Testing Differential Item Functioning in Small Samples

Abstract

Differential item functioning (DIF) is a pernicious statistical issue that can mask true group differences on a target latent construct. A considerable amount of research has focused on evaluating methods for testing DIF, such as using likelihood ratio tests in item response theory (IRT). Most of this research has focused on the asymptotic properties of DIF testing, in part because many latent variable methods require large samples to obtain stable parameter estimates. Much less research has evaluated these methods in small sample sizes despite the fact that many social and behavioral scientists frequently encounter small samples in practice. In this article, we examine the extent to which model complexity—the number of model parameters estimated simultaneously—affects the recovery of DIF in small samples. We compare three models that vary in complexity: logistic regression with sum scores, the 1-parameter logistic IRT model, and the 2-parameter logistic IRT model. We expected that logistic regression with sum scores and the 1-parameter logistic IRT model would more accurately estimate DIF because these models yielded more stable estimates despite being misspecified. Indeed, a simulation study and empirical example of adolescent substance use show that, even when data are generated from / assumed to be a 2-parameter logistic IRT, using parsimonious models in small samples leads to more powerful tests of DIF while adequately controlling for Type I error. We also provide evidence for minimum sample sizes needed to detect DIF, and we evaluate whether applying corrections for multiple testing is advisable. Finally, we provide recommendations for applied researchers who conduct DIF analyses in small samples.     

Modeling and Testing Differential Item Functioning in Unidimensional Binary Item Response Models with a Single Continuous Covariate: A Functional Data Analysis Approach

Differential item functioning (DIF), referring to between-group variation in item characteristics above and beyond the group-level disparity in the latent variable of interest, has long been regarded as an important item-level diagnostic. The presence of DIF impairs the fit of the single-group item response model being used, and calls for either model modification or item deletion in practice, depending on the mode of analysis. Methods for testing DIF with continuous covariates, rather than categorical grouping variables, have been developed; however, they are restrictive in parametric forms, and thus are not sufficiently flexible to describe complex interaction among latent variables and covariates. In the current study, we formulate the probability of endorsing each test item as a general bivariate function of a unidimensional latent trait and a single covariate, which is then approximated by a two-dimensional smoothing spline. The accuracy and precision of the proposed procedure is evaluated via Monte Carlo simulations. If anchor items are available, we proposed an extended model that simultaneously estimates item characteristic functions (ICFs) for anchor items, ICFs conditional on the covariate for non-anchor items, and the latent variable density conditional on the covariate—all using regression splines. A permutation DIF test is developed, and its performance is compared to the conventional parametric approach in a simulation study. We also illustrate the proposed semiparametric DIF testing procedure with an empirical example.     

认知诊断测验中基于信息矩阵的多群组DIF检验

基于改进的Wald统计量,将适用于两群组的DIF检测方法拓展至多群组的项目功能差异（DIF）检验;改进的Wald统计量将分别通过计算观察信息矩阵（Obs）和经验交叉相乘信息矩阵（XPD）而得到。模拟研究探讨了此二者与传统计算方法在多个群组下的DIF检验情况,结果表明：（1）Obs和XPD的一类错误率明显低于传统方法,DINA模型估计下Obs和XPD的一类错误率接近理论水平;（2）样本量和DIF量较大时,Obs和XPD具有与传统Wald统计量大体相同的统计检验力。     

Uncovering Substantive Patterns in Student Responses in International Large-Scale Assessments—Comparing a Latent Class to a Manifest DIF Approach

In this study, we contrast results from two differential item functioning (DIF) approaches (manifest and latent class) by the number of items and sources of items identified as DIF using data from an international reading assessment. The latter approach yielded three latent classes, presenting evidence of heterogeneity in examinee response patterns. It also yielded more DIF items with larger effect sizes and more consistent item response patterns by substantive aspects (e.g., reading comprehension processes and cognitive complexity of items). Based on our findings, we suggest empirically evaluating the homogeneity assumption in international assessments because international populations cannot be assumed to have homogeneous item response patterns. Otherwise, differences in response patterns within these populations may be under-detected when conducting manifest DIF analyses. Detecting differences in item responses across international examinee populations has implications on the generalizability and meaningfulness of DIF findings as they apply to heterogeneous examinee subgroups.     

Illustration of MIMIC-Model DIF Testing with the Schedule for Nonadaptive and Adaptive Personality

This research provides an example of testing for differential item functioning (DIF) using multiple indicator multiple cause (MIMIC) structural equation models. True/False items on five scales of the Schedule for Nonadaptive and Adaptive Personality (SNAP) were tested for uniform DIF in a sample of Air Force recruits with groups defined by gender and ethnicity. Uniform DIF exists when an item is more easily endorsed for one group than the other, controlling for group mean differences on the variable under study. Results revealed significant DIF for many SNAP items and some effects were quite large. Differentially-functioning items can produce measurement bias and should be either deleted or modeled as if separate items were administered to different groups. Future research should aim to determine whether the DIF observed here holds for other samples.     

Detecting Differential Item Functioning Using Multiple-Group Cognitive Diagnosis Models

This study proposes a multiple-group cognitive diagnosis model to account for the fact that students in different groups may use distinct attributes or use the same attributes but in different manners (e.g., conjunctive, disjunctive, and compensatory) to solve problems. Based on the proposed model, this study systematically investigates the performance of the likelihood ratio (LR) test and Wald test in detecting differential item functioning (DIF). A forward anchor item search procedure was also proposed to identify a set of anchor items with invariant item parameters across groups. Results showed that the LR and Wald tests with the forward anchor item search algorithm produced better calibrated Type I error rates than the ordinary LR and Wald tests, especially when items were of low quality. A set of real data were also analyzed to illustrate the use of these DIF detection procedures.     

Detecting differential item functioning with confirmatory factor analysis and item response theory: toward a unified strategy

In this article, the authors developed a common strategy for identifying differential item functioning (DIF) items that can be implemented in both the mean and covariance structures method (MACS) and item response theory (IRT). They proposed examining the loadings (discrimination) and the intercept (location) parameters simultaneously using the likelihood ratio test with a free-baseline model and Bonferroni corrected critical p values. They compared the relative efficacy of this approach with alternative implementations for various types and amounts of DIF, sample sizes, numbers of response categories, and amounts of impact (latent mean differences). Results indicated that the proposed strategy was considerably more effective than an alternative approach involving a constrained-baseline model. Both MACS and IRT performed similarly well in the majority of experimental conditions. As expected, MACS performed slightly worse in dichotomous conditions but better than IRT in polytomous cases where sample sizes were small. Also, contrary to popular belief, MACS performed well in conditions where DIF was simulated on item thresholds (item means), and its accuracy was not affected by impact.     

Item-focussed Trees for the Identification of Items in Differential Item Functioning

A novel method for the identification of differential item functioning (DIF) by means of recursive partitioning techniques is proposed. We assume an extension of the Rasch model that allows for DIF being induced by an arbitrary number of covariates for each item. Recursive partitioning on the item level results in one tree for each item and leads to simultaneous selection of items and variables that induce DIF. For each item, it is possible to detect groups of subjects with different item difficulties, defined by combinations of characteristics that are not pre-specified. The way a DIF item is determined by covariates is visualized in a small tree and therefore easily accessible. An algorithm is proposed that is based on permutation tests. Various simulation studies, including the comparison with traditional approaches to identify items with DIF, show the applicability and the competitive performance of the method. Two applications illustrate the usefulness and the advantages of the new method.     

Random Item IRT Models

It is common practice in IRT to consider items as fixed and persons as random. Both, continuous and categorical person parameters are most often random variables, whereas for items only continuous parameters are used and they are commonly of the fixed type, although exceptions occur. It is shown in the present article that random item parameters make sense theoretically, and that in practice the random item approach is promising to handle several issues, such as the measurement of persons, the explanation of item difficulties, and trouble shooting with respect to DIF. In correspondence with these issues, three parts are included. All three rely on the Rasch model as the simplest model to study, and the same data set is used for all applications. First, it is shown that the Rasch model with fixed persons and random items is an interesting measurement model, both, in theory, and for its goodness of fit. Second, the linear logistic test model with an error term is introduced, so that the explanation of the item difficulties based on the item properties does not need to be perfect. Finally, two more models are presented: the random item profile model (RIP) and the random item mixture model (RIM). In the RIP, DIF is not considered a discrete phenomenon, and when a robust regression approach based on the RIP difficulties is applied, quite good DIF identification results are obtained. In the RIM, no prior anchor sets are defined, but instead a latent DIF class of items is used, so that posterior anchoring is realized (anchoring based on the item mixture). It is shown that both approaches are promising for the identification of DIF.     

汉语词汇测验中的项目功能差异初探

该文运用两种不同的方法对实际的汉语词汇测验中的３６个词汇进行了ＤＩＦ探测。对于１４００多劬的初三学生分别作了男女生与城郊学生间的比较。在男女组分析中检出７个属于一致性ＤＩＦ的项目;对于城郊学生组经两种方法同时确定的ＤＩＦ项目有７个,其中５个是一致性ＤＩＦ,２个是非一致性ＤＩＦ的项目。该文还讨论了产生ＤＩＦ的可能因素。     

Examining differential item functioning due to item difficulty and alternative attractiveness

A method for analyzing test item responses is proposed to examine differential item functioning (DIF) in multiple-choice items through a combination of the usual notion of DIF, for correct/incorrect responses and information about DIF contained in each of the alternatives. The proposed method uses incomplete latent class models to examine whether DIF is caused by the attractiveness of the alternatives, difficulty of the item, or both. DIF with respect to either known or unknown subgroups can be tested by a likelihood ratio test that is asymptotically distributed as a chi-square random variable.     

Differential item functioning by sex and race in the Hogan Personality Inventory

The authors examined measurement bias in the Hogan Personality Inventory by investigating differential item functioning (DIF) across sex and two racial groups (Caucasian and Black). The sample consisted of 1,579 Caucasians (1,023 men, 556 women) and 523 Blacks (321 men, 202 women) who were applying for entry-level, unskilled jobs in factories. Although the group mean differences were trivial, more than a third of the items showed DIF by sex (38.4%) and by race (37.3%). A content analysis of potentially biased items indicated that the themes of items displaying DIF were slightly more cohesive for sex than for race. The authors discuss possible explanations for differing clustering tendencies of items displaying DIF and some practical and theoretical implications of DIF in the development and interpretation of personality inventories.     

基于多维题组反应模型的项目功能差异检验探究

本文将多维题组反应模型（MTRM）应用到多维题组测验的项目功能差异（DIF）检验中,通过模拟研究和应用研究探究MTRM在DIF检验中的准确性、有效性和影响因素,并与忽略题组效应的多维随机系数多项Logistic模型（MRCMLM）进行对比。结果表明：（1）随着样本量的增大,MTRM对有效DIF值检出率增高,错误率降低,在不同条件下结果的稳定性更高;（2）与MRCMLM相比,基于MTRM的DIF检验模型检验率更高,受到其他因素的影响更小;（3）当测验中题组效应较小时,MTRM与MRCMLM结果差异较小,但是MTRM模型拟合度更高。     

运用均数与协方差结构模型侦查项目功能差异

阐释了运用多组均数与协方差结构(MACS)模型侦查多级反应项目的一致性与非一致性项目功能差异(DIF)的原理与程序, 以道德自我概念量表DIF的侦查进行示例, 并对该方法进行了评价。与项目反应理论比照, MACS采用系统的、迭代的方式利用修正指数来侦查DIF, 并提供多个拟合指数协同评价模型拟合;与标准验证性因素分析相较, MACS不仅能侦查非一致性DIF, 而且能侦查一致性DIF。运用MACS侦查DIF是一种值得推荐的方法。     

Penalization approaches in the conditional maximum likelihood and Rasch modelling context

Recent detection methods for Differential Item Functioning (DIF) include approaches like Rasch Trees, DIFlasso, GPCMlasso and Item Focussed Trees, all of which - in contrast to well established methods - can handle metric covariates inducing DIF. A new estimation method shall address their downsides by mainly aiming at combining three central virtues: the use of conditional likelihood for estimation, the incorporation of linear influence of metric covariates on item difficulty and the possibility to detect different DIF types: certain items showing DIF, certain covariates inducing DIF, or certain covariates inducing DIF in certain items. Each of the approaches mentioned lacks in two of these aspects. We introduce a method for DIF detection, which firstly utilizes the conditional likelihood for estimation combined with group Lasso-penalization for item or variable selection and L1-penalization for interaction selection, secondly incorporates linear effects instead of approximation through step functions, and thirdly provides the possibility to investigate any of the three DIF types. The method is described theoretically, challenges in implementation are discussed. A dataset is analysed for all DIF types and shows comparable results between methods. Simulation studies per DIF type reveal competitive performance of cmlDIFlasso, particularly when selecting interactions in case of large sample sizes and numbers of parameters. Coupled with low computation times, cmlDIFlasso seems a worthwhile option for applied DIF detection.     

Differential item functioning in an international 360-degree assessment: Evidence of gender stereotype,environmental complexity,and organizational contingency

This research used logistic regression to model item responses from a popular 360-degree-for-development survey used in a leadership development programme given to middle and upper level European managers in Brussels. The survey contained 106 items on 16 scales. The model used gender of ratee and rater group to identify items that exhibited differential item functioning (DIF). The rater groups were self, boss, peer, and direct report. The sample consisted of 356 survey families where a survey family consisted of a matched set of four surveys: one self, one boss, one peer, and one direct report. The sample contained 88% male and 12% female raters. The sample contained 1424 total surveys. The procedure for flagging items exhibiting differential functioning used effect size computed from Wald chi-square statistics rather than statistical significance, resulting in fewer flagged items. One item exhibited rating anomalies due to the gender of the ratee; 55 items exhibited DIF attributable to rater group. The apparent effect of the DIF was small with each item. An examination of the maximum likelihood parameter estimates suggested the rater group DIF was the result of either hierarchical complexity or organizational contingency. The DIF due to gender conformed to prior expectations of gender-related stereotypical interpretations. This research further suggested that DIF due to environmental complexity or organizational contingency could be a naturally occurring phenomenon in some 360-degree assessment, and that the interpretation of some 360-degree feedback could need to include the potential for such DIF to exist.     

Identification of Differential Item Functioning in Multiple-Group Settings: A Multivariate Outlier Detection Approach

We focus on the identification of differential item functioning (DIF) when more than two groups of examinees are considered. We propose to consider items as elements of a multivariate space, where DIF items are outlying elements. Following this approach, the situation of multiple groups is a quite natural case. A robust statistics technique is proposed to identify DIF items as outliers in the multivariate space. For low dimensionalities, up to 2–3 groups, a simple graphical tool is derived. We illustrate our approach with a reanalysis of data from Kim, Cohen, and Park (1995) on using calculators for a mathematics test.     

The Effect of Propensity Scores on DIF Analysis: Inference on the Potential Cause of DIF

Differential item functioning (DIF) analysis is important in terms of test fairness. While DIF analyses have mainly been conducted with manifest grouping variables, such as gender or race/ethnicity, it has been recently claimed that not only the grouping variables but also contextual variables pertaining to examinees should be considered in DIF analyses. This study adopted propensity scores to incorporate the contextual variables into the gender DIF analysis. In this study, propensity scores were used to control for the contextual variables that potentially affect the gender DIF. Subsequent DIF analyses with the Mantel-Haenszel (MH) procedure and the Logistic Regression (LR) model were run with the propensity score applied reference (males) and focal groups (females) through propensity score matching. The propensity score embedded MH model and LR model detected fewer number of gender DIF than the conventional MH and LR models. The propensity score embedded models, as a confirmatory approach in DIF analysis, could contribute to hypothesizing an inference on the potential cause of DIF. Also, salient advantages of propensity score embedded DIF analysis models are discussed.     

Effects of Differential Item Functioning on Examinees' Test Performance and Reliability of Test

Simulations were conducted to examine the effect of differential item functioning (DIF) on measurement consequences such as total scores, item response theory (IRT) ability estimates, and test reliability in terms of the ratio of true-score variance to observed-score variance and the standard error of estimation for the IRT ability parameter. The objective was to provide bounds of the likely DIF effects on these measurement consequences. Five factors were manipulated: test length, percentage of DIF items per form, item type, sample size, and level of group ability difference. Results indicate that the greatest DIF effect was less than 2 points on the 0 to 60 total score scale and about 0.15 on the IRT ability scale. DIF had a limited effect on the ratio of true-score variance to observed-score variance, but its influence on the standard error of estimation for the IRT ability parameter was evident for certain ability values.