Showing papers in "Psychometrika in 1973"

A reliability coefficient for maximum likelihood factor analysis

Abstract: Maximum likelihood factor analysis provides an effective method for estimation of factor matrices and a useful test statistic in the likelihood ratio for rejection of overly simple factor models. A reliability coefficient is proposed to indicate quality of representation of interrelations among attributes in a battery by a maximum likelihood factor analysis. Usually, for a large sample of individuals or objects, the likelihood ratio statistic could indicate that an otherwise acceptable factor model does not exactly represent the interrelations among the attributes for a population. The reliability coefficient could indicate a very close representation in this case and be a better indication as to whether to accept or reject the factor solution.

Linear programming techniques for multidimensional analysis of preferences

Abstract: This paper offers a new methodology for analyzing individual differences in preference judgments with regard to a set of stimuli prespecified in a multidimensional attribute space. The individual is modelled as possessing an “ideal point” denoting his most preferred stimulus location in this space and a set of weights which reveal the relative saliences of the attributes. He prefers those stimuli which are “closer” to his ideal point (in terms of a weighted Euclidean distance measure). A linear programming model is proposed for “external analysis”i.e., estimation of the coordinates of his ideal point and the weights (involved in the Euclidean distance measure) by analyzing his paired comparison preference judgments on a set of stimuli, prespecified by their coordinate locations in the multidimensional space. A measure of “poorness of fit” is developed and the linear programming model minimizes this measure overall possible solutions. The approach is fully nonmetric, extremely flexible, and uses paired comparison judgments directly. The weights can either be constrained nonnegative or left unconstrained. Generalizations of the model to consider ordinal or interval preference data and to allow an orthogonal transformation of the attribute space are discussed. The methodology is extended to perform “internal analysis,”i.e., to determine the stimuli locations in addition to weights and ideal points by analyzing the preference judgments of all subjects simultaneously. Computational results show that the methodology for external analysis is “unbiased”—i.e., on an average it recovers the “true” ideal point and weights. These studies also indicate that the technique performs satisfactorily even when about 20 percent of the paired comparison judgments are incorrectly specified.

A goodness of fit test for the Rasch model.

Abstract: The Rasch model is an item analysis model with logistic item characteristic curves of equal slope,i.e. with constant item discriminating powers. The proposed goodness of fit test is based on a comparison between difficulties estimated from different scoregroups and over-all estimates. Based on the within scoregroup estimates and the over-all estimates of item difficulties a conditional likelihood ratio is formed. It is shown that—2 times the logarithm of this ratio isx 2-distributed when the Rasch model is true. The power of the proposed goodness of fit test is discussed for alternative models with logistic item characteristic curves, but unequal discriminating items from a scholastic aptitude test.

Generalized Least Squares Estimators in the Analysis of Covariance Structures.

Abstract: SUMMARY Let S represent the usual unbiased estimator of a covariance matrix, Σ0, whose elements are functions of a parameter vector . A generalized least squares (G.L.S) estimate, of may be obtained by minimizing where V is some positive definite matrix. Asymptotic properties of the G.L.S. estimators are investigated assuming only that satisfies certain regularity conditions and that the limiting distribution of S is multivariate normal with specified parameters. The estimator of which is obtained by maximizing the Wishart likelihood function (M.W.L. estimator) is shown to be a member of the class of G.L.S. estimators with minimum asymptotic variances. When is linear in a G.L.S. estimator which converges stochastically to the M.W.L. estimator involves far less computation. Methods for calculating estimates of , estimates of the dispersion matrix of , and test statistics, are given for certain linear models.

Estimating the weights for multiple attributes in a composite criterion using pairwise judgments

Abstract: This paper presents a new methodology for estimating the weights or saliences of subcriteria (attributes) in a composite criterion measure. The inputs to the estimation procedure consist of (i) a set of stimuli or objects with each stimulus defined by its subcriteria profile (set of attribute values) and (ii) the set of paired comparison dominance (e.g., preference) judgments on the stimuli made by a single judge (expert) in terms of the global criterion. A criterion of fit is developed and its optimization via linear programming is illustrated with an example. The procedure is generalized to estimate a common set of weights when the pairwise judgments on the stimuli are made by more than one judge. The procedure is computationally efficient and has been applied in developing a composite criterion of managerial success yielding high concurrent validity. This methodology can also be used to perform ordinal multiple regression—i.e., multiple regression with an ordinally scaled dependent variable and a set of intervally scaled predictor variables. The approach is further extended to “internal analysis” (unfolding) using the vector model of preference and to the additive model of “conjoint measurement.”

The role of cognitive style in academic performance and in teacher‐student relations12

Abstract: The cognitive characteristics involved in a relatively field-dependent or field-independent cognitive style and the personal characteristics associated with these contrasting styles have been shown to play a role in students' selection of electives and majors, in vocational preferences they experience early in their academic careers, and in the vocational choices they make later on. They have been related as well to performance in different subject-matter areas in school and in vocation chosen afterwards; they have also been implicated in shifts in major during the college years. Finally, they have been found to influence teachers' ways of teaching and students' ways of learning; in addition, teachers and students matched in cognitive style show positive mutual evaluation, whereas teachers and students who are mismatched tend to evaluate each other negatively.

Homogeneous case of the continuous response model

Abstract: In line with the latent trait model, the continuous response level is defined and considered, in contrast to the discrete response levels, which have already been explored by the author. Discussions are mainly focused on the homogeneous case and the open response situation. The operating density characteristic of the continuous item score is defined. Also the basic function, information functions and the positive-exponent family are discussed on the continuous response level, in connection with the sufficient condition that a unique maximum estimate is provided for the response pattern, which consists of the continuous item scores.

The effect of number of categories in rating scales on precision of estimation of scale values

Abstract: The precision of scale value estimates using maximum likelihood estimation is examined for varying numbers of categories, varying discriminal dispersions and with and without category boundary estimation. It is concluded that this type of estimation is somewhat unsatisfactory if boundaries are estimated unless these are of specific interest. If they are not estimated, it is concluded that using seven or more categories provides very nearly as much precision of estimate as a corresponding continuous judgment task.

Min and max hierarchical clustering using asymmetric similarity measures

Abstract: The min and the max hierarchical clustering methods discussed by Johnson are extended to include the use of asymmetric similarity values. The first part of the paper presents the basic min and max procedures but in the context of graph theory; this description is then generalized to directed graphs as a way of introducing the less restrictive characterization of the original clustering techniques.

The effectiveness of alternative instructional media: a survey*

Abstract: This survey provides an overview of research on the effectiveness of alternative instructional media. The media discussed are traditional classroom instruction (TI), instructional radio (IR), instructional television (ITV), programmed instruction (PI), and computer-assisted instruction (CAI). The effectiveness of these media is examined from a reasonably macroscopic point of view; the psychology of pupil-teacher interaction or the 'content variables' of ITV, to take two examples, are at a micro-level not considered. Achievement test scores constitute the measure of effectiveness most frequently used in this survey though, where available, results concerning the affective impact of the various media of instruction are included. Achievement test data, in most cases, were collected only on an annual basis, so they reveal no fine-grained detail about the learning process. Since this survey is relatively brief and its scope broad, a few caveats are in order. First, where literature surveys are available, their results have been cited to the extent possible and, frequently, original sources remain unchecked. Second, available knowledge of the effectiveness of the various media varies considerably; much more is known about TI and ITV than about the others. For this reason a survey such as this is inherently spotty in its conclusions. The third warning, related to the second, is that many of the evaluations fall short of (or lack entirely) scientific standards of analysis and reporting. For this reason, it was sometimes necessary to attempt to cull conclusions from essentially journalistic accounts of projects. Fourth, it should be noted that this survey is limited to instruction within a school setting. Finally, the survey excludes information on costs. Before beginning the literature survey, we present a paradigm for measurement of effectiveness and then discuss several less desirable alternatives that have actually been employed. In the surveys of the individual methods where adequate prior surveys are unavailable, results from a representative sample of individual evaluations are discussed.

A Comment On Birnbaum's Three-Parameter Logistic Model In The Latent Trait Theory.

Abstract: Birnbaum's three-parameter logistic model for the multiple-choice item in the latent trait theory is considered with respect to the item response information function and the unique maximum condition. It is clarified that with models of knowledge or random guessing nature, which include the three-parameter logistic model, the unique maximum condition is not satisfied for the correct answer, and the item response information function is negative for the interval (− ∞,θ g ). It is suggested that we should useθ g as a criterion in selecting optimal items for a specified group of examinees, so that we can practically avoid the possibility of non-unique maxima of the likelihood function on the response pattern given by an examinee in the group.

A Rank Test For Two Group Concordance.

Abstract: A test statistic is introduced which allows one to test the hypothesis of agreement of several judges on the ranking of items within each of two groups and between the two groups. The groups of judges may be unequal in size. A normal approximation for the test statistic is developed. The relationship to existing techniques given by Kendall, Friedman, Page, Spearman, and Lyerly is discussed. A generalization of the coefficient of concordance is presented and the extension of the method to multi-group problems is suggested.

Standard errors for obliquely rotated factor loadings

Abstract: In a manner similar to that used in the orthogonal case, formulas for the aymptotic standard errors of analytically rotated oblique factor loading estimates are obtained. This is done by finding expressions for the partial derivatives of an oblique rotation algorithm and using previously derived results for unrotated loadings. These include the results of Lawley for maximum likelihood factor analysis and those of Girshick for principal components analysis. Details are given in cases including direct oblimin and direct Crawford-Ferguson rotation. Numerical results for an example involving maximum likelihood estimation with direct quartimin rotation are presented. They include simultaneous tests for significant loading estimates.

Monotone invariant clustering procedures

Abstract: A major justification for the hierarchical clustering methods proposed by Johnson is based upon their invariance with respect to monotone increasing transformations of the original similarity measures Several alternative procedures are presented in this paper that also share in the same property of invariance One of these techniques constructs a hierarchy of partitions by sequentially minimizing a monotone invariant goodness-of-fit statistic; the other techniques construct a hierarchy of partitions by successively subdividing the complete set of objects until one partition class is defined for each individual member in the set A numerical example comparing these alternative procedures with Johnson's two methods is duscussed in terms of a simplified computational scheme for obtaining the necessary hierarchies

Dialectic operations: the final period of cognitive development1

Abstract: Arguments for an extension of Piaget's theory of cognitive development have been derived from philosophical and historical consideration of modern natural sciences. Implicit contradictions, which characterize these sciences as well as common thought, can be systematically apprehended only through a dialectic reinterpretation. The dialectic basis of Piaget's theory is expressed in his assimilation-accommodation paradigm. But development is interpreted as a continuing alienation from this basis culminating in the noncontradictory thinking of formal operations. Although Piaget's interpretations capture a rich variety of performances during childhood they fail to represent adequately the thought and emotions of mature and creative persons. For an interpretation of adulthood and aging, a return to the dialectic basis is necessary. Such a reorganization can proceed from any of the four major levels of development. It introduces intra- and interindividual variations into Piaget's theory. Individuals may operate simultaneously or in short succession at different cognitive levels. The ceaseless striving toward formal operations becomes inappropriate and ineffective for the level of dialectic maturity.

Standard errors for rotated factor loadings

Abstract: Beginning with the results of Girshick on the asymptotic distribution of principal component loadings and those of Lawley on the distribution of unrotated maximum likelihood factor loadings, the asymptotic distribution of the corresponding analytically rotated loadings is obtained. The principal difficulty is the fact that the transformation matrix which produces the rotation is usually itself a function of the data. The approach is to use implicit differentiation to find the partial derivatives of an arbitrary orthogonal rotation algorithm. Specific details are given for the orthomax algorithms and an example involving maximum likelihood estimation and varimax rotation is presented.

A test of inclusion which allows for errors of measurement

Abstract: A test which allows for errors of measurement is derived for the hypothesis that all the members of a population who possess a certain skill are a sub-set of the members who possess another skill. Formulae are given for one particular case when two questions are used for each skill, and for when three questions are used for each skill. An illustrative example is given for the two-question case.

Pairwise nonmetric multidimensional scaling

Abstract: A method of nonmetric multidimensional scaling is described which minimizes pairwise departures from monotonicity. The procedure is relatively simple, both conceptually and computationally. Experience to date suggests that it produces solutions comparable to those of other methods.

On The Numerical Approximation Of The Bivariate Normal (Tetrachoric) Correlation Coefficient.

Abstract: In this paper a rapid and reliable method is found for estimating the value of the Bivariate Normal Correlation Coefficient, ρ, given values of the joint probability and the normal deviates,h andk, or the related areas. This technique finds useful application in the computational approximation of the tetrachoric correlation coefficient,r, when the underlying distributions may be assumed to be normal.

The estimation of proportions inm groups

Abstract: In many applications, it is desirable to estimate binomial proportions inm groups where it is anticipated that these proportions are similar but not identical. Following a general approach due to Lindley, a Bayesian Model II aposteriori modal estimate is derived that estimates the inverse sine transform of each proportion by a weighted average of the inverse sine transform of the observed proportion in the individual group and the average of the estimated values. Comparison with a classical method due to Jackson spotlights some desirable features of Model II analyses. The simplicity of the present formulation makes it possible to study the behavior of the Bayesian Model II approach more closely than in more complex formulations. Also, it is possible to estimate the amount of gain afforded by the Model II analyses.

A note on a sufficiency condition for uniqueness of a restricted factor matrix

Abstract: A counterexample is produced to a conjecture by K. G. Joreskog concerning sufficiency conditions for uniqueness of a restricted factor matrix. A substitute condition is stated and proved for the most common situation where the restricted elements are specified to be zero.

Measures of reliability for profiles and test batteries.

Abstract: A general index of reliability, termed “canonical reliability,” is developed for use with profiles, or more generally, for use with vectors of random variables. Canonical reliability is defined as the ratio of the average squared distance among true scores to the average squared distance among observed scores. Based on Mahalonobis distances, canonical reliability is shown to be consistent with multivariate analogues of parallel form correlations, squared correlation between true and observed scores, and an analysis of variance formulation. The index of reliability based on Cronbach and Gleser'sD 2 is also derived from the general formulation. A comparison of the Mahalonobis andD 2 approaches indicates that score vectors usingD 2 distances are more reliable; however, both methods of comparing profiles are useful depending on the nature of the information that is desired. Transforming the observed variables to independent canonical variates provides a basis for comparing profiles on maximally reliable profile dimensions. For illustrative purposes, profile reliability is calculated and interpreted for the WISC subscales for a 7 1/2 year age group.

Comments on the “lens model” equation and the analysis of multiple-cue judgment tasks

Abstract: It is the purpose of this paper to discuss the ‘Lens Model’ approach to the analysis of subject performance in multiple-cue judgment tasks embedded in probabilistic environments. Various characteristics of the linear lens model are presented in matrix notation and are formulated without reference to standardized variables in order to facilitate analysis and subsequent generalization. Some new results on ‘matching’ behavior are presented. A brief outline of current problems and related work is presented.

A Note on Lawley's Formulas for Standard Errors in Maximum Likelihood Factor Analysis.

Abstract: Evidence is given to indicate that Lawley's formulas for the standard errors of maximum likelihood loading estimates do not produce exact asymptotic results. A small modification is derived which appears to eliminate this difficulty.

Multivariate models of cognition and personality: the need for both process and structure in psychological theory and measurement1

Abstract: This paper calls for the development of sequential models of cognition and personality as a way of adding process to the primarily structural concerns of current multivariate models At the same time, it points to the results of factor analysis, particularly as summarized in a hierarchical extension of Guilford's structure-of-intellect system, as a source of component variables for such sequential formulations The need to take into account personality, developmental, and environmental variables in these sequential models is also emphasized

Maximizing The Coefficient Of Generalizability In Multi-Facet Decision Studies.

Abstract: For random-model, fully-crossed, two- and three-facet experimental designs the following two problems were considered. First, equations were developed for determining the optimal number of conditions of a facet for maximizing the coefficient of generalizability under the constraint that the total number of observations per subject is constant. Second, the problem of determining the minimum number of observations per subject for a specified generalizability coefficient is solved for the two-facet crossed design.