A Two-Tier Full-Information Item Factor Analysis Model with Applications

Motivated by Gibbons et al.’s (Appl. Psychol. Meas. 31:4–19, 2007) full-information maximum marginal likelihood item bifactor analysis for polytomous data, and Rijmen, Vansteelandt, and De Boeck’s (Psychometrika 73:167–182, 2008) work on constructing computationally efficient estimation algorithms for latent variable models, a two-tier item factor analysis model is developed in this research. The modeling framework subsumes standard multidimensional IRT models, bifactor IRT models, and testlet response theory models as special cases. Features of the model lead to a reduction in the dimensionality of the latent variable space, and consequently significant computational savings. An EM algorithm for full-information maximum marginal likelihood estimation is developed. Simulations and real data demonstrations confirm the accuracy and efficiency of the proposed methods. Three real data sets from a large-scale educational assessment, a longitudinal public health survey, and a scale development study measuring patient reported quality of life outcomes are analyzed as illustrations of the model’s broad range of applicability.     

双因子模型MCAT中多级评分项目选题策略的比较

双因子模型假设测验考察一个一般因子和多个组因子,符合很多教育和心理测验的因素结构。“维度缩减”方法将参数估计中多维积分计算化简为多个迭代二维积分,是双因子模型的重要特征。本文针对考察多级评分项目的计算机化自适应测验,首先推导双因子等级反应模型下Fisher信息量的计算,然后推导“维度缩减”方法在项目选择方法中的应用,最后在低、中、高双因子模式题库中比较D-优化方法、后验加权Fisher信息D优化方法(PDO)、后验加权Kullback-Leibler方法(PKL)、连续熵(CEM)和互信息(MI)方法在能力估计的相关、均方根误差、绝对值偏差和欧氏距离的表现。模拟研究表明：(1)双因子模式越强,即一般因子和组因子在项目上的区分度的差异越小,一般因子估计精度降低,组因子估计精度增加,整体能力的估计精度提高;(2)相同实验条件下,连续熵方法的测量精度最高,PKL方法的能力估计精度最低,其它方法的测量精度没有显著差异。     

Bifactor models and rotations: exploring the extent to which multidimensional data yield univocal scale scores

The application of psychological measures often results in item response data that arguably are consistent with both unidimensional (a single common factor) and multidimensional latent structures (typically caused by parcels of items that tap similar content domains). As such, structural ambiguity leads to seemingly endless "confirmatory" factor analytic studies in which the research question is whether scale scores can be interpreted as reflecting variation on a single trait. An alternative to the more commonly observed unidimensional, correlated traits, or second-order representations of a measure's latent structure is a bifactor model. Bifactor structures, however, are not well understood in the personality assessment community and thus rarely are applied. To address this, herein we (a) describe issues that arise in conceptualizing and modeling multidimensionality, (b) describe exploratory (including Schmid-Leiman [Schmid & Leiman, 1957] and target bifactor rotations) and confirmatory bifactor modeling, (c) differentiate between bifactor and second-order models, and (d) suggest contexts where bifactor analysis is particularly valuable (e.g., for evaluating the plausibility of subscales, determining the extent to which scores reflect a single variable even when the data are multidimensional, and evaluating the feasibility of applying a unidimensional item response theory (IRT) measurement model). We emphasize that the determination of dimensionality is a related but distinct question from either determining the extent to which scores reflect a single individual difference variable or determining the effect of multidimensionality on IRT item parameter estimates. Indeed, we suggest that in many contexts, multidimensional data can yield interpretable scale scores and be appropriately fitted to unidimensional IRT models.     

A general Bayesian multilevel multidimensional IRT model for locally dependent data

Many item response theory (IRT) models take a multidimensional perspective to deal with sources that induce local item dependence (LID), with these models often making an orthogonal assumption about the dimensional structure of the data. One reason for this assumption is because of the indeterminacy issue in estimating the correlations among the dimensions in structures often specified to deal with sources of LID (e.g., bifactor and two-tier structures), and the assumption usually goes untested. Unfortunately, the mere fact that assessing these correlations is a challenge for some estimation methods does not mean that data seen in practice support such orthogonal structure. In this paper, a Bayesian multilevel multidimensional IRT model for locally dependent data is presented. This model can test whether item response data violate the orthogonal assumption that many IRT models make about the dimensional structure of the data when addressing sources of LID, and this test is carried out at the dimensional level while accounting for sampling clusters. Simulations show that the model presented is effective at carrying out this task. The utility of the model is also illustrated on an empirical data set.     

Generalized Fiducial Inference for Binary Logistic Item Response Models

Generalized fiducial inference (GFI) has been proposed as an alternative to likelihood-based and Bayesian inference in mainstream statistics. Confidence intervals (CIs) can be constructed from a fiducial distribution on the parameter space in a fashion similar to those used with a Bayesian posterior distribution. However, no prior distribution needs to be specified, which renders GFI more suitable when no a priori information about model parameters is available. In the current paper, we apply GFI to a family of binary logistic item response theory models, which includes the two-parameter logistic (2PL), bifactor and exploratory item factor models as special cases. Asymptotic properties of the resulting fiducial distribution are discussed. Random draws from the fiducial distribution can be obtained by the proposed Markov chain Monte Carlo sampling algorithm. We investigate the finite-sample performance of our fiducial percentile CI and two commonly used Wald-type CIs associated with maximum likelihood (ML) estimation via Monte Carlo simulation. The use of GFI in high-dimensional exploratory item factor analysis was illustrated by the analysis of a set of the Eysenck Personality Questionnaire data.     

多维测验项目参数的估计：基于SEM与MIRT方法的比较

作者简要回顾了SEM框架下分类数据因素分析(CCFA)模型和MIRT框架下测验题目和潜在能力的关系模型, 对两种框架下的主要参数估计方法进行了总结。通过模拟研究, 比较了SEM框架下WLSc和WLSMV估计方法与MIRT框架下MLR和MCMC估计方法的差异。研究结果表明：(1) WLSc得到参数估计的偏差最大, 且存在参数收敛的问题; (2)随着样本量增大, 各种项目参数估计的精度均提高, WLSMV方法与MLR方法得到的参数估计精度差异很小, 大多数情况下不比MCMC方法差; (3)除WLSc方法外, 随着每个维度测验题目的增多参数估计的精度逐渐增高; (4)测验维度对区分度参数和难度参数的影响较大, 而测验维度对项目因素载荷和阈值的影响相对较小; (5)项目参数的估计精度受项目测量维度数的影响, 只测量一个维度的项目参数估计精度较高。另外文章还对两种方法在实际应用中应该注意的问题提供了一些建议。     

Parameter estimation of multiple item response profile model

Multiple item response profile (MIRP) models are models with crossed fixed and random effects. At least one between-person factor is crossed with at least one within-person factor, and the persons nested within the levels of the between-person factor are crossed with the items within levels of the within-person factor. Maximum likelihood estimation (MLE) of models for binary data with crossed random effects is challenging. This is because the marginal likelihood does not have a closed form, so that MLE requires numerical or Monte Carlo integration. In addition, the multidimensional structure of MIRPs makes the estimation complex. In this paper, three different estimation methods to meet these challenges are described: the Laplace approximation to the integrand; hierarchical Bayesian analysis, a simulation-based method; and an alternating imputation posterior with adaptive quadrature as the approximation to the integral. In addition, this paper discusses the advantages and disadvantages of these three estimation methods for MIRPs. The three algorithms are compared in a real data application and a simulation study was also done to compare their behaviour.     

Choice model analysis of the “pick any/n” type of binary data

In the “pick any/n” method, subjects are asked to choose any number of items from a list of n items according to some criterion. This kind of data can be analyzed as a special case of either multiple-choice data or successive categories data where the number of response categories is limited to two. An item response model was proposed for the latter case, which is a combination of an unfolding model and a choice model. The marginal maximum-likelihood estimation method was developed for parameter estimation to avoid incidental parameters, and an expectation-maximization algorithm used for numerical optimization. Two examples of analysis are given to illustrate the proposed method, which we call MAXSC.     

项目表述效应对人格测验信度和效标效度的影响——以核心自我评价量表(CSES)为例

项目表述效应是指由项目表述方式的差异引起的与测量内容无关的系统变异,项目表述效应模型的统计本质是一种双因子模型。本研究以核心自我评价量表(CSES)为例,探讨项目表述效应对人格测验信效度的影响。采用核心自我评价量表、生活满意度量表和积极情感消极情感量表对340名“蚁族”进行测查。结果表明,CSES在核心自我评价特质以外,还存在一个反向题项目表述效应因子;忽视项目表述效应对CSES的同质性信度和效标关联效度有重要影响：高估CSES的同质性信度,低估核心自我评价与生活满意度、积极情感的正相关,高估核心自我评价与消极情感的负相关。     

Full-information item bi-factor analysis

A plausibles-factor solution for many types of psychological and educational tests is one that exhibits a general factor ands − 1 group or method related factors. The bi-factor solution results from the constraint that each item has a nonzero loading on the primary dimension and at most one of thes − 1 group factors. This paper derives a bi-factor item-response model for binary response data. In marginal maximum likelihood estimation of item parameters, the bi-factor restriction leads to a major simplification of likelihood equations and (a) permits analysis of models with large numbers of group factors; (b) permits conditional dependence within identified subsets of items; and (c) provides more parsimonious factor solutions than an unrestricted full-information item factor analysis in some cases. Supported by the Cognitive Science Program, Office of Naval Research, Under grant #N00014-89-J-1104. We would like to thank Darrell Bock for several helpful suggestions.     

Simple imputation methods versus direct likelihood analysis for missing item scores in multilevel educational data

Missing data, such as item responses in multilevel data, are ubiquitous in educational research settings. Researchers in the item response theory (IRT) context have shown that ignoring such missing data can create problems in the estimation of the IRT model parameters. Consequently, several imputation methods for dealing with missing item data have been proposed and shown to be effective when applied with traditional IRT models. Additionally, a nonimputation direct likelihood analysis has been shown to be an effective tool for handling missing observations in clustered data settings. This study investigates the performance of six simple imputation methods, which have been found to be useful in other IRT contexts, versus a direct likelihood analysis, in multilevel data from educational settings. Multilevel item response data were simulated on the basis of two empirical data sets, and some of the item scores were deleted, such that they were missing either completely at random or simply at random. An explanatory IRT model was used for modeling the complete, incomplete, and imputed data sets. We showed that direct likelihood analysis of the incomplete data sets produced unbiased parameter estimates that were comparable to those from a complete data analysis. Multiple-imputation approaches of the two-way mean and corrected item mean substitution methods displayed varying degrees of effectiveness in imputing data that in turn could produce unbiased parameter estimates. The simple random imputation, adjusted random imputation, item means substitution, and regression imputation methods seemed to be less effective in imputing missing item scores in multilevel data settings.     

Gaussian variational estimation for multidimensional item response theory

Multidimensional item response theory (MIRT) is widely used in assessment and evaluation of educational and psychological tests. It models the individual response patterns by specifying a functional relationship between individuals' multiple latent traits and their responses to test items. One major challenge in parameter estimation in MIRT is that the likelihood involves intractable multidimensional integrals due to the latent variable structure. Various methods have been proposed that involve either direct numerical approximations to the integrals or Monte Carlo simulations. However, these methods are known to be computationally demanding in high dimensions and rely on sampling data points from a posterior distribution. We propose a new Gaussian variational expectation--maximization (GVEM) algorithm which adopts variational inference to approximate the intractable marginal likelihood by a computationally feasible lower bound. In addition, the proposed algorithm can be applied to assess the dimensionality of the latent traits in an exploratory analysis. Simulation studies are conducted to demonstrate the computational efficiency and estimation precision of the new GVEM algorithm compared to the popular alternative Metropolis–Hastings Robbins–Monro algorithm. In addition, theoretical results are presented to establish the consistency of the estimator from the new GVEM algorithm.     

The challenges of fitting an item response theory model to the Social Anhedonia Scale

This study explored the application of latent variable measurement models to the Social Anhedonia Scale (SAS; Eckblad, Chapman, Chapman, & Mishlove, 1982), a widely used and influential measure in schizophrenia-related research. Specifically, we applied unidimensional and bifactor item response theory (IRT) models to data from a community sample of young adults (n = 2,227). Ordinal factor analyses revealed that identifying a coherent latent structure in the 40-item SAS data was challenging due to (a) the presence of multiple small content clusters (e.g., doublets); (b) modest relations between those clusters, which, in turn, implies a general factor of only modest strength; (c) items that shared little variance with the majority of items; and (d) cross-loadings in bifactor solutions. Consequently, we conclude that SAS responses cannot be modeled accurately by either unidimensional or bifactor IRT models. Although the application of a bifactor model to a reduced 17-item set met with better success, significant psychometric and substantive problems remained. Results highlight the challenges of applying latent variable models to scales that were not originally designed to fit these models.     

Using bifactor models to identify faking on Big Five questionnaires

To identify faking, bifactor models were applied to Big Five personality data in three studies of laboratory and applicant samples using within‐subjects designs. The models were applied to homogenous data sets from separate honest, instructed faking, applicant conditions, and to simulated applicant data sets containing random individual responses from honest and faking conditions. Factor scores from the general factor in a bifactor model were found to be most highly related to response condition in both types of data sets. Domain factor scores from the faking conditions were found less affected by faking in measurement of Big Five domains than summated scale scores across studies. We conclude that bifactor models are efficacious in assessing the Big Five domains while controlling for faking.     

Exploratory Bifactor Analysis: The Schmid-Leiman Orthogonalization and Jennrich-Bentler Analytic Rotations

Analytic bifactor rotations have been recently developed and made generally available, but they are not well understood. The Jennrich-Bentler analytic bifactor rotations (bi-quartimin and bi-geomin) are an alternative to, and arguably an improvement upon, the less technically sophisticated Schmid-Leiman orthogonalization. We review the technical details that underlie the Schmid-Leiman and Jennrich-Bentler bifactor rotations, using simulated data structures to illustrate important features and limitations. For the Schmid-Leiman, we review the problem of inaccurate parameter estimates caused by the linear dependencies, sometimes called “proportionality constraints,” that are required to expand a p correlated factors solution into a (p + 1) (bi)factor space. We also review the complexities involved when the data depart from perfect cluster structure (e.g., item cross-loading on group factors). For the Jennrich-Bentler rotations, we describe problems in parameter estimation caused by departures from perfect cluster structure. In addition, we illustrate the related problems of (a) solutions that are not invariant under different starting values (i.e., local minima problems) and (b) group factors collapsing onto the general factor. Recommendations are made for substantive researchers including examining all local minima and applying multiple exploratory techniques in an effort to identify an accurate model.     

Model-Based Measures for Detecting and Quantifying Response Bias

This paper proposes a model-based family of detection and quantification statistics to evaluate response bias in item bundles of any size. Compensatory (CDRF) and non-compensatory (NCDRF) response bias measures are proposed, along with their sample realizations and large-sample variability when models are fitted using multiple-group estimation. Based on the underlying connection to item response theory estimation methodology, it is argued that these new statistics provide a powerful and flexible approach to studying response bias for categorical response data over and above methods that have previously appeared in the literature. To evaluate their practical utility, CDRF and NCDRF are compared to the closely related SIBTEST family of statistics and likelihood-based detection methods through a series of Monte Carlo simulations. Results indicate that the new statistics are more optimal effect size estimates of marginal response bias than the SIBTEST family, are competitive with a selection of likelihood-based methods when studying item-level bias, and are the most optimal when studying differential bundle and test bias.     

A Multilevel Nonlinear Profile Analysis Model for Dichotomous Data

This study linked nonlinear profile analysis (NPA) of dichotomous responses with an existing family of item response theory models and generalized latent variable models (GLVM). The NPA method offers several benefits over previous internal profile analysis methods: (a) NPA is estimated with maximum likelihood in a GLVM framework rather than relying on the choice of different dissimilarity measures that produce different results, (b) item and person parameters are computed during the same estimation step with an appropriate distribution for dichotomous variables, (c) the model estimates profile coordinate standard errors, and (d) additional individual-level variables can be included to model relationships with the profile parameters. An application examined experimental differences in topographic map comprehension among 288 subjects. The model produced a measure of overall test performance or comprehension in addition to pattern variables that measured the correspondence between subject response profiles and an item difficulty profile and an item-discrimination profile. The findings suggested that subjects who used 3-dimensional maps tended to correctly answer more items in addition to correctly answering items that were more discriminating indicators of map comprehension. The NPA analysis was also compared with results from a multidimensional item response theory model.     

Item factor analysis: current approaches and future directions

The rationale underlying factor analysis applies to continuous and categorical variables alike; however, the models and estimation methods for continuous (i.e., interval or ratio scale) data are not appropriate for item-level data that are categorical in nature. The authors provide a targeted review and synthesis of the item factor analysis (IFA) estimation literature for ordered-categorical data (e.g., Likert-type response scales) with specific attention paid to the problems of estimating models with many items and many factors. Popular IFA models and estimation methods found in the structural equation modeling and item response theory literatures are presented. Following this presentation, recent developments in the estimation of IFA parameters (e.g., Markov chain Monte Carlo) are discussed. The authors conclude with considerations for future research on IFA, simulated examples, and advice for applied researchers.     

适用于多维迫选测验的IRT计分模型

迫选(forced-choice, FC)测验由于可以控制传统李克特方法带来的反应偏差, 被广泛应用于非认知测验中, 而迫选测验的传统计分方式会产生自模式数据, 这种数据由于不适合于个体间的比较, 一直备受批评。近年来, 多种迫选IRT模型的发展使研究者能够从迫选测验中获得接近常模性的数据, 再次引起了研究者与实践人员对迫选IRT模型的兴趣。首先, 依据所采纳的决策模型和题目反应模型对6种较为主流的迫选IRT模型进行分类和介绍。然后, 从模型构建思路、参数估计方法两个角度对各模型进行比较与总结。其次, 从参数不变性检验、计算机化自适应测验(computerized adaptive testing, CAT)和效度研究3个应用研究方面进行述评。最后提出未来研究可以在模型拓展、参数不变性检验、迫选CAT测验和效度研究4个方向深入。