Showing papers in "Psychometrika in 1986"
TL;DR: In this paper, it was shown that the non-normed fit index is inversely related to sample size, and a simple alternative fit measure was proposed that removes this dependency.
Abstract: Bentler and Bonett's nonnormed fit index is a widely used measure of goodness of fit for the analysis of covariance structures. This note shows that contrary to what has been claimed the nonnormed fit index is dependent on sample size. Specifically for a constant value of a fitting function, the nonnormed index is inversely related to sample size. A simple alternative fit measure is proposed that removes this dependency. In addition, it is shown that this new measure as well as the old nonnormed fit index can be applied to any fitting function that measures the deviation of the observed covariance matrix from the covariance matrix implied by the parameter estimates for a model.
385 citations
TL;DR: The taxonomy and models for categorical item response data may usefully be organized as members of only three distinct classes, within which the models are distinguished only by assumptions and constraints on their parameters.
Abstract: A number of models for categorical item response data have been proposed in recent years. The models appear to be quite different. However, they may usefully be organized as members of only three distinct classes, within which the models are distinguished only by assumptions and constraints on their parameters. “Difference models” are appropriate for ordered responses, “divide-by-total” models may be used for either ordered or nominal responses, and “left-side added” models are used for multiple-choice responses with guessing. The details of the taxonomy and the models are described in this paper.
381 citations
TL;DR: In this article, a Bayesian framework for estimation in item response models, with two-stage prior distributions on both item and examinee populations, is described, and a general procedure based on the EM algorithm is presented.
Abstract: This article describes a Bayesian framework for estimation in item response models, with two-stage prior distributions on both item and examinee populations. Strategies for point and interval estimation are discussed, and a general procedure based on the EM algorithm is presented. Details are given for implementation under one-, two-, and three-parameter binary logistic IRT models. Novel features include minimally restrictive assumptions about examinee distributions and the exploitation of dependence among item parameters in a population of interest. Improved estimation in a moderately small sample is demonstrated with simulated data.
290 citations
TL;DR: In this paper, the reproducing kernel for the Hilbert space of functions plays a central role, and defines the best interpolating functions, which are generalized spline functions, for principal component analysis of longitudinal data.
Abstract: This paper describes a technique for principal components analysis of data consisting ofn functions each observed atp argument values. This problem arises particularly in the analysis of longitudinal data in which some behavior of a number of subjects is measured at a number of points in time. In such cases information about the behavior of one or more derivatives of the function being sampled can often be very useful, as for example in the analysis of growth or learning curves. It is shown that the use of derivative information is equivalent to a change of metric for the row space in classical principal components analysis. The reproducing kernel for the Hilbert space of functions plays a central role, and defines the best interpolating functions, which are generalized spline functions. An example is offered of how sensitivity to derivative information can reveal interesting aspects of the data.
268 citations
TL;DR: The field of linear structural equation modeling with continuous variables is reviewed in this paper, where several major conceptual achievements involving general covariance structure representations, multiple population models, and moment structures are reviewed.
Abstract: The field of linear structural equation modeling with continuous variables is reviewed. Trends in psychometric theory and data analysis across the five decades of publication ofPsychometrika are discussed, especially the clarification of concepts of population and sample, explication of the parametric structure of models, delineation of concepts of exploratory and confirmatory data analysis, expansion of statistical theory in psychometrics, estimation via optimization of an explicit objective function, and implementation of general function minimization methods. Developments in the ideas of factor analysis, latent variables, as well as structural and causal modeling are noted. Some major conceptual achievements involving general covariance structure representations, multiple population models, and moment structures are reviewed. The major statistical achievements of normal theory generalized least squares estimation, elliptical and distribution-free estimation, and higher-moment estimation are discussed. Computer programs that implement some of the theoretical developments are described.
151 citations
TL;DR: In this article, an extension of Lastovicka's four-mode components analysis is developed using a convenient notation, both a canonical and a least squares solution are derived The relation between both solutions and their computational aspects are discussed
Abstract: As an extension of Lastovicka's four-mode components analysis ann-mode components analysis is developed Using a convenient notation, both a canonical and a least squares solution are derived The relation between both solutions and their computational aspects are discussed
120 citations
TL;DR: A computer program (EXTREE) that constructs extended trees is described and applied to several sets of conceptual and perceptual proximity data.
Abstract: Proximity data can be represented by an extended tree, which generalizes traditional trees by including marked segments that correspond to overlapping clusters. An extended tree is a graphical representation of the distinctive features model. A computer program (EXTREE) that constructs extended trees is described and applied to several sets of conceptual and perceptual proximity data.
92 citations
TL;DR: In this paper, general procedures are proposed to analyze agreements and disagreements among observers in experiments where each sample of subjects is assigned to one of C categories separately by each of a fixed or varying group of observers.
Abstract: Experiments are considered where each of a sample of subjects is assigned to one of C categories separately by each of a fixed or varying group of observers. Building on earlier publications, general procedures are proposed to analyze agreements and disagreements among observers. In the case of a varying group of observers, it is shown that it is not necessary to demand a constant number of observers per subject. In the case of a fixed group of observers, the problem of missing data is considered. The procedures are illustrated within the context of two clinical diagnosis examples. In the first example it is investigated which categories are relatively hard to distinguish from one another; a new theorem is applied that shows a useful property of the statistic kappa. In the second example it is investigated if a subgroup of observers can be found with a significantly higher degree of interobserver agreement.
87 citations
TL;DR: Conjunctive item response models are introduced in this article such that sufficient statistics for latent traits are not additive in item scores; items are not necessarily locally independent; and existing compensatory (additive) item response model including the binomial, Rasch, logistic, and general locally independent model are special cases.
Abstract: Conjunctive item response models are introduced such that (a) sufficient statistics for latent traits are not necessarily additive in item scores; (b) items are not necessarily locally independent; and (c) existing compensatory (additive) item response models including the binomial, Rasch, logistic, and general locally independent model are special cases. Simple estimates and hypothesis tests for conjunctive models are introduced and evaluated as well. Conjunctive models are also identified with cognitive models that assume the existence of several individually necessary component processes for a global ability. It is concluded that conjunctive models and methods may show promise for constructing improved tests and uncovering conjunctive cognitive structure. It is also concluded that conjunctive item response theory may help to clarify the relationships between local dependence, multidimensionality, and item response function form.
83 citations
TL;DR: In this paper, item response curves for a set of binary responses are studied from a Bayesian viewpoint of estimating the item parameters, and the EM algorithm is used to compute the posterior mode.
Abstract: Item response curves for a set of binary responses are studied from a Bayesian viewpoint of estimating the item parameters. For the two-parameter logistic model with normally distributed ability, restricted bivariate beta priors are used to illustrate the computation of the posterior mode via the EM algorithm. The procedure is illustrated by data from a mathematics test.
83 citations
TL;DR: In this paper, a direct method for handling incomplete data in general covariance structural models is investigated, and asymptotic statistical properties of the generalized least squares method are developed.
Abstract: A direct method in handling incomplete data in general covariance structural models is investigated. Asymptotic statistical properties of the generalized least squares method are developed. It is shown that this approach has very close relationships with the maximum likelihood approach. Iterative procedures for obtaining the generalized least squares estimates, the maximum likelihood estimates, as well as their standard error estimates are derived. Computer programs for the confirmatory factor analysis model are implemented. A longitudinal type data set is used as an example to illustrate the results.
TL;DR: In this article, the compatibility of the polychotomous Rasch model with dichotomization of the response continuum is discussed. But it is argued that in the case of graded responses, the response categories presented to the subject are essentially an arbitrary poly-chotomization, ranging from total rejection or disagreement to total acceptance or agreement of an item or statement.
Abstract: This paper discusses thecompatibility of the polychotomous Rasch model with dichotomization of the response continuum. It is argued that in the case of graded responses, the response categories presented to the subject are essentially an arbitrary polychotomization of the response continuum, ranging for example from total rejection or disagreement to total acceptance or agreement of an item or statement. Because of this arbitrariness, the measurement outcome should be independent of the specific polychotomization applied, for example, presenting a specific multicategory response format should not affect the measurement outcome. When such is the case, the original polychotomous model is called “compatible” with dichotomization.
TL;DR: In this article, the authors present eleven statistical procedures which test the equality of m-coefficient alphas when the sample alpha coefficients are dependent, and evaluate the accuracy of the procedures for sample sizes of 50, 100, and 200.
Abstract: In a variety of measurement situations, the researcher may wish to compare the reliabilities of several instruments administered to the same sample of subjects. This paper presents eleven statistical procedures which test the equality ofm coefficient alphas when the sample alpha coefficients are dependent. Several of the procedures are derived in detail, and numerical examples are given for two. Since all of the procedures depend on approximate asymptotic results, Monte Carlo methods are used to assess the accuracy of the procedures for sample sizes of 50, 100, and 200. Both control of Type I error and power are evaluated by computer simulation. Two of the procedures are unable to control Type I errors satisfactorily. The remaining nine procedures perform properly, but three are somewhat superior in power and Type I error control.
TL;DR: In this paper, the authors compare the corresponding estimated latent distributions obtained using the scaling model applied to the different groups, and compare the estimated item reliabilities (or item response error rates) for the different group, and test whether a scaling model applying to the several groups can be replaced by a more parsimonious scaling model that includes various homogeneity constraints.
Abstract: Statistical methods are presented to facilitate a more complete analysis of results obtained when a scaling model is applied to data from two or more groups. These methods can be used to (a) compare the corresponding estimated latent distributions obtained using the scaling model applied to the different groups, (b) compare the corresponding estimated item reliabilities (or item response error rates) for the different groups, and (c) test whether the scaling model applied to the several groups can be replaced by a more parsimonious scaling model that includes various homogeneity constraints (i.e., constraints that describe which parameters in the model are the same for the several groups). Various kinds of scaling models are considered here in the multiple-group context.
TL;DR: The Thurstonians' emphasis on the development of factor analysis as an exploratory methodology was not new with them but was taken from British statisticians and psychologists who preceded them as discussed by the authors.
Abstract: Papers on factor analysis appearing inPsychometrika reflect the initial efforts of the Thurstonians to reformulate psychology as a quantitative science. The Thurstonians' emphasis on the development of factor analysis as an exploratory methodology was not new with them but was taken from British statisticians and psychologists who preceded them, whose literature the Thurstonians otherwise tended to ignore. The Thurstonians' rejection of general factors and focus on rotation to simple structure reflected an attempt to avoid statistical artifact and to identify factors with psychological substance. Much of the literature on factor analysis inPsychometrika concerned solving technical problems in the exploratory factor analysis method. Factor analysis took a major shift in direction in the 1970's with the development of confirmatory methodologies, many of which now receive greater attention than the method of exploratory factor analysis, most of the problems of which are now resolved.
TL;DR: In this article, a closed form estimator of the uniqueness (unique variance) in factor analysis is proposed, which has analytically desirable properties such as consistency, asymptotic normality and scale invariance.
Abstract: A closed form estimator of the uniqueness (unique variance) in factor analysis is proposed. It has analytically desirable properties—consistency, asymptotic normality and scale invariance. The estimation procedure is given through the application to the two sets of Emmett's data and Holzinger and Swineford's data. The new estimator is shown to lead to values rather close to the maximum likelihood estimator.
TL;DR: In this paper, a bimatrix structure for examining ordinal partial rankings is presented, and a set of axioms is given similar to those of Kemeny and Snell (1962) and Bogart (1973), which uniquely determines the distance between any pair of such rankings.
Abstract: This paper presents a bimatrix structure for examining ordinal partial rankings. A set of axioms is given similar to those of Kemeny and Snell (1962) and Bogart (1973), which uniquely determines the distance between any pair of such rankings. Thel
1 norm is shown to satisfy this set of axioms, and to be equivalent to the Kemeny and Snell distance on their subspace of weak orderings. Consensus formation is discussed.
TL;DR: In this paper, a multivariate normal model with one of the component variables observable only in polytomous form is considered and the maximum likelihood approach is used for estimation of the parameters in the model.
Abstract: This paper considers a multivariate normal model with one of the component variables observable only in polytomous form. The maximum likelihood approach is used for estimation of the parameters in the model. The Newton-Raphson algorithm is implemented to obtain the solution of the problem. Examples based on real and simulated data are reported.
TL;DR: In this paper, an alternating procedure is proposed which fixes the row or column configuration in turn and finds the global optimum of the objective criterion with respect to the free parameters, iterating in this fashion until convergence is reached.
Abstract: We consider the problem of least-squares fitting of squared distances in unfolding. An alternating procedure is proposed which fixes the row or column configuration in turn and finds the global optimum of the objective criterion with respect to the free parameters, iterating in this fashion until convergence is reached. A considerable simplification in the algorithm results, namely that this conditional global optimum is identified by performing a single unidimensional search for each point, irrespective of the dimensionality of the unfolding solution.
TL;DR: A chance-corrected version of the family of association coefficients for metric scales proposed by Zegers and ten Berge is presented in this article, where it is shown that a matrix with chancecorrected coefficients between a number of variables is Gramian.
Abstract: A chance-corrected version of the family of association coefficients for metric scales proposed by Zegers and ten Berge is presented. It is shown that a matrix with chance-corrected coefficients between a number of variables is Gramian. The members of the chance-corrected family are shown to be partially ordered.
TL;DR: It is concluded that (a) research in test theory is in a healthy state and (b)Psychometrika is an important source of information about that research.
Abstract: On the occasion ofPsychometrika's fiftieth anniversary, the past twenty-five years' developments in mental test theory are reviewed, with special emphasis on the topics receiving attention in the pages of this journal. (Analogous reviews forPsychometrika's first quarter century were given by Gulliksen and Guilford in 1961.) Much of the recent progress in test theory (and in other branches of psychometrics as well) has been made by treating the problems in this field as being essentially ones of statistical inference. It is concluded that (a) research in test theory is in a healthy state and (b)Psychometrika is an important source of information about that research.
TL;DR: In this article, the identifiability and estimation of a general dynamic structural model under indirect observation is considered from a system theoretic perspective, using concepts and methods from system theory, such as the observability and controllability concept, the (steady-state) Kalman filter and a general nonlinear ML_estimation procedure known as prediction error estimation.
Abstract: In this—partly—expository paper the parameter identifiability and estimation of a general dynamic structural model under indirect observation will be considered from a system theoretic perspective. The general dynamic model covers (dynamic) factor analytic models, (dynamic) MIMIC models and Joreskog's linear structural model as special cases. Its reduced form is—under a slightly different specification—known in system theory and econometrics as the stochastic, stationary version of the state-space model. By using concepts and methods from system theory, such as the observability and controllability concept, the (steady-state) Kalman filter and a general nonlinear ML_estimation procedure known as prediction-error estimation the general dynamic model will be identified. It will be shown that Joreskog's LISREL-procedure is a special case of the prediction-error estimation procedure.
TL;DR: In this article, it was shown that problems of rotational equivalence of restricted factor loading matrices in orthogonal factor analysis are equivalent to problems of identification in simultaneous equations systems with covariance restrictions.
Abstract: It is shown that problems of rotational equivalence of restricted factor loading matrices in orthogonal factor analysis are equivalent to problems of identification in simultaneous equations systems with covariance restrictions. A necessary (under a regularity assumption) and sufficient condition for local uniqueness is given and a counterexample is provided to a theorem by J. Algina concerning necessary and sufficient conditions for global uniqueness.
TL;DR: In this paper, a test for the equality of the variances of k ≥ 2 correlated variables is proposed, which is valid for any correlation structure between the k normally distributed variables and can be extended to eliminate nuisance parameters by a bootstrap procedure.
Abstract: A test is proposed for the equality of the variances ofk ≥ 2 correlated variables. Pitman's test fork = 2 reduces the null hypothesis to zero correlation between their sum and their difference. Its extension, eliminating nuisance parameters by a bootstrap procedure, is valid for any correlation structure between thek normally distributed variables. A Monte Carlo study for several combinations of sample sizes and number of variables is presented, comparing the level and power of the new method with previously published tests. Some nonnormal data are included, for which the empirical level tends to be slightly higher than the nominal one. The results show that our method is close in power to the asymptotic tests which are extremely sensitive to nonnormality, yet it is robust and much more powerful than other robust tests.
TL;DR: Browne et al. as discussed by the authors presented alternative models for the perfect circumplex and quasi-circumplex that avoid these difficulties, and that includes the important model for a patterned correlation matrix.
Abstract: Joreskog (1974) developed a latent variable model for the covariance structure of the circumplex which, under certain conditions, includes a model for a patterned correlation matrix (Browne, 1977). This model is of limited usefulness, however, in that it employs a known matrix that is rank deficient for many problems. Furthermore, the model is inappropriate for the circumplex which contains negative covariances. This paper presents alternative models for the perfect circumplex and quasi-circumplex that avoids these difficulties, and that includes the important model for a patterned correlation circumplex matrix. Two numerical examples are provided.
TL;DR: A statistical model for interpreting psychological scaling research, based on the heuristic work of Reynolds (1983), is developed, which has certain advantages over the standard property fitting approach currently used to interpret multidimensional scaling spaces.
Abstract: A statistical model for interpreting psychological scaling research, based on the heuristic work of Reynolds (1983), is developed. This new approach has certain advantages over the standard property fitting approach (Chang and Carroll, 1969) currently used to interpret multidimensional scaling spaces (Shepard, 1962; Torgerson, 1965). These advantages are (a) the ability to directly assess the correspondence of a descriptor vector(s) to a symmetric matrix, and (b) to provide a method in which only ordinal properties of such descriptors are required: thus standard rating, ranking, or sorting data collection methods can be used as the basis to interpret the multidimensional space resulting from the distance data.
TL;DR: In this article, it is shown that for equal parameters similar explicit formulas do exist, 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.
Abstract: 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.
TL;DR: This article showed that if the decision to respond is correlated with a substantive variable of interest, then regression or analysis of variance methods based upon the questionnaire results may be adversely affected by self-selection bias.
Abstract: It is commonly held that even where questionnaire response is poor, correlational studies are affected only by loss of degrees of freedom or precision. We show that this supposition is not true. If the decision to respond is correlated with a substantive variable of interest, then regression or analysis of variance methods based upon the questionnaire results may be adversely affected by self-selection bias. Moreover such bias may arise even where response is 100%. The problem in both cases arises where selection information is passed to the score indirectly via the disturbance or individual effects, rather than entirely via the observable explanatory variables. We suggest tests for the ensuing self-selection bias and possible ways of handling the ensuing problems of inference.
TL;DR: In this article, an algorithm for assessing additivity conjunctively via both axiomatic conjoint analysis and numerical conjoint scaling is described, which first assesses the degree of individual differences among sets of rankings of stimuli, and subsequently examines either individual or averaged data for violations of axioms necessary for an additive model.
Abstract: An algorithm for assessing additivity conjunctively via both axiomatic conjoint analysis and numerical conjoint scaling is described. The algorithm first assesses the degree of individual differences among sets of rankings of stimuli, and subsequently examines either individual or averaged data for violations of axioms necessary for an additive model. The axioms are examined at a more detailed level than has been previously done. Violations of the axioms are broken down into different types. Finally, a nonmetric scaling of the data can be done based on either or both of two different badness-of-fit scaling measures. The advantages of combining all of these features into one algorithm for improving the diagnostic value of axiomatic conjoint measurement in evaluating additivity are discussed.
TL;DR: The scoring algorithm for maximum likelihood estimation in exploratory factor analysis can be developed in a way that is many times more efficient than a direct development based on information matrices and score vectors.
Abstract: It is shown that the scoring algorithm for maximum likelihood estimation in exploratory factor analysis can be developed in a way that is many times more efficient than a direct development based on information matrices and score vectors. The algorithm offers a simple alternative to current algorithms and when used in one-step mode provides the simplest and fastest method presently available for moving from consistent to efficient estimates. Perhaps of greater importance is its potential for extension to the confirmatory model. The algorithm is developed as a Gauss-Newton algorithm to facilitate its application to generalized least squares and to maximum likelihood estimation.