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.     

A Mixture Model Analysis of Differential Item Functioning

Once a differential item functioning (DIF) item has been identified, little is known about the examinees for whom the item functions differentially. This is because DIF focuses on manifest group characteristics that are associated with it, but do not explain why examinees respond differentially to items. We first analyze item response patterns for gender DIF and then illustrate, through the use of a mixture item response theory (IRT) model, how the manifest characteristic associated with DIF often has a very weak relationship with the latent groups actually being advantaged or disadvantaged by the item(s). Next, we propose an alternative approach to DIF assessment that first uses an exploratory mixture model analysis to define the primary dimension(s) that contribute to DIF, and secondly studies examinee characteristics associated with those dimensions in order to understand the cause(s) of DIF. Comparison of academic characteristics of these examinees across classes reveals some clear differences in manifest characteristics between groups.     

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.     

Raju's Differential Functioning of Items and Tests (DFIT)

Nambury S. Raju (1937–2005) developed two model‐based indices for differential item functioning (DIF) during his prolific career in psychometrics. Both methods, Raju's area measures ( Raju, 1988 ) and Raju's DFIT ( Raju, van der Linden, & Fleer, 1995 ), are based on quantifying the gap between item characteristic functions (ICFs). This approach provides an intuitive and flexible methodology for assessing DIF. The purpose of this tutorial is to explain DFIT and show how this methodology can be utilized in a variety of DIF applications.     

A Generalized DIF Effect Variance Estimator for Measuring Unsigned Differential Test Functioning in Mixed Format Tests

One approach to measuring unsigned differential test functioning is to estimate the variance of the differential item functioning (DIF) effect across the items of the test. This article proposes two estimators of the DIF effect variance for tests containing dichotomous and polytomous items. The proposed estimators are direct extensions of the noniterative estimators developed by Camilli and Penfield (1997) for tests composed of dichotomous items. A small simulation study is reported in which the statistical properties of the generalized variance estimators are assessed, and guidelines are proposed for interpreting values of DIF effect variance estimators.     

Analyzing Fairness Among Linguistic Minority Populations Using a Latent Class Differential Item Functioning Approach

ABSTRACT

Differential item functioning (DIF) analyses have been used as the primary method in large-scale assessments to examine fairness for subgroups. Currently, DIF analyses are conducted utilizing manifest methods using observed characteristics (gender and race/ethnicity) for grouping examinees. Homogeneity of item responses is assumed denoting that all examinees respond to test items using a similar approach. This assumption may not hold with all groups. In this study, we demonstrate the first application of the latent class (LC) approach to investigate DIF and its sources with heterogeneous (linguistic minority groups). We found at least three LCs within each linguistic group, suggesting the need to empirically evaluate this assumption in DIF analysis. We obtained larger proportions of DIF items with larger effect sizes when LCs within language groups versus the overall (majority/minority) language groups were examined. The illustrated approach could be used to improve the ways in which DIF analyses are typically conducted to enhance DIF detection accuracy and score-based inferences when analyzing DIF with heterogeneous populations.     

A Nested Logit Approach for Investigating Distractors as Causes of Differential Item Functioning

In multiple‐choice items, differential item functioning (DIF) in the correct response may or may not be caused by differentially functioning distractors. Identifying distractors as causes of DIF can provide valuable information for potential item revision or the design of new test items. In this paper, we examine a two‐step approach based on application of a nested logit model for this purpose. The approach separates testing of differential distractor functioning (DDF) from DIF, thus allowing for clearer evaluations of where distractors may be responsible for DIF. The approach is contrasted against competing methods and evaluated in simulation and real data analyses.     

Aggregating Polytomous DIF Results Over Multiple Test Administrations

In typical differential item functioning (DIF) assessments, an item's DIF status is not influenced by its status in previous test administrations. An item that has shown DIF at multiple administrations may be treated the same way as an item that has shown DIF in only the most recent administration. Therefore, much useful information about the item's functioning is ignored. In earlier work, we developed the Bayesian updating (BU) DIF procedure for dichotomous items and showed how it could be used to formally aggregate DIF results over administrations. More recently, we extended the BU method to the case of polytomously scored items. We conducted an extensive simulation study that included four “administrations” of a test. For the single‐administration case, we compared the Bayesian approach to an existing polytomous‐DIF procedure. For the multiple‐administration case, we compared BU to two non‐Bayesian methods of aggregating the polytomous‐DIF results over administrations. We concluded that both the BU approach and a simple non‐Bayesian method show promise as methods of aggregating polytomous DIF results over administrations.     

Assessing Differential Step Functioning in Polytomous Items Using a Common Odds Ratio Estimator

Many statistics used in the assessment of differential item functioning (DIF) in polytomous items yield a single item-level index of measurement invariance that collapses information across all response options of the polytomous item. Utilizing a single item-level index of DIF can, however, be misleading if the magnitude or direction of the DIF changes across the steps underlying the polytomous response process. A more comprehensive approach to examining measurement invariance in polytomous item formats is to examine invariance at the level of each step of the polytomous item, a framework described in this article as differential step functioning (DSF). This article proposes a nonparametric DSF estimator that is based on the Mantel-Haenszel common odds ratio estimator ( Mantel & Haenszel, 1959 ), which is frequently implemented in the detection of DIF in dichotomous items. A simulation study demonstrated that when the level of DSF varied in magnitude or sign across the steps underlying the polytomous response options, the DSF-based approach typically provided a more powerful and accurate test of measurement invariance than did corresponding item-level DIF estimators.     

RIM: A Random Item Mixture Model to Detect Differential Item Functioning

In this paper we present a new methodology for detecting differential item functioning (DIF). We introduce a DIF model, called the random item mixture (RIM), that is based on a Rasch model with random item difficulties (besides the common random person abilities). In addition, a mixture model is assumed for the item difficulties such that the items may belong to one of two classes: a DIF or a non-DIF class. The crucial difference between the DIF class and the non-DIF class is that the item difficulties in the DIF class may differ according to the observed person groups while they are equal across the person groups for the items from the non-DIF class. Statistical inference for the RIM is carried out in a Bayesian framework. The performance of the RIM is evaluated using a simulation study in which it is compared with traditional procedures, like the likelihood ratio test, the Mantel-Haenszel procedure and the standardized p -DIF procedure. In this comparison, the RIM performs better than the other methods. Finally, the usefulness of the model is also demonstrated on a real life data set.     

Identifying the Causes of DIF in Translated Verbal Items

Translated tests are being used increasingly for assessing the knowledge and skills of individuals who speak different languages. There is little research exploring why translated items sometimes function differently across languages. If the sources of differential item functioning (DIF) across languages could be predicted, it could have important implications on test development, scoring and equating. This study focuses on two questions: “Is DIF related to item type?”, “What are the causes of DIF?” The data were taken from the Israeli Psychometric Entrance Test in Hebrew (source) and Russian (translated). The results indicated that 34% of the items functioned differentially across languages. The analogy items were the most problematic with 65% showing DIF, mostly in favor of the Russian-speaking examinees. The sentence completion items were also a problem (45% D1F). The main reasons for DIF were changes in word difficulty, changes in item format, differences in cultural relevance, and changes in content.     

Exploring the Utility of Background and Cognitive Variables in Explaining Latent Differential Item Functioning: An Example of the PISA 2009 Reading Assessment

Differential item functioning (DIF) may be caused by an interaction of multiple manifest grouping variables or unexplored manifest variables, which cannot be detected by conventional DIF detection methods that are based on a single manifest grouping variable. Such DIF may be detected by a latent approach using the mixture item response theory model and subsequently explained by multiple manifest variables. This study facilitates the interpretation of latent DIF with the use of background and cognitive variables. The PISA 2009 reading assessment and student survey are analyzed. Results show that members in manifest groups were not homogenously advantaged or disadvantaged and that a single manifest grouping variable did not suffice to be a proxy of latent DIF. This study also demonstrates that DIF items arising from the interaction of multiple variables can be effectively screened by the latent DIF analysis approach. Background and cognitive variables jointly well predicted latent class membership.     

An Empirical Bayes Approach to Mantel-Haenszel DIF Analysis

We developed an empirical Bayes (EB) enhancement to Mantel-Haenszel (MH) DIF analysis in which we assume that the MH statistics are normally distributed and that the prior distribution of underlying DIF parameters is also normal. We use the posterior distribution of DIF parameters to make inferences about the item's true DIF status and the posterior predictive distribution to predict the item's future observed status. DIF status is expressed in terms of the probabilities associated with each of the five DIF levels defined by the ETS classification system: C–, B–, A, B+, and C+. The EB methods yield more stable DIF estimates than do conventional methods, especially in small samples, which is advantageous in computer-adaptive testing. The EB approach may also convey information about DIF stability in a more useful way by representing the state of knowledge about an item's DIF status as probabilistic.     

An NCME Instructional Module on Using Differential Step Functioning to Refine the Analysis of DIF in Polytomous Items

Traditional methods for examining differential item functioning (DIF) in polytomously scored test items yield a single item‐level index of DIF and thus provide no information concerning which score levels are implicated in the DIF effect. To address this limitation of DIF methodology, the framework of differential step functioning (DSF) has recently been proposed, whereby measurement invariance is examined within each step underlying the polytomous response variable. The examination of DSF can provide valuable information concerning the nature of the DIF effect (i.e., is the DIF an item‐level effect or an effect isolated to specific score levels), the location of the DIF effect (i.e., precisely which score levels are manifesting the DIF effect), and the potential causes of a DIF effect (i.e., what properties of the item stem or task are potentially biasing). This article presents a didactic overview of the DSF framework and provides specific guidance and recommendations on how DSF can be used to enhance the examination of DIF in polytomous items. An example with real testing data is presented to illustrate the comprehensive information provided by a DSF analysis.     

A SIBTEST Approach to Testing DIF Hypotheses Using Experimentally Designed Test Items

This paper considers a modification of the DIF procedure SIBTEST for investigating the causes of differential item functioning (DIF). One way in which factors believed to be responsible for DIF can be investigated is by systematically manipulating them across multiple versions of an item using a randomized DIF study (Schmitt, Holland, & Dorans, 1993). In this paper: it is shown that the additivity of the index used for testing DIF in SIBTEST motivates a new extension of the method for statistically testing the effects of DIF factors. Because an important consideration is whether or not a studied DIF factor is consistent in its effects across items, a methodology for testing item x factor interactions is also presented. Using data from the mathematical sections of the Scholastic Assessment Test (SAT), the effects of two potential DIF factors—item format (multiple-choice versus open-ended) and problem type (abstract versus concrete)—are investigated for gender Results suggest a small but statistically significant and consistent effect of item format (favoring males for multiple-choice items) across items, and a larger but less consistent effect due to problem type.     

An NCME Instructional Module on Latent DIF Analysis Using Mixture Item Response Models

The purpose of this ITEMS module is to provide an introduction to differential item functioning (DIF) analysis using mixture item response models. The mixture item response models for DIF analysis involve comparing item profiles across latent groups, instead of manifest groups. First, an overview of DIF analysis based on latent groups, called latent DIF analysis, is provided and its applications in the literature are surveyed. Then, the methodological issues pertaining to latent DIF analysis are described, including mixture item response models, parameter estimation, and latent DIF detection methods. Finally, recommended steps for latent DIF analysis are illustrated using empirical data.     

Differential Item Functioning Effect Size From the Multigroup Confirmatory Factor Analysis for a Meta-Analysis: A Simulation Study

This study presents a new approach to synthesizing differential item functioning (DIF) effect size: First, using correlation matrices from each study, we perform a multigroup confirmatory factor analysis (MGCFA) that examines measurement invariance of a test item between two subgroups (i.e., focal and reference groups). Then we synthesize, across the studies, the differences in the estimated factor loadings between the two subgroups, resulting in a meta-analytic summary of the MGCFA effect sizes (MGCFA-ES). The performance of this new approach was examined using a Monte Carlo simulation, where we created 108 conditions by four factors: (1) three levels of item difficulty, (2) four magnitudes of DIF, (3) three levels of sample size, and (4) three types of correlation matrix (tetrachoric, adjusted Pearson, and Pearson). Results indicate that when MGCFA is fitted to tetrachoric correlation matrices, the meta-analytic summary of the MGCFA-ES performed best in terms of bias and mean square error values, 95% confidence interval coverages, empirical standard errors, Type I error rates, and statistical power; and reasonably well with adjusted Pearson correlation matrices. In addition, when tetrachoric correlation matrices are used, a meta-analytic summary of the MGCFA-ES performed well, particularly, under the condition that a high difficulty item with a large DIF was administered to a large sample size. Our result offers an option for synthesizing the magnitude of DIF on a flagged item across studies in practice.     

Effect Size Measures for Differential Item Functioning in a Multidimensional IRT Model

This study adapted an effect size measure used for studying differential item functioning (DIF) in unidimensional tests and extended the measure to multidimensional tests. Two effect size measures were considered in a multidimensional item response theory model: signed weighted P‐difference and unsigned weighted P‐difference. The performance of the effect size measures was investigated under various simulation conditions including different sample sizes and DIF magnitudes. As another way of studying DIF, the χ² difference test was included to compare the result of statistical significance (statistical tests) with that of practical significance (effect size measures). The adequacy of existing effect size criteria used in unidimensional tests was also evaluated. Both effect size measures worked well in estimating true effect sizes, identifying DIF types, and classifying effect size categories. Finally, a real data analysis was conducted to support the simulation results.     

Detection of Differential Item Functioning with Nonlinear Regression: A Non‐IRT Approach Accounting for Guessing

In this article we present a general approach not relying on item response theory models (non‐IRT) to detect differential item functioning (DIF) in dichotomous items with presence of guessing. The proposed nonlinear regression (NLR) procedure for DIF detection is an extension of method based on logistic regression. As a non‐IRT approach, NLR can be seen as a proxy of detection based on the three‐parameter IRT model which is a standard tool in the study field. Hence, NLR fills a logical gap in DIF detection methodology and as such is important for educational purposes. Moreover, the advantages of the NLR procedure as well as comparison to other commonly used methods are demonstrated in a simulation study. A real data analysis is offered to demonstrate practical use of the method.     

Distinguishing between Net and Global DIF in Polytomous Items

In this article, I address two competing conceptions of differential item functioning (DIF) in polytomously scored items. The first conception, referred to as net DIF, concerns between-group differences in the conditional expected value of the polytomous response variable. The second conception, referred to as global DIF, concerns the conditional dependence of group membership and the polytomous response variable. The distinction between net and global DIF is important because different DIF evaluation methods are appropriate for net and global DIF; no currently available method is universally the best for detecting both net and global DIF. Net and global DIF definitions are presented under two different, yet compatible, modeling frameworks: a traditional item response theory (IRT) framework, and a differential step functioning (DSF) framework. The theoretical relationship between the IRT and DSF frameworks is presented. Available methods for evaluating net and global DIF are described, and an applied example of net and global DIF is presented.