Simulation Studies of the Effects of Small Sample Size and Studied Item Parameters on SIBTEST and Mantel-Haenszel Type I Error Performance

Two simulation studies investigated Type I error performance of two statistical procedures for detecting differential item functioning (DIF): SIBTEST and Mantel-Haenszel (MH). Because MH and SIBTEST are based on asymptotic distributions requiring "large" numbers of examinees, the first study examined Type 1 error for small sample sizes. No significant Type I error inflation occurred for either procedure. Because MH has the potential for Type I error inflation for non-Rasch models, the second study used a markedly non-Rasch test and systematically varied the shape and location of the studied item. When differences in distribution across examinee group of the measured ability were present, both procedures displayed inflated Type 1 error for certain items; MH displayed the greater inflation. Also, both procedures displayed statistically biased estimation of the zero DIF for certain items, though SIBTEST displayed much less than MH. When no latent distributional differences were present, both procedures performed satisfactorily under all conditions.     

An Investigation of the Power of the Likelihood Ratio Goodness-of-Fit Statistic in Detecting Differential Item Functioning

The purpose of this study was to investigate the power and Type I error rate of the likelihood ratio goodness-of-fit (LR) statistic in detecting differential item functioning (DIF) under Samejima's (1969, 1972) graded response model. A multiple-replication Monte Carlo study was utilized in which DIF was modeled in simulated data sets which were then calibrated with MULTILOG (Thissen, 1991) using hierarchically nested item response models. In addition, the power and Type I error rate of the Mantel (1963) approach for detecting DIF in ordered response categories were investigated using the same simulated data, for comparative purposes. The power of both the Mantel and LR procedures was affected by sample size, as expected. The LR procedure lacked the power to consistently detect DIF when it existed in reference/focal groups with sample sizes as small as 500/500. The Mantel procedure maintained control of its Type I error rate and was more powerful than the LR procedure when the comparison group ability distributions were identical and there was a constant DIF pattern. On the other hand, the Mantel procedure lost control of its Type I error rate, whereas the LR procedure did not, when the comparison groups differed in mean ability; and the LR procedure demonstrated a profound power advantage over the Mantel procedure under conditions of balanced DIF in which the comparison group ability distributions were identical. The choice and subsequent use of any procedure requires a thorough understanding of the power and Type I error rates of the procedure under varying conditions of DIF pattern, comparison group ability distributions.–or as a surrogate, observed score distributions–and item characteristics.     

Transforming SIBTEST to Account for Multilevel Data Structures

SIBTEST is a differential item functioning (DIF) detection method that is accurate and effective with small samples, in the presence of group mean differences, and for assessment of both uniform and nonuniform DIF. The presence of multilevel data with DIF detection has received increased attention. Ignoring such structure can inflate Type I error. This simulation study examines the performance of newly developed multilevel adaptations of SIBTEST in the presence of multilevel data. Data were simulated in a multilevel framework and both uniform and nonuniform DIF were assessed. Study results demonstrated that naïve SIBTEST and Crossing SIBTEST, ignoring the multilevel data structure, yield inflated Type I error rates, while certain multilevel extensions provided better error and accuracy control.     

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.     

Detecting DIF for Polytomously Scored Items: An Adaptation of the SIBTEST Procedure

Shealy and Stout (1993) proposed a DIF detection procedure called SIBTEST and demonstrated its utility with both simulated and real data sets'. Current versions of SIBTEST can be used only for dichotomous items. In this article, an extension to handle polytomous items is developed. Two simulation studies are presented which compare the modified SIBTEST procedure with the Mantel and standardized mean difference (SMD) procedures. The first study compares the procedures under conditions in which the Mantel and SMD procedures have been shown to perform well (Zwick, Donoghue, & Grima, 1993). Results of Study I suggest that SIBTEST performed reasonably well, but that the Mantel and SMD procedures performed slightly better. The second study uses data simulated under conditions in which observed-score DIF methods for dichotomous items have not performed well. The results of Study 2 indicate that under these conditions the modified SIBTEST procedure provides better control of impact-induced Type I error inflation than the other procedures.     

Evaluating Type I Error and Power Rates Using an Effect Size Measure With the Logistic Regression Procedure for DIF Detection

The logistic regression (LR) procedure for differential item functioning (DIF) detection is a model-based approach designed to identify both uniform and nonuniform DIF. However, this procedure tends to produce inflated Type I errors. This outcome is problematic because it can result in the inefficient use of testing resources, and it may interfere with the study of the underlying causes of DIF. Recently, an effect size measure was developed for the LR DIF procedure and a classification method was proposed. However, the effect size measure and classification method have not been systematically investigated. In this study, we developed a new classification method based on those established for the Simultaneous Item Bias Test. A simulation study also was conducted to determine if the effect size measure affects the Type I error and power rates for the LR DIF procedure across sample sizes, ability distributions, and percentage of DIF items included on a test. The results indicate that the inclusion of the effect size measure can substantially reduce Type I error rates when large sample sizes are used, although there is also a reduction in power.     

A Comparison of the Logistic Regression and Contingency Table Methods for Simultaneous Detection of Uniform and Nonuniform DIF

In this study, we investigate the logistic regression (LR), Mantel-Haenszel (MH), and Breslow-Day (BD) procedures for the simultaneous detection of both uniform and nonuniform differential item functioning (DIF). A simulation study was used to assess and compare the Type I error rate and power of a combined decision rule (CDR), which assesses DIF using a combination of the decisions made with BD and MH to those of LR. The results revealed that while the Type I error rate of CDR was consistently below the nominal alpha level, the Type I error rate of LR was high for the conditions having unequal ability distributions. In addition, the power of CDR was consistently higher than that of LR across all forms of DIF.     

Simulated Tests of Differential Item Functioning Using SIBTEST With and Without Impact

Monte Carlo simulations with 20,000 replications are reported to estimate the probability of rejecting the null hypothesis regarding DIF using SIBTEST when there is DIF present and/or when impact is present due to differences on the primary dimension to be measured. Sample sizes are varied from 250 to 2000 and test lengths from 10 to 40 items. Results generally support previous findings for Type I error rates and power. Impact is inversely related to test length. The combination of DIF and impact, with the focal group having lower ability on both the primary and secondary dimensions, results in impact partially masking DIF so that items biased toward the reference group are less likely to be detected.     

Hierarchical Logistic Regression: Accounting for Multilevel Data in DIF Detection

The purpose of this study was to examine the performance of differential item functioning (DIF) assessment in the presence of a multilevel structure that often underlies data from large-scale testing programs. Analyses were conducted using logistic regression (LR), a popular, flexible, and effective tool for DIF detection. Data were simulated using a hierarchical framework, such as might be seen when examinees are clustered in schools, for example. Both standard and hierarchical LR (accounting for multilevel data) approaches to DIF detection were employed. Results highlight the differences in DIF detection rates when the analytic strategy matches the data structure. Specifically, when the grouping variable was within clusters, LR and HLR performed similarly in terms of Type I error control and power. However, when the grouping variable was between clusters, LR failed to maintain the nominal Type I error rate of .05. HLR was able to maintain this rate. However, power for HLR tended to be low under many conditions in the between cluster variable case.     

Differential Item Functioning Assessment in Cognitive Diagnostic Modeling: Application of the Wald Test to Investigate DIF in the DINA Model

Analyzing examinees’ responses using cognitive diagnostic models (CDMs) has the advantage of providing diagnostic information. To ensure the validity of the results from these models, differential item functioning (DIF) in CDMs needs to be investigated. In this article, the Wald test is proposed to examine DIF in the context of CDMs. This study explored the effectiveness of the Wald test in detecting both uniform and nonuniform DIF in the DINA model through a simulation study. Results of this study suggest that for relatively discriminating items, the Wald test had Type I error rates close to the nominal level. Moreover, its viability was underscored by the medium to high power rates for most investigated DIF types when DIF size was large. Furthermore, the performance of the Wald test in detecting uniform DIF was compared to that of the traditional Mantel‐Haenszel (MH) and SIBTEST procedures. The results of the comparison study showed that the Wald test was comparable to or outperformed the MH and SIBTEST procedures. Finally, the strengths and limitations of the proposed method and suggestions for future studies are discussed.     

认知诊断模型题目功能差异检验方法的健壮性比较

在认知诊断模型中进行题目功能差异(DIF)的检测,目的在于保证测验的质量与效果。在以往研究的基础上,本研究重点探索在CDMs框架下,MH、LR、CSIBTEST、WObs、WSw、WXPD 6种DIF检测方法在Q矩阵是否正确设定以及有关DIF影响因素等条件下的表现。结果表明:在Q矩阵正确设定时,WObs、WSw和WXPD统计量表现要好于MH、LR和CSIBTEST方法;在Q矩阵错误设定时,6种方法都会出现Ⅰ类错误率膨胀和统计检验力较低的现象。相对而言,MH、LR和CSIBTEST方法的表现比较稳定,WObs、WSw和WXPD统计量的表现变化较大,WObs、WSw和WXPD统计量的Ⅰ类错误率和统计检验力的结果依然好于MH、LR、CSIBTEST方法。     

Logistic Regression Procedure Using Penalized Maximum Likelihood Estimation for Differential Item Functioning

In the logistic regression (LR) procedure for differential item functioning (DIF), the parameters of LR have often been estimated using maximum likelihood (ML) estimation. However, ML estimation suffers from the finite-sample bias. Furthermore, ML estimation for LR can be substantially biased in the presence of rare event data. The bias of ML estimation due to small samples and rare event data can degrade the performance of the LR procedure, especially when testing the DIF of difficult items in small samples. Penalized ML (PML) estimation was originally developed to reduce the finite-sample bias of conventional ML estimation and also was known to reduce the bias in the estimation of LR for the rare events data. The goal of this study is to compare the performances of the LR procedures based on the ML and PML estimation in terms of the statistical power and Type I error. In a simulation study, Swaminathan and Rogers's Wald test based on PML estimation (PSR) showed the highest statistical power in most of the simulation conditions, and LRT based on conventional PML estimation (PLRT) showed the most robust and stable Type I error. The discussion about the trade-off between bias and variance is presented in the discussion section.     

Effect of Sample Size Ratio and Model Misfit When Using the Difficulty Parameter Differences Procedure to Detect DIF

This study examined the effect of sample size ratio and model misfit on the Type I error rates and power of the Difficulty Parameter Differences procedure using Winsteps. A unidimensional 30-item test with responses from 130,000 examinees was simulated and four independent variables were manipulated: sample size ratio (20/100/250/500/1000); model fit/misfit (1 PL and 3PLc =. 15 models); impact (no difference/mean differences/variance differences/mean and variance differences); and percentage of items with uniform and nonuniform DIF (0%/10%/20%). In general, the results indicate the importance of ensuring model fit to achieve greater control of Type I error and adequate statistical power. The manipulated variables produced inflated Type I error rates, which were well controlled when a measure of DIF magnitude was applied. Sample size ratio also had an effect on the power of the procedure. The paper discusses the practical implications of these results.     

Identifying differential item functioning in multi-stage computer adaptive testing

The purpose of this study is to evaluate the performance of CATSIB (Computer Adaptive Testing-Simultaneous Item Bias Test) for detecting differential item functioning (DIF) when items in the matching and studied subtest are administered adaptively in the context of a realistic multi-stage adaptive test (MST). MST was simulated using a 4-item module in a 7-panel administration. Three independent variables, expected to affect DIF detection rates, were manipulated: item difficulty, sample size, and balanced/unbalanced design. CATSIB met the acceptable criteria, meaning that the Type I error and power rates met 5% and 80%, respectively, for the large reference/moderate focal sample and the large reference/large focal sample conditions. These results indicate that CATSIB can be used to consistently and accurately detect DIF on an MST, but only with moderate to large samples.     

A Note on Three Statistical Tests in the Logistic Regression DIF Procedure

Although logistic regression became one of the well‐known methods in detecting differential item functioning (DIF), its three statistical tests, the Wald, likelihood ratio (LR), and score tests, which are readily available under the maximum likelihood, do not seem to be consistently distinguished in DIF literature. This paper provides a clarifying note on those three tests when logistic regression is applied for DIF detection.     

Detecting Individual Differences in Change: Methods and Comparisons

This study examined and compared various statistical methods for detecting individual differences in change. Considering 3 issues including test forms (specific vs. generalized), estimation procedures (constrained vs. unconstrained), and nonnormality, we evaluated 4 variance tests including the specific Wald variance test, the generalized Wald variance test, the specific likelihood ratio (LR) variance test, and the generalized LR variance test under both constrained and unconstrained estimation for both normal and nonnormal data. For the constrained estimation procedure, both the mixture distribution approach and the alpha correction approach were evaluated for their performance in dealing with the boundary problem. To deal with the nonnormality issue, we used the sandwich standard error (SE) estimator for the Wald tests and the Satorra–Bentler scaling correction for the LR tests. Simulation results revealed that testing a variance parameter and the associated covariances (generalized) had higher power than testing the variance solely (specific), unless the true covariances were zero. In addition, the variance tests under constrained estimation outperformed those under unconstrained estimation in terms of higher empirical power and better control of Type I error rates. Among all the studied tests, for both normal and nonnormal data, the robust generalized LR and Wald variance tests with the constrained estimation procedure were generally more powerful and had better Type I error rates for testing variance components than the other tests. Results from the comparisons between specific and generalized variance tests and between constrained and unconstrained estimation were discussed.     

DIF Detection Using Multiple‐Group Categorical CFA With Minimum Free Baseline Approach

The aim of this study is to assess the efficiency of using the multiple‐group categorical confirmatory factor analysis (MCCFA) and the robust chi‐square difference test in differential item functioning (DIF) detection for polytomous items under the minimum free baseline strategy. While testing for DIF items, despite the strong assumption that all but the examined item are set to be DIF‐free, MCCFA with such a constrained baseline approach is commonly used in the literature. The present study relaxes this strong assumption and adopts the minimum free baseline approach where, aside from those parameters constrained for identification purpose, parameters of all but the examined item are allowed to differ among groups. Based on the simulation results, the robust chi‐square difference test statistic with the mean and variance adjustment is shown to be efficient in detecting DIF for polytomous items in terms of the empirical power and Type I error rates. To sum up, MCCFA under the minimum free baseline strategy is useful for DIF detection for polytomous items.     

Testing Features of Graphical DIF: Application of a Regression Correction to Three Nonparametric Statistical Tests

Inspection of differential item functioning (DIF) in translated test items can be informed by graphical comparisons of item response functions (IRFs) across translated forms. Due to the many forms of DIF that can emerge in such analyses, it is important to develop statistical tests that can confirm various characteristics of DIF when present. Traditional nonparametric tests of DIF (Mantel-Haenszel, SIBTEST) are not designed to test for the presence of nonuniform or local DIF, while common probability difference (P-DIF) tests (e.g., SIBTEST) do not optimize power in testing for uniform DIF, and thus may be less useful in the context of graphical DIF analyses. In this article, modifications of three alternative nonparametric statistical tests for DIF, Fisher's χ ² test, Cochran's Z test, and Goodman's U test ( Marascuilo & Slaughter, 1981 ), are investigated for these purposes. A simulation study demonstrates the effectiveness of a regression correction procedure in improving the statistical performance of the tests when using an internal test score as the matching criterion. Simulation power and real data analyses demonstrate the unique information provided by these alternative methods compared to SIBTEST and Mantel-Haenszel in confirming various forms of DIF in translated tests.     

Simultaneous DIF Amplification and Cancellation: Shealy-Stout's Test for DIF

The present study investigates the phenomena of simultaneous DIF amplification and cancellation and SIBTEST's role in detecting such. A variety of simulated test data were generated for this purpose. In addition, real test data from various sources were analyzed. The results from both simulated and real test data, as Sheafy and Stout's theory (1993a, 1993b) suggests, show that the SIBTEST is effective in assessing DIF amplification and cancellation (partially or fully) at the test score level. Finally, methodological and substantive implications of DIF amplification and cancellation are discussed.     

第二语言学习者专业背景对HSK阅读成绩影响的项目功能差异检验

本文旨在考察HSK应试者的专业背景是否会对他们的阅读成绩产生影响。运用MH方法和SIBTEST方法对2009年HSK(初中等)考试阅读题目进行DIF筛查,把专业背景为自然科学的HSK考生设为目标组,专业背景为人文社会科学的HSK考生设为参照组。MH方法的结果是没有找到含有DIF的题目;SIBTEST方法的结果如下:第一轮DIF筛查检测到一个题目,第二轮DBF筛查检测到一组题目。这组题目有利于人文社会学科专业背景的被试。就检测DIF的方法而言,本研究认为SIBTEST方法更加敏感,DBF检验更加适合像阅读理解测验这样的一组或多组相互关联的题目。