bluuebunny/crossref_metadata_2025 huggingface dataset:
A General Nonmetric Technique for Finding the Smallest Coordinate Space for a Configuration of Points
Authors: Guttman Louis
Published: January 2006

Let A1, A2, ..., An be any n objects, such as variables, categories, people, social groups, ideas, physical objects, or any other. The empirical data to be analyzed are coefficients of similarity or distance within pairs (Ai, Ai), such as correlation coefficients, conditional probabilities or likelihoods, psychological choice or confusion, etc. It is desired to represent these data parsimoniously in a coordinate space, by calculating m coordinates {xia} for each Ai for a semi-metric d of preassigned form dij = d(|xi1 -xj1|, |xi2 -xj2|, ..., |xim - xjm|). The dimensionality m is sought to be as small as possible, yet satisfy the monotonicity condition that dij <dkl whenever the observed data indicate that Ai is “closer” to Aj than Ak is to Al. Minkowski and Euclidean spaces are special metric examples of d. A general coefficient of monotonicity μ is defined, whose maximization is equivalent to optimal satisfaction of the monotonicity condition, and which allows various options both for treatment of ties and for weighting error-of-fit. A general rationale for algorithm construction is derived for maximizing μ by gradient-guided iterations; this provides a unified mathematical solution to the basic operational problems of norming the gradient to assure proper convergence, of trading between speed and robustness against undesired stationary values, and of a rational first approximation. Distinction is made between single-phase (quadratic) and two-phase (bilinear) strategies for algorithm construction, and between “hard-squeeze” and “soft-squeeze” tactics within these strategies. Special reference is made to the rank-image and related transformational principles, as executed by current Guttman-Lingoes families of computer programs.
Item Characteristic Curves Estimated without Knowledge of their Mathematical form–a Confrontation of Birnbaum's Logistic Model
Authors: Lord Frederic M.
Published: January 2006

Item characteristic curves are estimated without restrictive assumptions about their mathematical form. The resulting curves are compared with estimates obtained under the assumption that all curves are of logistic form. Surprising agreement is found between the curves obtained by the two unrelated methods.
Retrospective Anticipation
Authors: Harman Harry H.
Published: January 2006

The objectives of the Psychometric Society are coincident with those of its journal, namely, “the development of psychology as a quantitative rational science.” This general theme has frequently been referred to by the past presidents in their annual addresses. They did so by employing several different approaches: some by proposing directions of future research of the Society; some by making critical-instructive appraisals of past research; some by presenting reports of their own ongoing research; and some by viewing the present scene from an historical or philosophical perspective. Furthermore, the tone of these addresses has run the gamut from the frivolous to the profoundly scholarly.

A brief analysis of the previous addresses disclosed some interesting items. First, there have been eight instances, in our 34 year history, when the address was skipped entirely, beginning with Thurstone’s [1937] published in Science rather than in Psychometrika. Of the addresses that subsequently appeared in Psychometrika, the average was 13 pages (including references) with a range from 5 to 26 pages. (Hopefully, the longer ones were not read in their entirety.) More than half the papers were expository in nature. Although 12 were reports on serious research problems, only four contained heavy mathematical developments. Five addresses could be characterized as facetious or humorous, but nonetheless carried a moral for the Society. In general, suggestions were made in 13 instances for future research or directions of emphasis for the Society, and 9 addressed themselves explicitly to its aims. There was also a sprinkling of criticism, philosophy, and history relating to the Society or to specific psychometric methods. Of course, the foregoing classifications were not mutually exclusive, and merely constitute one man’s opinions.
Evaluation of Factor Analytic Research Procedures by means of Simulated Correlation Matrices
Authors: Tucker Ledyard R , Koopman Raymond F. , Linn Robert L.
Published: January 2006

In order to study the effectiveness of factor analytic methods, a procedure was developed for computing simulated correlation matrices which are more similar to real data correlation matrices than are those matrices computed from the factor analysis structural model. In the present investigation, three methods of factor extraction were studied as applied to 54 simulated correlation matrices which varied in proportion of variance derived from a major factor domain, number of factors in the major domain, and closeness of the simulation procedure to the factor analysis structural model. While the factor extraction methods differed little from one another in quality of results for matrices more dissimilar to the factor analytic model, major differences in quality of results were associated with fewer factors in the major domain, higher proportion of variance from the major domain, and closeness of the simulation procedure to the factor analysis structural model.
An Analysis of Some Conditions for Representing N State Markov Processes as General all or None Models
Authors: Steiner Theodore E. , Greeno James G.
Published: January 2006

Recently Markov learning models with two unidentifiable presolution success states, an error state, and an absorbing learned state, have been suggested to handle certain aspects of data better than the three state Markov models of the General All or None model type. In attempting to interpret psychologically, and evaluate statistically the adequacy of various classes of Markov models, a knowledge of the relationship between the classes of models would be helpful. This paper considers some aspects of the relationship between the class of General All or None models and the class of Stationary Absorbing Markov models with N error states, and M presolution success states.
Control Charts for Sequential Testing
Authors: Levitt H. , Treisman M.
Published: January 2006

Control charts facilitating the use of sequential strategies of the updown or staircase type are described. The charts are easy to use and the relevant estimates may be derived very simply from the chart. An example is given in which both the 29.3% and 70.7% points on a response curve are estimated using the Up-Down-Transformed-Response (UDTR) procedure. The charts can be used in psychophysics and are also generally applicable to experiments or control processes involving binary responses.
Prediction in Future Samples Studied in Terms of the Gain from Selection
Authors: Gross Alan L.
Published: January 2006

The gain from selection (GS) is defined as the standardized average performance of a group of subjects selected in a future sample using a regression equation derived on an earlier sample. Expressions for the expected value, density, and distribution function (DF) of GS are derived and studied in terms of sample size, number of predictors, and the prior distribution assigned to the population multiple correlation. The DF of GS is further used to determine how large sample sizes must be so that with probability .90 (.95), the expected GS will be within 90 percent of its maximum possible value. An approximately unbiased estimator of the expected GS is also derived.
Maximizing the Coefficient of Generalizability in Multi-Facet Decision Studies
Authors: Woodward J. Arthur , Joe George W.
Published: January 2006

For random-model, fully-crossed, two- and three-facet experimental designs the following two problems were considered. First, equations were developed for determining the optimal number of conditions of a facet for maximizing the coefficient of generalizability under the constraint that the total number of observations per subject is constant. Second, the problem of determining the minimum number of observations per subject for a specified generalizability coefficient is solved for the two-facet crossed design.
Probabilistic Unfolding
Authors: Sixtl Friedrich
Published: January 2006

A probabilistic unfolding model is presented which makes it possible to scale stimuli and subjects simultaneously, without assumptions concerning the distribution of subjects or of stimuli. The applicability of the model is demonstrated in areas of attitude measurement where the assumption of normality is inadequate.
A Reliability Coefficient for Maximum Likelihood Factor Analysis
Authors: Tucker Ledyard R , Lewis Charles
Published: January 2006

Maximum likelihood factor analysis provides an effective method for estimation of factor matrices and a useful test statistic in the likelihood ratio for rejection of overly simple factor models. A reliability coefficient is proposed to indicate quality of representation of interrelations among attributes in a battery by a maximum likelihood factor analysis. Usually, for a large sample of individuals or objects, the likelihood ratio statistic could indicate that an otherwise acceptable factor model does not exactly represent the interrelations among the attributes for a population. The reliability coefficient could indicate a very close representation in this case and be a better indication as to whether to accept or reject the factor solution.
The Estimation of Proportions in m Groups
Authors: Novick Melvin R. , Lewis Charles , Jackson Paul H.
Published: January 2006

In many applications, it is desirable to estimate binomial proportions in m groups where it is anticipated that these proportions are similar but not identical. Following a general approach due to Lindley, a Bayesian Model II aposteriori modal estimate is derived that estimates the inverse sine transform of each proportion by a weighted average of the inverse sine transform of the observed proportion in the individual group and the average of the estimated values. Comparison with a classical method due to Jackson spotlights some desirable features of Model II analyses. The simplicity of the present formulation makes it possible to study the behavior of the Bayesian Model II approach more closely than in more complex formulations. Also, it is possible to estimate the amount of gain afforded by the Model II analyses.
Comments on the “Lens Model” Equation and Analysis of Multiple-Cue Judgment Tasks
Authors: Castellan N. John
Published: January 2006

It is the purpose of this paper to discuss the ‘Lens Mode’ approach to the analysis of subject performance in multiple-cue judgment tasks embedded in probabilistic environments. Various characteristics of the linear lens model are presented in matrix notation and are formulated without reference to standardized variables in order to facilitate analysis and subsequent generalization. Some new results on ‘matching’ behavior are presented. A brief outline of current problems and related work is presented.
Testing a Linear Relation between True Scores of Two Measures
Authors: Kristof Walter
Published: January 2006

We concern ourselves with the hypothesis that two variables have a perfect disattenuated correlation, hence measure the same trait except for errors of measurement. This hypothesis is equivalent to saying, within the adopted model, that true scores of two psychological tests satisfy a perfect linear relation. Statistical tests of this hypothesis are derived when the relation is specified with the exception of the additive constant. Two approaches are presented and various assumptions concerning the error parameters are used. Then the results are reinterpreted in terms of the possible existence of an unspecified perfect linear relation between true scores of two psychological tests. A numerical example is appended by way of illustration.
On a Statistic Arising in Testing Correlation
Authors: Kristof Walter
Published: January 2006

This paper is devoted to the study of a certain statistic, u, defined on samples from a bivariate population with variances σ11, σ22 and correlation ρ. Let the parameter corresponding to u be υ. Under binormality assumptions the following is demonstrated. (i) If σ11 = σ22, then the distribution of u can be obtained rapidly from the F distribution. Statistical inferences about ρ = υ may be based on F. (ii) In the general case, allowing for σ11 ≠ σ22, a certain quantity involving u, r (sample correlation between the variables) and υ follows a t distribution. Statistical inferences about υ may be based on t. (iii) In the general case a quantity t′ may be constructed which involves only the statistic u and only the parameter υ. If treated like a t distributed magnitude, t′ admits conservative statistical inferences. (iv) The F distributed quantity mentioned in (i) is equivalent to a certain t distributed quantity as follows from an appropriate transformation of the variable. (v) Three test statistics are given which can be utilized in making statistical inferences about ρ = υ in the case σ11 = σ22. A comparison of expected lengths of confidence intervals for ρ obtained from the three test statistics is made. (vi) The use of the formulas derived is illustrated by means of an application to coefficient alpha.
The Relation between Sample and Population Characteristic Vectors
Authors: Cliff Norman
Published: January 2006

Data are reported which show the statistical relation between the sample and population characteristic vectors of correlation matrices with squared multiple correlations as communality estimates. Sampling fluctuations were found to relate only to differences in the square roots of characteristic roots and to sample size. A principle for determining the number of factors to rotate and interpret after rotation is suggested.
A Comparison of Three Predictor Selection Techniques in Multiple Regression
Authors: McCornack Robert L.
Published: January 2006

Three methods for selecting a few predictors from the many available are described and compared with respect to shrinkage in cross-validation. From 2 to 6 predictors were selected from the 15 available in 100 samples ranging in size from 25 to 200. An iterative method was found to select predictors with slightly, but consistently, higher cross-validities than the popularly used stepwise method. A gradient method was found to equal the performance of the stepwise method only in the larger samples and for the largest predictor subsets.
Simultaneous Factor Analysis in Several Populations
Authors: Jöreskog K. G.
Published: January 2006

This paper is concerned with the study of similarities and differences in factor structures between different groups. A common situation occurs when a battery of tests has been administered to samples of examinees from several populations.

A very general model is presented, in which any parameter in the factor analysis models (factor loadings, factor variances, factor covariances, and unique variances) for the different groups may be assigned an arbitrary value or constrained to be equal to some other parameter. Given such a specification, the model is estimated by the maximum likelihood method yielding a large sample x2 of goodness of fit. By computing several solutions under different specifications one can test various hypotheses.

The method is capable of dealing with any degree of invariance, from the one extreme, where nothing is invariant, to the other extreme, where everything is invariant. Neither the number of tests nor the number of common factors need to be the same for all groups, but to be at all interesting, it is assumed that there is a common core of tests in each battery that is the same or at least content-wise comparable.
Econometrics and Psychometrics: A Survey of Communalities
Authors: Goldberger Arthur S.
Published: January 2006

Several themes which are common to both econometrics and psychometrics are surveyed. The themes are illustrated by reference to permanent income hypotheses, simultaneous equation models, adaptive expectations and partial adjustment schemes, and by reference to test score theory, factor analysis, and time-series models.
A Comparative Evaluation of Several Prominent Methods of Oblique Factor Transformation
Authors: Hakstian A. Ralph
Published: January 2006

The oblimax, promax, maxplane, and Harris-Kaiser techniques are compared. For five data sets, of varying reliability and factorial complexity, each having a graphic oblique solution (used as criterion), solutions obtained using the four methods are evaluated on (1) hyperplane-counts, (2) agreement of obtained with graphic within-method primary factor correlations and angular separations, (3) angular separations between obtained and corresponding graphic primary axes. The methods are discussed and ranked (descending order): Harris-Kaiser, promax, oblimax, maxplane. The Harris-Kaiser procedure—independent cluster version for factorially simple data, P'P proportional to Φ, with equamax rotations, for complex—is recommended.
Some Boundary Conditions for a Monotone Analysis of Symmetric Matrices
Authors: Lingoes James C.
Published: January 2006

This paper gives a rigorous and greatly simplified proof of Guttman's theorem for the least upper-bound dimensionality of arbitrary real symmetric matrices S, where the points embedded in a real Euclidean space subtend distances which are strictly monotone with the off-diagonal elements of S. A comparable and more easily proven theorem for the vector model is also introduced. At most n-2 dimensions are required to reproduce the order information for both the distance and vector models and this is true for any choice of real indices, whether they define a metric space or not. If ties exist in the matrices to be analyzed, then greatest lower bounds are specifiable when degenerate solutions are to be avoided. These theorems have relevance to current developments in nonmetric techniques for the monotone analysis of data matrices.
Relations between Multidimensional Scaling and Three-Mode Factor Analysis
Authors: Tucker Ledyard R
Published: January 2006

A combination is achieved of two lines of psychometric interest: a) multidimensional scaling and b) factor analysis. This is accomplished with the use of three-mode factor analysis of scalar product matrices, one for each subject. Two of the modes are the groups of objects scaled and the third mode is the sample of subjects. Results are an object space, a person space, and a system for changing weights given to dimensions and of angles between dimensions in the object space for individuals located at different places in the person space. The development is illustrated with data from an adjective similarity study.
Estimating Item Parameters and Latent Ability when Responses are Scored in Two or More Nominal Categories
Authors: Bock R. Darrell
Published: January 2006

A multivariate logistic latent trait model for items scored in two or more nominal categories is proposed. Statistical methods based on the model provide 1) estimation of two item parameters for each response alternative of each multiple choice item and 2) recovery of information from “wrong” responses when estimating latent ability. An application to a large sample of data for twenty vocabulary items shows excellent fit of the model according to a chi-square criterion. Item and test information curves are compared for estimation of ability assuming multiple category and dichotomous scoring of these items. Multiple scoring proves substantially more precise for subjects of less than median ability, and about equally precise for subjects above the median.
Some New Results on Factor Indeterminacy
Authors: Schönemann Peter H. , Wang Ming-Mei
Published: January 2006

Some relations between maximum likelihood factor analysis and factor indeterminacy are discussed. Bounds are derived for the minimum average correlation between equivalent sets of correlated factors which depend on the latent roots of the factor intercorrelation matrix Ψ. Empirical examples are presented to illustrate some of the theory and indicate the extent to which it can be expected to be relevant in practice.
The Fisher-Yates Exact Test and Unequal Sample Sizes
Authors: Johnson Edgar M.
Published: January 2006

A computational short cut suggested by Feldman and Klinger for the one-sided Fisher-Yates exact test is clarified and is extended to the calculation of probability values for certain two-sided tests when sample sizes are unequal.
The Minimum Average Correlation between Equivalent Sets of Uncorrelated Factors
Authors: Schönemann Peter H.
Published: January 2006

A simplified proof of a lemma by Ledermann [1938], which lies at the core of the factor indeterminacy issue, is presented. It leads to a representation of an orthogonal matrix T, relating equivalent factor solutions, which is different from Ledermann's [1938] and Guttman's [1955]. T is used to evaluate bounds on the average correlation between equivalent sets of uncorrelated factors. It is found that the minimum average correlation is independent of the data.
A Nonmetric Variety of Linear Factor Analysis
Authors: Kruskal Joseph B. , Shepard Roger N.
Published: January 2006

The numbers in each column of an n × m matrix of multivariate data are interpreted as giving the measured values of all n of the objects studied on one of m different variables. Except for random error, the rank order of the numbers in such a column is assumed to be determined by a linear rule of combination of latent quantities characterizing each row object with respect to a small number of underlying factors. An approximation to the linear structure assumed to underlie the ordinal properties of the data is obtained by iterative adjustment to minimize an index of over-all departure from monotonicity. The method is “nonmetric” in that the obtained structure in invariant under monotone transformations of the data within each column. Except in certain degenerate cases, the structure is nevertheless determined essentially up to an affine transformation. Tests show (a) that, when the assumed monotone relationships are strictly linear, the recovered structure tends closely to approximate that obtained by standard (metric) factor analysis but (b) that, when these relationships are severely nonlinear, the nonmetric method avoids the inherent tendency of the metric method to yield additional, spurious factors. From the practical standpoint, however, the usefulness of the nonmetric method is limited by its greater computational cost, vulnerability to degeneracy, and sensitivity to error variance.
A Comparative Study of Association Measures
Authors: Särndal Carl Erik
Published: January 2006

This paper discusses the general problem of measuring the association between an independent nominal-scaled variable X and a dependent variable Y whose scale of measurement may be interval, ordinal or nominal. The theoretical foundations of a wide range of asymmetric association measures are discussed. Some new measures are also suggested. Fifteen of these association measures, some previously suggested, some new, are singled out for a computer-assisted numerical study in which we compute the value actually taken by each measure under a wide variety of conditions. This comparative study provides important insights into the behavior of the measures.
Models for the Analysis of Alternative Sources of Growth in Correlated Stochastic Variables
Authors: Frederiksen Carl H.
Published: January 2006

An approach to the analysis of multivariate time series is presented in which linear structural relationships among multiple stochastic variables are investigated. A number of alternative structural models are considered for the case of two stochastic variables. Each model represents a possible hypothesis concerning the relationship of growth in one variable to growth in the second. Both symmetric and asymmetric models are considered. Extensions of two of the models to three variables are illustrated by means of a numerical example. Implications of the models for the problem of detecting change in multivariate time series are discussed.
Determining Parameter Values in the Generalized Image System
Authors: Koopman Raymond F.
Published: January 2006

Generalized image analysis is considered as a logical algebraic extension of Guttmanian image analysis. Under the assumption that a reduced-rank description of the images is desired, a procedure is developed which achieves the scale-free property by simultaneously rescaling in the metrics of both the images and anti-images, and which produces images and anti-images that are maximally independent in terms of the dimensions needed to account for them.
Empirical Bayes Point Estimates of Latent Trait Scores without Knowledge of the Trait Distribution
Authors: Meredith William , Kearns Jack
Published: January 2006

In this paper, recent developments in empirical Bayes procedures are tied in with current work in mental test theory. Point estimators of “true scores” are derived for the binomial and Rasch test models. These estimators are shown to be asymptotically optimal. Smoothing and an empirical study of the behavior of empirical Bayes estimates are taken up in the final section.
Metric Unfolding: Data Requirement for Unique Solution & Clarification of Schönemann's Algorithm
Authors: Gold E. Mark
Published: January 2006

The object of this paper is to clarify Schönemann's unfolding algorithm and, in particular, to make it clear that the equations numbered (3.2) in Schönemann's [1970] article, which define Schönemann's solutions, are not a complete set of restraints for the purpose of defining metric unfoldings. Namely, Schönemann has transformed the original equations which define an unfolding to a set of linear and non-linear equations of which he uses only the linear equations to define his solutions. Given infallible data (solution(s) exist) Schönemann's solutions will include the correct solutions. If enough data are available so that there are enough linear equations to uniquely determine a single solution, then Schönemann's solution will coincide with the correct solution. Let P and Q denote the number of elements in the two sets of points, the interset distances of which are specified by the data in the unfolding problem. Let m denote the dimensionality of the Euclidean space into which these points are to be imbedded. If only the linear equations, numbered (18) herein, are to be used, then Schönemann gives the following data requirement for the solution to be uniquely determined: If the full set of linear and nonlinear equations (18–20) are used, then the amount of data required for a solution to be locally unique is relaxed to Both of these results assume that the equations are independent, which has not been proved.
Standard Errors for Rotated Factor Loadings
Authors: Archer Claude O. , Jennrich Robert I.
Published: January 2006

Beginning with the results of Girshick on the asymptotic distribution of principal component loadings and those of Lawley on the distribution of unrotated maximum likelihood factor loadings, the asymptotic distribution of the corresponding analytically rotated loadings is obtained. The principal difficulty is the fact that the transformation matrix which produces the rotation is usually itself a function of the data. The approach is to use implicit differentiation to find the partial derivatives of an arbitrary orthogonal rotation algorithm. Specific details are given for the orthomax algorithms and an example involving maximum likelihood estimation and varimax rotation is presented.
Standard Errors for Obliquely Rotated Factor Loadings
Authors: Jennrich Robert I.
Published: January 2006

In a manner similar to that used in the orthogonal case, formulas for the aymptotic standard errors of analytically rotated oblique factor loading estimates are obtained. This is done by finding expressions for the partial derivatives of an oblique rotation algorithm and using previously derived results for unrotated loadings. These include the results of Lawley for maximum likelihood factor analysis and those of Girshick for principal components analysis. Details are given in cases including direct oblimin and direct Crawford-Ferguson rotation. Numerical results for an example involving maximum likelihood estimation with direct quartimin rotation are presented. They include simultaneous tests for significant loading estimates.
A Test Theory Model for Ordinal Measurements
Authors: Schulman Robert S. , Haden Richard L.
Published: January 2006

A model is proposed for the description of ordinal test scores based on the definition of true score as expected rank. Derivations from the model are compared with results from clasiscal test theory as developed by Lord and Novick, in particular with respect to parallel tests and composites. An unbiased estimator of population true score from sample data is derived and its variance is shown to decrease with increasing sample size. Population reliability is shown to be analytically related to expected sample reliability, and methods of reliability estimation are discussed.
On Estimation and Hypothesis Testing Problems for Correlation Coefficients
Authors: Kraemer Helena Chmura
Published: January 2006

A selection of statistical problems commonly encountered in psychological or psychiatric research concerning correlation coefficients are re-evaluated in the light of recently developed simplifications in the forms of the distribution theory of the intraclass correlation coefficient (exact theory), of the product-moment correlation coefficient and of the Spearman rank correlation coefficient (approximate).
Bayesian Estimation in Unrestricted Factor Analysis: A Treatment for Heywood Cases
Authors: Martin James K. , McDonald Roderick P.
Published: January 2006

A Bayesian procedure is given for estimation in unrestricted common factor analysis. A choice of the form of the prior distribution is justified. It is shown empirically that the procedure achieves its objective of avoiding inadmissible estimates of unique variances, and is reasonably insensitive to certain variations in the shape of the prior distribution.
The Use of Extreme Groups in Assessing Relationships
Authors: Alf. Edward F. , Abrahams Norman M.
Published: January 2006

The relationship between variables in applied and experimental research is often investigated by the use of extreme (i.e., upper and lower) groups. Earlier analytical work has demonstrated that the extreme groups procedure is more powerful than the standard correlational approach for some values of the correlation and extreme group size. The present article provides methods for using the covariance information that is usually discarded in the classical extreme groups approach. Essentially, then, the new procedure combines the extreme groups approach and the correlational approach. Consequently, it includes the advantages of each and is shown to be more powerful than either approach used alone.
An Empirical Comparison of Several Interval Estimation Procedures for Coefficient Alpha
Authors: Pandey Tej N. , Hubert Lawrence
Published: January 2006

Following a general introduction to Tukey’s Jackknife technique and the construction of approximate confidence intervals, ten variants of the procedure are defined for giving interval estimates for coefficient alpha. A data base of known population characteristics was generated and used to compare the robustness of the ten Jackknife alternatives with Feldt’s well-known sampling theory based upon the assumptions of normality. The empirical results indicate that out of the ten variants, defined by five transformations and two methods of data subdivision, only two are justifiable competitors to Feldt’s approach.
A Relationship between Harris Factors and Guttman’s Sixth Lower Bound to Reliability
Authors: Nicewander W. Alan
Published: January 2006

Using an approach nearly identical to one adopted by Guttman, it is established that within the framework of classical test theory the squared multiple correlation for predicting an element of a composite measure from the n − 1 remaining elements is a lower-bound to the reliability of the element. The relationship existing between the reliabilities of the elements of a composite measure and their squared-multiple correlations with remaining elements is used to derive Guttman’s sixth lower bound (λ6) to the reliability of a composite measure. It is shown that Harris factors of a correlation matrix R are associated with a set of (observable) uncorrelated latent variables having maximum coefficients λ6.
On Accuracy in Reliability Estimation
Authors: Kristof Walter
Published: January 2006

This study in parametric test theory deals with the statistics of reliability estimation when scores on two parts of a test follow a binormal distribution with equal (Case 1) or unequal (Case 2) expectations. In each case biased maximum-likelihood estimators of reliability are obtained and converted into unbiased estimators. Sampling distributions are derived. Second moments are obtained and utilized in calculating mean square errors of estimation as a measure of accuracy. A rank order of four estimators is established. There is a uniformly best estimator. Tables of absolute and relative accuracies are provided for various reliability parameters and sample sizes.
An Index of Factorial Simplicity
Authors: Kaiser Henry F.
Published: January 2006

An index of factorial simplicity, employing the quartimax transformational criteria of Carroll, Wrigley and Neuhaus, and Saunders, is developed. This index is both for each row separately and for a factor pattern matrix as a whole. The index varies between zero and one. The problem of calibrating the index is discussed.
On the Determination of Appropriate Dimensionality in Data with Error
Authors: Isaac Paul D. , Poor David D. S.
Published: January 2006

The study deals with the problem of determining true dimensionality of data-with-error scaled by Kruskal's multidimensional scaling technique. Artificial data was constructed for 6, 8, 12, 16, and 30 point configurations of 1, 2, or 3 true dimensions by adding varying amounts of error to the true distances. Results show how stress is affected by error, number of points, and number of dimensions, and indicate that stress and the “elbow” criterion are inadequate for purposes of identifying true dimensionality when there is error in the data. The Wagenaar-Padmos procedure for identifying true dimensionality and error level is discussed. A simplified technique, involving a measure called Constraint, is suggested.
The Guttman-Harris Uniqueness Estimates
Authors: Meyer Edward P.
Published: January 2006

It is shown that, under very general conditions, uniqueness estimates proposed independently by Guttman [1957] and by Harris [1963] provide tighter upper bounds on the unknown uniqueness values of factor analysis than do existing estimates.
Nonlinear Submodels of Orthogonal Linear Models
Authors: Bechtel Gordon G.
Published: January 2006

A model containing linear and nonlinear parameters (e. g., a spatial multidimensional scaling model) is viewed as a linear model with free and constrained parameters. Since the rank deficiency of the design matrix for the linear model determines the number of side conditions needed to identify its parameters, the design matrix acts as a guide in identifying the parameters of the nonlinear model. Moreover, if the design matrix and the uniqueness conditions constitute an orthogonal linear model, then the associated error sum of squares may be expressed in a form which separates the free and constrained parameters. This immediately provides least squares estimates of the free parameters, while simplifying the least squares problem for those which are constrained. When the least squares estimates for a nonlinear model are obtained in this way, i.e. by conceptualizing it as a submodel, the final error sum of squares for the nonlinear model will be a restricted minimum whenever the side conditions of the model become real restrictions upon its submodel. In this case the design matrix for the embracing orthogonal model serves as a guide in introducing parameters into the nonlinear model as well as in identifying these parameters. The method of overwriting a nonlinear model with an orthogonal linear model is illustrated with two different spatial analyses of a three-way preference table.
Measures of Reliability for Profiles and Test Batteries
Authors: Conger Anthony J. , Lipshitz Raanan
Published: January 2006

A general index of reliability, termed “canonical reliability,” is developed for use with profiles, or more generally, for use with vectors of random variables. Canonical reliability is defined as the ratio of the average squared distance among true scores to the average squared distance among observed scores. Based on Mahalonobis distances, canonical reliability is shown to be consistent with multivariate analogues of parallel form correlations, squared correlation between true and observed scores, and an analysis of variance formulation. The index of reliability based on Cronbach and Gleser's D2 is also derived from the general formulation. A comparison of the Mahalonobis and D2 approaches indicates that score vectors using D2 distances are more reliable; however, both methods of comparing profiles are useful depending on the nature of the information that is desired. Transforming the observed variables to independent canonical variates provides a basis for comparing profiles on maximally reliable profile dimensions. For illustrative purposes, profile reliability is calculated and interpreted for the WISC subscales for a 7 1/2 year age group.
Estimation of Reliability and True Score Variance from a Split of a Test into Three Arbitrary Parts
Authors: Kristof Walter
Published: January 2006

This paper gives a method of estimating the reliability of a test which has been divided into three parts. The parts do not have to satisfy any statistical criteria like parallelism or τ-equivalence. If the parts are homogeneous in content (congeneric), i.e., if their true scores are linearly related and if sample size is large then the method described in this paper will give the precise value of the reliability parameter. If the homogeneity condition is violated then underestimation will typically result. However, the estimate will always be at least as accurate as coefficient α and Guttman's lower bound λ3 when the same data are used. An application to real data is presented by way of illustration. Seven different splits of the same test are analyzed. The new method yields remarkably stable reliability estimates across splits as predicted by the theory. One deviating value can be accounted for by a certain unsuspected peculiarity of the test composition. Both coefficient α and λ3 would not have led to the same discovery.
Interpretation of Canonical Analysis: Rotated vs. Unrotated Solutions
Authors: Cliff Norman , Krus David J.
Published: January 2006

Orthogonal rotation of canonical variates is shown to preserve the major properties of the canonical solution and may increase its interpretability.
A Monte Carlo Evaluation of Interactive Multidimensional Scaling
Authors: Girard Roger A. , Cliff Norman
Published: January 2006

Interactive Scaling with Individual Subjects (ISIS) developed by Young & Cliff [1972], is a method involving interaction between subject and computer in real time to determine which judgments made by the subject are critical to the definition of a dimensional structure. The procedure is based on the mathematical fact that it is possible to define a space of R dimensions in terms of only the interpoint distances between all stimuli being scaled and a subset of (R +1) of these stimuli. For errorless judgments, any subset of (R + 1) stimuli is appropriate. However, fallible data require that the subset consist of stimuli that are maximally dissimilar, and the ISIS procedure is designed to obtain such an optimum subset (a “basis”).

This research evaluates a modified version of ISIS with respect to (a) a metric MDS analysis based on all possible pairs of the stimuli, and (b) a metric MDS analysis based on a subset of one-third of the possible pairs, or about the same number as that required by ISIS. Results show that the ISIS method achieves better fit than (b) at low error levels, and may also achieve better fit than (b) at higher error levels if the size of the basis is increased. The more stimuli in the basis the more indices of fit approach those of (a).

A new method of introducing error in MDS studies is used in the evaluation.
Statistical Consideration of Mastery Scores
Authors: Huynh Huynh
Published: January 2006

The kind of mastery score problem often encountered in educational and psychological testing may be roughly described as follows: given that a certain unit of instruction must be mastered by a subject, a test is administered at the end of this unit and, on the basis of the observed score or an estimate of the true score, a decision is made about the grant of the mastery status. If the subject is declared master of the unit, he will be allowed to proceed to the next unit of instruction. Otherwise, remedial work will be given. Given relevant consequences of misclassification, the problem is to determine an optimum decision rule.
An Extended Class of Matched Pairs Tests based on Powers of Ranks
Authors: Mielke Paul W. , Berry Kenneth J.
Published: January 2006

A large class of rank tests, which includes the familiar sign test and the Wilcoxon signed-ranks test, is described and discussed. This class of distribution-free tests provides a flexible basis for testing research hypotheses of various forms. Exact small sample and approximate large sample procedures are considered. Applications of these procedures are presented, including simple numerical examples.
Components of Correlation and Extensions of the Lens Model Equation
Authors: Stewart Thomas R.
Published: January 2006

By applying a general procedure for analyzing a correlation coefficient into components, the lens model equation is extended to a) analyze the effects of different types of variation and b) analyze the relations between judgmental systems that are not based on the same set of cues.
Jackknifing Disattenuated Correlations
Authors: Rogers W. Todd
Published: January 2006

The utility of the jackknife for constructing confidence intervals and testing hypotheses about the disattenuated correlation is evaluated for small samples. Computer simulations were used to generate the empirical sampling distributions of jackknife statistics for two sample sizes (30, 60), five values of the disattenuated correlation coefficient (1.00, .90, .80, .50, .00) and three pairs of reliabilities (.90, .80; .80, .80; .90, .50). The theoretical and cumulative proportions of jackknife statistics were compared at selected points in the appropriate t-distributions. The results obtained support the claim that the jackknife can be used to construct sensible confidence intervals. However, the jackknife possesses limited utility for testing hypotheses about the disattenuated correlation coefficient.
A Simple Proof of the Spearman-Brown Formula for Continuous Test Lengths
Authors: Allison Paul D.
Published: January 2006

A direct proof is given for the generalized Spearman-Brown formula for any real multiple of test length.
Some Applications of Graph Theory to Clustering
Authors: Hubert Lawrence J.
Published: January 2006

This paper attempts to review and expand upon the relationship between graph theory and the clustering of a set of objects. Several graphtheoretic criteria are proposed for use within a general clustering paradigm as a means of developing procedures “in between” the extremes of complete-link and single-link hierarchical partitioning; these same ideas are then extended to include the more general problem of constructing subsets of objects with overlap. Finally, a number of related topics are surveyed within the general context of reinterpreting and justifying methods of clustering either through standard concepts in graph theory or their simple extensions.
Testing that K Independent Random Samples were Drawn from K Normal Populations
Authors: Levy Kenneth J.
Published: January 2006

Recently, D’Agostino and Cureton {1972] discussed a test of normality against skewed alternatives; the test was based upon g1, the third standardized sample moment. In an earlier paper, D’Agostino [1970] suggested that an experimenter might be interested in testing the underlying assumption of normality associated with an analysis of variance. For such a test, one would be required to decide if k independent random samples were drawn from k normal populations. The purpose of this note is to suggest a simple extension of the test based upon g1, to the case of k independent random samples.
Ordmet: A General Algorithm for Constructing All Numerical Solutions to Ordered Metric Structures
Authors: McClelland Gary H. , Coombs Clyde H.
Published: January 2006

The algorithm is applicable to structures such as are obtained from additive conjoint measurement designs, unfolding theory, general Fechnerian scaling, some special types of multidimensional scaling, and ordinal multiple regression. A description is obtained of the space containing all possible numerical representations which can satisfy the structure, the size and shape of which is informative. The Abelson-Tukey maximin r2 solution is provided.
Nonmetric Grouping: Clusters and Cliques
Authors: Peay Edmund R.
Published: January 2006

A class of related nonmetric (“monotone invariant”) hierarchical grouping methods is presented. The methods are defined in terms of generalized cliques, based on a systematically varying specification of the degree of indirectness of permitted relationships (i.e., degree of “chaining”). This approach to grouping is shown to provide a useful framework for grouping methods based on an a priori specification of the properties of the desired subsets, and includes a natural generalization for “complete linkage” and “single linkage” clustering, such as the methods of Johnson [1967]. The central feature of the class of methods is a simple iterative matrix operation on the original disparities (“inverse-proximities” or “dissimilarities”) matrix, and one of the methods also constitutes a very efficient single linkage clustering procedure.
Probability Spaces, Hilbert Spaces, and the Axioms of Test Theory
Authors: Zimmerman Donald W.
Published: January 2006

A branch of probability theory that has been studied extensively in recent years, the theory of conditional expectation, provides just the concepts needed for mathematical derivation of the main results of the classical test theory with minimal assumptions and greatest economy in the proofs. The collection of all random variables with finite variance defined on a given probability space is a Hilbert space; the function that assigns to each random variable its conditional expectation is a linear operator; and the properties of the conditional expectation needed to derive the usual test-theory formulas are general properties of linear operators in Hilbert space. Accordingly, each of the test-theory formulas has a simple geometric interpretation that holds in all Hilbert spaces.
Paired Comparisons with Individual Differences
Authors: Andersen Erling B.
Published: January 2006

A model of Bradley-Terry type for paired comparisons is considered. In addition to the usual parameters, the model allows for individual parameters and corresponding parameters for the choice-alternatives. The model is applied to a set of data from a Danish investigation of the attitude of blue collar workers towards alternative social gains. The proposed model is finally compared to a model recently suggested by Schönemann and Wang.
Part and Bipartial Canonical Correlation Analysis
Authors: Timm Neil H. , Carlson James E.
Published: January 2006

Extending the definitions of part and bipartial correlation to sets of variates, the notion of part and bipartial canonical correlation analysis are developed and illustrated.