Showing papers in "Psychometrika in 1962"

The analysis of proximities: Multidimensional scaling with an unknown distance function. I.

Abstract: A computer program is described that is designed to reconstruct the metric configuration of a set of points in Euclidean space on the basis of essentially nonmetric information about that configuration. A minimum set of Cartesian coordinates for the points is determined when the only available information specifies for each pair of those points—not the distance between them—but some unknown, fixed monotonic function of that distance. The program is proposed as a tool for reductively analyzing several types of psychological data, particularly measures of interstimulus similarity or confusability, by making explicit the multidimensional structure underlying such data.

A recruitment theory of simple behavior

Abstract: A statistical theory of choice is developed using a sequential sampling assumption. Response latency distributions for certain simple reaction-time situations are derived and tested. Both response probability and response latency measures are developed for a two-alternative judgment situation and the relationship between the two measures explored. The sampling parameter is proposed as a means of representing incentive conditions in choice situations and ROC curves are obtained by appropriate manipulations of this parameter. A solution to the overlap problem in simple discrimination-learning situations is also derived.

Sample and population score matrices and sample correlation matrices from an arbitrary population correlation matrix

Abstract: A method for generating sample and population score matrices and sample correlation matrices from a given population correlation matrix is developed. An example giving the desired matrices for a population Guttman simplex correlation matrix is presented.

Some Rao-Guttman relationships.

Abstract: An examination of the determinantal equation associated with Rao's canonical factors suggests that Guttman's best lower bound for the number of common factors corresponds to the number of positive canonical correlations when squared multiple correlations are used as the initial estimates of communality. When these initial communality estimates are used, solving Rao's determinantal equation (at the first stage) permits expressing several matrices as functions of factors that differ only in the scale of their columns; these matrices include the correlation matrix with units in the diagonal, the correlation matrix with squared multiple correlations as communality estimates, Guttman's image covariance matrix, and Guttman's anti-image covariance matrix. Further, the factor scores associated with these factors can be shown to be either identical or simply related by a scale change. Implications for practice are discussed, and a computing scheme which would lead to an exhaustive analysis of the data with several optional outputs is outlined.

A theoretical distribution for mental test scores

Abstract: The negative hypergeometric distribution of raw scores on mental tests is derived from certain assumptions relating to test theory. This result is checked empirically in a number of examples. Further derivations lead to the bivariate distribution of parallel tests which is also verified with actual data. The bivariate distribution of raw score and true score is also derived from a further assumption. This distribution is used to set confidence limits for true scores for persons with a given raw score.

A general approach to nonlinear factor analysis

Abstract: The method presented attempts to allow for nonlinear, possibly nonmonotonic relations between manifest and latent variates. An attempt is made to provide a workable criterion for choosing between alternative models on the basis of observable data as well as for constructing the appropriate function. An idealized numerical example is given.

Optimal scaling for ordered categories

Abstract: This paper deals with the determination of optimal weights for points on scoring scales for subjective comparative experiments. A scoring scale with a specific number of points is considered, and it is assumed that verbal or other indications imply an order to the scale points. The optimal spacing for the scale points is obtained in the sense that treatment or item differences are maximized relative to error or within-treatment variation. The method is presented in sufficiently generalized form to be used directly with any experimental design leading to the analysis of variance. An iterative procedure, suitable for computer use, yields the optimal differences among the ordered scale points. Properties of this procedure are discussed.

Random fluctuations of response rate

Abstract: A simple model for fluctuating interresponse times is developed and studied. It involves a mechanism that generates regularly spaced excitations, each of which can trigger off a response after a random delay. The excitations are not observable, but their periodicity is reflected in a regular patterning of responses. The probability distribution of the time between responses is derived and its properties are analyzed. Several limiting cases are also examined.

Formulas for component scores

Abstract: General formulas for obtaining scores for individuals on components (factors derived from correlation matrices with unit communality estimates) are given. They are specialized to give Hotelling's formula for principal component scores. Formulas for scores on components rotated from principal components are developed.

An assessment of some structural properties of the jungian personality typology1

Abstract: Jung's typology implies that the attitudes and functions are (a) stable; (b) categorical or qualitatively dichotomous; (c) interacting; and (d) giving rise to different compounds of surface traits. In addition, it implies that type indeterminacy produces ineffective and maladjusted behavior. A program of studies aimed at assessing these structural properties of Jung's typology was designed and executed, using a self-report inventory, the Myers-Briggs Type Indicator. These studies found that (a) the type classifications had moderate temporal stability; (b) Indicator score distributions showed no marked evidence of bimodality; (c) with one exception, the regressions of other variables on Indicator scales did not change in either level or slope at the zero point of the Indicator scales; (d) the Indicator scales did not interact in relation to other variables; (e) the Indicator scales did not moderate the regression of other variables on one another; and (f) type indeterminacy was unrelated to measures of ineffective behavior and maladjustment. It was concluded that these findings offer little support for any of the structural properties attributed to the typology. Two possible interpretations of these negative findings were suggested: (a) Jung's typology is not consistent with the real world; or (b) the Indicator does not correspond to the theoretical formulation of the typology; that is, the Indicator does not operationally define the typology. Verification of these interpretations awaits further research.

On the statistical treatment of residuals in factor analysis

Abstract: A method for estimation in factor analysis is presented. The method is based on the assumption that the residual (specific and error) variances are proportional to the reciprocal values of the diagonal elements of the inverted covariance (correlation) matrix. The estimation is performed by a modification of Whittle's least squares technique. The method is independent of the unit of scoring in the tests. Applications are given in the form of nine reanalyses of data of various kinds found in earlier literature.

Use of the wilcoxon statistic for a generalized behrens-fisher problem1

Abstract: 0. Summary. Heretofore, the ordinary Wilcoxon statistic for the two-sample problem [9], [5] has been used only to test the null hypothesis that the two parent populations are identical. This paper presents a technique for utilizing the Wilcoxon statistic to test a broader type of null hypothesis, like that encountered in the Behrens-Fisher problem: we show that the usual Wilcoxon test, with (m + n + 1)/12mn replaced by 1/[4 min(m, n)], may be used to test the null hypothesis of the equality of the medians of two symmetrical (continuous) distributions which are of the same form but which have different (unknown) scale parameters; more generally, the test still works for testing the equality of the medians of any two symmetrical distributions. 1. Introduction and statement of results. We have a sample X m Xi's and a sample of n Yj's from parent populations with c.d.f.'s G(x) anld H(y) respectively. It is known that

A stochastic model for free recall

Abstract: A statistical model for verbal learning is presented and tested against experimental data. The model describes a Markov process with a realizable absorbing state, allowing complete learning on some finite trial as well as imperfect retention prior to this trial.

The relation between factor loadings and biserial correlations in item analysis

Abstract: The first centroid factor loadings obtained from various interitem relations are compared with item discrimination indices commonly used in item analysis. Depending upon what type of matrix is factored, the factor loadings are shown to be related to point biserial and biserial correlations.

Cutting scores and errors of measurement

Abstract: Arguments for using multiple cutting scores are theoretically inapplicable when the selection measures are fallible. The effect of errors of measurement in altering the shape of some optimum selection regions is here investigated mathematically, with numerical illustrations, for the case of two selection variables.

Scaling a simplex

Abstract: A least-squares solution for scaling the variables of a Guttman simplex is developed. The procedure yields a ratio scale, two varieties of interval scale, and orders the variables. A measure of the goodness of fit of the scale to the data is suggested. An example of the application of the method is given. The problem of non-positive correlations is discussed.

A note on the derivation of the general latent class model

Abstract: A general formulation of the latent structure principle is suggested, from which it is possible to derive Lazarsfeld's accounting equations in their most general form. The basic equations of Gibson's latent profile model can thence be derived in a single step.

On the least-squares orthogonalization of an oblique transformation

Abstract: After proving a special case of a theorem stated by Eckart and Young, namely, that an oblique transformationG is the product of two different orthogonal transformations and an intervening diagonal, this note shows that the best fitting orthogonal approximation toG is obtained simply by replacing the intervening diagonal by the identity matrix. This result is shown to be identical with two earlier orthogonalizing procedures whenG is of full rank. A multiplicity of solutions is shown for the case of a singularG.

Simulation of human problem-solving behavior

Abstract: In the simulation of human behavior on a digital computer, one first attempts to discover the manner in which subjects (Ss) internally represent the environment and the rules that they employ for acting upon this representation. The interaction between the rules and the environmental representation over a period of time constitutes a set of processes. Processes can be expressed as flow charts which, in turn, are stated formally in terms of a computer program. The program serves as a theory which is tested by executing the program on a computer and comparing the machine's performance with S's behavior.

Formula scoring and validity

Abstract: Formulas are derived for a rather restricted, situation showing the decrement in test validity that might be expected from random guessing and the increment in validity that might be expected from a priori formula scoring. Numerical illustrations are given. The reduction in validity that occurs when formula scoring is abandoned is generally found to be small in terms of the usual correlational scale, but not necessarily small when considered in terms of the reduction in test length that could be achieved with the same sacrifice of validity.

Analytic rotation to a functional relationship

Abstract: It is suggested that a number of theoretical models for non-test data can be tested by factor analysis. A traditional method for transforming principal-axis factor solutions is used as an aid in making such tests; a modification of this method is suggested. Means of using all of the information in a model for the purpose of determining the transformation of the arbitrary factor solution are presented, and an illustrative example is given.

Regression analysis of linear composite variance

Abstract: The problem of defining and determining the effective contribution of a component variable to the variance of a composite is briefly reviewed. Another method of dealing with this problem is proposed and illustrated with analysis of a three-component problem. The proposed method is based on systematic application of a series of regressions and is restricted in application to positive manifold systems.

A method of determining unbiased distribution in the latin square

Abstract: The paper describes a method of constructing Latin-square designs in which treatment sequences are unbiased with regard to serial order as well as position of treatment. The procedure is useful for those Latin squares in which the number of cells in each column (or row) is an even number, which, when written in ascending and descending series, contain corresponding ordered numbers prime to each other. Such numbers are 2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, etc.

Mathematical analysis of the durations of reinforced inter-response times during variable interval reinforcement

Abstract: In a free-responding situation in which reinforcements are scheduled by a variable interval program, the durations of reinforced inter-response times are shown to be a function of the durations of all inter-response times produced by S and of the frequency of reinforcement.

Comparative prediction of success and satisfaction in high school curricula

Abstract: Comparative Prediction Project. As a project intended to investigate the possibility of comparative prediction in school and college by means of pure-factor tests, this investigation started out with three factor studies to complete the establishment of factors in areas thought to be important in academic work (RB-53-16, RB-55-22, RB-54-7). A ground-laying bulletin discussed other comparative and differential prediction batteries, statistical methodology, and the presentation of results to counselors (RB-55-4). At the college level, a preliminary experiment was carried out with West Point cadets to select the most appropriate tests (RB-51-12, RB-55-6 or French, 1955) and a principal validity study involved tryout of the battery against major-field grades at eight colleges (RB-59-10, RB-59-15 or French, 1961b, RB-61-7). At the high-school level, a preliminary experiment was carried out with New Jersey vocational schools (RB-56-1). A memorandum has been issued on evaluation and presentation of results for the college and high-school levels (RM-61-13). The present bulletin is the complete technical report on the high-school study. High-School Study. A battery, consisting of 13 “pure-factor” aptitude tests, 7 information (interest?) scores, and 10 personality factor scales, was given to beginning tenth graders in 49 Massachusetts high schools. The scores were validated against grades in the various academic and trade courses given during the tenth, eleventh, and twelfth grades. They were also validated against satisfaction with chosen curriculum. Multiple validities were computed to show the effectiveness of the tests for absolute prediction; those for academic subjects ranged from .34 for boys in social science through .60 for girls in English; those for trade courses ranged from .26 for auto repair through .71 for home economics. Validities for satisfaction were all very low. Estimated validities of predicted differences between course grades had a median of .31 for academic boys, .40 for academic girls, and .50 for trade courses. Corrections of these values for shrinkage and for the short length of the experimental tests are considered in the Appendix. It is concluded that these corrections can be ignored. The results forecast that a test battery of this kind by itself would have only moderate value for differential and comparative prediction in practice, but, when used in conjunction with school grades and other information, it would make a substantial contribution. Better differential prediction obtained in the college-level study was recognized to have been brought about by the use of self-report interest scales. These could be used equally well at the high-school level.

A functional equation analysis of two learning models

Abstract: One-absorbing barrier random walks arising from Luce's nonlinear beta model for learning and a linear commuting-operator model (called the alpha model) are considered. Functional equations for various statistics are derived from the branching processes defined by the two models. Solutions to general functional equations, satisfied by statistics of the alpha and beta models, are obtained. The methods presented have application to other learning models.

Extending latent class solutions to other variables

Abstract: A procedure is derived for estimating the latent parameters of items not initially included in a latent class solution, on the assumption that the relations between those additional items and the original ones are accounted for by the same latent structure. A chi-square test is proposed for evaluating the relatedness of the additional items to the latent structure. The extension is generalized to associate continuous outside variables with the original solution, and this is accompanied by the suggestion that a simple analysis of variance be used to assess the results. Finally, this latent structure extension is compared with the Dwyer extension of factor analysis.

A note on mathematical models for learning

Abstract: It is shown that a rational learning curve developed by Estes for paired associate learning is a special case of a more general function. The latter is the product of two functions and assumes that the discovery and fixation aspects of learning are independent. The indications are that the form of one of the functions has not been tested sharply by Estes in the paired associate learning setting.

Matrix reduction and approximation to principal axes

Abstract: A modification of Hotelling's iteration method of factor analysis is presented which is much more rapid and almost as accurate. At any stage of the approximation for a factor vector its major product moment reduces the rank of the residual matrix by precisely one. Each approximation to an eigenvalue is larger than the preceding one. By observing the decline in these increments one can often stop the iterations at early stages without serious loss. If subsequent rotational procedures are used, the method gives practically the same results as the more exact methods and in a small fraction of the time.

The role of memory and color in group and individual embedded‐figures measures of field independence1

Abstract: Three group-administered embedded-figures tests were designed to evaluate the possibility of developing economical and reliable measures of perceptual field-independence. These were administered together with individual and group forms of Witkin's EFT in a factorial design to study the effects of color, memory, and item format. Embedded-figures test performances were also studied in relation to perceptual speed, spatial orientation, and Adjective Preference Scale scores. Within-group correlations between group- and individually-administered embedded-figures tests were sufficiently high to warrant the use of the group forms as measures of perceptual field-independence. A comparison of the correlational patterns between sexes indicated greater differentiation in cognitive performances for males than for females.