The Kernel Method of Test Equating (Statistics for Social and Behavioral Sciences)

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In addition, the difficulty levels of tests at two adjacent grade levels are typically different, so the tests cannot be parallel as required for a formal equating. Most of the designs used to accomplish vertical linking do, however, involve some content that is appropriate for adjacent grade levels.

Linking , equating , and calibrating refer to a series of statistical methods for comparing scores from tests scales, measures, etc. An interest in averaging two or more equating functions can arise in various settings. As the motivation for the angle bisector method described later in this paper, Angoff mentioned situations with multiple estimates of the same linear equating function for which averaging the different estimates may be appropriate. In the nonequivalent groups with anchor test NEAT equating design, several possible linear and nonlinear equating methods are available.

It might be useful to average the results of some of the options for a final compromise method.

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In his discussion of the angle bisector, Angoff implicitly weighted the two linear functions equally. The idea of weighting the two functions differently is a natural and potentially useful added flexibility to the averaging process that we use throughout our discussion. The purpose of this chapter is to introduce the reader to some recent innovations intended to solve the problem of equating test scores on the basis of data from small numbers of test takers.

We begin with a brief description of the problem and of the techniques that psychometricians now use in attempting to deal with it. We then describe three new approaches to the problem, each dealing with a different stage of the equating process: 1 data collection, 2 estimating the equating relationship from the data collected, and 3 using collateral information to improve the estimate. We begin with Stage 2, describing a new method of estimating the equating transformation from small-sample data. We also describe the type of research studies we are using to evaluate the effectiveness of this new method.

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Then we move to Stage 3, describing some procedures for using collateral information from other equatings to improve the accuracy of an equating based on small-sample data. Finally, we turn to Stage 1, describing a new data collection plan in which the new form is introduced in a series of stages rather than all at once.

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The use of continuous distributions with positive densities facilitates the equating process, for such distributions have continuous and strictly increasing distribution functions on intervals of interest, so conversion functions can be constructed based on the principles of equipercentile equating. When the density functions are also continuous, as is the case in kernel equating, the further gain is achieved that the conversion functions are differentiable. This gain permits derivation of normal approximations for the distribution of the conversion function, so estimated asymptotic standard deviations EASDs can be derived.

In this framework, the presmoothing step is usually done with log-linear smoothing. Step 2 is to transform smoothed distribution into two marginal distributions for the target population sometimes called synthetic population. In their framework, Step 3 is done with an adjusted Gaussian kernel procedure.

  • Castilla y León (Ciudades en 48 horas nº 3) (Spanish Edition).
  • Statistical Models for Test Equating, Scaling, and Linking |
  • The Kernel Method of Test Equating (Statistics for Social Science and Behavorial Sciences).
  • Statistics for Social and Behavioral Sciences Series by Therese D. Pigott.

The need for test equating arises when two or more test forms measure the same construct and can yield different scores for the same examinee. The most common example involves multiple forms of a test within a testing program, as opposed to a single testing instrument.

In a testing program, different test forms that are similar in content and format typically contain completely different test items. Consequently, the tests can vary in difficulty depending on the degree of control available in the test development process. We introduce a Bayesian nonparametric model for test score equating, which can be applied to any of the major equating designs.

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It provides a flexible model for the continuized distribution of test scores, by means of a mixture of beta distributions with an unknown number of mixture components. Also, the model can be specified to account for dependence between score distributions from the tests to be equated.

This dependence can be accounted for even under an equivalent-groups design, where typically the questionable assumption of independence is made. Moreover, unlike the current methods of observed score equating, the Bayesian nonparametric model provides symmetric equating and always equates scores that fall within the correct range of test scores. Thus, the Bayesian model fully accounts for the uncertainty in the equated scores, for any sample size. In contrast, current approaches to test score equating only provide large-sample approximations to estimate the confidence interval of the equated score.

This Bayesian equating model is illustrated through the analysis of two data sets. The purpose of this chapter is to introduce generalized equating functions for the equating of test scores through an anchor. Depending on the choice of parameter values, the generalized equating function can perform either linear equating or equipercentile equating, either by poststratification on the anchor or by chained linking through the anchor. One of the highlights in the observed-score equating literature is a theorem by Lord in his monograph, Applications of Item Response Theory to Practical Testing Problems.

The theorem states that observed scores on two different tests cannot be equated unless the scores are perfectly reliable or the forms are strictly parallel Lord, , Chapter 13, Theorem Because the first condition is impossible and equating under the second condition is unnecessary, the theorem is rather sobering. The need for equating arises when two or more tests on the same construct or subject area can yield different scores for the same examinee.

The goal of test equating is to allow the scores on different forms of the same tests to be used and interpreted interchangeably. Using IRT in the equating process usually also requires some sort of linking procedure to place the IRT parameter estimates on a common scale. In educational testing, it is a common practice to produce test forms under a nonequivalent groups with anchor test NEAT design. In this design, the two test forms share a certain number of common items, while the populations who take the test forms might not be equivalent to each other.

Test linking is conducted to establish the equivalency of ability scales from separate item response theory IRT calibrations of two test forms. The purpose of this chapter is to have asymptotic expansions of the distributions of the estimators of linking coefficients using item response theory IRT; see, e. In IRT linking, usually item parameters are available only as their estimates. Consequently, the parameters in IRT linking, that is, linking coefficients, are subject to sampling variation. This second edition expands upon the coverage of the first edition by providing a new chapter on test scaling and a second on test linking.

Test scaling is the process of developing score scales that are used when scores on standardized tests are reported. In test linking, scores from two or more tests are related to one another.

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  8. Linking has received much recent attention, due largely to investigations of linking similarly named tests from different test publishers or tests constructed for different purposes. The expanded coverage in the second edition also includes methodology for using polytomous item response theory in equating. Michael J.