On the Bock-Aitkin procedure—From an EM algorithm perspective

The relationship between the EM algorithm and the Bock-Aitkin procedure is described with a continuous distribution of ability (latent trait) from an EM-algorithm perspective. Previous work has been restricted to the discrete case from a probit-analysis perspective. The author is grateful to Bradley A. Hanson for valuable discussion and comments. Thanks also go to Terry A. Ackerman, Meichu Fan, Subrata Kundu, and Robert K. Tsutakawa for their help and encouragement in this study.     

Logistic positive exponent family of models: Virtue of asymmetric item characteristic curves

The paper addresses and discusses whether the tradition of accepting point-symmetric item characteristic curves is justified by uncovering the inconsistent relationship between the difficulties of items and the order of maximum likelihood estimates of ability. This inconsistency is intrinsic in models that provide point-symmetric item characteristic curves, and in this paper focus is put on the normal ogive model for observation. It is also questioned if in the logistic model the sufficient statistic has forfeited the rationale that is appropriate to the psychological reality. It is observed that the logistic model can be interpreted as the case in which the inconsistency in ordering the maximum likelihood estimates is degenerated.The paper proposes a family of models, called the logistic positive exponent family, which provides asymmetric item chacteristic curves. A model in this family has a consistent principle in ordering the maximum likelihood estimates of ability. The family is divided into two subsets each of which has its own principle, and includes the logistic model as a transition from one principle to the other. Rationale and some illustrative examples are given.     

On the loss of information in conditional maximum likelihood estimation of item parameters

In item response models of the Rasch type (Fischer & Molenaar, 1995), item parameters are often estimated by the conditional maximum likelihood (CML) method. This paper addresses the loss of information in CML estimation by using the information concept of F-information (Liang, 1983). This concept makes it possible to specify the conditions for no loss of information and to define a quantification of information loss. For the dichotomous Rasch model, the derivations will be given in detail to show the use of the F-information concept for making comparisons for different estimation methods. It is shown that by using CML for item parameter estimation, some information is almost always lost. But compared to JML (joint maximum likelihood) as well as to MML (marginal maximum likelihood) the loss is very small. The reported efficiency in the use of information of CML to JML and to MML in several comparisons is always larger than 93%, and in tests with a length of 20 items or more, larger than 99%.     

An improved stochastic EM algorithm for large-scale full-information item factor analysis

In this paper, we explore the use of the stochastic EM algorithm (Celeux & Diebolt (1985) Computational Statistics Quarterly, 2, 73) for large-scale full-information item factor analysis. Innovations have been made on its implementation, including an adaptive-rejection-based Gibbs sampler for the stochastic E step, a proximal gradient descent algorithm for the optimization in the M step, and diagnostic procedures for determining the burn-in size and the stopping of the algorithm. These developments are based on the theoretical results of Nielsen (2000, Bernoulli, 6, 457), as well as advanced sampling and optimization techniques. The proposed algorithm is computationally efficient and virtually tuning-free, making it scalable to large-scale data with many latent traits (e.g. more than five latent traits) and easy to use for practitioners. Standard errors of parameter estimation are also obtained based on the missing-information identity (Louis, 1982, Journal of the Royal Statistical Society, Series B, 44, 226). The performance of the algorithm is evaluated through simulation studies and an application to the analysis of the IPIP-NEO personality inventory. Extensions of the proposed algorithm to other latent variable models are discussed.     

A hierarchical latent response model for inferences about examinee engagement in terms of guessing and item-level non-response

In low-stakes assessments, test performance has few or no consequences for examinees themselves, so that examinees may not be fully engaged when answering the items. Instead of engaging in solution behaviour, disengaged examinees might randomly guess or generate no response at all. When ignored, examinee disengagement poses a severe threat to the validity of results obtained from low-stakes assessments. Statistical modelling approaches in educational measurement have been proposed that account for non-response or for guessing, but do not consider both types of disengaged behaviour simultaneously. We bring together research on modelling examinee engagement and research on missing values and present a hierarchical latent response model for identifying and modelling the processes associated with examinee disengagement jointly with the processes associated with engaged responses. To that end, we employ a mixture model that identifies disengagement at the item-by-examinee level by assuming different data-generating processes underlying item responses and omissions, respectively, as well as response times associated with engaged and disengaged behaviour. By modelling examinee engagement with a latent response framework, the model allows assessing how examinee engagement relates to ability and speed as well as to identify items that are likely to evoke disengaged test-taking behaviour. An illustration of the model by means of an application to real data is presented.     

多维对数正态作答时间模型：对潜在加工速度多维性的探究

在心理与教育测量中, 潜在加工速度反映学生运用潜在能力解决问题的效率。为在多维测验中探究潜在加工速度的多维性并实现参数估计, 本研究提出多维对数正态作答时间模型。实证数据分析及模拟研究结果表明：(1)潜在加工速度具有与潜在能力相匹配的多维结构; (2)新模型可精确估计个体水平的多维潜在加工速度及与作答时间有关的题目参数; (3)冗余指定潜在加工速度具有多维性带来的负面影响低于忽略其多维性所带来的。     

Stimulus affectivity of the Danish Word Association Test as measured by response heterogeneity and Rasch scaled number of prolonged reaction times

With the purpose of increasing the knowledge of the psychometric properties of the 70-item Danish Word Association Test, data from three samples of non-patients and psychiatric patients (N = 326) were used to provide two measures of affectivity of the stimulus words, response heterogeneity and reaction time prolongation. It was possible to fit an item response theory one-parameter measurement (Rasch) model to the number of reaction time prolongations (> or =3 seconds) for 54 of the stimulus words. Correlation between Rasch-model item parameters and response heterogeneity was high (r = 0.86), while no correlation was found between either of these measures and frequency of the stimulus words in the Danish language. Both measures of stimulus affectivity supported a theoretically based classification of stimulus words as emotional or neutral. Response heterogeneity measures and Rasch measurement item and person parameters for reaction time prolongations are provided.     

MCMC estimation and some model-fit analysis of multidimensional IRT models

A Bayesian procedure to estimate the three-parameter normal ogive model and a generalization of the procedure to a model with multidimensional ability parameters are presented. The procedure is a generalization of a procedure by Albert (1992) for estimating the two-parameter normal ogive model. The procedure supports analyzing data from multiple populations and incomplete designs. It is shown that restrictions can be imposed on the factor matrix for testing specific hypotheses about the ability structure. The technique is illustrated using simulated and real data. The authors would like to thank Norman Verhelst for his valuable comments and ACT, CITO group and SweSAT for the use of their data.     

Asymptotic identifiability of nonparametric item response models

The identifiability of item response models with nonparametrically specified item characteristic curves is considered. Strict identifiability is achieved, with a fixed latent trait distribution, when only a single set of item characteristic curves can possibly generate the manifest distribution of the item responses. When item characteristic curves belong to a very general class, this property cannot be achieved. However, for assessments with many items, it is shown that all models for the manifest distribution have item characteristic curves that are very near one another and pointwise differences between them converge to zero at all values of the latent trait as the number of items increases. An upper bound for the rate at which this convergence takes place is given. The main result provides theoretical support to the practice of nonparametric item response modeling, by showing that models for long assessments have the property of asymptotic identifiability. The research was partially supported by the National Institute of Health grant R01 CA81068-01.     

Identifiability of nonlinear logistic test models

The linear logistic test model (LLTM) specifies the item parameters as a weighted sum of basic parameters. The LLTM is a special case of a more general nonlinear logistic test model (NLTM) where the weights are partially unknown. This paper is about the identifiability of the NLTM. Sufficient and necessary conditions for global identifiability are presented for a NLTM where the weights are linear functions, while conditions for local identifiability are shown to require a model with less restrictions. It is also discussed how these conditions are checked using an algorithm due to Bekker, Merckens, and Wansbeek (1994). Several illustrations are given.This article was written while the first author was a post doctoral fellow at the university of Twente. He gratefully acknowledges the university's hospitality and the financial support by NWO (project nr. 30002).