bluuebunny/crossref_metadata_2025 huggingface dataset:
The Approximation of One Matrix by Another of Lower Rank
Authors: Eckart Carl , Young Gale
Published: January 2006

The mathematical problem of approximating one matrix by another of lower rank is closely related to the fundamental postulate of factor-theory. When formulated as a least-squares problem, the normal equations cannot be immediately written down, since the elements of the approximate matrix are not independent of one another. The solution of the problem is simplified by first expressing the matrices in a canonic form. It is found that the problem always has a solution which is usually unique. Several conclusions can be drawn from the form of this solution.

A hypothetical interpretation of the canonic components of a score matrix is discussed.
The Determination of the Factor Loadings of a given Test from the Known Factor Loadings of Other Tests
Authors: Dwyer Paul S.
Published: January 2006

A technique is indicated by which approximations to the factor loadings of a new test may be obtained if factor loadings of a given group of tests and the correlations of the new test with the other tests are known. The technique is applicable to any orthogonal system and is especially adapted to cases in which Σajiajk = 0 when i ≠ k. Application is also made to the simultaneous determination of the factor weights of a group of tests in which no additional common factor is present. The technique is useful in adding tests to a completed factorial solution and in using factorial solutions involving errors to give results which are approximately correct.
A Comparative Study of Scales Constructed by Three Psychophysical Methods
Authors: Saffir Milton A.
Published: January 2006

A comparison is made between the scales constructed by the Method of Paired Comparison, Rank Order, and the Method of Successive Intervals. Application of the three psychophysical methods to handwriting specimens and to nationality preferences results in mutually linear scales. Choice of scaling methods becomes, then, a matter of practical convenience rather than of relative validity.
The Unit Hierarchy and its Properties
Authors: Burt Cyril
Published: January 2006

A correlation matrix may be expanded as the weighted sum of a series of ‘unit hierarchies’. The properties of the ‘unit hierarchy’ are not only of theoretical interest for themselves, but lead to simpler modes of practical calculation. The analysis is analogous to a spectral set of projective operations in quantum-theory: and the analogy itself suggests many further problems and solutions.
Note on Factor Analysis
Authors: Wilson Edwin B. , Worcester Jane
Published: January 2006

Certain assumptions and procedures basic to factor analysis are examined from the point of view of the mathematician. It is demonstrated that the Hotelling method does not yield meaningful traits, and an example from the theory of gas mixtures with convertible components is cited as evidence. The justification of current methods for determining the adequacy of the reproduction of a correlation matrix by a factorial matrix is questioned, and a x2 criterion, practical only for a small matrix, is proposed. By means of a hypothetical example from geometry, it is shown that results of a Hotelling analysis are necessarily relative to the population at hand. The factorial effects of the adjunction of a “total test” to a group of tests are considered. Some of the general considerations and questions raised are pertinent to types of analysis other than the Hotelling.
Determining a Simple Structure When Loadings for Certain Tests are Known
Authors: Mosier Charles I.
Published: January 2006

A rigorous and an approximate solution are found for the problem: Given a primary trait matrix for n tests and r1 traits, and a matrix for the same n tests and r2 reference axes, to discover the transformation which will transform the second matrix into the first, or primary trait matrix. Formulas for determining the limits of the effect of using the approximate solution are presented. The method is applied to a set of twenty hypothetical tests, defined by their loadings on four orthogonal primary traits. After factoring the inter-correlations of these variables by Thurstone's centroid method, approximating the diagonals, the original hypothetical matrix is reproduced with a root mean square discrepancy of .014 by assuming as known the primary trait loadings of only the first eight tests. The method is applied to the results of factoring two batteries of 14 tests, having 8 tests in common, to give the factor loadings of the two batteries on the same reference axes. The method provides a means of comparing directly and quantitatively the results of two different factor studies, provided they have tests in common, and of testing the stability of simple structure under changes in the battery. The relations of the method here developed to certain problems in multiple correlation are shown.
The Contribution of an Orthogonal Multiple Factor Solution to Multiple Correlation
Authors: Dwyer P. S.
Published: January 2006

A method is indicated by which multiple factor analysis may be used in determining a number, r, and then in selecting r “predicting” variables out of n variables so that each of the remaining n-r variables may be predicted almost as well from the r variables as it could be predicted from all the n−1 variables.
Factor Analysis and the Index of Clustering
Authors: Young Gale
Published: January 2006

The initial problem of factor analysis is described as a search for clustering of the test vectors. Curves are developed which give a visual picture of the clustering tendency, and an index of clustering is derived which provides a simple estimate for the number of factors.
A Factor Analysis of the Original Stanford-Binet Scale
Authors: Wright Ruth E.
Published: January 2006

From the original Stanford-Binet scale, those items passed by between 10 and 90 per cent of a group of ten-year-old children were analyzed by the centroid method. Upon rotation, there appeared a common factor, for which two explanatory hypotheses are offered, the more tenable being that it is an effect of maturation. Primary factors tentatively identified are Number, Space, Imagery, Verbal Relations and Induction. A sixth factor apparently involves a reasoning ability and a seventh can not be interpreted.
Studies in Mathematical Theory of Human Relations
Authors: Rashevsky N.
Published: January 2006

In continuation of previous studies, an abstract mathematical theory is developed for the interaction of several social classes, of which one influences and controls the behavior of the others. In some cases such interaction results in the existence of two configurations of equilibrium for the social structure, characterized by different types of behavior. Each configuration corresponds to one definite type of behavior. The transition from one configuration to the other, in other words, the transition from one behavior to another, occurs rather rapidly. Equations governing these transitions are given. In other cases, namely when the efforts of the individuals to influence others into a given behavior lessen as the success of this influence increases, there is only one stable configuration characterized by a mixed behavior. Due to the dissimilarity of parents and progeny, the composition of each social class changes with time, as generations change. This results in the appearance of instabilities of the social structure and in relatively sudden social changes. Possibilities of quasi-periodic alterations of different social structures are discussed. Finally, the developed equations are applied to the case of physical conflicts between groups of individuals, such as riots, wars, etc. Possible factors in addition to mere physical force which may determine the outcome of such conflicts are investigated.
Methods of Item Validation and Abacs for Item-Test Correlation and Critical Ratio of Upper-Lower Difference
Authors: Mosier Charles I. , McQuitty John V.
Published: January 2006

It is shown that by making the assumption that the knowledge of the test-item and the knowledge of the entire test are both distributed normally, the correlation coefficient between any item and the entire test can be expressed as a function solely of two proportions — the percentage of a high-scoring group passing the item and the percentage of a low-scoring group passing the item. This function is expressed graphically as a family of curves for each of two conditions — where the high-scoring and low-scoring groups are samples of the highest and the lowest quarters respectively, and where they are samples from the upper and lower halves. It is shown, moreover, that two other common measures of item validity, the upper-lower difference and the critical ratio of the upper-lower difference, may be drawn on the same coordinate axes.
Time Scores and Factor Analysis
Authors: Landahl H. D.
Published: January 2006

The assumption of specific operation times leads to equations similar to those which are the basis for a factor analysis but in which the time scores replace item scores. An analysis of time scores is shown to lead to results equivalent, in a first approximation, to those obtained from an analysis of item scores. It is suggested that by modifying the factor technique it may be possible to use additional information to check with the results from an analysis.
The Measurement of Speed in Motor Performance
Authors: McCloy C. H.
Published: January 2006

When the centroid method of factor analysis was applied to two sets of data on athletic performances, three significant factors emerged: strength, velocity, and dead weight. Scores on this speed factor were predicted by the multiple regression technique, the factor loadings on the speed factor being used as the criterion correlations, and these predicted scores were correlated with each of the other variables. When the original tables, augmented by the new speed variable, were refactored, the computed speed factor fell on the speed axis as a primary trait. It is thus shown that it is possible to isolate and measure a factor which appears in variables under consideration only as a compound.
Weber Laws, the Weber Law, and Psychophysical Analysis
Authors: Householder Alston S. , Young Gale
Published: January 2006

In failing to define the units in which the stimulus is to be measured, the Weber law might seem to make no definite assertion, and indeed, it is shown that any single empirical function, supposed to relate a given stimulus intensity with that intensity which is just noticeably greater, can be put into the Weber form by a suitable change of scale in which the stimulus intensity is to be measured. Nevertheless, it turns out that if different individuals have different Weber functions, when the intensities are measured on a given scale, then it is by no means always possible to transform the scale so that all of the functions can take on the Weber form. Some necessary conditions are given for the possibility of such a transformation when there is at hand a finite number of functions, and when the functions depend upon a single parameter the necessary and sufficient condition is easily derived. The same discussion leads to a generalization of Thurstone's psychophysical scale and shows that such a scale is always possible.
Scoring Weighted Multiple Keyed Tests on the IBM Counting Sorter
Authors: Bedell Ralph
Published: January 2006

Tests or personal inventories with differential item response weights may be scored by means of punch card equipment. Detailed instructions are given for preparing the cards and scoring the forms. The scoring speed is approximately four to eight times that attained by manual scoring.
The Evaluation of Multiple and Partial Correlation Coefficients from the Factorial Matrix
Authors: Dwyer P. S.
Published: January 2006

This paper shows how to compute multiple correlation coefficients, partial correlation coefficients, and regression coefficients from the factorial matrix. Special emphasis is given to computation technique and to approximation formulas. The method is extremely flexible in application since it may be applied to any subset of the original set of observed variables. It is also extremely useful when many of these coefficients are desired.
The Phi Coefficient and Chi Square as Indices of Item Validity
Authors: Guilford J. P.
Published: January 2006

Two new methods of item analysis are described. One involves the computation of the ϕ coefficient (correlation of a fourfold point distribution) and the other involves chi square. The only data required are the proportions of passing individuals in the upper and lower criterion groups, for the determination of ϕ, and in addition, N, for the determination of chi square. Abacs are presented for graphic solution of the two indices of validity, and tests of significance are provided.
On the Rectilinear Prediction of Oblique Factors
Authors: Harman Harry H.
Published: January 2006

The general problem of estimating correlated or uncorrelated factors is treated. It is specifically indicated wherein the prediction of oblique factors differs from that of orthogonal factors. A shortened method of estimation of correlated factors is developed.
Note on the Mathematical Theory of Interaction of Social Classes
Authors: Rashevsky N.
Published: January 2006

In continuation of previous studies, the interreaction of two social classes is investigated from a somewhat different point of view. An exchange of results of activities of the two classes is considered, and a mathematical approach is outlined based on the use of such psychological concepts as the satisfaction function.
Maximum Likelihood Estimation and Factor Analysis
Authors: Young Gale
Published: January 2006

Fisher's method of maximum likelihood is applied to the problem of estimation in factor analysis, as initiated by Lawley, and found to lead to a generalization of the Eckart matrix approximation problem. The solution of this in a special case is applied to show how test fallability enters into factor determination, it being noted that the method of communalities underestimates the number of factors.
Music Ability
Authors: Karlin J. E.
Published: January 2006

Two batteries of music tests were factored by the centroid method. From each battery three oblique factors were extracted and in each case were tentatively identified as tonal sensitivity, retentivity (memory for elements), and memory for form. The correlations of the music tests of one battery with subtests of Cattell's intelligence test and with tests of a literary nature are also reported.
A Factor Analysis of Verbal Abilities
Authors: Carroll John B.
Published: January 2006

A multiple-factor analysis was made of a battery of 42 tests of verbal abilities administered to 119 college adults. Where necessary, the distributions of test scores were normalized before the inter-test correlations were computed. Thurstone's M (Memory or Rote Learning) factor has been confirmed, but his V (Verbal Relations) factor seems to have been split into two or possibly three factors, C, J, and G; and his W (Word Fluency) factor has been split into two factors, A and E. The C factor seems to represent the richness of the individual's stock of linguistic responses, and the J factor seems to involve the ability to handle semantic relationships. No satisfactory interpretation can as yet be made of the G factor. The A factor seems to correspond to the speed of association for common words where there is a high degree of restriction as to appropriate responses. The E factor is described as an associational facility with verbal material where the only restriction is that the responses must be syntactically coherent. The new factors are: F, facility and fluency in oral speech; H, facility in attaching appropriate names or symbols to stimuli; and D, speed of articulatory movements.
On the Number of Factors
Authors: McNemar Quinn
Published: January 2006

A proposed criterion for the number of factors is developed on the basis of the similarity between a factorial residual and the partial correlation coefficient; something is known concerning the sampling error of the latter. Instead of computing the residuals as partials, a formula is presented for adjusting the standard deviation of the distribution of residuals so as to approximate the S.D. of the residuals as partial correlations. The criterion requires that factors be extracted until the adjusted S.D. reaches or falls below 1/√N. When tried out on six samples drawn from six universes of known factorial description, the criterion indicated the correct number of factors each time. The requisites of situations adequate for such empirical checks are discussed.
Tests of Statistical Hypotheses in the Case of Unequal or Disproportionate Numbers of Observations in the Subclasses
Authors: Tsao Fei
Published: January 2006

General solutions of the analysis of variance in the case of unequal numbers of observations in the subclasses are presented. If we have k criteria for the classification, there will be k! solutions in making a complete analysis and 2k−1 answers, bearing different meanings, for the sum of squares between subclasses of each criterion. The sum of squares for the interaction of any order, however, will be identical in different solutions of same problem.
Use of the Test Scoring Machine and the Graphic Item Counter for Statistical Work
Authors: Bloom Benjamin S. , Lubin Ardie
Published: January 2006

The graphic item-counter is described and its use as a statistical device is explained. Procedures are presented for obtaining Pearson product-moment correlations by means of the graphic item-counter.
Test Scores Examined with the Lexis Ratio
Authors: Edgerton Harold A. , Thomson Kenneth F.
Published: January 2006

The Lexis Ratio is discussed in its application to distributions of test scores where the items of the test can be assumed to be of equal difficulty. The ratio indicates the extent to which inter-individual variation operates as a source of the variance. The concept is related to the Lexis, Bernoulli, and Poisson distributions and illustrated by urn schemata. The Ratio is applied to the scores of 560 university freshmen on the Robinson Reading Test. The relation of the Lexis Ratio to the Kuder-Richardson estimation of reliability is also discussed and the latter authors' case IV is rewritten explicitly in terms of the Ratio.
Contribution to the Mathematical Theory of Human Relations: VI. Periodic Fluctuations in the Behavior of Social Groups
Authors: Rashevsky N.
Published: January 2006

A case of interaction between two groups of “active” and one group of “passive” individuals, in which the efforts of the influencing active groups decrease with increasing total success in the past, is studied. In that case the numbers of passive individuals, exhibiting respectively the two opposite behaviors, fluctuate periodically, with a positive damping.
Some Factors of Temperament: A Re-Examination
Authors: Reyburn H. A. , Taylor J. G.
Published: January 2006

By application of the technique presented earlier, the factors found by the Guilfords in two studies of personality factors are reanalyzed. The rotation technique for the isolation of meaningful factors relates the factors of the two studies to each other through three common items, and to other factors and traits appearing in the literature of personality measurement. For each of four of the factors resulting from the analysis of the first battery there is close agreement with its counterpart from the analysis of the second battery with respect to loadings of the three common items. These factors are interpreted as perseverance, surgency, flexibility, and tension. Other factors appearing in only one or the other of the batteries are tentatively identified.
Factorial Analysis of Thurstone's Seven Primary Abilities
Authors: Goodman Charles H.
Published: January 2006

Intercorrelation coefficients among Thurstone's seven primary mental abilities scores were obtained from scores of 170 freshmen engineering students on the experimental edition of the Primary Mental Abilities Tests. These correlation coefficients were factored to four factors, interpreted as a general factor, a reasoning factor, a verbal factor, and, tentatively, a specific memory factor. The finding of a general factor for a college population corroborates the Thurstones' finding of a general factor for eighth-grade children.
A Nomograph for Rapid Determination of Medians
Authors: Jurgensen Clifford E.
Published: January 2006

Directions are given for constructing a very simple nomograph for computing medians, which is entered with information from the cumulative frequency distribution. It gives a linear interpolation within the class interval containing the median.
Statistical Problems in the Evaluation of Army Tests
Authors: Burt Cyril
Published: January 2006

The introduction of psychological tests for personnel selection in the British forces has given rise to several novel problems in statistical procedure. The solutions proposed are in the main extensions of devices already familiar in educational psychology. The more important are: (i) where the criterion yields a threefold classification only, a method of triserial correlation or of biserial correlation assuming point-distributions for the extremes; (ii) where the data on which validation has to be based are drawn from a selected sample, a simplified form of Pearson's equations to correct for selection; (iii) where the best line of demarcation has to be deduced from theoretical rather than practical considerations, a formula based on the principle of minimal discrepancy.
Factor Pattern of Test Items and Tests as a Function of the Correlation Coefficient: Content, Difficulty, and Constant Error Factors
Authors: Wherry Robert J. , Gaylord Richard H.
Published: January 2006

A dilemma was created for factor analysts by Ferguson (Psychometrika, 1941, 6, 323–329) when he demonstrated that test items or sub-tests of varying difficulty will yield a correlation matrix of rank greater than 1, even though the material from which the items or sub-tests are drawn is homogeneous, although homogeneity of such material had been defined operationally by factor analysts as having a correlation matrix of rank 1. This dilemma has been resolved as a case of ambiguity, which lay in (1) failure to specify whether homogeneity was to apply to content, difficulty, or both, and (2) failure to state explicitly the kind of correlation to be used in obtaining the matrix. It is demonstrated that (1) if the material is homogeneous in both respects, the type of coefficient is immaterial, but (2) if content is homogeneous but difficulty is not, the homogeneity of the content can be demonstrated only by using the tetra chorie correlation coefficient in deriving the matrix; and that the use of the phi-coefficient (Pearsonian r) will disclose only the nonhomogeneity of the difficulty and lead to a series of constant error factors as contrasted with content factors. Since varying difficulty of items (and possibly of sub-tests) is desirable as well as practically unavoidable, it is recommended that all factor analysis problems be carried out with tetrachoric correlations. While no one would want to obtain the constant error factors by factor analysis (difficulty being more easily obtained by counting passes), their importance for test construction is pointed out.
A Note on Cluster-Directed Analysis
Authors: Adcock C. J.
Published: January 2006

Thurstone's multiple group method of factor analysis has been widely used as a basis for rotation to simple structure. To make the most of the economy offered by this method, factor axes may be located directly by the correlation clusters; and methods of doing this are here discussed.
The Closure Factors Related to Other Cognitive Processes
Authors: Pemberton Carol
Published: January 2006

Twenty-five group tests, assembled with certain hypotheses concerning the first and second closure factors in mind, were administered to 154 subjects, mostly graduate students. The intercorrelations were analyzed factorially, yielding eight factors that were rotated to an oblique simple structure. The factors were interpreted as: speed of closure, C1; flexibility of closure, C2; verbal closure, C3; word fluency, W; reasoning, R; perceptual speed, P; the first space factor, S1; and speed of handwriting, H. Four second-order factors were tentatively described as analytical ability, synthetic ability, speed of perception, and word fluency. Three of the reasoning tests had their highest loadings on C2 and one on C3, which seems to be evidence that flexibility of closure generalizes in the cognitive domain and is associated with analytical ability.
Notes on a Problem of Multiple Classification
Authors: Lord Frederic M.
Published: January 2006

A solution is developed in implicit form for the problem of assigning N men to n jobs, the proportion of men to be assigned to each job being specified in advance.
An Extended Table of Chi-Square for Two Degrees of Freedom, for use in Combining Probabilities from Independent Samples
Authors: Gordon Mordecai H. , Loveland Edward H. , Cureton Edward E.
Published: January 2006

A table of values of Chi-square for two degrees of freedom corresponding to values of P from .001 to .999 is presented, together with a description and an example of its use in combining probabilities from two or more independent samples to obtain an aggregate probability.
Orthogonal and Oblique Simple Structures
Authors: Gibson W. A.
Published: January 2006

A quick approximation of the best-fitting orthogonal simple structure from the known oblique simple structure is here developed.
The Relation of the Reliability of Multiple-Choice Tests to the Distribution of Item Difficulties
Authors: Lord Frederic M.
Published: January 2006

Under certain assumptions an expression, in terms of item difficulties and intercorrelations, is derived for the curvilinear correlation of test score on the “ability underlying the test,” this ability being defined as the common factor of the item tetrachoric intercorrelations corrected for guessing. It is shown that this curvilinear correlation is equal to the square root of the test reliability. Numerical values for these curvilinear correlations are presented for a number of hypothetical tests, defined in terms of their item parameters. These numerical results indicate that the reliability and the curvilinear correlation will be maximized by (1) minimizing the variability of item difficulty and (2) making the level of item difficulty somewhat easier than the halfway point between a chance percentage of correct answers and 100 per cent correct answers.
A Technique for Facilitating the Rotation of Factor Axes, Based on an Equivalence between Persons and Tests
Authors: Sandler Joseph
Published: January 2006

A technique is outlined which may facilitate the rotation of factor axes to a meaningful position. It is based on certain relationships between the results of test and person factor analysis, and consists essentially of supplementing the test factor space with tests which are the test-equivalents of persons or groups of persons. These persons may be, for instance, well-known “types” in the domain being investigated, or even “freaks.” The ways in which these persons may be selected and used to determine the final rotated position of the factor axes is discussed.
IBM Computation of Sums of Products for Positive and Negative Numbers
Authors: Burke Paul J.
Published: January 2006

The wiring of the plugboard of the IBM type 405 machine for the computation of sums of squares and cross products of positive and negative numbers is described. The method makes greater demands on the X distributor and class selector capacity of the machine than does the method of wiring the plugboard when the numbers all have the same sign.
Factor Analysis of the Army Air Forces Sheppard Field Battery of Experimental Aptitude Tests
Authors: Guilford J. P. , Fruchter Benjamin , Zimmerman Wayne S.
Published: January 2006

A factor analysis was made of 39 experimental printed aptitude tests and seven reference tests selected from the Army Air Forces Aircrew Classification Battery. Thirteen factors were extracted and two independent orthogonal rotational solutions were completed. Twelve factors were interpreted. Of these, seven were clearly identifiable with previously known factors: numerical, perceptual-speed, spatial-relations, visualization, visual-memory, paired-associates-memory, and length-estimation factors. A planning factor was not as clearly identifiable. A reasoning factor was probably a composite of two or more factors that failed to separate. A new factor possibly has to do with orientation with respect to the points of the compass. Two factors were doublets, each apparently specific to one kind of test. Better conceptions were gained of the spatial-relations and visualization factors and of the kinds of tests that measure them best. Efforts to improve measures of unique factors were not uniformly successful. The attempt to duplicate a psychomotor test rather directly by analogy in printed form failed almost completely.
The Factors in Factoring Behavior
Authors: McNemar Quinn
Published: January 2006

Twenty years have now elapsed since Professor Thurstone’s ingenuity pulled the factor problem out of its tetrad difference quagmire. Most of us have watched Thurstone’s brain-child grow. One might say that the fledgling was so precocious as to reach maturity during the first ten years of its life. Its development was fostered not only by Thurstone but also by the resistance it encountered. Indeed, even as an infant, it was forced to throw aside its swaddling clothes in order to kick back at the Spearmans, the Tryons, the Anastasis, and others who sometimes rightly, ofttimes wrongly, pointed a finger at supposed imperfections. The youngster weathered more or less successfully the many storms it had stirred up, and by the end of its tenth year of life had in the hands of Thurstone made significant contributions and was ready for the sober coming-of-age evaluation of its strengths and weaknesses by Dael Wolfle. We suspect that the facts of life pointed out by Wolfle had already been whispered to the growing adolescent by none other than Thurstone himself.
Estimation of Parameters in a Truncated Trivariate Normal Distribution
Authors: Votaw D. F. , Rafferty J. A. , Deemer W. L.
Published: January 2006

This paper gives maximum-likelihood estimators for certain parameters in a truncated trivariate normal distribution when the values of the other parameters are known. The estimators are functions of a random sample. Approximate variances and covariances of the estimators, when the sample size is large, are also given. The type of truncation considered is merely restriction of the range of one of the variates, whose true mean and variance are assumed to be known. Two cases of such restriction are treated: (a) (δ ≤x < + ∞); (b) (- ∞ <x ≤ δ'), where δ and δ' are arbitrary “cutoff points” which are assumed to be known. A precise statement of the estimation problem is given in Section 1. Section 2 contains preliminary calculations. The estimators appear in Section 3. The asymptotic variances and covariances of the estimators are given in Section 4. The estimators and their asymptotic variances and covariances can be easily specialized to be suitable for the case of a certain truncated bivariate normal distribution (Sections 3 and 4).
The Johnson-Neyman Technique, its Theory and Application
Authors: Johnson Palmer O. , Fay Leo C.
Published: January 2006

The theoretical basis for the Johnson-Neyman Technique is here presented for the first time in an American journal. In addition, a simplified working procedure is outlined, step-by-step, for an actual problem. The determination of significance is arrived at early in the analysis; and where no significant difference is found, the problem is complete at this point. The plotting of the region of significance where a significant difference does exist has also been simplified by using the procedure of rotation and translation of axes.
The Variance Error of the P50-Discriminant
Authors: Betts Gilbert L.
Published: January 2006

The P50-discriminant has been reported elsewhere in connection with its use in predicting whether selective service registrants if inducted would become normal operative soldiers or would commit offenses causing their imprisonment. The standard error of the P50-discriminant is a good measure to use in determining how far to the side of this statistic a particular case falls. The standard error formula itself has also been published elsewhere; but its derivation, as the variance error, is given here.
A Multiple Group Method of Factoring the Correlation Matrix
Authors: Thurstone L. L.
Published: January 2006

There are a number of methods of factoring the correlation matrix which require the calculation of a table of residual correlations after each factor has been extracted. This is perhaps the most laborious part of factoring. The method to be described here avoids the computation of residuals after each factor has been computed. Since the method turns on the selection of a set of constellations or clusters of test vectors, it will be called a multiple group method of factoring. The method can be used for extracting one factor at a time if that is desired but it will be considered here for the more interesting case in which a number of constellations are selected from the correlation matrix at the start. The result of this method of factoring is a factor matrix F which satisfies the fundamental relation FF'=R.
The Relation of Item Difficulty and Inter-Item Correlation to Test Variance and Reliability
Authors: Gulliksen Harold
Published: January 2006

Under assumptions that will hold for the usual test situation, it is proved that test reliability and variance increase (a) as the average inter-item correlation increases, and (b) as the variance of the item difficulty distribution decreases. As the average item variance increases, the test variance will increase, but the test reliability will not be affected. (It is noted that as the average item variance increases, the average item difficulty approaches .50). In this development, no account is taken of the effect of chance success, or the possible effect on student attitude of different item difficulty distributions. In order to maximize the reliability and variance of a test, the items should have high intercorrelations, all items should be of the same difficulty level, and the level should be as near to 50% as possible.
Maximum Validity of a Test with Equivalent Items
Authors: Tucker Ledyard R
Published: January 2006

It is assumed that a scale of true scores on a function exists and that the probability of answering an item correctly is a curve of the type of the integral of the normal curve. The product moment correlation between the test score and true score is derived for a normal distribution of subjects and a test composed of equivalent items. Numerical examples demonstrate that the maximum correlation between test scores and true scores occurs for a one hundred item test when the point correlation between items is less than three tenths.
A Simple Graphical Method for Orthogonal Rotation of Axes
Authors: Zimmerman Wayne S.
Published: January 2006

A simple method of making orthogonal rotations by projecting coordinates from one plot to another is presented. An easily constructed apparatus for rapid and accurate work is described in detail. By the technique proposed 12 to 15 rotations of pairs of axes involving 20 test variables can be completed per hour by a practiced technician.
A Statistical Description of Verbal Learning
Authors: Miller George A. , McGill William J.
Published: January 2006

Free-recall verbal learning is analyzed in terms of a probability model. The general theory assumes that the probability of recalling a word on any trial is completely determined by the number of times the word has been recalled on previous trials. Three particular cases of this general theory are examined. In these three cases, specific restrictions are placed upon the relation between probability of recall and number of previous recalls. The application of these special cases to typical experimental data is illustrated. An interpretation of the model in terms of set theory is suggested but is not essential to the argument.
Multidimensional Scaling: I. Theory and Method
Authors: Torgerson Warren S.
Published: January 2006

Multidimensional scaling can be considered as involving three basic steps. In the first step, a scale of comparative distances between all pairs of stimuli is obtained. This scale is analogous to the scale of stimuli obtained in the traditional paired comparisons methods. In this scale, however, instead of locating each stimulus-object on a given continuum, the distances between each pair of stimuli are located on a distance continuum. As in paired comparisons, the procedures for obtaining a scale of comparative distances leave the true zero point undetermined. Hence, a comparative distance is not a distance in the usual sense of the term, but is a distance minus an unknown constant. The second step involves estimating this unknown constant. When the unknown constant is obtained, the comparative distances can be converted into absolute distances. In the third step, the dimensionality of the psychological space necessary to account for these absolute distances is determined, and the projections of stimuli on axes of this space are obtained. A set of analytical procedures was developed for each of the three steps given above, including a least-squares solution for obtaining comparative distances by the complete method of triads, two practical methods for estimating the additive constant, and an extension of Young and Householder's Euclidean model to include procedures for obtaining the projections of stimuli on axes from fallible absolute distances.
The Significance of Difference Between Means Without Reference to the Frequency Distribution Function
Authors: Festinger Leon
Published: January 2006

Most existing tests for significance of difference between means require specific assumptions concerning the distribution function in the parent population. The need for a test which can be applied without making any such assumption is stressed. Such a statistical test is derived. The application of the test involves converting scores to rank orders. The exact probabilities may then be calculated for specified differences between samples by means of which the null hypothesis may be tested. The application of the test is simple and requires a minimum of calculation. The test loses in precision because of the conversion to rank orders but gains in generality since it may be safely used with any kind of distribution.
A New Method for Analyzing Aesthetic Preferences: Some Theoretical Considerations
Authors: Peel E. A.
Published: January 2006

The aesthetic preferences of a group of persons are obtained from their orders of sets of pictures and patterns according to “liking.” The same pictures are ordered independently by a team of experts, according to certain artistic criteria such as naturalism, composition, color, rhythm, etc. The orders of preference and orders according to the criteria are compared by correlation and matrices of correlation formed from (1) correlations between the persons' orders of preference; (2) correlations between the orders of preference and orders according to artistic criteria; and (3) correlations between the criterion orders. These matrices are symbolised by Rp, R0, and Rc, respectively, and combined to form a single matrix \documentclass[12pt]{minimal}
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$$\left[ \begin{gathered} R_p R_o \hfill \\ R'_o R_c \hfill \\ \end{gathered} \right]$$
\end{document}. Three interesting analyses of this matrix are suggested: Analysis of the whole matrix into its factors and rotation of the factors about the criteria, regression estimates of individual preferences on the artistic criteria, and regression estimates of the person preference factors on the same criteria. Theoretical conditions and consequences of these analyses are then discussed by the use of matrix notation.
Two-Alternative Learning Situations with Partial Reinforcement
Authors: Hanania Mary I.
Published: January 2006

The comparative effects of reward and nonreward on learning are considered in connection with a two-alternative learning situation. Conditions are more general in nature than those discussed in an earlier article. In the statistical model proposed, the question of whether reward and nonreward are equivalent in their effects on learning reduces to testing a composite hypothesis on a multivariate probability distribution. An asymptotic test of this hypothesis is described, and its use is illustrated with data from psychological experiments.
Study of Some Factors Related to Intelligence
Authors: Rimoldi H. J. A.
Published: January 2006

From the factorial analysis of a battery of nineteen individual tests seven factors were obtained, six of them being interpreted. Some of these factors are probably related to similar ones isolated in different batteries and by different authors. The test Progressive Matrices here analyzed is loaded in some of the factors, as was expected according to the nature of its problems and to previous experience with the test. The loadings of the tests and of the factors in the second-order factors are given. It is expected that this extension of factor theory will prove useful for the isolation of more fundamental psychological parameters, although at present it is difficult to evaluate precisely the theoretical and practical implications of this aspect of factor analysis.
Separation of Data as a Principle in Factor Analysis
Authors: Harris Chester W.
Published: January 2006

Two systems of factor analysis—factoring correlations with units in the diagonal cells and factoring correlations with communalities in the diagonal cells—are considered in relation to the commonly used statistical procedure of separating a set of data (scores) into two or more parts. It is shown that both systems of factor analysis imply the separation of the observed data into two orthogonal parts. The matrices used to achieve the separation differ for the two systems of factor analysis.
A Rational Curve Relating Length of Rest Period and Length of Subsequent Work Period for an Ergographic Experiment
Authors: Tucker Ledyard R
Published: January 2006

A rational function is developed relating the length of a rest period and length of subsequent work period in an ergographic situation. Simple energistic postulates are used for a critical organ or neuromuscular structure whose failure to perform adequately results in a stoppage of the work period. Experimental results for two subjects using a finger ergograph indicate that the function yields the general trend of the data but that there seem to be some systematic deviations of the data from the present rational function. One parameter determined from the data represents rate of recovery from moderate fatigue. It is hoped that this development will aid in studies of motor functions as related to such other variables as age, motivation, and effects of drugs.
An IBM Method for Computing Intraserial Correlations
Authors: Payne M. Carr , Staugas Leonard
Published: January 2006

A method for computing intraserial correlations using a 602-A Calculating Punch, an 077 Collator, a 513 Gang Punch, and a 403 Tabulator is described. An example of the use of the procedure and an estimate of the time needed with each machine are given. This procedure is compared with another method, which makes use of a more powerful IBM machine.
New Problems for Old Solutions
Authors: Brogden Hubert E.
Published: January 2006

Some of the methodologies that have become standard tools in psychometrics suffer from neglect. They are taken too much for granted and are not given the attention that seems appropriate to the important role they play in research advances. I propose to make some suggestions which may, in a modest way, assist in alleviating this difficulty. In a very general way I would like to suggest that effort expended in examining a variety of restatements of a methodological problem may lead to new methodologies of real value.
Optimal Test Length for Multiple Prediction: The General Case
Authors: Horst Paul , MacEwan Charlotte
Published: January 2006

The concepts of differential prediction and multiple absolute prediction were developed in earlier papers [2, 3]. Methods for determining optimal distribution of testing time for each type of prediction are available [4, 5] and are appropriate for use provided that no altered time allotment approaches zero. In this article the methods developed in [4, 5] are extended to include cases where the altered time allotment for one or more tests may approach zero. The procedures developed are illustrated by numerical examples, after which the mathematical rationales are provided.
Stimulus and Response Generalization: A Stochastic Model Relating Generalization to Distance in Psychological Space
Authors: Shepard Roger N.
Published: January 2006

A mathematical model is developed in an attempt to relate errors in multiple stimulus-response situations to psychological inter-stimulus and inter response distances. The fundamental assumptions are (a) that the stimulus and response confusions go on independently of each other, (b) that the probability of a stimulus confusion is an exponential decay function of the psychological distance between the stimuli, and (c) that the probability of a response confusion is an exponential decay function of the psychological distance between the responses. The problem of the operational definition of psychological distance is considered in some detail.
The Inclusion of Response Times within a Stochastic Description of the Learning Behavior of Individual Subjects
Authors: Audley R. J.
Published: January 2006

A stochastic process applicable to the learning behavior of an individual subject is discussed. The process describes both the response times and the sequence of choices obtained from a situation involving two alternatives. Parameter estimates and techniques for assessing goodness of fit are considered.
Determining the Degree of Inconsistency in a Set of Paired Comparisons
Authors: Gerard Harold B. , Shapiro Harold N.
Published: January 2006

Consistency in paired comparison data is defined. Two types of inconsistency which may arise are defined. Computational formulas for these types of inconsistency are derived, and examples illustrating the use of these formulas are presented.
The Measurement of Function Fluctuation
Authors: Garside R. F.
Published: January 2006

A method of measuring function fluctuation is suggested and an appropriate test of significance is indicated. The proposed method is compared with bi-factor analysis and with some other suggested methods of measuring function fluctuation.
Predetermination of Test Weights
Authors: Hoffman Paul J.
Published: January 2006

Derivations are presented relating the length of a test to its weight in a composite. Tests of varying length are constructed so that their weights will be of predetermined magnitudes, and the results compared with expectations. Weighting schemes involving standard deviations of raw scores and of true scores are compared. An important secondary derivation is presented from which it is possible to estimate test reliability knowing only the relative length of a test, its shortened form, and the standard deviation of each.