Showing papers in "Psychometrika in 1996"
TL;DR: A large class of models, including several generalizations of stochastic block models, as well as models parameterizing global tendencies towards clustering and centralization, and individual differences in such tendencies are described and extended.
Abstract: Spanning nearly sixty years of research, statistical network analysis has passed through (at least) two generations of researchers and models. Beginning in the late 1930's, the first generation of research dealt with the distribution of various network statistics, under a variety of null models. The second generation, beginning in the 1970's and continuing into the 1980's, concerned models, usually for probabilities of relational ties among very small subsets of actors, in which various simple substantive tendencies were parameterized. Much of this research, most of which utilized log linear models, first appeared in applied statistics publications. But recent developments in social network analysis promise to bring us into a third generation. The Markov random graphs of Frank and Strauss (1986) and especially the estimation strategy for these models developed by Strauss and Ikeda (1990; described in brief in Strauss, 1992), are very recent and promising contributions to this field. Here we describe a large class of models that can be used to investigate structure in social networks. These models include several generalizations of stochastic blockmodels, as well as models parameterizing global tendencies towards clustering and centralization, and individual differences in such tendencies. Approximate model fits are obtained using Strauss and Ikeda's (1990) estimation strategy. In this paper we describe and extend these models and demonstrate how they can be used to address a variety of substantive questions about structure in social networks.
1,250 citations
TL;DR: In this article, an alternative 2SLS estimator of the parameters in LISREL type models is proposed, which allows observed and latent variables to originate from nonnormal distributions, is consistent, has a known asymptotic covariance matrix, and is estimable with standard statistical software.
Abstract: The Maximum-likelihood estimator dominates the estimation of general structural equation models. Noniterative, equation-by-equation estimators for factor analysis have received some attention, but little has been done on such estimators for latent variable equations. I propose an alternative 2SLS estimator of the parameters in LISREL type models and contrast it with the existing ones. The new 2SLS estimator allows observed and latent variables to originate from nonnormal distributions, is consistent, has a known asymptotic covariance matrix, and is estimable with standard statistical software. Diagnostics for evaluating instrumental variables are described. An empirical example illustrates the estimator.
331 citations
TL;DR: A demonstration using simulated response data illustrates that multidimensional adaptive testing can provide equal or higher reliabilities with about one-third fewer items than are required by one-dimensional adaptive testing (OAT).
Abstract: Maximum likelihood and Bayesian procedures for item selection and scoring of multidimensional adaptive tests are presented. A demonstration using simulated response data illustrates that multidimensional adaptive testing (MAT) can provide equal or higher reliabilities with about one-third fewer items than are required by one-dimensional adaptive testing (OAT). Furthermore, holding test-length constant across the MAT and OAT approaches, substantial improvements in reliability can be obtained from multidimensional assessment. A number of issues relating to the operational use of multidimensional adaptive testing are discussed.
231 citations
TL;DR: Crossing SIBTEST as mentioned in this paper is a hypothesis testing and estimation procedure for detecting crossing differences in item response theory, where the difference in the probabilities of a correct answer for the two examinee groups changes signs as ability level is varied.
Abstract: The purpose of this paper is to present a hypothesis testing and estimation procedure, Crossing SIBTEST, for detecting crossing DIF. Crossing DIF exists when the difference in the probabilities of a correct answer for the two examinee groups changes signs as ability level is varied. In item response theory terms, crossing DIF is indicated by two crossing item characteristic curves. Our new procedure, denoted as Crossing SIBTEST, first estimates the matching subtest score at which crossing occurs using least squares regression analysis. A Crossing SIBTEST statistic then is used to test the hypothesis of crossing DIF. The performance of Crossing SIBTEST is evaluated in this study.
114 citations
TL;DR: This article presents a proof of unique axis orientation for a considerably more general parallel profiles model which incorporates interacting dimensions and does not assume symmetry in the data or in the interactions among factors.
Abstract: Some existing three-way factor analysis and MDS models incorporate Cattell's “Principle of Parallel Proportional Profiles”. These models can—with appropriate data—empirically determine a unique best fitting axis orientation without the need for a separate factor rotation stage, but they have not been general enough to deal with what Tucker has called “interactions” among dimensions. This article presents a proof of unique axis orientation for a considerably more general parallel profiles model which incorporates interacting dimensions. The model, Xk=AADk HBDk B', does not assume symmetry in the data or in the interactions among factors. A second proof is presented for the symmetrically weighted case (i.e., whereADk=BDk). The generality of these models allows one to impose successive restrictions to obtain several useful special cases, including PARAFAC2 and three-way DEDICOM.
111 citations
TL;DR: In this article, the authors focus on the problem of local minima of the STRESS function and introduce the tunneling method for global minimization, and adjust it for multidimensional scaling with general Minkowski distances.
Abstract: This paper focuses on the problem of local minima of the STRESS function. It turns out that unidimensional scaling is particularly prone to local minima, whereas full dimensional scaling with Euclidean distances has a local minimum that is global. For intermediate dimensionality with Euclidean distances it depends on the dissimilarities how severe the local minimum problem is. For city-block distances in any dimensionality many different local minima are found. A simulation experiment is presented that indicates under what conditions local minima can be expected. We introduce the tunneling method for global minimization, and adjust it for multidimensional scaling with general Minkowski distances. The tunneling method alternates a local search step, in which a local minimum is sought, with a tunneling step in which a different configuration is sought with the same STRESS as the previous local minimum. In this manner successively better local minima are obtained, and experimentation so far shows that the last one is often a global minimum.
80 citations
TL;DR: In this paper, the item response model was developed on the multinomial distribution and asymptotic variances were obtained for residuals associated with response patterns and first-, and second-order marginal frequencies of manifest variables.
Abstract: Using the item response model as developed on the multinomial distribution, asymptotic variances are obtained for residuals associated with response patterns and first-, and second-order marginal frequencies of manifest variables. When the model does not fit well, an examination of these residuals may reveal the source of the poor fit. Finally, a limited-information test of fit for the model is developed by using residuals defined for the first-, and second-order marginals. Model evaluation based on residuals for these marginals is particularly useful when the response pattern frequencies are sparse.
75 citations
TL;DR: In this paper, it was shown that for polytomous items, the unweighted total score has monotone likelihood ratio (MLR) in the latent trait θ.
Abstract: In a broad class of item response theory (IRT) models for dichotomous items the unweighted total score has monotone likelihood ratio (MLR) in the latent traitθ. In this study, it is shown that for polytomous items MLR holds for the partial credit model and a trivial generalization of this model. MLR does not necessarily hold if the slopes of the item step response functions vary over items, item steps, or both. MLR holds neither for Samejima's graded response model, nor for nonparametric versions of these three polytomous models. These results are surprising in the context of Grayson's and Huynh's results on MLR for nonparametric dichotomous IRT models, and suggest that establishing stochastic ordering properties for nonparametric polytomous IRT models will be much harder.
69 citations
TL;DR: This paper derives marginal maximum likelihood (MML) estimation equations for the structural parameters of the Saltus model and suggests a computing approximation based on the EM algorithm.
Abstract: Item response theory models posit latent variables to account for regularities in students' performances on test items. Wilson's “Saltus” model extends the ideas of IRT to development that occurs in stages, where expected changes can be discontinuous, show different patterns for different types of items, or even exhibit reversals in probabilities of success on certain tasks. Examples include Piagetian stages of psychological development and Siegler's rule-based learning. This paper derives marginal maximum likelihood (MML) estimation equations for the structural parameters of the Saltus model and suggests a computing approximation based on the EM algorithm. For individual examinees, empirical Bayes probabilities of learning-stage are given, along with proficiency parameter estimates conditional on stage membership. The MML solution is illustrated with simulated data and an example from the domain of mixed number subtraction.
64 citations
TL;DR: In this paper, the sensitivity analysis of structural equation model when minor perturbation is introduced is investigated, and an influence measure that based on the general case weight perturbations is derived for the generalized least squares estimation.
Abstract: The main purpose of this paper is to investigate the sensitivity analysis of structural equation model when minor perturbation is introduced. Some influence measure that based on the general case weight perturbation is derived for the generalized least squares estimation. An influence measure that related to the Cook's distance is also developed for the special case deletion perturbation scheme. Using the proposed methodology, the influential observation in a data set can be detected. Moreover, the general theory can be applied to detect the influential parameters in a model. Finally, some illustrative artificial and real examples are presented.
62 citations
TL;DR: In this article, correspondence analysis for three-way contingency tables is presented using threeway generalisations of the singular value decomposition, and a detailed analysis is possible of the deviations from independence.
Abstract: In this paper correspondence analysis for three-way contingency tables is presented using three-way generalisations of the singular value decomposition. It is shown that in combination with Lancaster's (1951) additive decomposition of interactions in three-way tables, a detailed analysis is possible of the deviations from independence. Finally, biplots are shown to produce powerful graphical representations of the results from three-way correspondence analyses. An example from child development is used to illustrate the theoretical developments.
TL;DR: How much more needs to be clarified for routine use of dual scaling and similar quantification methods to arrive at valid conclusions is demonstrated.
Abstract: Some historical background and preliminary technical information are first presented, and then a number of hidden, but important, methodological aspects of dual scaling are illustrated and discussed: normed versus projected weights, the amount of information accounted for by each solution, a perfect solution to the problem of multidimensional unfolding, multidimensional quantification space, graphical display, number-of-option problems, option standardization versus item standardization, and asymmetry of symmetric (dual) scaling. Contrary to the common perception that dual scaling and similar quantification methods are now mathematically transparent, the present study demonstrates how much more needs to be clarified for routine use of the method to arrive at valid conclusions. Data analysis must be carried out in such a way that common sense, intuition and sound logic will prevail.
TL;DR: In this article, loglinear unidimensional and multidimensional Rasch models are considered for the analysis of repeated observations of polytomous indicators with ordered response categories, which facilitate specification of a variety of hypotheses about latent processes of change.
Abstract: Loglinear unidimensional and multidimensional Rasch models are considered for the analysis of repeated observations of polytomous indicators with ordered response categories. Reparameterizations and parameter restrictions are provided which facilitate specification of a variety of hypotheses about latent processes of change. Models of purely quantitative change in latent traits are proposed as well as models including structural change. A conditional likelihood ratio test is presented for the comparison of unidimensional and multiple scales Rasch models. In the context of longitudinal research, this renders possible the statistical test of homogeneity of change against subject-specific change in latent traits. Applications to two empirical data sets illustrate the use of the models.
TL;DR: In this paper, the uniqueness of the rank two case of the PARAFAC2 model for covariance matrices has been examined, and it has been shown that three matrices are enough to have uniqueness in the rank 2 case.
Abstract: Whereas the unique axes properties of PARAFAC1 have been examined extensively, little is known about uniqueness properties for the PARAFAC2 model for covariance matrices. This paper is concerned with uniqueness in the rank two case of PARAFAC2. For this case, Harshman and Lundy have recently shown, subject to mild assumptions, that PARAFAC2 is unique when five (covariance) matrices are analyzed. In the present paper, this result is sharpened. PARAFAC2 is shown to be usually unique with four matrices. With three matrices it is not unique unless a certain additional assumption is introduced. If, for instance, the diagonal matrices of weights are constrained to be non-negative, three matrices are enough to have uniqueness in the rank two case of PARAFAC2.
TL;DR: In this article, the authors present a matrix formulation of the Social Relations model and use the formulation to derive exact and estimated standard errors for round-robin estimates of Social Relations parameters.
Abstract: Kenny has proposed a variance-components model for dyadic social interaction. His Social Relations model estimates variances and covariances from a round-robin of two-person interactions. The current paper presents a matrix formulation of the Social Relations model. It uses the formulation to derive exact and estimated standard errors for round-robin estimates of Social Relations parameters.
TL;DR: An algorithm for the computation of the maximum likelihood and the maximum a posteriori estimates of the parameters of PMD models is presented and is a special case of a more general algorithm that can be used for the whole class of LRMs.
Abstract: In this paper, we consider a class of models for two-way matrices with binary entries of 0 and 1. First, we considerBoolean matrix decomposition, conceptualize it as alatent response model (LRM) and, by making use of this conceptualization, generalize it to a larger class of matrix decomposition models. Second,probability matrix decomposition (PMD) models are introduced as a probabilistic version of this larger class of deterministic matrix decomposition models. Third, an algorithm for the computation of the maximum likelihood (ML) and the maximum a posteriori (MAP) estimates of the parameters of PMD models is presented. This algorithm is an EM-algorithm, and is a special case of a more general algorithm that can be used for the whole class of LRMs. And fourth, as an example, a PMD model is applied to data on decision making in psychiatric diagnosis.
TL;DR: In this paper, the authors introduce dynamic latent-class models for the analysis and interpretation of stability and change in recurrent choice data, which provide a nonparametric representation of individual taste differences.
Abstract: This paper introduces dynamic latent-class models for the analysis and interpretation of stability and change in recurrent choice data. These latent-class models provide a nonparametric representation of individual taste differences. Changes in preferences are modeled by allowing for individual-level transitions from one latent class to another over time. The most general model facilitates a saturated representation of class membership changes. Several special cases are presented to obtain a parsimonious description of latent change mechanisms. An easy to implement EM algorithm is derived for parameter estimation. The approach is illustrated by a detailed analysis of a purchase incidence data set.
TL;DR: For each Rasch (Masters) partial credit item, there exists a set of independent Rasch binary and indecomposable trinary items for which the sum of the scores and the partial credit score have identical probability density functions.
Abstract: For each Rasch (Masters) partial credit item, there exists a set of independent Rasch binary and indecomposable trinary items for which the sum of the scores and the partial credit score have identical probability density functions. If each indecomposable trinary item is further expressed as the sum of two binary items, then the binary items are positively dependent and cannot be both of the Rasch type.
TL;DR: A Bayesian approach for simultaneous optimization of test-based decisions is presented using the example of a selection decision for a treatment followed by a mastery decision, showing that under mild conditions on the test score distributions and utility functions, weak rules are always compensatory by nature.
Abstract: A Bayesian approach for simultaneous optimization of test-based decisions is presented using the example of a selection decision for a treatment followed by a mastery decision. A distinction is made between weak and strong rules where, as opposed to strong rules, weak rules use prior test scores as collateral data. Conditions for monotonicity of optimal weak and strong rules are presented. It is shown that under mild conditions on the test score distributions and utility functions, weak rules are always compensatory by nature.
TL;DR: In this paper, an information index that combines both the global and local information is proposed for adaptive testing applications, and the relationship between the global information function and the currently used information functions is discussed.
Abstract: Chang and Stout (1993) presented a derivation of the asymptotic posterior normality of the latent trait given examinee responses under nonrestrictive nonparametric assumptions for dichotomous IRT models. This paper presents an extention of their results to polytomous IRT models in a fairly straightforward manner. In addition, a global information function is defined, and the relationship between the global information function and the currently used information functions is discussed. An information index that combines both the global and local information is proposed for adaptive testing applications.
TL;DR: It is shown that statistics based on the true score theory are virtually unbiased if the number of items presented to each examinee is larger than fifteen, and three types of estimators are compared: maximum likelihood, weightedmaximum likelihood, and Bayes modal.
Abstract: The quality of approximations to first and second order moments (e.g., statistics like means, variances, regression coefficients) based on latent ability estimates is being discussed. The ability estimates are obtained using either the Rasch, or the two-parameter logistic model. Straightforward use of such statistics to make inferences with respect to true latent ability is not recommended, unless we account for the fact that the basic quantities are estimates. In this paper true score theory is used to account for the latter; the counterpart of observed/true score being estimated/true latent ability. It is shown that statistics based on the true score theory are virtually unbiased if the number of items presented to each examinee is larger than fifteen. Three types of estimators are compared: maximum likelihood, weighted maximum likelihood, and Bayes modal. Furthermore, the (dis)advantages of the true score method and direct modeling of latent ability is discussed.
TL;DR: An iterative majorization algorithm is proposed which is guaranteed to converge from every starting point, and it is proven that the function value converges monotonically, and that the difference between subsequent iterates converges to zero.
Abstract: Brokken has proposed a method for orthogonal rotation of one matrix such that its columns have a maximal sum of congruences with the columns of a target matrix. This method employs an algorithm for which convergence from every starting point is not guaranteed. In the present paper, an iterative majorization algorithm is proposed which is guaranteed to converge from every starting point. Specifically, it is proven that the function value converges monotonically, and that the difference between subsequent iterates converges to zero. In addition to the better convergence properties, another advantage of the present algorithm over Brokken's one is that it is easier to program. The algorithms are compared on 80 simulated data sets, and it turned out that the new algorithm performed well in all cases, whereas Brokken's algorithm failed in almost half the cases. The derivation of the algorithm is given in full detail because it involves a series of inequalities that can be of use to derive similar algorithms in different contexts.
TL;DR: In this article, an approach based on constrained variants of components analysis (CA) is proposed where improper solutions are ruled out altogether and convergence is guaranteed, and the approach is illustrated by means of simulated data, as well as empirical data sets.
Abstract: Multitrait-Multimethod (MTMM) matrices are often analyzed by means of confirmatory factor analysis (CFA). However, fitting MTMM models often leads to improper solutions, or non-convergence. In an attempt to overcome these problems, various alternative CFA models have been proposed, but with none of these the problem of finding improper solutions was solved completely. In the present paper, an approach is proposed where improper solutions are ruled out altogether and convergence is guaranteed. The approach is based on constrained variants of components analysis (CA). Besides the fact that these methods do not give improper solutions, they have the advantage that they provide component scores which can later on be used to relate the components to external variables. The new methods are illustrated by means of simulated data, as well as empirical data sets.
TL;DR: In this paper, a stochastic multidimensional unfolding (MDU) procedure is used to spatially represent individual differences in phased or sequential decision processes, where consumers form judgments sequentially in their awareness, consideration, and choice set compositions.
Abstract: This paper presents a stochastic multidimensional unfolding (MDU) procedure to spatially represent individual differences in phased or sequential decision processes. The specific application or scenario to be discussed involves the area of consumer psychology where consumers form judgments sequentially in their awareness, consideration, and choice set compositions in a phased or sequential manner as more information about the alternative brands in a designated product/service class are collected. A brief review of the consumer psychology literature on these nested congnitive sets as stages in phased decision making is provided. The technical details of the proposed model, maximum likelihood estimation framework, and algorithm are then discussed. A small scale Monte Carlo analysis is presented to demonstrate estimation proficiency and the appropriateness of the proposed model selection heuristic. An application of the methodology to capture awareness, consideration, and choice sets in graduate school applicants is presented. Finally, directions for future research and other potential applications are given.
TL;DR: A three-step procedure is proposed to analyze data sets by generalized bilinear models, including the ability to treat response and explanatory variables differently in the models, and the incorporation of external information about the variables directly into the analysis.
Abstract: Generalized bilinear models are presented for the statistical analysis of two-way arrays. These models combine bilinear models and generalized linear modeling, and yield a family of models that includes many existing models, as well as suggest other potentially useful ones. This approach both unifies and extends models for two-way arrays, including the ability to treat response and explanatory variables differently in the models, and the incorporation of external information about the variables directly into the analysis. A unifying framework for the generalized bilinear models is provided by considering four particular cases which have been proposed and used in the existing statistical literature. A three-step procedure is proposed to analyze data sets by generalized bilinear models. Two data sets of different nature are analyzed.
TL;DR: The 3-mode association model proposed in this article is a generalization of the RC(M) association model for 3-way tables, which is useful for analyzing the relationship between the variables of a 2-way cross-classification.
Abstract: TheRC(M) association model (Goodman, 1979, 1985, 1986, 1991) is useful for analyzing the relationship between the variables of a 2-way cross-classification. The models presented here are generalizations of theRC(M) association model for 3-way tables. The family of models proposed here, “3-mode association” models, use Tucker's 3-mode components model (Tucker, 1964, 1966; Kroonenberg, 1983) to represent either the three factor interaction or the combined effects of two and three factor interactions. An example from a study in developmental psychology (Kramer & Gottman, 1992) is provided to illustrate the usefulness of the proposed models.
TL;DR: In this article, a general approach to the analysis of subjective categorical data is considered, in which agreement matrices of two or more raters are directly expressed in terms of error and agreement parameters.
Abstract: A general approach to the analysis of subjective categorical data is considered, in which agreement matrices of two or more raters are directly expressed in terms of error and agreement parameters. The method provides focused analyses of ratings from several raters for whom ratings have measurement error distributions that may induce bias in the evaluation of substantive questions of interest. Each rater's judgment process is modeled as a mixture of two components: an error variable that is unique for the rater in question as well as an agreement variable that operationalizes the “true” values of the units of observation. The statistical problems of identification, estimation, and testing of such measurement models are discussed.
TL;DR: In this article, the geometric properties of dissimilarity coefficients defined on finite sets and especially with their Euclidean nature were investigated, through the study of a one-parameter family.
Abstract: This paper is concerned with the geometric properties of dissimilarity coefficients defined on finite sets and especially with their Euclidean nature. We present several particular transformations which preserve Euclideanarity and we complete, through the study of a one-parameter family, the current knowledge of the metric and Euclidean structure of coefficients based on binary data. These results are directly deduced from two theorems which prove the positive semi-definite status of some quadratic forms which play a large role in some definitions of dissimilarity commonly used.
TL;DR: In this paper, the authors extended the multiplicative Poisson model of Rasch's multiplicative poisson model such that the parameters for individuals in the prior gamma distribution have continuous covariates.
Abstract: As a multivariate model of the number of events, Rasch's multiplicative Poisson model is extended such that the parameters for individuals in the prior gamma distribution have continuous covariates. The parameters for individuals are integrated out and the hyperparameters in the prior distribution are estimated by a numerical method separately from difficulty parameters that are treated as fixed parameters or random variables. In addition, a method is presented for estimating parameters in Rasch's model with missing values.
TL;DR: In this paper, a Taylor expansion of the equations that define the two-step estimator for polychoric correlations is used to derive the asymptotic covariance matrix for the estimated correlations.
Abstract: By using a Taylor expansion of the equations that define the two step estimator for polychoric correlations, the asymptotic covariance matrix for the estimated correlations can be derived in a simple and straightforward way.