Conditional Standard Errors of Measurement for Scale Scores

Standard errors of measurement of scale scores by score level (conditional standard errors of measurement) can be valuable to users of test results. In addition, the Standards for Educational and Psychological Testing (AERA, APA, & NCME, 1985) recommends that conditional standard errors be reported by test developers. Although a variety of procedures are available for estimating conditional standard errors of measurement for raw scores, few procedures exist for estimating conditional standard errors of measurement for scale scores from a single test administration. In this article, a procedure is described for estimating the reliability and conditional standard errors of measurement of scale scores. This method is illustrated using a strong true score model. Practical applications of this methodology are given. These applications include a procedure for constructing score scales that equalize standard errors of measurement along the score scale. Also included are examples of the effects of various nonlinear raw-to-scale score transformations on scale score reliability and conditional standard errors of measurement. These illustrations examine the effects on scale score reliability and conditional standard errors of measurement of (a) the different types of raw-to-scale score transformations (e.g., normalizing scores), (b) the number of scale score points used, and (c) the transformation used to equate alternate forms of a test. All the illustrations use data from the ACT Assessment testing program.     

Conditional Standard Errors of Measurement for Scale Scores Using IRT

An IRT method for estimating conditional standard errors of measurement of scale scores is presented, where scale scores are nonlinear transformations of number-correct scores. The standard errors account for measurement error that is introduced due to rounding scale scores to integers. Procedures for estimating the average conditional standard error of measurement for scale scores and reliability of scale scores are also described. An illustration of the use of the methodology is presented, and the results from the IRT method are compared to the results from a previously developed method that is based on strong true-score theory.     

Generating Dichotomous Item Scores with the Four-Parameter Beta Compound Binomial Model

A Monte Carlo simulation technique for generating dichotomous item scores is presented that implements (a) a psychometric model with different explicit assumptions than traditional parametric item response theory (IRT) models, and (b) item characteristic curves without restrictive assumptions concerning mathematical form. The four-parameter beta compound-binomial (4PBCB) strong true score model (with two-term approximation to the compound binomial) is used to estimate and generate the true score distribution. The nonparametric item-true score step functions are estimated by classical item difficulties conditional on proportion-correct total score. The technique performed very well in replicating inter-item correlations, item statistics (point-biserial correlation coefficients and item proportion-correct difficulties), first four moments of total score distribution, and coefficient alpha of three real data sets consisting of educational achievement test scores. The technique replicated real data (including subsamples of differing proficiency) as well as the three-parameter logistic (3PL) IRT model (and much better than the 1PL model) and is therefore a promising alternative simulation technique. This 4PBCB technique may be particularly useful as a more neutral simulation procedure for comparing methods that use different IRT models.     

Confidence Intervals for Weighted Composite Scores Under the Compound Binomial Error Model

Reporting confidence intervals with test scores helps test users make important decisions about examinees by providing information about the precision of test scores. Although a variety of estimation procedures based on the binomial error model are available for computing intervals for test scores, these procedures assume that items are randomly drawn from a undifferentiated universe of items, and therefore might not be suitable for tests developed according to a table of specifications. To address this issue, four interval estimation procedures that use category subscores for the computation of confidence intervals are presented in this article. All four estimation procedures assume that subscores instead of test scores follow a binomial distribution (i.e., compound binomial error model). The relative performance of the four compound binomial–based interval estimation procedures is compared to each other and to the better known normal approximation and Wilson score procedures based on the binomial error model.     

Stabilizing Conditional Standard Errors of Measurement in Scale Score Transformations

The focus of this article is on scale score transformations that can be used to stabilize conditional standard errors of measurement (CSEMs). Three transformations for stabilizing the estimated CSEMs are reviewed, including the traditional arcsine transformation, a recently developed general variance stabilization transformation, and a new method proposed in this article involving cubic transformations. Two examples are provided and the three scale score transformations are compared in terms of how well they stabilize CSEMs estimated from compound binomial and item response theory (IRT) models. Advantages of the cubic transformation are demonstrated with respect to CSEM stabilization and other scaling criteria (e.g., scale score distributions that are more symmetric).     

IRT Approaches to Modeling Scores on Mixed-Format Tests

This article considers psychometric properties of composite raw scores and transformed scale scores on mixed-format tests that consist of a mixture of multiple-choice and free-response items. Test scores on several mixed-format tests are evaluated with respect to conditional and overall standard errors of measurement, score reliability, and classification consistency and accuracy under three item response theory (IRT) frameworks: unidimensional IRT (UIRT), simple structure multidimensional IRT (SS-MIRT), and bifactor multidimensional IRT (BF-MIRT) models. Illustrative examples are presented using data from three mixed-format exams with various levels of format effects. In general, the two MIRT models produced similar results, while the UIRT model resulted in consistently lower estimates of reliability and classification consistency/accuracy indices compared to the MIRT models.     

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 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.     

A Comparison of Score Level Estimates of the Standard Error of Measurement

Numerous methods have been proposed and investigated for estimating · the standard error of measurement (SEM) at specific score levels. Consensus on the preferred method has not been obtained, in part because there is no standard criterion. The criterion procedure in previous investigations has been a single test occasion procedure. This study compares six estimation techniques. Two criteria were calculated by using test results obtained from a test-retest or parallel forms design. The relationship between estimated score level standard errors and the score scale was similar for the six procedures. These relationships were also congruent to findings from previous investigations. Similarity between estimates and criteria varied over methods and criteria. For test-retest conditions, the estimation techniques are interchangeable. The user's selection could be based on personal preference. However, for parallel forms conditions, the procedures resulted in estimates that were meaningfully different. The preferred estimation technique would be Feldt's method (cited in Gupta, 1965; Feldt, 1984).     

Generalization of the Lord‐Wingersky Algorithm to Computing the Distribution of Summed Test Scores Based on Real‐Number Item Scores

With known item response theory (IRT) item parameters, Lord and Wingersky provided a recursive algorithm for computing the conditional frequency distribution of number‐correct test scores, given proficiency. This article presents a generalized algorithm for computing the conditional distribution of summed test scores involving real‐number item scores. The generalized algorithm is distinct from the Lord‐Wingersky algorithm in that it explicitly incorporates the task of figuring out all possible unique real‐number test scores in each recursion. Some applications of the generalized recursive algorithm, such as IRT test score reliability estimation and IRT proficiency estimation based on summed test scores, are illustrated with a short test by varying scoring schemes for its items.     

Classification Consistency and Accuracy for Complex Assessments Using Item Response Theory

In this article, procedures are described for estimating single-administration classification consistency and accuracy indices for complex assessments using item response theory (IRT). This IRT approach was applied to real test data comprising dichotomous and polytomous items. Several different IRT model combinations were considered. Comparisons were also made between the IRT approach and two non-IRT approaches including the Livingston-Lewis and compound multinomial procedures. Results for various IRT model combinations were not substantially different. The estimated classification consistency and accuracy indices for the non-IRT procedures were almost always lower than those for the IRT procedures.     

Generalizability in Item Response Modeling

An approach called generalizability in item response modeling (GIRM) is introduced in this article. The GIRM approach essentially incorporates the sampling model of generalizability theory (GT) into the scaling model of item response theory (IRT) by making distributional assumptions about the relevant measurement facets. By specifying a random effects measurement model, and taking advantage of the flexibility of Markov Chain Monte Carlo (MCMC) estimation methods, it becomes possible to estimate GT variance components simultaneously with traditional IRT parameters. It is shown how GT and IRT can be linked together, in the context of a single-facet measurement design with binary items. Using both simulated and empirical data with the software WinBUGS, the GIRM approach is shown to produce results comparable to those from a standard GT analysis, while also producing results from a random effects IRT model.     

Psychometric Properties of Scale Scores and Performance Levels for Performance Assessments Using Polytomous IRT

With a focus on performance assessments, this paper describes procedures for calculating conditional standard error of measurement (CSEM) and reliability of scale scores and classification consistency of performance levels. Scale scores that are transformations of total raw scores are the focus of these procedures, although other types of raw scores are considered as well. Polytomous IRT models provide the psychometric foundation for the procedures that are described. The procedures are applied using test data from ACT's Work Keys Writing Assessment to demonstrate their usefulness. Two polytomous IRT models were compared, as were two different procedures for calculating scores. One simulation study was done using one of the models to evaluate the accuracy of the proposed procedures. The results suggest that the procedures provide quite stable estimates and have the potential to be useful in a variety of performance assessment situations.     

An empirical investigation of two variance estimation procedures for use with criterion-referenced tests

This study investigated two procedures for estimating the population standard deviation of nonnormed tests. Two normed tests, both whose population standard deviation was known, were administered to 272 students in grades 3–6. One of the normed tests was treated as a criterion-referenced test; the two variance estimation procedures were applied to the scores from this test. Substantial differences were found between both estimated statistics and the actual standard deviation. The first estimation procedure estimated the standard deviation systematically higher, whereas the second procedure's estimation was systematically lower. These results are discussed in terms of using such procedures for program evaluation.     

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.     

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.     

The Effects of a Student Sampling Plan on Estimates of the Standard Errors for Student Passing Rates

Examined in this study were three procedures for estimating the standard errors of school passing rates using a generalizability theory model. Also examined was how these procedures behaved for student samples that differed in size. The procedures differed in terms of their assumptions about the populations from which students were sampled, and it was found that student sample size generally had a notable effect on the size of the standard error estimates they produced. Also the three procedures produced markedly different standard error estimates when student sample size was small.     

Monitoring Items in Real Time to Enhance CAT Security

An IRT‐based sequential procedure is developed to monitor items for enhancing test security. The procedure uses a series of statistical hypothesis tests to examine whether the statistical characteristics of each item under inspection have changed significantly during CAT administration. This procedure is compared with a previously developed CTT‐based procedure through simulation studies. The results show that when the total number of examinees is fixed both procedures can control the rate of type I errors at any reasonable significance level by choosing an appropriate cutoff point and meanwhile maintain a low rate of type II errors. Further, the IRT‐based method has a much lower type II error rate or more power than the CTT‐based method when the number of compromised items is small (e.g., 5), which can be achieved if the IRT‐based procedure can be applied in an active mode in the sense that flagged items can be replaced with new items.     

Classification Consistency and Accuracy for Mixed-Format Tests

This study explores classification consistency and accuracy for mixed-format tests using real and simulated data. In particular, the current study compares six methods of estimating classification consistency and accuracy for seven mixed-format tests. The relative performance of the estimation methods is evaluated using simulated data. Study results from real data analysis showed that the procedures exhibited similar patterns across various exams, but some tended to produce lower estimates of classification consistency and accuracy than others. As data became more multidimensional, unidimensional and multidimensional item response theory (IRT) methods tended to produce different results, with the unidimensional approach yielding lower estimates than the multidimensional approach. Results from simulated data analysis demonstrated smaller estimation error for the multidimensional IRT methods than for the unidimensional IRT method. The unidimensional approach yielded larger error as tests became more multidimensional, whereas a reverse relationship was observed for the multidimensional IRT approach. Among the non-IRT approaches, the normal approximation and Livingston-Lewis methods performed well, whereas the compound multinomial method tended to produce relatively larger error.     

Classification Consistency and Accuracy With Atypical Score Distributions

The current study aims to evaluate the performance of three non-IRT procedures (i.e., normal approximation, Livingston-Lewis, and compound multinomial) for estimating classification indices when the observed score distribution shows atypical patterns: (a) bimodality, (b) structural (i.e., systematic) bumpiness, or (c) structural zeros (i.e., no frequencies). Under a bimodal distribution, the normal approximation procedure produced substantially large bias. For a distribution with structural bumpiness, the compound multinomial procedure tended to introduce larger bias. Under a distribution with structural zeroes, the relative performance of selected estimation procedures depended on cut score location and the sample-size conditions. In general, the differences in estimation errors among the three procedures were not substantially large.