The Circle-Arc Method for Equating in Small Samples

This article suggests a method for estimating a test-score equating relationship from small samples of test takers. The method does not require the estimated equating transformation to be linear. Instead, it constrains the estimated equating curve to pass through two pre-specified end points and a middle point determined from the data. In a resampling study with two test forms that differed substantially in difficulty, the proposed method compared favorably with other equating methods, especially for equating scores below the 10th percentile and above the 90th percentile.     

A General Linear Method for Equating With Small Samples

Research on equating with small samples has shown that methods with stronger assumptions and fewer statistical estimates can lead to decreased error in the estimated equating function. This article introduces a new approach to linear observed‐score equating, one which provides flexible control over how form difficulty is assumed versus estimated to change across the score scale. A general linear method is presented as an extension of traditional linear methods. The general method is then compared to other linear and nonlinear methods in terms of accuracy in estimating a criterion equating function. Results from two parametric bootstrapping studies based on real data demonstrate the usefulness of the general linear method.     

Evaluating Equating Accuracy and Assumptions for Groups That Differ in Performance

Accurate equating results are essential when comparing examinee scores across exam forms. Previous research indicates that equating results may not be accurate when group differences are large. This study compared the equating results of frequency estimation, chained equipercentile, item response theory (IRT) true‐score, and IRT observed‐score equating methods. Using mixed‐format test data, equating results were evaluated for group differences ranging from 0 to .75 standard deviations. As group differences increased, equating results became increasingly biased and dissimilar across equating methods. Results suggest that the size of group differences, the likelihood that equating assumptions are violated, and the equating error associated with an equating method should be taken into consideration when choosing an equating method.     

Futility1 of Log‐Linear Smoothing when Equating with Unrepresentative Small Samples

The impact of log‐linear presmoothing on the accuracy of small sample chained equipercentile equating was evaluated under two conditions . In the first condition the small samples differed randomly in ability from the target population. In the second condition the small samples were systematically different from the target population. Results showed that equating with small samples (e.g., N < 25 or 50) using either raw or smoothed score distributions led to considerable large random equating error (although smoothing reduced random equating error). Moreover, when the small samples were not representative of the target population, the amount of equating bias also was quite large. It is concluded that although presmoothing can reduce random equating error, it is not likely to reduce equating bias caused by using an unrepresentative sample. Other alternatives to the small sample equating problem (e.g., the SiGNET design) which focus more on improving data collection are discussed.     

Collateral Information for Equating in Small Samples: A Preliminary Investigation

This article describes a preliminary investigation of an empirical Bayes (EB) procedure for using collateral information to improve equating of scores on test forms taken by small numbers of examinees. Resampling studies were done on two different forms of the same test. In each study, EB and non-EB versions of two equating methods—chained linear and chained mean—were applied to repeated small samples drawn from a large data set collected for a common-item equating. The criterion equating was the chained linear equating in the large data set. Equatings of other forms of the same test provided the collateral information. New-form sample size was varied from 10 to 200; reference-form sample size was constant at 200. One of the two new forms did not differ greatly in difficulty from its reference form, as was the case for the equatings used as collateral information. For this form, the EB procedure improved the accuracy of equating with new-form samples of 50 or fewer. The other new form was much more difficult than its reference form; for this form, the EB procedure made the equating less accurate.     

Comparison of Item Preequating and Random Groups Equating Using IRT and Equipercentile Methods

An item-preequating design and a random groups design were used to equate forms of the American College Testing (ACT) Assessment Mathematics Test. Equipercentile and 3-parameter logistic model item-response theory (IRT) procedures were used for both designs. Both pretest methods produced inadequate equating results, and the IRT item preequating method resulted in more equating error than had no equating been conducted. Although neither of the item preequating methods performed well, the results from the equipercentile preequating method were more consistent with those from the random groups method than were the results from the IRT item pretest method. Item context and position effects were likely responsible, at least in part, for the inadequate results for item preequating. Such effects need to be either controlled or modeled, and the design further researched before the item preequating design can be recommended for operational use.     

Random-Groups Equating with Samples of 50 to 400 Test Takers

Five methods for equating in a random groups design were investigated in a series of resampling studies with samples of 400, 200, 100, and 50 test takers. Six operational test forms, each taken by 9,000 or more test takers, were used as item pools to construct pairs of forms to be equated. The criterion equating was the direct equipercentile equating in the group of all test takers. Equating accuracy was indicated by the root-mean-squared deviation, over 1,000 replications, of the sample equatings from the criterion equating. The methods investigated were equipercentile equating of smoothed distributions, linear equating, mean equating, symmetric circle-arc equating, and simplified circle-arc equating. The circle-arc methods produced the most accurate results for all sample sizes investigated, particularly in the upper half of the score distribution. The difference in equating accuracy between the two circle-arc methods was negligible.     

Equating Error and Statistical Bias in Small Sample Linear Equating

A resampling study was conducted to compare the statistical bias and standard errors of nonequivalent-groups linear test equating in small samples of examinees. Sample sizes of 15, 25, 50, and 100 were examined. One thousand samples of each size were drawn with replacement from each of 5 archival data files from teacher subject area tests. For each test, data files from 2 parallel forms were used. Results suggest trivial levels of equating bias even with small samples, but substantial increases in standard errors as sample size decreases. Results were interpreted in terms of applications to testing situations in which small numbers of examinees are available.     

A New Statistic for Selecting the Smoothing Parameter for Polynomial Loglinear Equating Under the Random Groups Design

Smoothing is designed to yield smoother equating results that can reduce random equating error without introducing very much systematic error. The main objective of this study is to propose a new statistic and to compare its performance to the performance of the Akaike information criterion and likelihood ratio chi-square difference statistics in selecting the smoothing parameter for polynomial loglinear equating under the random groups design. These model selection statistics were compared for four sample sizes (500, 1,000, 2,000, and 3,000) and eight simulated equating conditions, including both conditions where equating is not needed and conditions where equating is needed. The results suggest that all model selection statistics tend to improve the equating accuracy by reducing the total equating error. The new statistic tended to have less overall error than the other two methods.     

Adjoined Piecewise Linear Approximations (APLAs) for Equating: Accuracy Evaluations of a Postsmoothing Equating Method

The purpose of this study was to evaluate the use of adjoined and piecewise linear approximations (APLAs) of raw equipercentile equating functions as a postsmoothing equating method. APLAs are less familiar than other postsmoothing equating methods (i.e., cubic splines), but their use has been described in historical equating practices of large‐scale testing programs. This study used simulations to evaluate APLA equating results and compare these results with those from cubic spline postsmoothing and from several presmoothing equating methods. The overall results suggested that APLAs based on four line segments have accuracy advantages similar to or better than cubic splines and can sometimes produce more accurate smoothed equating functions than those produced using presmoothing methods.     

Comparisons among Small Sample Equating Methods in a Common-Item Design

Score equating based on small samples of examinees is often inaccurate for the examinee populations. We conducted a series of resampling studies to investigate the accuracy of five methods of equating in a common-item design. The methods were chained equipercentile equating of smoothed distributions, chained linear equating, chained mean equating, the symmetric circle-arc method, and the simplified circle-arc method. Four operational test forms, each containing at least 110 items, were used for the equating, with new-form samples of 100, 50, 25, and 10 examinees and reference-form samples three times as large. Accuracy was described in terms of the root-mean-squared difference (over 1,000 replications) of the sample equatings from the criterion equating. Overall, chained mean equating produced the most accurate results for low scores, but the two circle-arc methods produced the most accurate results, particularly in the upper half of the score distribution. The difference in equating accuracy between the two circle-arc methods was negligible.     

Two-Level Dynamic Structural Equation Models with Small Samples

Advances in data collection have made intensive longitudinal data easier to collect, unlocking potential for methodological innovations to model such data. Dynamic structural equation modeling (DSEM) is one such methodology but recent studies have suggested that its small N performance is poor. This is problematic because small N data are omnipresent in empirical applications due to logistical and financial concerns associated with gathering many measurements on many people. In this paper, we discuss how previous studies considering small samples have focused on Bayesian methods with diffuse priors. The small sample literature has shown that diffuse priors may cause problems because they become unintentionally informative. Instead, we outline how researchers can create weakly informative admissible-range-restricted priors, even in the absence of previous studies. A simulation study shows that metrics like relative bias and non-null detection rates with these admissible-range-restricted priors improve small N properties of DSEM compared to diffuse priors.     

Standard Deviation for Small Samples

Neater representations for variance are given for small sample sizes, especially for 3 and 4. With these representations, variance can be calculated without a calculator if sample sizes are small and observations are integers, and an upper bound for the standard deviation is immediate. Accessible proofs of lower and upper bounds are presented for a broad spectrum of readers.     

基于共同题非等组设计的等值结果评价标准研究综述

国内外关于等值结果的评价标准尚无统一的认识。本文以共同题非等组设计为例,根据评价对象的不同,将等值结果的评价标准划分为两种类型:一类是用于评价等值分数的标准,一类是用于评价等值参数的标准。本文通过对两类等值结果的评价标准进行对比和梳理,提出了不同等值结果评价标准的适用范围及其局限性,以期为研究者今后选用合理的等值结果评价标准提供参考。     

Selection Strategies for Univariate Loglinear Smoothing Models and Their Effect on Equating Function Accuracy

In this study, we compared 12 statistical strategies proposed for selecting loglinear models for smoothing univariate test score distributions and for enhancing the stability of equipercentile equating functions. The major focus was on evaluating the effects of the selection strategies on equating function accuracy. Selection strategies' influence on the estimation of cumulative test score distributions was also assessed. The results of this simulation study differentiate the selection strategies and define the situations where their use has the most important implications for equating function accuracy. The recommended strategy for estimating test score distributions and for equating is AIC minimization.     

Effectiveness of Equating at the Passing Score for Exams With Small Sample Sizes

This article explores the amount of equating error at a passing score when equating scores from exams with small samples sizes. This article focuses on equating using classical test theory methods of Tucker linear, Levine linear, frequency estimation, and chained equipercentile equating. Both simulation and real data studies were used in the investigation. The results of the study supported past findings that as the sample sizes increase, the amount of bias in the equating at the passing score decreases. The research also highlights the importance for practitioners to understand the data, to have an informed expectation of the results, and to have a documented rationale for an acceptable amount of equating error.     

A Comparison of IRT Equating and Beta 4 Equating

Four equating methods (3PL true score equating, 3PL observed score equating, beta 4 true score equating, and beta 4 observed score equating) were compared using four equating criteria: first-order equity (FOE), second-order equity (SOE), conditional-mean-squared-error (CMSE) difference, and the equipercentile equating property. True score equating more closely achieved estimated FOE than observed score equating when the true score distribution was estimated using the psychometric model that was used in the equating. Observed score equating more closely achieved estimated SOE, estimated CMSE difference, and the equipercentile equating property than true score equating. Among the four equating methods, 3PL observed score equating most closely achieved estimated SOE and had the smallest estimated CMSE difference, and beta 4 observed score equating was the method that most closely met the equipercentile equating property.     

Observed Score Linear Equating with Covariates

This paper examined observed score linear equating in two different data collection designs, the equivalent groups design and the nonequivalent groups design, when information from covariates (i.e., background variables correlated with the test scores) was included. The main purpose of the study was to examine the effect (i.e., bias, variance, and mean squared error) on the estimators of including this additional information. A model for observed score linear equating with covariates first was suggested. As a second step, the model was used in a simulation study to show that the use of covariates such as gender and education can increase the accuracy of an equating by reducing the mean squared error of the estimators. Finally, data from two administrations of the Swedish Scholastic Assessment Test were used to illustrate the use of the model.     

A Comparison of the Kernel Equating Method with Traditional Equating Methods Using SAT® Data

This study applied kernel equating (KE) in two scenarios: equating to a very similar population and equating to a very different population, referred to as a distant population, using SAT^® data. The KE results were compared to the results obtained from analogous traditional equating methods in both scenarios. The results indicate that KE results are comparable to the results of other methods. Further, the results show that when the two populations taking the two tests are similar on the anchor score distributions, different equating methods yield the same or very similar results, even though they have different assumptions.     

论伦理即道德的合理性

在许多人看来伦理就是道德,道德也就是伦理,两者是同义异词,是一回事为什么会存在这样的现象呢?文章从二者研究的对象,解答的问题,对社会的意义,作用的方式四个方面探讨究竟在哪个层面上伦理可以等同于道德的问题