational forms of a statewide science assessment. Two testlets contained regular field-test items, but in a linguistically simplified condition. The testlets were randomly assigned to LEP and non-LEP students through the spiraling of test booklets. For non-LEP students, in 4 t-test analyses of the differences in means for each corresponding testlet, 3 of the mean score comparisons were not significantly different, and the 4th showed the regular version to be slightly easier than the simplified version. Analysis of variance (ANOVA), followed by pairwise comparisons of the testlets, showed no significant differences in the scores of non-LEP students across the 2 item types. Among the 40 items administered in both regular and simplified format, item difficulty did not vary consistently in favor of either format. Qualitative analyses of items that displayed significant differences in p values were not informative, because the differences were typically very small. For LEP students, there was 1 significant difference in student means, and it favored the regular version. However, because the study was conducted in a state with a small number of LEP students, the analyses of LEP student responses lacked statistical power. The results of this study show that linguistic simplification is not helpful to monolingual English-speaking students who receive the accommodation. Therefore, the results provide evidence that linguistic simplification is not a threat to the comparability of scores of LEP and monolingual English-speaking students when offered as an accommodation to LEP students. The study findings may also have implications for the use of linguistic simplification accommodations in science assessments in other states and in content areas other than science.     

斯坦福(Stanford)成就阅读理解考试的原本和客户本的结构等价

本文是第一篇探索斯坦福成就阅读考试(第十版)的原本及其客户化版本的结构相似性的文章。研究分析是跨年级在多个观测变量(个别题目,题组,题包)上进行的。分析方法主要包括线性和非线性的探索性和实证性因素分析。分析结果表明在所有文章内的试题,都有不同程度的题组效应。在所有的模型当中,个别题目作为观测变量的模型的拟合度最低,题组作为观测变量的模型的拟合;其次,题包作为观测变量的模型的拟合度最高。在三种结构等性等级:同性等性(congenric),陶性等性(tau-equivalent)和并行等性(parallel)中,斯坦福成就阅读考试原本与其客户化版本的结构具有同性相似。     

Examining type I error and power for detection of differential item and testlet functioning

In this study, the effectiveness of detection of differential item functioning (DIF) and testlet DIF using SIBTEST and Poly-SIBTEST were examined in tests composed of testlets. An example using data from a reading comprehension test showed that results from SIBTEST and Poly-SIBTEST were not completely consistent in the detection of DIF and testlet DIF. Results from a simulation study indicated that SIBTEST appeared to maintain type I error control for most conditions, except in some instances in which the magnitude of simulated DIF tended to increase. This same pattern was present for the Poly-SIBTEST results, although Poly-SIBTEST demonstrated markedly less control of type I errors. Type I error control with Poly-SIBTEST was lower for those conditions for which the ability was unmatched to test difficulty. The power results for SIBTEST were not adversely affected, when the size and percent of simulated DIF increased. Although Poly-SIBTEST failed to control type I errors in over 85% of the conditions simulated, in those conditions for which type I error control was maintained, Poly-SIBTEST demonstrated higher power than SIBTEST.     

Item and testlet position effects in computer-based alternate assessments for students with disabilities

Item positions in educational assessments are often randomized across students to prevent cheating. However, if altering item positions results in any significant impact on students’ performance, it may threaten the validity of test scores. Two widely used approaches for detecting position effects – logistic regression and hierarchical generalized linear modelling – are often inconvenient for researchers and practitioners due to some technical and practical limitations. Therefore, this study introduced a structural equation modeling (SEM) approach for examining item and testlet position effects. The SEM approach was demonstrated using data from a computer-based alternate assessment designed for students with cognitive disabilities from three grade bands (3–5, 6–8, and high school). Item and testlet position effects were investigated in the field-test (FT) items that were received by each student at different positions. Results indicated that the difficulty of some FT items in grade bands 3–5 and 6–8 differed depending on the positions of the items on the test. Also, the overall difficulty of the field-test task in grade bands 6–8 increased as students responded to the field-test task in later positions. The SEM approach provides a flexible method for examining different types of position effects.     

A comparison of the approaches of generalizability theory and item response theory in estimating the reliability of test scores for testlet-composed tests

Previous assessments of the reliability of test scores for testlet-composed tests have indicated that item-based estimation methods overestimate reliability. This study was designed to address issues related to the extent to which item-based estimation methods overestimate the reliability of test scores composed of testlets and to compare several estimation methods for different measurement models using simulation techniques. Three types of estimation approach were conceptualized for generalizability theory (GT) and item response theory (IRT): item score approach (ISA), testlet score approach (TSA), and item-nested-testlet approach (INTA). The magnitudes of overestimation when applying item-based methods ranged from 0.02 to 0.06 and were related to the degrees of dependence among within-testlet items. Reliability estimates from TSA were lower than those from INTA due to the loss of information with IRT approaches. However, this could not be applied in GT. Specified methods in IRT produced higher reliability estimates than those in GT using the same approach. Relatively smaller magnitudes of error in reliability estimates were observed for ISA and for methods in IRT. Thus, it seems reasonable to use TSA as well as INTA for both GT and IRT. However, if there is a relatively large dependence among within-testlet items, INTA should be considered for IRT due to nonnegligible loss of information.