bluuebunny/crossref_metadata_2025 huggingface dataset:
Verification of Hypothesized Factors in One Hundred and Fifteen Objective Personality Test Designs
Authors: Cattell R. B. , Dubin S. S. , Saunders D. R.
Published: January 2006

This paper reports on the third of a series of four experiments using similar concepts and methods for objective personality measurement, and overlapping test batteries. One hundred students were measured with 115 tests. The scores were correlated and factored by a re-iterated multiple group centroid method. The 17 factors thus obtained were rotated toward a clear simple structure. The relation of the rotated factors to earlier ones is indicated, but no extensive interpretation is attempted.
rp and other Coefficients of Pattern Similarity
Authors: Cattell Raymond B.
Published: January 2006

A pattern consists of fundaments and relations. Its similarity to other patterns can be expressed either in terms of relations alone—in which case a modified correlation coefficient, rs, suffices—or in terms of fundaments and relations, where new coefficients, rp and r′p are required. The derivation of a coefficient of pattern similarity, rp, is presented and its properties are investigated, both deductively and in terms of actual data on culture patterns.
Test Reliability and Correction for Attenuation
Authors: Johnson Helmer G.
Published: January 2006

Evidence is cited to show that specificity, or lack of equivalence, in the comparable forms of tests has a tendency to lower the value of reliability coefficients but has no tendency to lower the value of observed trait coefficients. This implies that the greater the lack of equivalence, the higher will be coefficients corrected for attenuation. Errors of measurement are supposed to reduce the magnitude of observed trait coefficients. Since specificity does not lower the correlation between two tests and since the split-half and equivalent-form reliability coefficients treat specificity as error, it follows that these two coefficients cannot legitimately be used in Spearman's correction-for-attenuation formula.
Connectivity and Generalized Cliques in Sociometric Group Structure
Authors: Luce R. Duncan
Published: January 2006

By using the concepts of antimetry and n-chain it is possible to define and to investigate some properties of connectivity in a sociometric group. It is shown that the number of elements in a group, the number of antimetries, and the degree of connectivity must satisfy certain inequalities. Using the ideas of connectivity, a generalized concept of clique, called an n-clique, is introduced. n-cliques are shown to have a very close relationship to the existence of cliques in an artificial structure defined on the same set of elements, thus permitting the determination of n-cliques by means of the same simple matrix procedures used to obtain the clique structures. The presence of two or more m-cliques, where m is the number of elements in the group, is proved to mean an almost complete splitting of the group.
On the Effect of Selection Performed on Some Coordinates of a Multi-Dimensional Population
Authors: Birnbaum Z. W. , Paulson E. , Andrews F. C.
Published: January 2006

Samples are often obtained under circumstances which make it less likely to draw individuals from some parts of a population than from others. In this paper a method is presented which makes it possible, under certain assumptions, to correct for the resulting bias and thus reconstruct the means, standard deviations, and correlation coefficients of the original population.
On the Ranking Problem
Authors: Blumen Isadore
Published: January 2006

Observed rankings of objects can be treated as arising from a time dependent probability process. Under such circumstances, associations observed are an indication of the character of this underlying process. In the particular example treated in some detail here, a quantity related to Kendall's tau is found to have an important role and its properties are examined.
The Detailed Method of Optimal Regions
Authors: Dwyer Paul S.
Published: January 2006

The detailed method of optimal regions is an extended form of the method of optimal regions which has been found effective in solving the personnel classification problem when the number of job categories is small. The automatic determination of the successive values of the vi, made possible by the more exact techniques of the detailed method, provide easier solutions for the more complex problems and provide solutions, which, for the most part, can be mechanized. In a sense the detailed method of optimal regions is more than a detailed form of the method of optimal regions. It is essentially a method of transformations by which the original matrix is reduced to a matrix from which the solution is easily obtained.
A Theory of Pattern Analysis for the Prediction of a Quantitative Criterion
Authors: Lubin Ardie , Osburn Hobart G.
Published: January 2006

A method of pattern analysis is presented for the case of dichotomous items and a quantitative criterion. This “configural scale” has maximum validity in the least squares sense. A technique for computing the configural scale as a polynomial function of the item scores is given. Tests of significance are outlined for such questions as: Is there a linear or non-linear relation between the quantitative criterion and the item scores? Does the addition of certain items to the test increase the validity of the configural scale? Are all the items in the configural scale fully effective?
A Necessary and Sufficient Formula for Matric Factoring
Authors: Guttman Louis
Published: January 2006

For the purpose of extracting factors from matrices, it is proved that a certain formula is both necessary and sufficient. In factor analysis, the formula may be applied either to the correlation matrix, or directly to the score matrix (assuming the communality problem is solved). As many factors as desired can be extracted in one operation. Having such a compact formulation is useful for teaching as well as computing purposes, since it includes all techniques of factor extraction as special cases.
Exact Probabilities for Contingency Tables Using Binomial Coefficients
Authors: Sakoda James M. , Cohen Burton H.
Published: January 2006

The use of binomial coefficients in place of factorials to shorten the calculation of exact probabilities for 2 × 2 and 2 × r contingency tables is discussed. A useful set of inequalities for estimating the cumulative probabilities in the tail of the distribution from the probability of a single table is given. A table of binomial coefficients with four significant places and n through 60 is provided.
Iterative Inverse Factor Analysis—A Rapid Method for Clustering Persons
Authors: Bass Bernard M.
Published: January 2006

By interchanging persons and items, iterative inverse factor analysis provides a relatively inexpensive way of clustering persons according to their patterns of response to the items. In addition to permitting the clustering of large numbers of persons, the technique enables one to determine the bases for such clustering. The items of behavior used can be heterogeneous in content and form.
Factor Analysis I: Some Effects of Chance Error
Authors: Saunders D. R.
Published: January 2006

Ignorance concerning the standard error of individual factor loadings and their differences has been a major obstacle to the proper interpretation of factorial results. The effects of three types of experimental error (selection of variables, selection of subjects and selection of scores) are here reported. In order to handle the errors of rotation systematically, it has been necessary to introduce a new semi-analytical criterion for the attainment of simple structure. Variability in results which may theoretically be eliminated is discussed under the heading of non-essential error.
The Maximum Expected Correlation Between Two Multiple-Choice Tests
Authors: Horst Paul
Published: January 2006

A formula is derived which gives the maximum expected correlation between two multiple-choice tests as a function of the distributions of proportions correct for the items in the two tests and the probability of chance success. The formula is similar to one derived by Carroll based on “true” item difficulties. A numerical example is provided.
The Reliability of a Point Biserial Coefficient of Correlation
Authors: Perry Norman C. , Michael William B.
Published: January 2006

An attempt is made to elaborate upon Lev's results concerning the reliability of the point biserial coefficient of correlation in a manner that will be helpful to the psychological statistician. Procedures required in the use of the non-central t tables prepared by Johnson and Welch are described as they relate to the determination of the fiducial limits for a point biserial coefficient. A normal approximation technique for the estimation of fiducial limits is also suggested. Numerical evidence is presented which shows that relative to a given level of significance the width of the fiducial interval estimated from a point biserial coefficient of any size is smaller than that of the fiducial interval corresponding to an ordinary Pearsonian coefficient of the same magnitude.
The Varimax Criterion for Analytic Rotation in Factor Analysis
Authors: Kaiser Henry F.
Published: January 2006

An analytic criterion for rotation is defined. The scientific advantage of analytic criteria over subjective (graphical) rotational procedures is discussed. Carroll's criterion and the quartimax criterion are briefly reviewed; the varimax criterion is outlined in detail and contrasted both logically and numerically with the quartimax criterion. It is shown that the normal varimax solution probably coincides closely to the application of the principle of simple structure. However, it is proposed that the ultimate criterion of a rotational procedure is factorial invariance, not simple structure –although the two notions appear to be highly related. The normal varimax criterion is shown to be a two-dimensional generalization of the classic Spearman case, i.e., it shows perfect factorial invariance for two pure clusters. An example is given of the invariance of a normal varimax solution for more than two factors. The oblique normal varimax criterion is stated. A computational outline for the orthogonal normal varimax is appended.
Power Function Charts for Specification of Sample Size in Analysis of Variance
Authors: Feldt Leonard S. , Mahmoud Moharram W.
Published: January 2006

The specification of sample size is an important aspect of the planning of every experiment. When the investigator intends to use the techniques of analysis of variance in the study of treatments effects, he should, in specifying sample size, take into consideration the power of the F tests which will be made. The charts presented in this paper make possible a simple and direct estimate of the sample size required for F tests of specified power.
A Comparative Study of Three Methods of Rotation
Authors: Fruchter Benjamin , Novak Edwin
Published: January 2006

Three methods of rotation (the graphical, the Thurstone analytical, and the direct-rotational) were applied to the matrix of centroid loadings for 35 variables, to determine which method is the most efficient from theoretical and practical standpoints. The direct-rotational method provided the most information for determining the rank of the configuration and was most economical with respect to time required to reach a rotational solution. The analytical method required the least number of judgmental decisions and was the most objective. The graphical method was the most laborious but had a slight advantage with regard to the number of near-zero loadings in the rotational solution.
Attenuation and Interaction
Authors: McNemar Quinn
Published: January 2006

A significance test is proposed for determining whether a correlation coefficient is less than unity by an amount greater than that attributable to errors of measurement.
Multiple Bi-Serial and Multiple Point Bi-Serial Correlation
Authors: Wherry Robert J.
Published: January 2006

Normal equations, using data in various forms, are presented for securing the regression weights for prediction of a dichotomized criterion, and a simplified equation for the estimation of the multiple bi-serial or multiple point bi-serial, depending upon the proper assumption as to the nature of the distribution of the criterion, on the basis of these maximal weights is given also. The weights, unaffected by the assumption as to the nature of the criterion, are identical (or proportional) to those found by the discriminant function approach based upon analysis of variance. The author holds that the present multiple correlation approach is both easier and more informative than the discriminant function (analysis of variance) approach and suggests that the discriminant function be abandoned in favor of multiple bi-serial and/or multiple point bi-serial correlation and regression.
A Generalization of Thurstone's Learning Function
Authors: Gulliksen Harold
Published: January 2006

Thurstone's equation giving the probability of a correct response (p) as a function of practice time (t) when punishment and reward have equal effects has been generalized to the case where the effect of punishment is not necessarily equal to the effect of reward. Since the general equation is somewhat unwieldy, three special cases are considered, where reward has no effect, where punishment has no effect, and where these effects are equal. Equations are given together with tables for making a rectified plot for each of the three special cases.
Some Applications of Attribute Prediction Theory to Psychophysics
Authors: Perry Norman C.
Published: January 2006

The following development is a coordination of certain aspects of mental test methods and psychophysics. The biserial and triserial prediction equations of the Katzell-Cureton theory are utilized to reformulate the determination of absolute and differential limens from data produced by the constant method.
Notes on an Approximation Method for Fitting Parabolic Equations to Experimental Data
Authors: Chapanis A.
Published: January 2006

When a numerical transformation of raw data is used only to simplify the arithmetic of curve fitting, the transformation may lead to undesirable and even highly distorted results. This principle is illustrated with an approximation method of fitting parabolic equations to experimental data, as described recently in texts by Johnson and Lewis. Although the approximation method will never yield as good fits as the exact, least-squares method, satisfactory results are in general achieved whenever the transformed scores yield a linear plot as a function of X. The principal difficulty with the method is that some data which fall along a parabola may not yield a linear plot of the transformed scores versus X, and so cannot be fitted satisfactorily by the approximation method.
Test Selection with Integral Gross Score Weights
Authors: Wherry Robert J. , Gaylord Richard H.
Published: January 2006

A new method of test selection, which attempts to combine the merits of the Toops L-Method with those of the Wherry-Doolittle Method, is presented. It results in integral (unit if desired) positive and/or negative (optional) weights. This flexibility makes the method applicable to all kinds of material and for both selecting items for tests and tests for batteries. An explicit solution of one test construction problem is presented. Necessary changes in method for the solution of five other types of test construction problem are presented. A few cautions are provided for potential users.
Extensions of Factorial Solutions
Authors: Harman Harry H.
Published: January 2006

A method is developed for extending any type of factor solution to new tests. The theoretical basis for this approximating scheme is thoroughly investigated, and then a simplification in the technique is introduced for practical purposes. An example is presented which illustrates the procedure of extending a factor solution to three new tests simultaneously.
The Application of Multiple Factorial Methods to the Study of Motor Abilities
Authors: Buxton Claude
Published: January 2006

A battery of thirteen tests of motor abilities are analyzed by means of the Thurstone factor methods. Interpretations of the nature of the six factors extracted are offered, with reservations. The high degree of specificity of motor abilities results in very narrow group factors.
A Note on Invariance in Three-Mode Factor Analysis
Authors: Bloxom Bruce
Published: January 2006

Previous results of the application of Lawley's selection theorem to the common factor analysis model are extended to a revision of Tucker's three-mode principal components model. If the regression of the three-mode manifest variates on variates used to select subpopulations is both linear and homoscedastic, the two factor pattern matrices, the core matrix, and the residual variance-covariance matrix in the three-mode model can all be assumed to be invariant across subpopulations. The implication of this finding for simple structure is discussed.
Symmetric Limiting Distributions of Response Probabilities
Authors: Marliss G. S. , McGregor J. R.
Published: January 2006

Necessary and sufficient conditions for the symmetry of the limiting distributions of response probabilities are derived for the Bush and Mosteller two experimenter-controlled events learning model [Bush and Mosteller, 1955].
Some Statistical Considerations in Multidimensional Scaling
Authors: Ramsay J. O.
Published: January 2006

Some shortcomings of current methods of estimating the magnitude of perceived difference are considered. A statistical model for perceived difference is derived which avoids these difficulties and employs judgments of ratios of differences as data. Three estimators of squared difference are developed.
Derivations of Learning Statistics from Absorbing Markov Chains
Authors: Millward Richard B.
Published: January 2006

Learning-process statistics for absorbing Markov-chain models are developed by using matrix methods exclusively. The paper extends earlier work by Bernbach by deriving the distribution of the total number of errors, u-tuples, autocorrelation of errors, sequential statistics, and the expectation and variance of all statistics presented. The technique is then extended to latency derivations including the latencies of sequential statistics. Suggestions are made for using the sequential-statistic algorithm in a maximum-likelihood estimation procedure. The technique is important because statistics for very large absorbing matrices can be easily computed without going through tedious theoretical calculations to find explicit mathematical expressions.
On the oblique Rotation of a Factor Matrix to a Specified Pattern
Authors: Browne Michael , Kristof Walter
Published: January 2006

This paper presents a procedure for rotating an arbitrary factor matrix to maximum similarity with a specified factor pattern. The sum of squared distances between specified vectors and rotated vectors in oblique Euclidian space is minimized. An example of the application of the procedure is given.
Estimating True-Score Distributions in Psychological Testing (an Empirical Bayes Estimation Problem)
Authors: Lord Frederic M.
Published: January 2006

The following problem is considered: Given that the frequency distribution of the errors of measurement is known, determine or estimate the distribution of true scores from the distribution of observed scores for a group of examinees. Typically this problem does not have a unique solution. However, if the true-score distribution is “smooth,” then any two smooth solutions to the problem will differ little from each other. Methods for finding smooth solutions are developed a) for a population and b) for a sample of examinees. The results of a number of tryouts on actual test data are summarized.
Best Linear Composites with a Specified Structure
Authors: Green Bert F.
Published: January 2006

Least squares linear composites of predictors for estimating several criteria are derived, satisfying the restriction that the composites have an arbitrary specified intercorrelation matrix. These composites are compared with the usual unrestricted regression composites. An illustrative example is provided. The derivation depends on a general result, given in an appendix, about best-fitting orthonormal transformations.
A Monte Carlo Investigation of the Statistical Significance of Kruskal's Nonmetric Scaling Procedure
Authors: Klahr David
Published: January 2006

Recent advances in computer based psychometric techniques have yielded a collection of powerful tools for analyzing nonmetric data. These tools, although particularly well suited to the behavioral sciences, have several potential pitfalls. Among other things, there is no statistical test for evaluating the significance of the results. This paper provides estimates of the statistical significance of results yielded by Kruskal's nonmetric multidimensional scaling. The estimates, obtained from attempts to scale many randomly generated sets of data, reveal the relative frequency with which apparent structure is erroneously found in unstructured data. For a small number of points (i.e., six or seven) it is very likely that a good fit will be obtained in two or more dimensions when in fact the data are generated by a random process. The estimates presented here can be used as a bench mark against which to evaluate the significance of the results obtained from empirically based nonmetric multidimensional scaling.
Testing Significance of differences between Two Chi-Squares
Authors: Knepp Dennis L. , Entwisle Doris R.
Published: January 2006

This paper presents a way to test the difference between two X2's. The test requires evaluating the difference with respect to the Tm(x) Bessel function. Included is a table of the 5 percent and 1 percent points for the Bessel function with degrees of freedom up to 100.
A Test of the Hypothesis that Cronbach's Alpha or Kuder-Richardson Coefficent Twenty is the Same for Two Tests
Authors: Feldt Leonard S.
Published: January 2006

An approximate statistical test is derived for the hypothesis that the reliability coefficients (Cronbach's α) associated with two measurement procedures are equal. Control of Type I error is investigated by comparing empirical sampling distributions of the test statistic with the theoretical model derived for it. The effect of platykurtosis in the test-score distribution on the test statistic is also considered.
A Significance Test for One Parameter Isosensitivity Functions
Authors: Gourevitch Vivian , Galanter Eugene
Published: January 2006

A large-sample test for the significance of the difference between two detection data points is developed based upon the assumptions of a one-parameter signal detectability model. In essence, the null hypothesis tested is that two observed data points belong to the same d′ function.
Testing Independence in Two-Way Contingency Tables with Data Subject to Misclassification
Authors: Assakul Kwanchai , Proctor Charles H.
Published: January 2006

The misclassification process is represented by a stochastic matrix containing the probabilities that an individual who belongs in one cell is counted as belonging to another (or perhaps the same) cell. These probabilities are supposed known. If misclassification in the row direction is independent of that along the column variable then the size of the usual chi-square test is unchanged. It is shown how to calculate loss of power in this case and also how to calculate the change in size of the test if the errors are not independent. A modified test criterion is suggested when errors are not independent and a numerical example is included.
On Oblique Procrustes Rotation
Authors: Browne Michael W.
Published: January 2006

The equations involved in the rotation of an arbitrary factor matrix to a least squares fit to a specified factor structure have been known for many years. These equations, in general, cannot be solved by purely algebraic means, and an approximate solution has previously been used in practical applications.

In this paper an effective iterative method for obtaining the exact solution is developed. By algebraic manipulation the set of equations is expressed in the form of one polynomial equation in one unknown. Newton's method is suggested for solving this equation. Practical applications of the procedure indicate that convergence within small tolerance limits is generally attained after few iterations.
The Number of Ways to Label a Structure
Authors: Harary Frank , Palmer Edgar M. , Read Ronald C.
Published: January 2006

It has been observed that the number of different ways in which a graph with p points can be labelled is p! divided by the number of symmetries, and that this holds regardless of the species of structure at hand. In this note, a simple group-theoretic proof is provided.
Difficulty Scaling of Recognition Items
Authors: Cureton Edward E.
Published: January 2006

Usual system: take as scale value the normal deviate corresponding to the area p″ = (p − p0)/(1 − p0), where p is the proportion of correct answers and p0 is the chance proportion. If p ≦ p0, the item difficulty is unscalable. Proposed system: take as scale value the normal deviate corresponding to the area p′ = px, with x so chosen that p′ = .5 when p is half-way between p0 and 1. Table gives D (=500 + 100z) for p = .00(.01)1.00 and for a (= number of alternatives) 2,3,4,5,6,7, ∞.
A Rationale and Test for the Number of Factors in Factor Analysis
Authors: Horn John L.
Published: January 2006

It is suggested that if Guttman’s latent-root-one lower bound estimate for the rank of a correlation matrix is accepted as a psychometric upper bound, following the proofs and arguments of Kaiser and Dickman, then the rank for a sample matrix should be estimated by subtracting out the component in the latent roots which can be attributed to sampling error, and least-squares “capitalization” on this error, in the calculation of the correlations and the roots. A procedure based on the generation of random variables is given for estimating the component which needs to be subtracted.
Components of Reliability
Authors: LaForge Rolfe
Published: January 2006

As an alternative to the analysis of variance approach to reliability a multiple-factor analysis approach is illustrated. The one-factor and the multiple-factor models for reliability are compared. Tests on the latent roots associated with the principal components of intercorrelation matrices are used to determine the number of components to be retained. Conditions under which one or more of the principal components should be utilized are discussed.
A Method for Clustering Eigenvalues
Authors: Irwin Larry
Published: January 2006

A method of computing communalities is presented which attempts to cluster eigenvalues about zero. Results are exhibited which show reduction of rank using computed communalities.
Orthogonal Rotation to Congruence
Authors: Cliff Norman
Published: January 2006

Two problems are considered. The first is that of rotating two factor solutions orthogonally to a position where corresponding factors are as similar as possible. A least-squares solution for transformations of the two factor matrices is developed. The second problem is that of rotating a factor matrix orthogonally to a specified target matrix. The solution to the second problem is related to the first. Applications are discussed.
On the Estimation of the Tetrachoric Correlation Coefficient
Authors: Castellan N. John
Published: January 2006

This paper presents briefly the rationale of the tetrachoric correlation coefficient. Pearson's results are outlined and several estimates of the coefficient are given. These estimates are compared with Pearson's expressions to determine the relative accuracy of the various approximations in determining the tetrachoric correlation coefficient.
Comparative Scaling of Unidimensional Discrimination and Similarity Data
Authors: Bechtel Gordon G.
Published: January 2006

Scales constructed from paired-comparison designs under Thurstone's Case V model are discussed in relation to those derived from similarity data by means of the Unilateral Law of Comparative Judgment [3]. Factors inherent in the triadic similarity task are then considered with respect to scale invariance across experimental designs. Illustrative data, although revealing the influence of these factors upon similarity response consistency, indicate the similarity scale to be somewhat robust to their biasing effects.
An Empirical Study of a Logistic Mental Test Model
Authors: Ross John
Published: January 2006

A logistic model developed by Birnbaum was tested in two ways. First, plots of proportions of subjects in different score categories were examined for consistency with the assumption of a logistic trace line, and especially for departures from the logistic which seemed due to guessing in multiple choice items. The results showed that guessing seemed to have little effect. Second, an attempt was made to predict the obtained score distributions of samples of subjects on six tests from item parameters estimated on independent samples. The fits were good in all cases, despite considerable differences between the tests, and some extremely odd distributions.
Predicting Effectiveness of Bayesian Classification Systems
Authors: Herman Louis M. , Dollinger Michael B.
Published: January 2006

A model is presented for evaluating potential effectiveness of a Bayesian classification system using the expected value of the posterior probability for true classifications as an evaluation metric. For a given set of input parameters, the value of this complex metric is predictable from a simply computed row variance metric. Prediction equations are given for several representative sets of input parameters.
A Strong True-Score Theory, with Applications
Authors: Lord Frederic M.
Published: January 2006

A “strong” mathematical model for the relation between observed scores and true scores is developed. This model can be used

1. To estimate the frequency distribution of observed scores that will result when a given test is lengthened.

2. To equate true scores on two tests by the equipercentile method.

3. To estimate the frequencies in the scatterplot between two parallel (nonparallel) tests of the same psychological trait, using only the information in a (the) marginal distribution(s).

4. To estimate the frequency distribution of a test for a group that has taken only a short form of the test (this is useful for obtaining norms).

5. To estimate the effects of selecting individuals on a fallible measure.

6. To effect matching of groups with respect to true score when only a fallible measure is available.

7. To investigate whether two tests really measure the same psychological function when they have a nonlinear relationship.

8. To describe and evaluate the properties of a specific test considered as a measuring instrument.

The model has been tested empirically, using it to estimate bivariate distributions from univariate distributions, with good results, as checked by chi-square tests.
The Connection Between Magnitude and Discrimination Scales and Direct and Indirect Scaling Methods
Authors: Eisler Hannes
Published: January 2006

(1) An analysis of the method of paired comparisons shows that Case V and Case VI, the latter characterized by log-normal distributions and Weber’s law for subjective continua, are fundamentally indistinguishable. Case VI produces a log-arithmetic interval scale of subjective magnitude. (2) It is demonstrated that the difference between discrimination scales according to Case V and from category rating is due to the difference between intra- and interindividual variability yielding different Weber functions.
The Average Error of a Learning Model, Estimation and Use in Testing the Fit of Models
Authors: Kraemer Helena Chmura
Published: January 2006

A measure of the discrepancy between observed transition frequencies and those predicted by a learning model, an “average error” of a learning model, is presented. The maximum-likelihood estimator of the average error is derived and its use in a modified test of goodness of fit is demonstrated.
Matrix Formulas for Part and Partial Correlation
Authors: Jennings Earl
Published: January 2006

General formulas for part and partial correlation of any order are derived in terms of multiple correlation coefficients, standard partial regression weights, and validities. The relationship between part correlation and the “independent contribution of a predictor” is discussed.
The Approximate Sampling Distribution of Kuder-Richardson Reliability Coefficient Twenty
Authors: Feldt Leonard S.
Published: January 2006

An approximation to the sampling distribution of Kuder-Richardson reliability formula 20 is derived, using its algebraic equivalent obtained through an items-by-subjects analysis of variance. The theoretical distribution is compared to empirical estimates of the sampling distribution to assess how crucial certain assumptions are. The use of the theoretical distribution for testing hypotheses and deriving confidence intervals is illustrated. A table of equations for approximating 80, 90, and 95 per cent confidence intervals is presented with N ranging from 40 to 500.
A Note on the Normal Ogive or Logistic Curve in Item Analysis
Authors: Lord Frederic M.
Published: January 2006

It is common to assume that the proportion of correct answers to an item has a normal-ogive or logistic relationship to total test score. However, this is shown to be a mistaken and an undesirable notion.
Varisim: A New Machine Method for Orthogonal Rotation
Authors: Schönemann Peter H.
Published: January 2006

The derivative of Kaiser's Varimax criterion, if set to zero, yields a set of equations which are quite similar to those obtained for a least-squares problem of the “Procrustes” type. This similarity suggested an iterative technique for orthogonal rotation, dubbed “Varisim,” which was programmed for the IBM 7094 in FORTRAN. An empirical comparison between Varimax and Varisim, which was based on a number of data sets taken from the literature yielded three major results so far: (i) Varisim is slower than Varimax, roughly by a factor of 3, (ii) Varisim yields factors which in general contribute more evenly to the common test variance than Varimax factors, and which (iii) line up more closely with oblique configurations obtained with Binormamin than Varimax factors.
The Probability of a given 1-Choice Structure
Authors: Harary Frank , Read Ron
Published: January 2006

A 1-choice structure arises whenever each person in a group chooses exactly one other person according to some criterion. Our purpose is to study the situation in which these choices are made at random. As noted in Harary, Norman and Cartwright [2], such a structure can be represented by a directed graph in which the points represent people and the directed lines their choices. We first describe the shape of such a 1-choice structure, and define its symmetry number. With the help of these properties we are then able to develop and prove a formula which gives a probability of obtaining a given structure in the random choice situation. In order to supply data for these results, we include in the Appendix the diagrams of all 1-choice structures with at most 7 points and the probability of each.
The Computer Revolution in Psychometrics
Authors: Green Bert F.
Published: January 2006

We have said that psychometric methods involving algorithms are completely objective—at least they are if the algorithm is in the form of a program for a digital computer. These objective procedures need Monte Carlo and other computer runs to determine their properties, but so do many equation-oriented techniques. The objective algorithms are flexible but not flaccid. They offer a way of dealing with complexities that formerly seemed beyond our grasp. As the computer revolution continues in psychometrics, we can expect objective algorithmic methods to become the rule rather than the exception.
A Base-Free Measure of Change
Authors: Tucker Ledyard R , Damarin Fred , Messick Samuel
Published: January 2006

A model for the measurement of the discrepancy between two scores is presented and discussed as a paradigm for the study of growth or experimentally produced change. The model assumes two tests or measures differing in complexity, and it analyzes the true difference between the test scores into a component that is entirely dependent on the first or base-line test and a second component that is entirely independent of it. Equations for estimating both components are given and these are compared with other measurement efforts with similar goals.
Some Results Based on a General Stochastic Model for Mental Tests
Authors: Meredith William
Published: January 2006

There have been a number of important developments by several authors which are based on the assumption that test taking behavior is stochastic. That is, it is assumed that the probability that an individual will “endorse” or “pass” a dichotomous item can be represented by some function of an underlying, “latent” attribute (possibly multivariate) called the item trace line or item characteristic curve. Various forms for trace lines have been postulated such as polynomial [9, 11], step function [9], normal ogive [13, 19], and logistic [1, 2]. An interesting model due to Rasch [18] utilizes only a subject parameter and item parameter to determine the probability of passing an item. The only model which does not make some formal assumptions about trace lines is Lazarsfeld's latent class model. All of these models for test behavior are unified by two basic assumptions. One is the stochastic assumption of a probability of passing or endorsing an item.
A Set of Inequalities in Factor Analysis
Authors: Darroch J. N.
Published: January 2006

Inequalities relating the communalities to the multiple-correlation coefficients are derived. They are stronger than the well-known inequalities which have played an important role in factor analysis for the past thirty years. Necessary and sufficient conditions for equality are obtained.
On Admissible Communalities in Factor Analysis
Authors: Madansky Albert
Published: January 2006

A set of inequalities on communalities developed by Darroch [1] is used to provide some rules for determining admissible sets of communalities in factor analysis.
Estimating the Accuracy of Dichotomous Judgments
Authors: Fleiss Joseph L.
Published: January 2006

A reliability study is assumed to be carried out with each of a number of observers making a dichotomous judgment concerning each of a sample of subjects. A nonparametric model is proposed for the errors underlying such judgments, and conditions are given under which Cochran's Q statistic is valid for testing the hypothesis of no systematic differences among the judgments of the different observers. Inferences concerning the probabilities of error are shown to be possible in terms of the intraclass correlation coefficient. A numerical example is given.
A Special Latin Square for the Use of Each Subject “As His Own Control”
Authors: Benjamin Lorna S.
Published: January 2006

When the purpose of the experiment is to compare treatments, the Sequences × Positions Latin Square has been employed to control unwanted effects attributable to individuals, position, and sequence. This particular Latin Square has been subjected to criticism on the grounds there is confounding due to structure, random variables, and subject interactions. Special Latin Square, a subclass of the Sequences × Positions Latin Square, is basically a p ×p factorial design in blocks of size p. The two factors are treatments (T) and positions (P). Sequence is one component of the TP interaction, and square uniqueness is the sum of the remaining components. This completely replicated factorial design has no structural or random variable confounding; if subject interactions are present, square uniqueness may be used as the error term and the bias in the test of treatments will be conservative.
The Use of Factor Analysis in the Statistical Analysis of Multiple Time Series
Authors: Anderson T. W.
Published: January 2006

Factor analysis has been proposed and used as a method of statistical analysis of several measurements made on one individual repeatedly over time. This paper discusses some difficulties in applying factor analysis to multiple time series and attempts to indicate to what extent such methods can accomplish the goals of time series analysis. Some other methods are suggested.
A Response Time Model for Choice Behavior
Authors: Kintsch Walter
Published: January 2006

A subject in a two-choice situation characteristically makes several observing responses before performing the final choice. This behavior can be described by means of a random walk model. The present paper explores some possibilities as to how this model can be extended to include choice time. The assumption is made that the duration of each step in the random walk is a random variable which is exponentially distributed. With this assumption, one can predict the probability distributions of the choice times as well as the moments of these distributions.
A Note Concerning the Creager—Valentine Paper
Authors: Hazewinkel Aart
Published: January 2006

The Ui statistic, discussed by Creager and Valentine in a recent paper, is defined as a standard error of estimate for the regression of a composite, consisting of n components, on the n − 1 components, exclusive of the jth component. It is shown that this standard error equals the standard error of estimate for the prediction of the jth component from the other n − 1 components. This equality is used to suggest an efficient procedure to compute all primary uniquenesses Ui.
On the Symmetric Treatment of an Asymmetric Approach to Factor Analysis
Authors: Gibson W. A.
Published: January 2006

On the assumption that a partitioning can be found such that three mutually exclusive test vector configurations span the same factor space, a procedure is derived whereby symmetric parts of the correlation matrix are estimated from functions of asymmetric parts treated symmetrically. This yields an explicit matrix formula for communality estimation which generalizes earlier work by Albert. Conventional factoring methods, with all their computational and fitting advantages, can be applied once the symmetric portions of the correlation matrix have been estimated. Extension to four subgroups of test vectors allows for a matrix generalization of the old tetrad difference criterion to the multiple-factor case.
Mean Performance and the Factor Analysis of Learning Data
Authors: Ross John
Published: January 2006

The difference in factor structure resulting from the factorization of correlations, covariances, and cross products is discussed. Factoring cross products has the advantage of retaining information on both means and variances; this method of factoring learning data is recommended. The conditions under which factoring covariances and cross products lead to the same essential structure are investigated.
The Statistical Theory of Stepped-Up Reliability Coefficients when a Test has been divided into Several Equivalent Parts
Authors: Kristof Walter
Published: January 2006

This paper is a presentation of the statistical sampling theory of stepped-up reliability coefficients when a test has been divided into any number of equivalent parts. Maximum-likelihood estimators of the reliability are obtained and shown to be biased. Their sampling distributions are derived and form the basis of the definition of new unbiased estimators with known sampling distributions. These unbiased estimators have a smaller sampling variance than the maximum-likelihood estimators and are, because of this and some other favorable properties, recommended for general use. On the basis of the variances of the unbiased estimators the gain in accuracy in estimating reliability connected with further division of a test can be expressed explicitly. The limits of these variances and thus the limits of accuracy of estimation are derived. Finally, statistical small sample tests of the reliability coefficient are outlined. This paper also covers the sampling distribution of Cronbach's coefficient alpha.
The Prediction of Averages
Authors: Meredith William
Published: January 2006

Criterion measures are frequently obtained by averaging ratings, but the number and kind of ratings available may differ from individual to individual. This raises issues as to the appropriateness of any single regression equation, about the relation of variance about regression to number and kind of criterion observations, and about the preferred estimate of regression parameters. It is shown that if criterion ratings all have the same true score the regression equation for predicting the average is independent of the number and kind of criterion scores averaged.

Two cases are distinguished, one where criterion measures are assumed to have the same true score, and the other where criterion measures have the same magnitude of error of measurement as well. It is further shown that the variance about regression is a function of the number and kind of criterion ratings averaged, generally decreasing as the number of measures averaged increases. Maximum likelihood estimates for the regression parameters are derived for the two cases, assuming a joint normal distribution for predictors and criterion average within each subpopulation of persons for whom the same type of criterion average is available.
On a Theorem Stated by Eckart and Young
Authors: Johnson Richard M.
Published: January 2006

Proof is given for a theorem stated but not proved by Eckart and Young in 1936, which has assumed considerable importance in the theory of lower-rank approximations to matrices, particularly in factor analysis.
Short Cut Calculation of the Fisher-Yates “Exact Test”
Authors: Feldman Solomon E. , Klinger Eric
Published: January 2006

The Fisher-Yates “Exact Test” has become widely known to researchers in psychology as the significance test of choice for comparing pairs of proportions in 2 × 2 contingency tables, particularly when the expected frequency of one or more of the cells is small. Less widely recognized are some readily available tabular aids for calculating the “Exact Test” in a substantial number of the cases in which it might be applied. Thus, Finney [1] has tabled probability levels at or beyond the .05 level for those 2 × 2 tables in which neither of the two row marginals exceeds 15. Since it is a matter of arbitrary choice as to which pair of marginals of an obtained table is designated the row pair and which the column pair, Finney’s table has broad applicability. It has been incorporated into the Biometrika Tables for Statisticians [3], which also present extensive discussion of its applications and extensions. Finney’s table has been extended by Latscha [2] to accommodate those contingency tables in which neither of the two row marginals exceeds 20. The Biometrika Tables provide additional, if less direct, help in their tables of n! for values of n up to 250, and of log10n! for values of n up to 1000.
Hierarchical Clustering Schemes
Authors: Johnson Stephen C.
Published: January 2006

Techniques for partitioning objects into optimally homogeneous groups on the basis of empirical measures of similarity among those objects have received increasing attention in several different fields. This paper develops a useful correspondence between any hierarchical system of such clusters, and a particular type of distance measure. The correspondence gives rise to two methods of clustering that are computationally rapid and invariant under monotonic transformations of the data. In an explicitly defined sense, one method forms clusters that are optimally “connected,” while the other forms clusters that are optimally “compact.”