bluuebunny/crossref_metadata_2025 huggingface dataset:
The Geometry of Factorial Indeterminancy
Authors: Heermann Emil F.
Published: January 2006

The N-dimensional geometry of a Spearman-Thurstone factor solution reveals two sources for the indeterminancy of factor scores: indeterminancy of total factor space and a rotational indeterminancy within a given total factor space. The analytical papers of Ledermann [4] and Guttman [2] on indeterminancy of factor scores are related to these findings and a simple vector model is developed to reveal the properties of rotational indeterminancy. The significance of factor-score indeterminancy is discussed in light of these findings.
Simplified Regression Patterns for Classification
Authors: Brogden Hubert E.
Published: January 2006

A transformation of a set of partial regression equations to a potentially simple form is shown to be permissible for personnel classification purposes when all individuals are assigned to jobs. Simplification is achieved by maximizing the number of near zero partial regression weights.
The Mean and Standard Deviation of the Distribution of Group Assembly Sums
Authors: Dwyer Paul S.
Published: January 2006

An interesting problem in linear programming is the group assembly problem which is mathematically equivalent to the general transportation problem of economics. Computer programs designed for the determination of exact and approximate optimal group assemblies have been available for some time. This paper presents formulas for the mean and squared standard deviation of the distribution of all possible group assembly sums. Computational techniques are presented and the results are related to those of the analysis of variance of a k-factor problem with n levels of each factor.
Comparisons of Power for Some Exact Multinomial Significance Tests
Authors: Gurian Joan M. , Cornfield Jerome , Mosimann James E.
Published: January 2006

Two small-sample tests (proposed by Tate and Clelland and by Chapanis respectively) of hypotheses about the parameters of the multinomial distribution, where
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$$f(x|p) = n!\prod\limits_{i = 1}^k {\frac{{p_i^{x_i } }}{{x_i !}}} $$
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are described. A proof is given that a test of the form
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$$R = \frac{{\smallint f(x|p)g(p)dp}}{{f(x|p_0 )}}$$
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where g(p) ≥ 0 and p0 is the vector of p's specified by the hypothesis, is admissible. It is shown that the Tate and Clelland test is obtained by setting g(p) equal to a constant for all p and is therefore admissible. The Chapanis test is shown to have power less than or equal to the power of the Tate and Clelland test for (k = n = 3) and (k = 3, n = 4).
Statistical Inferences about the Error Variance
Authors: Kristof Walter
Published: January 2006

This paper is a presentation of an essential part of the sampling theory of the error variance and the standard error of measurement. An experimental assumption is that several equivalent tests with equal variances are available. These may be either final forms of the same test or obtained by dividing one test into several parts. The simple model of independent and normally distributed errors of measurement with zero mean is employed. No assumption is made about the form of the distributions of true and observed scores. This implies unrestricted freedom in defining the population. First, maximum-likelihood estimators of the error variance and the standard error of measurement are obtained, their sampling distributions given, and their properties investigated. Then unbiased estimators are defined and their distributions derived. The accuracy of estimation is given special consideration from various points of view. Next, rigorous statistical tests are developed to test hypotheses about error variances on the basis of one and two samples. Also the construction of confidence intervals is treated. Finally, Bartlett's test of homogeneity of variances is used to provide a multi-sample test of equality of error variances.
Comments on Parameter Estimation for Restle's Theory of Cue Learning
Authors: Kraemer Helena Chmura
Published: January 2006

Improved estimation procedures are suggested and a more complete distribution theory is presented for Restle's Cue Learning Model.
The Analysis of Proximities: Multidimensional Scaling with an Unknown Distance Function. II
Authors: Shepard Roger N.
Published: January 2006

The first in the present series of two papers described a computer program for multidimensional scaling on the basis of essentially nonmetric data. This second paper reports the results of two kinds of test applications of that program. The first application is to artificial data generated by monotonically transforming the interpoint distances in a known spatial configuration. The purpose is to show that the recovery of the original metric configuration does not depend upon the particular transformation used. The second application is to measures of interstimulus similarity and confusability obtained from some actual psychological experiments.
Analytic Rotation to a Functional Relationship
Authors: Cliff Norman
Published: January 2006

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.
Informational Coverage and Correlational Analysis
Authors: Ross John
Published: January 2006

The question “How much does a set of variables tell us?” is formulated in terms of information measures, and a set of formulas is established which allows appropriate information measures to be derived from correlations. Particular attention is given to the question in the context of factor analysis and to formulas involving the latent roots of correlation matrices.
A Method of Determining Unbiased Distribution in the Latin Square
Authors: Alimena Benjamin S.
Published: January 2006

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.
An Observable Property Equivalent to a Choice Model for Discrimination Experiments
Authors: Luce R. Duncan
Published: January 2006

The choice model considered by Luce in an earlier work is stated in terms of unobservable parameters. In this paper a consequence of the model, involving only observables, is shown to be equivalent to the model.
On the Least-Squares Orthogonalization of an Oblique Transformation
Authors: Gibson W. A.
Published: January 2006

After proving a special case of a theorem stated by Eckart and Young, namely, that an oblique transformation G is the product of two different orthogonal transformations and an intervening diagonal, this note shows that the best fitting orthogonal approximation to G is obtained simply by replacing the intervening diagonal by the identity matrix. This result is shown to be identical with two earlier orthogonalizing procedures when G is of full rank. A multiplicity of solutions is shown for the case of a singular G.
Optimal Scaling for Ordered Categories
Authors: Bradley R. A. , Katti S. K. , Coons Irma J.
Published: January 2006

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.
A Recruitment Theory of Simple Behavior
Authors: LaBerge David
Published: January 2006

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.
Properties and Applications of Gramian Factoring
Authors: Gibson W. A.
Published: January 2006

The Gramian factorization G of a Gramian R is square and symmetric and has no negative characteristic roots. It is shown to be that square factorization that is, in the least-squares sense, most isomorphic to R, most like a scalar K, and most highly traced, and to be the necessary and sufficient relation between the oblique vectors of an oblique transformation and the orthogonal vectors of the least-squares orthogonal counterpart. A slightly modified Gramian factorization is shown to be the factorization that is most isomorphic to a specified diagonal D, and to be the main part of an iterative procedure for obtaining “simplimax,” a square factor matrix with simple structure maximized in the sense of having the largest sum of squared diagonal loadings. Several published applications of Gramian factoring are cited.
Some Contributions to Maximum Likelihood Factor Analysis
Authors: Jöreskog K. G.
Published: January 2006

A new computational method for the maximum likelihood solution in factor analysis is presented. This method takes into account the fact that the likelihood function may not have a maximum in a point of the parameter space where all unique variances are positive. Instead, the maximum may be attained on the boundary of the parameter space where one or more of the unique variances are zero. It is demonstrated that such improper (Heywood) solutions occur more often than is usually expected. A general procedure to deal with such improper solutions is proposed. The proposed methods are illustrated using two small sets of empirical data, and results obtained from the analyses of many other sets of data are reported. These analyses verify that the new computational method converges rapidly and that the maximum likelihood solution can be determined very accurately. A by-product obtained by the method is a large sample estimate of the variance-covariance matrix of the estimated unique variances. This can be used to set up approximate confidence intervals for communalities and unique variances.
Extending Latent Class Solutions to Other Variables
Authors: Gibson W. A.
Published: January 2006

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.
Formulas for Component Scores
Authors: Kaiser Henry F.
Published: January 2006

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.
Some Uses of the Square Root Method as a Tool in Multiple Prediction
Authors: Hazewinkel Aart
Published: January 2006

It is shown how the square root method can be used in the derivation of formulas in multiple prediction and in the computations necessary for their application. A formula derived by Horst indicates the validity a test must possess in order to contribute a specified amount to the multiple correlation of a battery. Then a formula is derived which relates the length of one test in a battery to the multiple correlation of the battery as a whole. The formulas hold for population parameters; in this paper, sampling theory is not discussed. A numerical example is presented; it is shown that one square root analysis provides all the data needed for the formulas.
A Statistical Model which Combines Features of Factor Analytic and Analysis of Variance Techniques
Authors: Gollob Harry F.
Published: January 2006

This paper describes a method of matrix decomposition which retains the ability of factor analytic techniques to summarize data in terms of a relatively low number of coordinates; but at the same time, does not sacrifice the useful analysis of variance heuristic of partitioning data matrices into independent sources of variation which are relatively simple to interpret. The basic model is essentially a two-way analysis of variance model which requires that the matrix of interaction parameters be decomposed by using factor analytic techniques. Problems of judging statistical significance are discussed; and an illustrative example is presented.
Linear and Multidimensional Scaling
Authors: Gulliksen Harold
Published: January 2006

Fechner’s work [12] was motivated by an interest in the mind-body problem concentrating particularly upon the question, “What is the relationship between the measurable characteristics of the physical stimulation and the subjective characteristics of the sensations produced?” Some of Fechner’s work was devoted to experiments in areas where the physical measurements that can be made on the object are not functionally related to the psychological characteristics.
Measurement of Learning and Mental Abilities
Authors: Gulliksen Harold
Published: January 2006

About 27 years ago, a small group of students met with Professor Thurstone in Chicago to discuss methods of encouraging quantitative work in psychology. The initial group that was concerned about the slow rate of development of quantitative work in psychology included Jack Dunlap, Al Kurtz, Marion Richardson, John Stalnaker, G. Frederic Kuder, and Paul Horst. They had discussed the problem, had been helped a bit by Donald Paterson, and had decided that possibly if a magazine were set up to publish quantitative psychological material this would facilitate the development of the field. Persons who did good quantitative work, either theoretical or experimental, would thus have a forum where it would be accepted because it was high quality quantitative work, rather than being rejected because it was quantitative and hence “not of too great interest” to the readers.
Psychological Measurement a Hundred and Twenty-Five Years Later
Authors: Guilford J. P.
Published: January 2006

When writing this paper as well as during our commemorative luncheon, I could not help going back in memory to two experiences. One of these occurred during the fall semester of 1935 while I was visiting professor at Northwestern University and took advantage of the opportunity and had the rare privilege of attending Thurstone’s seminar. The atmosphere at the University of Chicago at that time clearly reflected the birth and early infancy of both the Psychometric Society and its Journal Psychometrika.

The other experience comes back in the form of an image of our first annual dinner in the Inn at Dartmouth College in September 1936. The banquet had an unexpectedly large and enthusiastic attendance. Thurstone provided one of his memorable addresses, which was later published in Science, entitled “Psychology as a quantitative rational science” [35]. The address is well worth rereading.

I might spend the time alloted to me in declaiming the merits of quantitative psychology and what it has done for the progress of psychology in general. But in this group I shall take our substantial contribution for granted and try instead to help measurement psychologists look at themselves and their own progress and some of their own problems.
Estimation in the Three-State Markov Learning Model
Authors: Kraemer Helena Chmura
Published: January 2006

The problem of estimation of the parameters of the Bower three-state learning model is discussed for three forms of the model. In the simplest case, a minimum variance unbiased estimator is found and is presented with its asymptotic distribution theory and with a method of obtaining approximate confidence intervals. In the other cases, methods are discussed for obtaining the maximum likelihood estimators at least approximately. All estimation techniques are illustrated by application to a set of data obtained by Theios.
A Graph Theoretic Approach to Similarity Relations
Authors: Harary Frank
Published: January 2006

A similarity relation is reflexive and symmetric. A graph may be regarded as a geometric representation of an irreflexive symmetric relation; an r-graph then coincides with a similarity relation. For the realization of psychophysical phenomena, the points of a graph stand for the stimuli and a line for indistinguishability of a stimulus pair. Not all graphs, however, are likely to arise as similarity relations in a real situation. A special class of graphs, called interval graphs, does have this realistic property. These graphs are described and characterized.
Relations among m Sets of Measures
Authors: Horst Paul
Published: January 2006

The problem of determining linear functions for two sets of variables so as to maximize the correlation between the two functions has been solved by Hotelling. This article presents a more efficient computational solution for the case of two sets of variables and a generalized solution for any number of sets. Applications are discussed and a numerical example is included to demonstrate the solution for more than two sets.
A General Procedure for Obtaining Paired Comparisons from Multiple Rank Orders
Authors: Gulliksen Harold , Tucker Ledyard R
Published: January 2006

From a theoretical point of view, paired comparisons and the law of comparative judgment provide an excellent approach to the problem of psychological measurement. However, if a reasonably large number of stimuli are to be investigated, paired comparisons become extremely time-consuming and fatiguing to the subjects. A balanced incomplete block design, requiring multiple rank order judgments for each subject, provides an efficient experimental method for obtaining paired comparisons judgments. Features of the analysis proposed for this design are discussed in detail. A program for the analysis is available for the IBM 650 electronic computer.
A Multivariate Analysis of an Experimental Design Involving a Complete Set of Latin Squares
Authors: Chan Man-Wai
Published: January 2006

A model for an experimental design involving a complete set of Latin squares for testing the homogeneity of treatment effects was constructed and analyzed by Gourlay. In his analysis, however, if one or both of the preliminary F-tests are significant, the analysis cannot differentiate. He then suggests the use of a less desirable test which is biased and has fewer degrees of freedom, regardless of the number of replications (the d.f. cannot be increased by increasing the replications). Further, when heterogeneity of variance occurs, Gourlay's test procedures are in general invalid. The present paper reviews Gourlay's analysis and proposes a modified test procedure.
Measures of Invariance and Comparability in Factor Analysis for Fixed Variables
Authors: Pinneau Samuel R. , Newhouse Albert
Published: January 2006

New procedures are presented for measuring invariance and matching factors for fixed variables and for fixed or different subjects. Two of these, the coefficient of invariance for factor loadings and the coefficient of factor similarity, utilize factor scores computed from the different sets of factor loadings and one of the original standard score matrices. Another, the coefficient of subject invariance, is obtained by using one of the sets of factor loadings in conjunction with the different standard score matrices. These coefficients are correlations between factor scores of the appropriate matrices. When the best match of factors is desired, rather than degree of resemblance, the method of assignment is proposed.
Reliability Formulas for Independent Decision Data When Reliability Data are Matched
Authors: Rajaratnam Nageswari
Published: January 2006

A distinction is made between reliability data and decision data. Each of these sets of data may be matched or independent, depending on whether the same instruments (tests, judges, etc.) are applied to every individual in the group or the instruments to be applied to each individual are selected independently for him. Reliability formulas are developed (for both single observations and for composites of k observations) for the case where reliability data are matched but decision data are independent. Formulas previously reported in the literature are inappropriate for this case.
A Proposed Variation of the Matching Technique
Authors: Weinberg George H. , Fluckiger Fritz A. , Tripp Clarence A.
Published: January 2006

A matching procedure is proposed by which a judge may use the same items in different matches. The K items in one group most likely to include the correct match with each item in the other are selected. Inclusion of the correct match among the K items chosen is defined as a success. The distribution of the number of successes is discussed. Tables are presented showing the number of successes needed for significance for various values of K and of N, the number of items in each group.
Nonlinear Probit Analysis and its Application to Psychometric Data
Authors: Maritz J. S.
Published: January 2006

Although probit analysis has been applied in the field of psychometrics, its use has been restricted by the requirement of linear regression of probits on the independent variate. This paper shows that polynomial probit analysis, a natural extension of the idea of probit analysis, can sometimes overcome the difficulty of a nonlinear plot and does not involve excessive computational labor.
Generalizability of Stratified-Parallel Tests
Authors: Rajaratnam Nageswari , Cronbach Lee J. , Gleser Goldine C.
Published: January 2006

One of the major concerns of reliability theory has been the estimation of the reliability of a composite measure from the degree of agreement among its component parts. In the classical theory, formulas were developed under the assumption that the parts are strictly equivalent. It was later shown that the same formulas follow from various sets of weaker assumptions which require the composites to be strictly equivalent and require the parts to have a certain homogeneity of statistical properties, but not necessarily to be equivalent. An alternative model which has received increasing attention in recent years regards a given measure as a random sample from a universe of measures whose homogeneity or equivalence is not specified a priori, and a composite test as a random sample of items from a universe of not-necessarily-equivalent items. This too permits an internal-consistency estimate of reliability. Both the equivalent-composites model and the randomsampling model appear to be unduly restrictive and unrealistic; we propose here to develop the implications of a third model in which a test is considered to have been formed by stratified sampling of items.
Linear Correlations between Sets of Variables
Authors: Rozeboom William W.
Published: January 2006

While the traditional multiple correlation coefficient appears to be inherently an asymmetrical statistic, it is actually a special case of a more general measure of linear relationship between two sets of variables. Another symmetric generalization of linear correlation is to the total relatedness within a set of variables. Both of these developments rest upon the generalized variance of a multivariate distribution, which is seen to be the fundamental concept of linear correlational theory.
Investigations of the Simplex
Authors: Humphreys Lloyd G.
Published: January 2006

I am honored to be able to speak on the 25th anniversary of the founding of the Psychometric Society. It would be appropriate for your president on this occasion to review the 25-year history of our society and of mathematical methods in psychology. Fortunately our committee headed by Charles Wrigley has taken care of this need, and speakers better qualified than I will speak to these issues. I shall turn to a less demanding problem.

In limiting my topic I was faced with certain constraints arising from my training, background, and interests. One constraint arises from a misspent youth during which too many years were spent studying disciplines other than mathematics. A second constraint, which probably stems from my training in experimental psychology, is a tendency to be more interested in data than theory. Finally, there is the Air Force personnel research experience which developed in me an abiding interest in applied prediction problems. If you consider this combination to be an odd one for the President of the Psychometric Society, don't blame me. You elected me!
Remarks about a Paired Comparison Model
Authors: Noether Gottfried E.
Published: January 2006

A generalization of the customary model underlying the Law of Comparative Judgment is established. Methods for estimating location parameters and testing a certain hypothesis are discussed. The problem of the influence of the shape of the curve used to grade responses is raised and a possible approach to its solution is indicated. Two examples are provided.
Hierarchical Factor Solutions without Rotation
Authors: Wherry Robert J.
Published: January 2006

A method is presented for securing a hierarchical factor solution which achieves simple structure at each hierarchical level without rotation or even preliminary arbitrary orthogonal or oblique solutions. The method is based upon the assumption that if overlap is removed from clusters the remaining specifics will achieve simple structure automatically. The problem presented earlier by Schmid and Leiman, using oblique simple structural rotation as a basis, is reworked by this new approach.
A Generalization of the Bush-Mosteller Model with Some Significance Tests
Authors: Hanania Mary I.
Published: January 2006

A generalization of the Bush-Mosteller learning model is proposed in connection with two-alternative learning situations with continuous reinforcement. The problem is to test the hypothesis that reward and non-reward are equally effective in promoting learning. Statistically this reduces to testing a hypothesis about the value of a single parameter θ, while a set of other parameters remains unspecified. The test presented has the property of being asymptotically locally most powerful among all tests of the same size and asymptotically similar. The application of the test is illustrated.
Stochastic Learning Theories for a Response Continuum with Non-Determinate Reinforcement
Authors: Suppes Patrick , Zinnes Joseph L.
Published: January 2006

Continuous analogues of the finite linear and stimulus sampling theories are developed for non-determinate reinforcement schedules. Closed-form expressions are derived for the asymptotic response distribution and certain sequential statistics. Computations for a target experiment are given to illustrate the character of the theoretical results.
Two Learning Models for Responses Measured on a Continuous Scale
Authors: Anderson Norman H.
Published: January 2006

Two linear operator models are presented for a class of learning situations in which the response is on a numerical scale and the subject is given the magnitude of his error on some or all of the trials. Theoretical expressions are developed for sequential dependencies, mean learning curves, variances, and covariances, which permit a number of tests of goodness of fit.
Geometrical Representation of Two Methods of Linear Least Squares Multiple Correlation
Authors: Fruchter Benjamin , Anderson Harry E.
Published: January 2006

Geometrical properties and relationships of the Doolittle and square root methods of multiple correlation, as represented in the variable subspace of an orthogonal person space, are shown. The method of representation is also useful for depicting zero-order and partial correlations, as well as for the more general problem of the combination of variables.
Some Relations between Guttman's Principal Components of Scale Analysis and Other Psychometric Theory
Authors: Lord Frederic M.
Published: January 2006

Guttman's principal components for the weighting system are the item scoring weights that maximize the generalized Kuder-Richardson reliability coefficient. The principal component for any item is effectively the same as the factor loading of the item divided by the item standard deviation, the factor loadings being obtained from an ordinary factor analysis of the item intercorrelation matrix.
Remarks on the Test of Significance for the Method of Paired Comparisons
Authors: Bock R. Darrell
Published: January 2006

A three-component model for comparative judgment which allows for individual differences in preference is proposed. An implication of the model is that errors in the observed proportions due to sampling individuals in paired comparisons experiments are correlated. By neglecting this correlation, Mosteller's test for the method of paired comparisons tends to accept falsely the goodness of fit of the Case V solution. It is shown that bounds may be set for the correlation effect which make a valid test possible in some cases and provide useful standard errors for the estimated affective values.
The Single Latin Square Design in Psychological Research
Authors: Gaito John
Published: January 2006

The expected value of mean square concept is used to determine the effects of the presence of interactions in the single Latin square design on F tests. The results indicate that as the number of random effects included in the experiment increase, more F tests are unbiased, and that some of these are valid F tests. However, when F test bias does occur it is almost always of a negative nature so that the conclusions stated are conservative ones. Positive F test bias may occur when the triple interaction is extant and when zero or one random variate is included in the experiment.
The Effect of Difficulty and Chance Success on Correlations between Items or between Tests
Authors: Carroll John B.
Published: January 2006

A study is made of the extent to which correlations between items and between tests are affected by the difficulties of the items involved and by chance success through guessing. The Pearsonian product-moment coefficient does not necessarily give a correct indication of the relation between items or sets of items, since it tends to decrease as the items or tests become less similar in difficulty. It is suggested that the tetrachoric correlation coefficient can properly be used for estimating the correlation between the continua underlying items or sets of items even though they differ in difficulty, and a method for correcting a 2 × 2 table for the effect of chance is proposed.
A Note on Steady States and the Weber-Fechner Law
Authors: Spiegelman S. , Reiner J. M.
Published: January 2006

The steady state of a simple reaction system has been shown to have some of the properties of a psychophysical discrimination system, including the possibility of deducing a generalized Weber-Fechner Law, both in integral form and in difference form. The Weber ratio so deduced is not constant, and its dependence on stimulus intensity is exhibited. The dependence of the difference limen on the internal threshold is discussed; it is found that in general there is a finite value of this threshold for which response is impossible. This critical threshold is lower for higher values of the reference stimulus intensity. Similarly, it is shown that the difference limen and the Weber ratio, for a fixed value of the threshold, become infinite (i.e., discrimination is impossible) for a value of the stimulus intensity which in general is finite.
A Synthesis of Two Factor Analyses of Intermediate Algebra
Authors: Kline William E.
Published: January 2006

A battery of 18 tests of intermediate algebra and 20 reference tests was administered to two successive second-year algebra classes. Each battery was separately factor analyzed by Thurstone methods, and the two analyses were synthesized by the Tucker method. The five congruent factors obtained were identified as: Verbal Comprehension, Deductive Reasoning, Algebraic Manipulative Skill, Number Ability, and Adaptability to a New Task.
Redundancy in Task Assignments and Group Performance
Authors: Zajonc Robert B. , Smoke William H.
Published: January 2006

The problem of combining abilities of group members to maximize the performance of the group as a whole is examined in terms of redundancy in task assignments. In particular, ways of distributing a given number of items of information among a given number of individuals to obtain the maximum probability of each item being recalled by at least one individual are studied. It is shown that there exists an optimal distribution scheme which is independent of the amount of material originally given, the size of the group, and individual differences in ability.
Least Squares Estimation in Finite Markov Processes
Authors: Madansky Albert
Published: January 2006

The usual least squares estimate of the transitional probability matrix of a finite Markov process is given for the case in which, for each point in time, only the proportions of the sample in each state are known. The purpose of this paper is to give another estimate of this matrix and to investigate the properties of this estimate. It is shown that this estimate is consistent and asymptotically more efficient than the previously considered estimate in a sense defined in this paper.
A Model for Ordered Metric Scaling by Comparison of Intervals
Authors: Fagot Robert F.
Published: January 2006

This paper presents a model of individual choice behavior for application to experimental situations in which a subject is required to compare utility intervals (differences in subjective value). This model is contrasted with a weaker model, which is also derived. Both models generate ordered metric scales, but differ in predictive power. An experiment on the utility of grades, which provides a test and comparison of the models, is presented.
A Note on Factor Analysis: Arbitrary Orthogonal Transformations
Authors: Cureton Edward E.
Published: January 2006

A modification of the Gram-Schmidt process yields an easily constructed orthogonal transformation matrix which may be used to rotate a centroid, principal axis, or maximum likelihood factor matrix in a manner such that one of the new axes has predetermined direction. The procedure is illustrated by rotating a centroid factor matrix into an abbreviated bifactor matrix, the general factor being defined as the centroid of a specified subgroup of reasoning tests.
Randomly Parallel Tests and Lyerly's Basic Assumption for the Kuder-Richardson Formula (21)
Authors: Lord Frederic M.
Published: January 2006

The K-R (21) formula can be derived without recourse to the undesirable assumption that the test items are all indistinguishable.
A Metric and an Ordering on Sets
Authors: Restle Frank
Published: January 2006

Beginning with sets of arbitrary elements, concepts of distance and betweenness of sets are defined. Since betweenness as defined is not transitive, an investigation is made of the conditions which ensure desirable regularity. It is found that a straight line or linear array of sets is a generalization of nested sets (Guttman scales). Close relationships among the notions of distance, betweenness, and linear arrays are demonstrated. Parallel and perpendicular arrays, dimensions, and multidimensional spaces are characterized.
Strategies and Learning Models
Authors: Bryant Steven J. , Marica John G.
Published: January 2006

A class of strategies is defined, each member of which possesses a certain plausibility. If a subject follows any strategy in this class in a two-choice learning experiment of the type dealt with by the Estes model, the subject's long-run behavior will be the same as that predicted by the Estes model.
An Improved Procedure for the Wherry-Winer Method for Factoring Large Numbers of Items
Authors: Wolins Leroy
Published: January 2006

A technique is presented that differs from the previous one in that the use of variance terms is eliminated from the computations; thus some formulas are simplified. A rationale for the improved method is presented.
Generating Variables with Arbitrary Properties
Authors: Hoffman Paul J.
Published: January 2006

There are occasions in psychological research where it is desirable to have available sets of variables with arbitrary intercorrelations. A quite simple procedure is described for generating pairs of such variables.
A Note on the use of Triads for Paired Comparisons
Authors: Schucker R. E.
Published: January 2006

When scaling a large number of stimuli from comparative judgments, considerable savings in time and labor may be realized if stimuli are presented in triad form rather than in pairs. If, for N stimuli, the proper configuration of triads can be assembled so that all possible pairs appear once, the paired judgment matrix may be reproduced with one-third fewer judgments and two-thirds fewer presentations than would be required with complete pairing. A simple procedure is described for enumerating triad configurations for which N is an odd multiple of three.
Variation in Test Validity with Variation in the Distribution of Item Difficulties, Number of Items, and Degree of their Intercorrelation
Authors: Brogden Hubert E.
Published: January 2006

The relation between item difficulty distributions and the “validity” and reliability of tests is computed through use of normal correlation surfaces for varying numbers of items and varying degrees of item intercorrelations. Optimal or near optimal item difficulty distributions are thus identified for various possible item difficulty distributions. The results indicate that, if a test is of conventional length, is homogeneous as to content, and has a symmetrical distribution of item difficulties, correlation with a normally distributed perfect measure of the attribute common to the items does not vary appreciably with variation in the item difficulty distribution. Greater variation was evident in correlation with a second duplicate test (reliability). The general implications of these findings and their particular significance for evaluating techniques aimed at increasing reliability are considered.
Test Selection and Suppressor Variables
Authors: Wherry Robert J.
Published: January 2006

A theoretical discussion of the factor pattern of predictor tests and criterion shows that ordinary test selection methods break down under certain circumstances. It is shown that maximal results may not occur if suppressor variables are present among the predictors. Suggested solutions to the problem include: (1) prior item analysis of tests against the criterion, (2) selection of several trial batteries including some with suppressor variables on the basis of a factor analysis of tests and criterion, (3) modification of the usual test selection procedures to include separate solutions based upon each of several starting variables, or (4) the cumbersome and tedious solution of all possible combinations of predictors. The solutions are recommended in the order named above. Although all of the suggested solutions involve added labor and may not be necessary, the test or battery constructor should at least be aware of the problem.
On the Theory of Test Discrimination
Authors: Ferguson George A.
Published: January 2006

This paper discusses the properties of distributions of test scores and advances the view that the properties of the distribution should depend on the function which the test is intended to perform. A theory of test discrimination is developed which defines discriminatory capacity in terms of the number of relations of difference established by the operation of administering a test of k items to a sample of n individuals. A simple proof is presented which indicates that maximum discrimination between individuals is achieved when tests are constructed to yield distributions of the rectangular form. A coefficient of test discrimination is developed. The problem of obtaining in practice distributions approximating to the rectangular form is briefly discussed.
The Theory of Abstract Partials: An Introduction
Authors: Rozeboom William W.
Published: January 2006

It is customary in multivariate analysis to search out relational structure by partitioning measures of total variation or overall relatedness into components variously attributable to different sources. Although the multivariate systems most familiar today–notably, linear correlational analysis, analysis of variance, and Information theory–base their decompositions upon very different substantive quantities, they all embody the very same abstract mathematical structure which, moreover, is capable of generating potentially fruitful patterns of data analysis in many other substantive applications as well.
Comments on “Markovian Processes with Identifiable States: General Considerations and Applications to All-or-None Learning”
Authors: Greeno James G. , Steiner Theodore E.
Published: January 2006

When knowledge of the response sequence permits one to specify the sequence of theoretical states that occurred, we propose to say that the theory has observable states. Our earlier usage of “identifiable” conflicts with established usage in econometrics. We also present a revised definition of implication between theories that avoids a technical flaw in the earlier definition, and we correct a mistake in an example in our earlier paper.
A Remark on Bower's Estimate for the One-Element Markov Learning Model
Authors: Holland Paul W.
Published: January 2006

For the case of the one-element Markov learning model for which the guessing parameter p is assumed known the efficiency of Bower's method of moments estimator c+ for the learning parameter, c, is found. It seems that although c+ is not efficient, for some practical value of c it is not very inefficient. If p and c are small, ĉ, the maximum likelihood estimate, and c+ have approximately the same first term when expanded in powers of p.
The “Idealized Individual” Interpretation of Individual differences in Multidimensional Scaling
Authors: Cliff Norman
Published: January 2006

It is shown that several variants of the Tucker-Messick points of view procedure are interpretable in terms of their “idealized individual” concepts. It is contended that the main function of the method is that it allows the computation of several vectors of mean judgments instead of only one and that each of these may be used to stand for the actual judgments of a subgroup of subjects. Vectors that do not correspond to the judgments of any subjects are to be avoided. It is alleged that Ross's interpretation of “points of view” is quite different from that intended by Tucker and Messick.
A Note on Lower Bounds for the Number of Common Factors
Authors: Browne Michael W.
Published: January 2006

It is shown that the lower bounds for the number of common factors, established by Guttman [1954] and modified by Kaiser [1961], cannot decrease as the number of observed variates is increased. The result implies that the lower bounds cannot become weaker if the number of observed variates is increased and the number of factors remains constant.
A Note on a Cognitive Probability Model Incorporating the Serial Position Effect
Authors: Brewer James K. , King F. J.
Published: January 2006

A generalization of a cognitive probability model is proposed utilizing the serial position effect to account for decay in probability of recall. The general nature of the model permits the incorporation of almost any of the recent free-recall serial position curve results. Two examples, hypothesizing different serial position curves, are given.
The Error
Authors: Winer B. J.
Published: January 2006

After I had started to put some of my thoughts into a first draft for this talk, I selected “The Error” as a title. This selection gave me a great deal of flexibility since, no matter how I went wrong, I was still well within the limits of the title. After additional deliberation, I had more than a slight feeling that the title was more appropriate for a potential best-seller in the field of popular fiction than it was for a potential presidential address to this society. I have, on occasion, thought that I could write–in fact, I started my academic life in a School of Journalism. At least at the academic level, the journalistic style of writing rather quickly suppressed my interest therein as a means of making a livelihood.

Errors, random and otherwise, are often the heart of some rather undercover plots and subplots in best-sellers. But just as in the case of novels, in experimental designs errors also play a dominant role–unfortunately the experimental units, which are the players, often do not understand their roles as set forth in the design script. The latter is usually in the form of a mathematical model. A question might be raised in some instances as to whether it is more difficult to teach the experimental units what the model calls for, or to teach the designer what types of designs are potentially appropriate for expressing the behavior of experimental units, under specified experimental conditions.
The Influence of Communality and N on the Sampling Distributions of Factor Loadings
Authors: Pennell Roger
Published: January 2006

The effects of N and communality on the variability of zero and nonzero factor loadings were assessed using a Monte Carlo approach. It was found that increasing N or communality resulted in decreased sampling error of individual factor loadings, but for zero loadings N was found to have the greatest influence. It was also found that distributions of factor loadings become relatively elongated as communality increases.
A Central Limit Theorem for Families of Stochastic Processes Indexed by a Small Average Step Size Parameter, and Some Applications to Learning Models
Authors: Norman M. Frank , Graham Norma V.
Published: January 2006

Let θ > 0 be a measure of the average step size of a stochastic process {pn(θ) }n=1(∞). Conditions are given under which pn(θ) is approximately normally distributed when n is large and θ is small. This result is applied to a number of learning models where θ is a learning rate parameter and pn(θ) is the probability that the subject makes a certain response on the nth experimental trial. Both linear and stimulus sampling models are considered.