Showing papers in "Psychometrika in 1977"

Nonmetric individual differences multidimensional scaling: An alternating least squares method with optimal scaling features.

Abstract: A new procedure is discussed which fits either the weighted or simple Euclidian model to data that may (a) be defined at either the nominal, ordinal, interval or ratio levels of measurement; (b) have missing observations; (c) be symmetric or asymmetric; (d) be conditional or unconditional; (e) be replicated or unreplicated; and (f) be continuous or discrete. Various special cases of the procedure include the most commonly used individual differences multidimensional scaling models, the familiar nonmetric multidimensional scaling model, and several other previously undiscussed variants. The procedure optimizes the fit of the model directly to the data (not to scalar products determined from the data) by an alternating least squares procedure which is convergent, very quick, and relatively free from local minimum problems. The procedure is evaluated via both Monte Carlo and empirical data. It is found to be robust in the face of measurement error, capable of recovering the true underlying configuration in the Monte Carlo situation, and capable of obtaining structures equivalent to those obtained by other less general procedures in the empirical situation.

Redundancy analysis an alternative for canonical correlation analysis

Abstract: A component method is presented maximizing Stewart and Love's redundancy index. Relationships with multiple correlation and principal component analysis are pointed out and a rotational procedure for obtaining bi-orthogonal variates is given. An elaborate example comparing canonical correlation analysis and redundancy analysis on artificial data is presented.

Additive similarity trees

Abstract: Similarity data can be represented by additive trees. In this model, objects are represented by the external nodes of a tree, and the dissimilarity between objects is the length of the path joining them. The additive tree is less restrictive than the ultrametric tree, commonly known as the hierarchical clustering scheme. The two representations are characterized and compared. A computer program, ADDTREE, for the construction of additive trees is described and applied to several sets of data. A comparison of these results to the results of multidimensional scaling illustrates some empirical and theoretical advantages of tree representations over spatial representations of proximity data.

Sufficient Statistics and Latent Trait Models.

Abstract: For questionnaires with two answer categories, it has been proven in complete generality that if a minimal sufficient statistic exists for the individual parameter and if it is the same statistic for all values of the item parameters, then the raw score (or the number of correct answers) is the minimal sufficient statistic. It follows that the model must by of the Rasch type with logistic item characteristic curves and equal item-discriminating powers. This paper extends these results to multiple choice questionnaires. It is shown that the minimal sufficient statistic for the individual parameter is a function of the so-called score vector. It is also shown that the so-called equidistant scoring is the only scoring of a questionnaire that allows for a real valued sufficient statistic that is independent of the item parameters, if a certain ordering property for the sufficient statistic holds.

Orthogonal Procrustes Rotation for Two or More Matrices.

Abstract: Necessary and sufficient conditions for rotating matrices to maximal agreement in the least-squares sense are discussed. A theorem by Fischer and Roppert, which solves the case of two matrices, is given a more straightforward proof. A sufficient condition for a best least-squares fit for more than two matrices is formulated and shown to be not necessary. In addition, necessary conditions suggested by Kristof and Wingersky are shown to be not sufficient. A rotation procedure that is an alternative to the one by Kristof and Wingersky is presented. Upper bounds are derived for determining the extent to which the procedure falls short of attaining the best least-squares fit. The problem of scaling matrices to maximal agreement is discussed. Modifications of Gower's method of generalized Procrustes analysis are suggested.

Maximum likelihood estimation in multidimensional scaling

Abstract: A variety of distributional assumptions for dissimilarity judgments are considered, with the lognormal distribution being favored for most situations. An implicit equation is discussed for the maximum likelihood estimation of the configuration with or without individual weighting of dimensions. A technique for solving this equation is described and a number of examples offered to indicate its performance in practice. The estimation of a power transformation of dissimilarity is also considered. A number of likelihood ratio hypothesis tests are discussed and a small Monte Carlo experiment described to illustrate the behavior of the test of dimensionality in small samples.

On a Metric, Unidimensional Unfolding Model for Attitudinal and Developmental Data.

Abstract: If stimulus responses are linearly related to squared distances between stimulus scale values and person scores along a latent continuum, (a) the stimulus × stimulus correlation matrix will display a simplex-like pattern, (b) the signs of first-order partial correlations can be specified in an empirically testable manner, and (c) the variables will have a semicircular, two-factor structure. Along the semicircle, variables will be ordered by their positions on the latent dimension. The above results suggest procedures for examining the appropriateness of the model and procedures for ordering the stimuli. Applications to developmental and attitudinal data are discussed.

Lower bounds for the reliability of the total score on a test composed of non-homogeneous items: I: Algebraic lower bounds

Abstract: Let Σ x be the (population) dispersion matrix, assumed well-estimated, of a set of non-homogeneous item scores. Finding the greatest lower bound for the reliability of the total of these scores is shown to be equivalent to minimizing the trace of Σ x by reducing the diagonal elements while keeping the matrix non-negative definite. Using this approach, Guttman's bounds are reviewed, a method is established to determine whether his λ4 (maximum split-half coefficient alpha) is the greatest lower bound in any instance, and three new bounds are discussed. A geometric representation, which sheds light on many of the bounds, is described.

Lower bounds for the reliability of the total score on a test composed of non-homogeneous items: II: A search procedure to locate the greatest lower bound

Abstract: Finding the greatest lower bound for the reliability of the total score on a test comprisingn non-homogenous items with dispersion matrix Σ x is equivalent to maximizing the trace of a diagonal matrix Σ E with elements θ I , subject to Σ E and Σ T =Σ x − Σ E being non-negative definite. The casesn=2 andn=3 are solved explicity. A computer search in the space of the θ i is developed for the general case. When Guttman's λ4 (maximum split-half coefficient alpha) is not the g.l.b., the maximizing set of θ i makes the rank of Σ T less thann − 1. Numerical examples of various bounds are given.

Factor simplicity index and transformations

Abstract: A scale-invariant index of factorial simplicity is proposed as a summary statistic for principal components and factor analysis. The index ranges from zero to one, and attains its maximum when all variables are simple rather than factorially complex. A factor scale-free oblique transformation method is developed to maximize the index. In addition, a new orthogonal rotation procedure is developed. These factor transformation methods are implemented using rapidly convergent computer programs. Observed results indicate that the procedures produce meaningfully simple factor pattern solutions.

Estimating the parameters of the latent population distribution

Abstract: Under consideration is a test battery of binary items. The responses ofn individuals are assumed to follow a Rasch model. It is further assumed that the latent individual parameters are distributed within a given population in accordance with a normal distribution. Methods are then considered for estimating the mean and variance of this latent population distribution. Also considered are methods for checking whether a normal population distribution fits the data. The developed methods are applied to data from an achievement test and from an attitude test.

Thurstone's discriminal processes fifty years later

Abstract: Four issues are discussed concerning Thurstone's discriminal processes: the distributions governing the representation, the nature of the response decision rules, the relation of the mean representation to physical characteristics of the stimulus, and factors affecting the variance of the representation. A neural schema underlying the representation is proposed which involves samples in time of pulse trains on individual neural fibers, estimators of parameters of the several pulse trains, samples of neural fibers, and an aggregation of the estimates over the sample. The resulting aggregated estimate is the Thurstonian representation. Two estimators of pulse rate, which is monotonic with signal intensity, are timing and counting ratios and two methods of aggregation are averaging and maximizing. These lead to very different predictions in a speed-accuracy experiment; data indicate that both estimators are available and the aggregation is by averaging. Magnitude estimation data are then used both to illustrate an unusual response rule and to study the psychophysical law. In addition, the pattern of variability and correlation of magnitude estimates on successive trials is interpreted in terms of the sample size over which the aggregation takes place. Neural sample size is equated with selective attention, and is an important factor affecting the variability of the representation. It accounts for the magical number seven phenomenon in absolute identification and predicts the impact of nonuniform distributions of intensities on the absolute identification of two frequencies.

Psychological differentiation: current status

Abstract: The status of the differentiation hypothesis, derived from differentiation theory, is examined in light of the evidence that has accumulated since the hypothesis was proposed a decade and a half ago. Assuming that greater or less differentiation is an organismic attribute, the hypothesis postulates that behaviors reflecting extent of differentiation are likely to be interrelated, making for self-consistency in individual functioning across domains. The newer evidence has on the whole confirmed the linkages among the behaviors examined in the earlier research that was done to test the differentiation hypothesis. The differentiation hypothesis has also proven useful in generating predictions about linkages to behaviors in new domains (cognitive restructuring, interpersonal competencies and cerebral lateralization); these predictions have been tested and generally confirmed. The differentiation construct appears able to account for phenomena that cannot be accommodated by other lower-order constructs, such as field dependence-independence. On all these grounds the differentiation construct continues to serve a useful function. Whereas the differentiation hypothesis has dealt only with the interrelatedness among components of a cluster of behaviors subsumed under differentiation, the newer evidence carries suggestions for a hierarchical ordering of these components and for the nature of causal connections among them.

The Rasch Model, the Law of Comparative Judgment and Additive Conjoint Measurement.

Abstract: Relationships between the Rasch model and both the law of comparative judgment and additive conjoint measurement are discussed. The distance between the ability of Persona and the difficult of Itemi is, in the Rasch model, the baseline value corresponding to the probability thata will respond correctly toi, where this probability is interpreted as the area under a logistic curve (which is substantially equivalent to the normal curve) and is thus an application of the law of comparative judgment. Under certain assumptions, the Rasch model is also a special case of additive conjoint measurement and, properly reinterpreted, may be usefully applied in contexts other than individual differences.

Practical applications of item characteristic curve theory

Abstract: Much of classical test theory deals with an entire test; this theory is applicable even if the test is not composed of items. If a test consists of separate items and if test score is a (possibly weighted) sum of item scores, then statistics describing the test scores of a certain group of examinees can be expressed algebraically in terms of statistics describing the individual item scores for the same group of examinees. Insofar as it relates to tests, classical item theory (this is only a part of classical test theory) consists of such algebraic tautologies. Such a theory makes no assumptions about matters that are beyond the control of the psychometrician. This is actuarial science. It cannot predict how individuals will respond to items unless the items have previously been administered to similar individuals.

Some exact conditional tests of independence for, r × c cross-classification tables

Abstract: Exact conditional tests of independence in cross-classification tables are formulated based on theχ 2 statistic and statistics with stronger operational interpretations, such as some nominal and ordinal measures of association. Guidelines for the table dimensions and sample sizes for which the tests are economically implemented on a computer are given. Some selected sample sizes and marginal distributions are used in a numerical comparison between the significance levels of the approximate and exact conditional tests based on theχ 2 statistic.

On the Mean and Variance of the Tetrachoric Correlation Coefficient.

Abstract: Estimates of the mean and standard deviation of the tetrachoric correlation are compared with their expected values in several 2 × 2 tables. Significant bias in the mean is found when the minimum cell frequency is less than 5. Three formulas for the standard deviation are compared and guidelines given for their use.

Weakly parallel tests in latent trait theory with some criticisms of classical test theory

Abstract: A new concept of weakly parallel tests, in contrast to strongly parallel tests in latent trait theory, is proposed. Some criticisms of the fundamental concepts in classical test theory, such as the reliability of a test and the standard error of estimation, are given.

Effects of conditionality on INDSCAL and ALSCAL weights

Abstract: The role of conditionality in the INDSCAL and ALSCAL procedures is explained. The effects of conditionality on subject weights produced by these procedures is illustrated via a single set of simulated data. Results emphasize the need for caution in interpreting subject weights provided by these techniques.

A general solution to Mosier's oblique procrustes problem

Abstract: Browne provided a method for finding a solution to the normal equations derived by Mosier for rotating a factor matrix to a best least squares fit with a specified structure. Cramer showed that Browne's solution is not always valid, and proposed a modified algorithm. Both Browne and Cramer assumed the factor matrix to be of full rank. In this paper a general solution is derived, which takes care of rank deficient factor matrices as well. A new algorithm is offered.

A theory of consistency of ordering generalizable to tailored testing

Abstract: Measures of consistency and completeness of order relations derived from test-type data are proposed. The measures are generalized to apply to incomplete data such as from tailored testing. The measures are based on consideration of the items-plus-persons by items-plus-persons matrix as an adjacency matrix in which a 1 means that the row element, whether item or person, dominated the column element. From this the number of item-item and person-person dominance relations can be derived. Indices of consistency are developed which compare the number of dominance relations in a matrix to the number that would occur if it were perfectly consistent and to the number in a random matrix; relations to other indices are noted.

A generalization of coefficient alpha.

Abstract: It is well known that coefficient alpha can be used to estimate the reliability of a test even when the test is split into several parts It is also known that alpha can severely underestimate test reliability when the several parts have an unequal number of items A gernalization of alpha,β k, is proposed to correct this defect Several properties ofβ k are also presented

Group comparisons for the test of standard written english

Abstract: Group comparisons for male, female, majority, and minority students were conducted for the Test of Standard Written English (TSWE). Data for two academic years and from 18 different institutions were analyzed by pooling data across institutions within each of the academic years. Analyses of data from the first academic year focused on relationships between TSWE scores and grades. Analyses of the second academic year focused on relationships between TSWE scores and scores on graded writing samples. No important group differences in traditional correlational analyses for either grade or essay prediction were observed. No important group differences were observed in either correlational analyses or regression analyses of the second year essay data. Analyses of correct and incorrect placement decisions (hits and misses) at specific TSWE cut-off scores revealed no noteworthy group differences whether outcomes were based on English course grades or on freshman year writing performance. Whether grades or essays were used as the outcome, the proportion of incorrect decisions (misses) was less for minority students than for any of the groups. For all groups, the TSWE appeared to predict freshman year writing performance as well as or better than pre-course writing samples, high school English grades, or high school rank in class.

Origins of the field‐dependent and field‐independent cognitive styles

Abstract: The literature on sources of individual differences in field dependence-independence is reviewed and findings on ontogenetic development and cross-cultural differences are incorporated into the theory of psychological differentiation. During the growth years, individuals develop toward greater field independence. Hormonal and X-linked genetic factors may influence development of specific cognitive components of the field-dependence-independence dimension. It is clear from a wide variety of data that the development of field independence is enhanced by appropriate training programs, child-rearing practices which encourage separation from parental authority, and “loose” social structures which permit individual autonomy. Among subsistence-level societies, members of nomadic hunting groups tend to be more field independent than members of sedentary farming groups. Viewed from an ecocultural perspective, this finding suggests a progression from relative field independence toward relative field dependence as cultural forms changed from early hunting to later agricultural economies. That developmental changes may proceed in opposite directions in the course of ontogeny and during cultural history is compatible with the bipolar conception of cognitive styles; the cognitive restructuring skills of field-independent individuals and the social-interpersonal competencies of field-dependent individuals have adaptive value in different life circumstances.

Correctness of Kruskal's Algorithms for Monotone Regression with Ties.

Abstract: Kruskal has proposed two modifications of monotone regression that can be applied if there are ties in nonmetric scaling data. In this note we prove Kruskal's conjecture that his algorithms give the optimal least squares solution of these modified monotone regression problems. We also propose another (third) approach for dealing with ties.

Signal/noise ratios for domain-referenced tests

Abstract: Using the assumption of randomly parallel tests and concepts from generalizability theory, three signal/noise ratios for domain-referenced tests are developed, discussed, and compared. The three ratios have the same noise but different signals depending upon the kind of decision to be made as a result of measurement. It is also shown that these ratios incorporate a definition of noise or error which is different from the classical definition of noise typically used to characterize norm-referenced tests.

Two Simple Classes of Mastery Scores Based On the Beta-Binomial Model.

Abstract: Two simple classes of mastery scores which are suitable for hand calculations are presented for beta-binomial test score distributions combined with linear and cubic referral success. The models provide a simple way to explore the consequences of selecting an arbitrary mastery score. Such assessment would be useful whenever the test user is not willing to posta priori a loss ratio, but wishes to look at the various consequences before aiming at a particular score.

A method of estimating item characteristic functions using the maximum likelihood estimate of ability

Abstract: A method of estimating item characteristic functions is proposed, in which a set of test items, whose operating characteristics are known and which give a constant test information function for a substantially wide range of ability, are used. The method is based on the maximum likelihood estimates of ability for a group of several hundred examinees. Throughout the present study the Monte Carlo method is used.

Two-step weighted least squares factor analysis of dichotomized variables

Abstract: A two-step weighted least squares estimator for multiple factor analysis of dichotomized variables is discussed. The estimator is based on the first and second order joint probabilities. Asymptotic standard errors and a model test are obtained by applying the Jackknife procedure.

A monte carlo investigation of recovery of structure by alscal

Abstract: A Monte Carlo study was carried out to investigate the ability of ALSCAL to recover true structure inherent in simulated proximity measures. The nature of the simulated data varied according to (a) number of stimuli, (b) number of individuals, (c) number of dimensions, and (d) level of random error. Four aspects of recovery were studied: (a) SSTRESS, (b) recovery of true distances, (c) recovery of stimulus dimensions, and (d) recovery of individual weights. Results indicated that all four measures were rather strongly affected by random error. Also, SSTRESS improved with fewer stimuli in more dimensions, but the other three indices behaved in the opposite fashion. Most importantly, it was found that the number of individuals, over the range studied, did not have a substantial effect on any of the four measures of recovery. Practical implications and suggestions for further research are discussed.