Showing papers in "Psychometrika in 1979"
TL;DR: In this paper, the polychoric correlation is discussed as a generalization of the tetrachoric correlation coefficient to more than two classes and two estimation methods are discussed: maximum likelihood estimation, and what may be called "two-step maximum likelihood" estimation.
Abstract: The polychoric correlation is discussed as a generalization of the tetrachoric correlation coefficient to more than two classes. Two estimation methods are discussed: Maximum likelihood estimation, and what may be called “two-step maximum likelihood” estimation. For the latter method, the thresholds are estimated in the first step. For both methods, asymptotic covariance matrices for estimates are derived, and the methods are illustrated and compared with artificial and real data.
1,050 citations
TL;DR: In this article, the authors proposed a population definition of CoEfficient KCC as a measure of diagnostic reliability in characterizing an individual, and the effect of reliability, as measured by KCC, on estimation bias, precision, and test power.
Abstract: Coefficientκ is generally defined in terms of procedures of computation rather than in terms of a population. Here a population definition is proposed. On this basis, the interpretation ofκ as a measure of diagnostic reliability in characterizing an individual, and the effect of reliability, as measured byκ, on estimation bias, precision, and test power are examined. Factors influencing the magnitude ofκ are identified. Strategies to improve reliability are proposed, including that of combining multiple unreliable diagnoses.
289 citations
TL;DR: In this article, a sampling from the null distribution of tables having the same marginal frequencies as the observed table is used for approximating attained significance levels of exact conditional tests, which yields precise approximations for practically any table dimensions and sample size.
Abstract: A procedure is proposed for approximating attained significance levels of exact conditional tests. The procedure utilizes a sampling from the null distribution of tables having the same marginal frequencies as the observed table. Application of the approximation through a computer subroutine yields precise approximations for practically any table dimensions and sample size.
162 citations
TL;DR: The present method shows some advantage in accuracy of estimation over the heuristic methods but is considerably more costly computationally.
Abstract: A maximum likelihood method of estimating the parameters of the multiple factor model when data are missing from the sample is presented. A Monte Carlo study compares the method with 5 heuristic methods of dealing with the problem. The present method shows some advantage in accuracy of estimation over the heuristic methods but is considerably more costly computationally.
150 citations
TL;DR: In this article, the Gauss-Newton, Newton-Raphson, Fisher Scoring, and Fletcher-Reeves algorithms for covariance structure analysis are considered in addition to the Fletcher-Powell algorithm.
Abstract: Several algorithms for covariance structure analysis are considered in addition to the Fletcher-Powell algorithm. These include the Gauss-Newton, Newton-Raphson, Fisher Scoring, and Fletcher-Reeves algorithms. Two methods of estimation are considered, maximum likelihood and weighted least squares. It is shown that the Gauss-Newton algorithm which in standard form produces weighted least squares estimates can, in iteratively reweighted form, produce maximum likelihood estimates as well. Previously unavailable standard error estimates to be used in conjunction with the Fletcher-Reeves algorithm are derived. Finally all the algorithms are applied to a number of maximum likelihood and weighted least squares factor analysis problems to compare the estimates and the standard errors produced. The algorithms appear to give satisfactory estimates but there are serious discrepancies in the standard errors. Because it is robust to poor starting values, converges rapidly and conveniently produces consistent standard errors for both maximum likelihood and weighted least squares problems, the Gauss-Newton algorithm represents an attractive alternative for at least some covariance structure analyses.
145 citations
TL;DR: The issue of factor indeterminacy, and its meaning and significance for factor analysis, has been the subject of considerable debate in recent years as discussed by the authors, and there are strong parallels between the arguments in the early literature, and those which have appeared in recent papers.
Abstract: The issue of factor indeterminacy, and its meaning and significance for factor analysis, has been the subject of considerable debate in recent years. Interestingly, the identical issue was discussed widely in the literature of the late 1920's and early 1930's, but this early discussion was somehow lost or forgotten during the development and popularization of multiple factor analysis. There are strong parallels between the arguments in the early literature, and those which have appeared in recent papers. Here I review the history of this early literature, briefly survey the more recent work, and discuss these parallels where they are especially illuminating.
137 citations
TL;DR: In this paper, a distinction is drawn between redundancy measurement and the measurement of multivariate association for two sets of variables, and several measures of multi-dimensional association between the original set of variables are examined, and it is shown that all of these measures are generalizations of the squared-multiple correlation.
Abstract: A distinction is drawn between redundancy measurement and the measurement of multivariate association for two sets of variables. Several measures of multivariate association between two sets of variables are examined. It is shown that all of these measures are generalizations of the (univariate) squared-multiple correlation; all are functions of the canonical correlations, and all are invariant under linear transformations of the original sets of variables. It is further shown that the measures can be considered to be symmetric and are strictly ordered for any two sets of observed variables. It is suggested that measures of multivariate relationship may be used to generalize the concept of test reliability to the case of vector random variables.
137 citations
TL;DR: In this article, a new subroutine was developed for calculating the terachoric correlation coefficient, which is based on the inverse normal and bivariate normal distributions, and the iterative procedure was started with an approximation with an error less than±.0135.
Abstract: A new subroutine has been developed for calculating the terachoric correlation coefficient. Recent advances in computing inverse normal and bivariate normal distributions have been utilized. The iterative procedure is started with an approximation with an error less than±.0135.
89 citations
TL;DR: Johnson has shown that the single linkage and the complete linkage hierarchical clustering algorithms induce a metric on the data known as the ultrametric through the use of the Lance and Williams recurrence formula.
Abstract: Johnson has shown that the single linkage and the complete linkage hierarchical clustering algorithms induce a metric on the data known as the ultrametric. Through the use of the Lance and Williams recurrence formula, Johnson's proof is extended to four other common clustering algorithms. It is also noted that two additional methods produce hierarchical structures which can violate the ultrametric inequality.
76 citations
TL;DR: In this article, it was shown that the parameter estimates obtained with the iterative procedure cannot lie outside the allowed interval, and this was later confirmed in a follow-up paper.
Abstract: In this note, we describe the iterative procedure introduced earlier by Goodman to calculate the maximum likelihood estimates of the parameters in latent structure analysis, and we provide here a simple and direct proof of the fact that the parameter estimates obtained with the iterative procedure cannot lie outside the allowed interval. Formann recently stated that Goodman's algorithm can yield parameter estimates that lie outside the allowed interval, and we prove in the present note that Formann's contention is incorrect.
75 citations
TL;DR: In this paper, a centroid scaling technique is used to estimate multidimensional joint spaces from the pick-any data, where subjects are represented as points whose coordinates are proportional to the centroids of the points representing their choices.
Abstract: Social and naturally occurring choice phenomena are often of the “pick-any” type in which the number of choices made by a subject as well as the set of alternatives from which they are chosen is unconstrained. These data present a special analytical problem because the meaning of non-choice among pick-any choice data is always ambiguous: A non-chosen alternative may be either unacceptable, or acceptable but not considered, or acceptable and considered but not chosen. A model and scaling method for these data are introduced, allowing for this ambiguity of non-choice. Subjects are represented as points whose coordinates are proportional to the centroids of the points representing their choices. Alternatives are represented at points whose coordinates are proportional to the centroids of the points representing subjects who have chosen them. This centroid scaling technique estimates multidimensional joint spaces from the pick-any data.
TL;DR: In this paper, it was shown that for nonpositive values of the parameter γ in the oblimin criterion, the criterion achieves a minimum on the manifold of all possible oblique rotations of any given full rank initial loading matrix A.
Abstract: It is shown that for nonpositive values of the parameter γ in the oblimin criterion, the criterion achieves a minimum on the manifold of all possible oblique rotations of any given full rank initial loading matrixA. For every positive value of γ, on the other hand, it is shown that there exists a full rank initial loading matrixA for which the oblimin criterion does not achieve a minimum over the manifold of all oblique rotations ofA. These results help explain the sometimes divergent behavior that results from using direct oblimin algorithms with γ set to a positive value.
TL;DR: In this article, the similarity between weighted and unweighted composites of the same items is investigated for complex item configurations, where the item configuration's variance structure is complex and it is shown that equal item weights are often virtually as good for a particular predictive purpose as the item weights that are theoretically optimal.
Abstract: Many authors have demonstrated for idealized item configurations that equal item weights are often virtually as good for a particular predictive purpose as the item weights that are theoretically optimal. What has not been heretofore clear, however, is what happens to the similarity between weighted and unweighted composites of the same items when the item configuration's variance structure is complex.
TL;DR: It was found that while minimum STRESS was the criterion of fit, another statistic, predictive capability, provided a better diagnosis of the known generating model, and the additive model is the most robust of the fitted models.
Abstract: Synthetic data are used to examine how well axiomatic and numerical conjoint measurement methods, individually and comparatively, recover simple polynomial generators in three dimensions. The study illustrates extensions of numerical conjoint measurement (NCM) to identify and model distributive and dual-distributive, in addition to the usual additive, data structures. It was found that while minimum STRESS was the criterion of fit, another statistic, predictive capability, provided a better diagnosis of the known generating model. That NCM methods were able to better identify generating models conflicts with Krantz and Tversky's assertion that, in general, the direct axiom tests provide a more powerful diagnostic test between alternative composition rules than does evaluation of numerical correspondence. For all methods, dual-distributive models are most difficult to recover, while consistent with past studies, the additive model is the most robust of the fitted models.
TL;DR: In this paper, the authors used the jackknife by groups and modifications of the jackknifeword by groups to estimate standard errors of rotated factor loadings for selected populations in common factor model maximum likelihood factor analysis.
Abstract: The jackknife by groups and modifications of the jackknife by groups are used to estimate standard errors of rotated factor loadings for selected populations in common factor model maximum likelihood factor analysis. Simulations are performed in whicht-statistics based upon these jackknife estimates of the standard errors are computed. The validity of thet-statistics and their associated confidence intervals is assessed. Methods are given through which the computational efficiency of the jackknife may be greatly enhanced in the factor analysis model.
TL;DR: In this paper, a theorem is presented which gives the range of possible correlations between a common factor and an external variable (i.e., a variable not included in the test battery factor analyzed).
Abstract: A theorem is presented which gives the range of possible correlations between a common factor and an external variable (i.e., a variable not included in the test battery factor analyzed). Analogous expressions for component (and regression component) theory are also derived. Some situations involving external correlations are then discussed which dramatize the theoretical differences between components and common factors.
TL;DR: In this article, the authors trace the course of the consequences of viewing test responses as simply providing dichotomous data concerning ordinal relations and propose a test theory based on ordinal statistics and frequency counts.
Abstract: This paper traces the course of the consequences of viewing test responses as simply providing dichotomous data concerning ordinal relations. It begins by proposing that the score matrix is best considered to be items-plus-persons by items-plus-persons, and recording the wrongs as well as the rights. This shows how an underlying order is defined, and was used to provide the basis for a tailored testing procedure. It also was used to define a number of measures of test consistency. Test items provide person dominance relations, and the relations provided by one item can be in one of three relations with a second one: redundant, contradictory, or unique. Summary statistics concerning the number of relations of each kind are easy to get and provide useful information about the test, information which is related to but different from the usual statistics. These concepts can be extended to form the basis of a test theory which is based on ordinal statistics and frequency counts and which invokes the concept of true scores only in a limited sense.
TL;DR: In this paper, a matching reinterpretation for several problems of interest in the prediction analysis of cross-classifications is presented, and appropriate significance tests can be given that may differ from those justified by the more common multinomial models.
Abstract: Inference models motivated by the combinatorial chance literature and the concept of object matching may be used in the analysis of a contingency table if the conditional assumption of fixed row and column totals is imposed. More specifically, by developing a matching reinterpretation for several problems of interest in the prediction analysis of cross-classifications—as defined by Hildebrand, Laing and Rosenthal, appropriate significance tests can be given that may differ from those justified by the more common multinomial models. In the course of the paper the distinction between a degree-1 statistic (based on the relationship between single objects) and a degree-2 statistic (based on the relationship between object pairs) is reviewed in some detail. Also, several specializations are presented to topics of current methodological importance in psychology; for instance, a number of references are made to the measurement of nominal scale response agreement between two raters.
TL;DR: In this paper, a qualitative unfolding model is used to test whether responses conform to a unidimensional, qualitative unfolding, and to an a priori stimulus ordering, assuming that persons and stimulus variables are ordered along a single continuum and that subjects most prefer stimuli nearest their own position.
Abstract: Assuming that subject responses rank order stimuli by preference, statistical methods are presented for testing the hypothesis that responses conform to a unidimensional, qualitative unfolding model and to an a priori stimulus ordering. The model postulates that persons and stimulus variables are ordered along a single continuum and that subjects most prefer stimuli nearest their own position. The underlying continuum need not form an interval scale of the stimulus attribute. The general assumptions of the test for the unfolding model make it suitable for the analysis of structure in attitude responses, preference data, and developmental stage data.
TL;DR: A modified beta binomial model is presented for use in analyzing ramdom guessing multiple choice tests and certain forms of taste tests and is more suitable for predicting outcomes on similar tests.
Abstract: A modified beta binomial model is presented for use in analyzing ramdom guessing multiple choice tests and certain forms of taste tests. Detection probabilities for each item are distributed beta across the population subjects. Properties for the observable distribution of correct responses are derived. Two concepts of true score estimates are presented. One, analogous to Duncan's empirical Bayes posterior mean score, is appropriate for assessing the subject's performance on that particular test. The second is more suitable for predicting outcomes on similar tests.
TL;DR: In this article, the authors used a result of Box to find the appropriate multiplicative correction term for the degrees of freedom in the analysis of variance table for the case when there is equal variability per time point, and when the correlation between observationsu time units apart is equal to ρ ≥ 0.
Abstract: Consider a repeated measurements study in which a single observation is taken at each oft(t > 2) equally spaced time points for each ofn subjects. This paper uses a result of Box to find the appropriate multiplicative correction term for the degrees of freedom in the analysis of variance table for the case when there is equal variability per time point, and when the correlation between observationsu time units apart is equal to ρ
u
. For large values oft, the multiplicative correction has a lower bound approximately equal to 2.5/(t − 1).
TL;DR: In this paper, a Monte Carlo study was carried out in order to investigate the ability of ALSCAL to recover true structure inherent in simulated proximity measures when portions of the data are missing, and very accurate recovery of true distances, stimulus coordinates, and weight vectors could be achieved with as much as 60% missing data as long as sample size was sufficiently large and the level of random error was low.
Abstract: A Monte Carlo study was carried out in order to investigate the ability of ALSCAL to recover true structure inherent in simulated proximity measures when portions of the data are missing. All sets of simulated proximity measures were based on 30 stimuli and three dimensions, and selection of missing elements was done randomly. Properties of the simulated data varied according to (a) the number of individuals, (b) the level of random error, (c) the proportion of missing data, and (d) whether the same entries or different entries were deleted for each individual. Results showed that very accurate recovery of true distances, stimulus coordinates, and weight vectors could be achieved with as much as 60% missing data as long as sample size was sufficiently large and the level of random error was low.
TL;DR: In this article, a scoring system is devised that depends on the size of the subset and on whether or not the correct answer is eliminated, and the mean and variance of the score per item are obtained.
Abstract: On a multiple-choice test in which each item hask alternative responses, the test taker is permitted to choose any subset which he believes contains the one correct answer. A scoring system is devised that depends on the size of the subset and on whether or not the correct answer is eliminated. The mean and variance of the score per item are obtained. Methods are derived for determining the total number of items that should be included on the test so that the average score on all items can be regarded as a good measure of the subject's knowledge. Efficiency comparisons between conventional and the subset selection scoring procedures are made. The analogous problem ofr > 1 correct answers for each item (withr fixed and known) is also considered.
TL;DR: In this paper, it was demonstrated that tests of homogeneity of independent correlation coefficients based on simple forms of the normal andt approximations to the distribution of the correlation coefficients are comparable in terms of robustness, size and power.
Abstract: It is demonstrated that tests of homogeneity of independent correlation coefficients based on the simple forms of the normal andt approximations to the distribution of the correlation coefficients are comparable in terms of robustness, size and power.
TL;DR: In this article, conditions are given under which the stationary points and values of a ratio of quadratic forms in two singular matrices can be obtained by a series of simple matrix operations.
Abstract: Conditions are given under which the stationary points and values of a ratio of quadratic forms in two singular matrices can be obtained by a series of simple matrix operations. It is shown that two classes of optimal weighting problems, based respectively on the grouping of variables and on the grouping of observations, satisfy these conditions. The classical treatment of optimal scaling of forced-choice multicategory data is extended for these cases. It is shown that previously suggested methods based on reparameterization will work only under very special conditions.
TL;DR: In this article, an alternative probability model for the distribution of ordinal data is introduced and examined, and the conditional expectations of Spearman's rho and Kendall's tau under the uniform distribution are derived.
Abstract: To date, virtually all techniques appropriate for ordinal data are based on the uniform probability distribution over the permutations. In this paper we introduce and examine an alternative probability model for the distribution of ordinal data. Preliminary to deriving the expectations of Spearman's rho and Kendall's tau under this model, we show how to compute certain conditional expectations of rho and tau under the uniform distribution. The alternative probability model is then applied to ordinal test theory, and the calculation of true scores and test reliability are discussed.
TL;DR: In this article, a correct derivation of the initial position is obtained, and change correlation in terms of α is obtained and Tucker et al.'s discussion of the "law of initial values" is reconsidered.
Abstract: Tucker, Damarin and Messick proposed a “base-free” measure of change which involves the computation of residual scores that are uncorrelated with true scores on the pretest. The present note discusses this change measure and demonstrates that, because of an incorrect derivation by the authors, properties they attribute toα (the coefficient for the regression of true scores from the second testing on true scores from the first testing) are, in fact, properties ofb, the ordinary regression coefficient. A correct derivation of the initial position—change correlation in terms ofα is obtained, and Tucker et al.'s discussion of the “law of initial values” is reconsidered.
TL;DR: In this article, a general framework is described for deriving a lower bound on the probability of correctly classifying the examinees based on the number of items used and number of examinees tested.
Abstract: When comparing examinees to a control, the examiner usually does not know the probability of correctly classifying the examinees based on the number of items used and the number of examinees tested. Using ranking and selection techniques, a general framework is described for deriving a lower bound on this probability. We illustrate how these techniques can be applied to the binomial error model. New exact results are given for normal populations having unknown and unequal variances.
TL;DR: In this paper, the authors use the same term "split-half" for division of ann-item test into two subtests containing equal [Cronbach], and possibly unequal [Guttman], which leads to a misunderstanding about the relation between Guttman's maximum split-half bound and Cronbach's coefficient alpha Coefficient alpha is the average of splithalf bounds in the Cronbach sense and so is not larger than the maximum splithalf bound in either sense whenn is even Whenn is odd.
Abstract: Use of the same term “split-half” for division of ann-item test into two subtests containing equal [Cronbach], and possibly unequal [Guttman], numbers of items sometimes leads to a misunderstanding about the relation between Guttman's maximum split-half bound and Cronbach's coefficient alpha Coefficient alpha is the average of split-half bounds in the Cronbach sense and so is not larger than the maximum split-half bound in either sense whenn is even Whenn is odd, however, splithalf bounds exist only in the Guttman sense and the largest of these may be smaller than coefficient alpha