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%.     

多维题组效应Rasch模型

首先, 本文诠释了“题组”的本质即一个存在共同刺激的项目集合。并基于此, 将题组效应划分为项目内单维题组效应和项目内多维题组效应。其次, 本文基于Rasch模型开发了二级评分和多级评分的多维题组效应Rasch模型, 以期较好地处理项目内多维题组效应。最后, 模拟研究结果显示新模型有效合理, 与Rasch题组模型、分部评分模型对比研究后表明：(1)测验存在项目内多维题组效应时, 仅把明显的捆绑式题组效应进行分离而忽略其他潜在的题组效应, 仍会导致参数的偏差估计甚或高估测验信度; (2)新模型更具普适性, 即便当被试作答数据不存在题组效应或只存在项目内单维题组效应, 采用新模型进行测验分析也能得到较好的参数估计结果。     

Using the open-source statistical language R to analyze the dichotomous Rasch model

R, an open-source statistical language and data analysis tool, is gaining popularity among psychologists currently teaching statistics. R is especially suitable for teaching advanced topics, such as fitting the dichotomous Rasch model-a topic that involves transforming complicated mathematical formulas into statistical computations. This article describes R’s use as a teaching tool and a data analysis software program in the analysis of the Rasch model in item response theory. It also explains the theory behind, as well as an educator’s goals for, fitting the Rasch model with joint maximum likelihood estimation. This article also summarizes the R syntax for parameter estimation and the calculation of fit statistics. The results produced by R is compared with the results obtained from MINI STEP and the output of a conditional logit model. The use of R is encouraged because it is free, supported by a network of peer researchers, and covers both basic and advanced topics in statistics frequently used by psychologists.     

A dynamic generalization of the Rasch model

In the present paper a model for describing dynamic processes is constructed by combining the common Rasch model with the concept of structurally incomplete designs. This is accomplished by mapping each item on a collection of virtual items, one of which is assumed to be presented to the respondent dependent on the preceding responses and/or the feedback obtained. It is shown that, in the case of subject control, no unique conditional maximum likelihood (CML) estimates exist, whereas marginal maximum likelihood (MML) proves a suitable estimation procedure. A hierarchical family of dynamic models is presented, and it is shown how to test special cases against more general ones. Furthermore, it is shown that the model presented is a generalization of a class of mathematical learning models, known as Luce's beta-model.     

An item response model with internal restrictions on item difficulty

An IRT model based on the Rasch model is proposed for composite tasks, that is, tasks that are decomposed into subtasks of different kinds. There is one subtask for each component that is discerned in the composite tasks. A component is a generic kind of subtask of which the subtasks resulting from the decomposition are specific instantiations with respect to the particular composite tasks under study. The proposed model constrains the difficulties of the composite tasks to be linear combinations of the difficulties of the corresponding subtask items, which are estimated together with the weights used in the linear combinations, one weight for each kind of subtask. Although the model does not belong to the exponential family, its parameters can be estimated using conditional maximum likelihood estimation. The approach is demonstrated with an application to spelling tasks. We thank Eric Maris for his helpful comments.     

An approximation for the bias function of the maximum likelihood estimate of a latent variable for the general case where the item responses are discrete

Lord developed an approximation for the bias function for the maximum likelihood estimate in the context of the three-parameter logistic model. Using Taylor's expansion of the likelihood equation, he obtained an equation that includes the conditional expectation, given true ability, of the discrepancy between the maximum likelihood estimate and true ability. All terms of orders higher thann ^?1 are ignored wheren indicates the number of items. Lord assumed that all item and individual parameters are bounded, all item parameters are known or well-estimated, and the number of items is reasonably large. In the present paper, an approximation for the bias function of the maximum likelihood estimate of the latent trait, or ability, will be developed using the same assumptions for the more general case where item responses are discrete. This will include the dichotomous response level, for which the three-parameter logistic model has been discussed, the graded response level and the nominal response level. Some observations will be made for both dichotomous and graded response levels.     

Spurious Latent Class Problem in the Mixed Rasch Model: A Comparison of Three Maximum Likelihood Estimation Methods under Different Ability Distributions

Recent research has shown that over-extraction of latent classes can be observed in the Bayesian estimation of the mixed Rasch model when the distribution of ability is non-normal. This study examined the effect of non-normal ability distributions on the number of latent classes in the mixed Rasch model when estimated with maximum likelihood estimation methods (conditional, marginal, and joint). Three information criteria fit indices (Akaike information criterion, Bayesian information criterion, and sample size adjusted BIC) were used in a simulation study and an empirical study. Findings of this study showed that the spurious latent class problem was observed with marginal maximum likelihood and joint maximum likelihood estimations. However, conditional maximum likelihood estimation showed no overextraction problem with non-normal ability distributions.     

An irt-based model for dichotomous longitudinal data

The LLRA (linear logistic model with relaxed assumptions; Fischer, 1974, 1977a, 1977b, 1983a) was developed, within the framework of generalized Rasch models, for assessing change in dichotomous item score matrices between two points in time; it allows to quantify change on latent trait dimensions and to explain change in terms of treatment effects, treatment interactions, and a trend effect. A remarkable feature of the model is that unidimensionality of the item set is not required. The present paper extends this model to designs with any number of time points and even with different sets of items presented on different occasions, provided that one unidimensional subscale is available per latent trait. Thus unidimensionality assumptions within subscales are combined with multidimensionality of the item set. Conditional maximum likelihood methods for parameter estimation and hypothesis testing are developed, and a necessary and sufficient condition for unique identification of the model, given the data, is derived. Finally, a sample application is presented.To my friend Josef Roppert who has taught me how to apply statistical reasoning to substantive problems.This research was supported in part by Österreichische Forschungsgemeinschaft under grant No. 01/0054. The author wishes to thank B. Wild for the numerical computation of the sample application in section 5.     

Loss of Information in Estimating Item Parameters in Incomplete Designs

In this paper, the efficiency of conditional maximum likelihood (CML) and marginal maximum likelihood (MML) estimation of the item parameters of the Rasch model in incomplete designs is investigated. The use of the concept of F-information (Eggen, 2000) is generalized to incomplete testing designs. The scaled determinant of the F-information matrix is used as a scalar measure of information contained in a set of item parameters. In this paper, the relation between the normalization of the Rasch model and this determinant is clarified. It is shown that comparing estimation methods with the defined information efficiency is independent of the chosen normalization. The generalization of the method to other models than the Rasch model is discussed. In examples, information comparisons are conducted. It is found that for both CML and MML some information is lost in all incomplete designs compared to complete designs. A general result is that with increasing test booklet length the efficiency of an incomplete design, compared to a complete design, is increasing, as is the efficiency of CML compared to MML. The main difference between CML and MML is seen in the effect of the length of the test booklet. It will be demonstrated that with very small booklets, there is a substantial loss in information (about 35%) with CML estimation, while this loss is only about 10% in MML estimation. However, with increasing test length, the differences between CML and MML quickly disappear.     

Logistic latent trait models with linear constraints

Two linearly constrained logistic models which are based on the well-known dichotomous Rasch model, the ‘linear logistic test model’ (LLTM) and the ‘linear logistic model with relaxed assumptions’ (LLRA), are discussed. Necessary and sufficient conditions for the existence of unique conditional maximum likelihood estimates of the structural model parameters are derived. Methods for testing composite hypotheses within the framework of these models and a number of typical applications to real data are mentioned.     

A note on the computation of the second-order derivatives of the elementary symmetric functions in the rasch model

Jansen (1984) gave explicit formulas for the computation of the second-order derivatives of the elementary symmetric functions. But they are only applicable to those pairs of items which have unequal parameters. It is shown here that for equal parameters similar explicit formulas do exist, too, facilitating the application of the Newton-Raphson procedure to estimate the parameters in the Rasch model and related models according to the conditional maximum likelihood principle.The author wishes to thank the reviewers for their helpful comments on an earlier draft of this paper.     

A latent class model for rating data

A latent class model for rating data is presented which is the analogue of Andrich's binomial Rasch model for Lazarsfeld's latent class analysis (LCA). The response probabilities for the rating categories follow a binomial distribution and depend on class-specific item parameters. The EM-algorithm for parameter estimation as well as goodness of fit tests for the model are described. An example using questionnaire items on interest in physics illustrates the use of the model as an alternative to the latent trait approach of analyzing test data.I would like to thank Clifford Clogg and the anonymous reviewers for their helpful comments.     

A rasch model for continuous ratings

A unidimensional latent trait model for continuous ratings is developed. This model is an extension of Andrich's rating formulation which assumes that the response process at latent thresholds is governed by the dichotomous Rasch model. Item characteristic functions and information functions are used to illustrate that the model takes ceiling and floor effects into account. Both the dichotomous Rasch model and a linear model with normally distributed error can be derived as limiting cases. The separability of the structural and incidental parameters is demonstrated and a procedure for estimating the parameters is outlined.     

A marginal maximum likelihood method for the vector threshold model to analyze dichotomous choice data

Abstract: At least two types of models, the vector model and the unfolding model can be used for the analysis of dichotomous choice data taken from, for example, the pick any/ n method. The previous vector threshold models have a difficulty with estimation of the nuisance parameters such as the individual vectors and thresholds. This paper proposes a new probabilistic vector threshold model, where, unlike the former vector models, the angle that defines an individual vector is a random variable, and where the marginal maximum likelihood estimation method using the expectation-maximization algorithm is adopted to avoid incidental parameters. The paper also attempts to discuss which of the two models is more appropriate to account for dichotomous choice data. Two sets of dichotomous choice data are analyzed by the model.     

Marginal likelihood inference for a model for item responses and response times

Marginal maximum‐likelihood procedures for parameter estimation and testing the fit of a hierarchical model for speed and accuracy on test items are presented. The model is a composition of two first‐level models for dichotomous responses and response times along with multivariate normal models for their item and person parameters. It is shown how the item parameters can easily be estimated using Fisher's identity. To test the fit of the model, Lagrange multiplier tests of the assumptions of subpopulation invariance of the item parameters (i.e., no differential item functioning), the shape of the response functions, and three different types of conditional independence were derived. Simulation studies were used to show the feasibility of the estimation and testing procedures and to estimate the power and Type I error rate of the latter. In addition, the procedures were applied to an empirical data set from a computerized adaptive test of language comprehension.     

Sufficiency and Conditional Estimation of Person Parameters in the Polytomous Rasch Model

Rasch models are characterised by sufficient statistics for all parameters. In the Rasch unidimensional model for two ordered categories, the parameterisation of the person and item is symmetrical and it is readily established that the total scores of a person and item are sufficient statistics for their respective parameters. In contrast, in the unidimensional polytomous Rasch model for more than two ordered categories, the parameterisation is not symmetrical. Specifically, each item has a vector of item parameters, one for each category, and each person only one person parameter. In addition, different items can have different numbers of categories and, therefore, different numbers of parameters. The sufficient statistic for the parameters of an item is itself a vector. In estimating the person parameters in presently available software, these sufficient statistics are not used to condition out the item parameters. This paper derives a conditional, pairwise, pseudo-likelihood and constructs estimates of the parameters of any number of persons which are independent of all item parameters and of the maximum scores of all items. It also shows that these estimates are consistent. Although Rasch’s original work began with equating tests using test scores, and not with items of a test, the polytomous Rasch model has not been applied in this way. Operationally, this is because the current approaches, in which item parameters are estimated first, cannot handle test data where there may be many scores with zero frequencies. A small simulation study shows that, when using the estimation equations derived in this paper, such a property of the data is no impediment to the application of the model at the level of tests. This opens up the possibility of using the polytomous Rasch model directly in equating test scores.     

The derivation of some tests for the rasch model from the multinomial distribution

The present paper is concerned with testing the fit of the Rasch model. It is shown that this can be achieved by constructing functions of the data, on which model tests can be based that have power against specific model violations. It is shown that the asymptotic distribution of these tests can be derived by using the theoretical framework of testing model fit in general multinomial and product-multinomial models. The model tests are presented in two versions: one that can be used in the context of marginal maximum likelihood estimation and one that can be applied in the context of conditional maximum likelihood estimation.I am indebted to Norman Verhelst and Niels Veldhuijzen for their helpful comments. Requests for reprints should be sent to Cees A. W. Glas, Cito, PO Box 1034, 6801 MG Arnhem, THE NETHERLANDS.     

Random effects and extended generalized partial credit models

In this paper it is shown that under the random effects generalized partial credit model for the measurement of a single latent variable by a set of polytomously scored items, the joint marginal probability distribution of the item scores has a closed-form expression in terms of item category location parameters, parameters that characterize the distribution of the latent variable in the subpopulation of examinees with a zero score on all items, and item-scaling parameters. Due to this closed-form expression, all parameters of the random effects generalized partial credit model can be estimated using marginal maximum likelihood estimation without assuming a particular distribution of the latent variable in the population of examinees and without using numerical integration. Also due to this closed-form expression, new special cases of the random effects generalized partial credit model can be identified. In addition to these new special cases, a slightly more general model than the random effects generalized partial credit model is presented. This slightly more general model is called the extended generalized partial credit model. Attention is paid to maximum likelihood estimation of the parameters of the extended generalized partial credit model and to assessing the goodness of fit of the model using generalized likelihood ratio tests. Attention is also paid to person parameter estimation under the random effects generalized partial credit model. It is shown that expected a posteriori estimates can be obtained for all possible score patterns. A simulation study is carried out to show the usefulness of the proposed models compared to the standard models that assume normality of the latent variable in the population of examinees. In an empirical example, some of the procedures proposed are demonstrated.     

Loglinear representations of multivariate Bernoulli Rasch models

In this paper, the extended Rasch model for dichotomously scored items is derived from the general multivariate Bernoulli distribution. The necessary and sufficient conditions for the multivariate Bernoulli distribution to be equal to the extended Rasch model provide a new loglinear representation of the extended Rasch model. Conditions are also given under which the extended Rasch model is equal to the random effects Rasch model, and it is shown under what conditions the extended Rasch model is equal to a random effects Rasch model in which the underlying variable has a normal distribution. In addition, alternative models for the construction of likelihood ratio tests are proposed. One of these alternative models is Haberman's extended interaction model. Furthermore, it is shown how both the SPSS and SAS programs can be used to estimate and test loglinear representations of extended Rasch models.