A least squares solution for successive intervals assuming unequal standard deviations

A least squares solution has been presented for the Law of Categorical Judgment and the Method of Successive Intervals, which is formally equivalent to Horst's solution for the matrix of incomplete data. A simplified approximation which is recommended for use with any matrix of complete data is also given. A procedure, analogous to that originally devised by Thurstone and presented by Saffir, is outlined for the typical experimental matrix of incomplete data. It is essentially a point and slope method of fitting a straight line which comes rather directly from the least squares solution.This study was supported in part by Office of Naval Research Contract N6onr 270-20 with Princeton University.The author wishes to acknowledge helpful suggestions and comments received in discussions of this problem with Max Woodbury, Frederic Lord, Frederick Mosteller, Warren Torgerson, Robert Abelson, and Bert F. Green, Jr. Thanks are due to Mrs. Gertrude Diederich and Irving Abrams for work on the computing necessary for the illustrative applications.     

Multidimensional successive categories scaling: A maximum likelihood method

A single-step maximum likelihood estimation procedure is developed for multidimensional scaling of dissimilarity data measured on rating scales. The procedure can fit the euclidian distance model to the data under various assumptions about category widths and under two distributional assumptions. The scoring algorithm for parameter estimation has been developed and implemented in the form of a computer program. Practical uses of the method are demonstrated with an emphasis on various advantages of the method as a statistical procedure.The research reported here was partly supported by Grant A6394 to the author by Natural Sciences and Engineering Research Council of Canada. Portions of this research were presented at the Psychometric Society meeting in Uppsala, Sweden, in June, 1978. MAXSCAL-2.1, a program to perform the computations discussed in this paper may be obtained from the author. Thanks are due to Jim Ramsay for his helpful comments.     

A general least squares solution for successive intervals

A general least squares solution for successive intervals is presented, along with iterative procedures for obtaining stimulus scale values, discriminal dispersions, and category boundaries. Because provisions for weighting were incorporated into the derivation, the solution may be applied without loss of rigor to the typical experimental matrix of incomplete data, i.e., to a data matrix with missing entries, as well as to the rarely occurring matrix of complete data. The use of weights also permits adjustments for variations in the reliability of estimates obtained from the data. The computational steps involved in the solution are enumerated, the amount of labor required comparing favorably with other procedures. A quick, yet accurate, graphical approximation suggested by the least squares derivation is also described.This research was jointly supported in part by Princeton University, the Office of Naval Research under contract N6onr-270-20, and the National Science Foundation under grant NSF G-642, and in part by Educational Testing Service.     

A modification of the method of successive intervals

A modification of the method of successive intervals is presented which yields scale values correlating .995 with those from Thurstone's method described by Saffir. Values yielded by the present method can be obtained in 25 per cent of the time required by the older method and are shown to be, on apriori grounds, more reliable as well.     

A maximum likelihood method for fitting the wandering vector model

After introducing some extensions of a recently proposed probabilistic vector model for representing paired comparisons choice data, an iterative procedure for obtaining maximum likelihood estimates of the model parameters is developed. The possibility of testing various hypotheses by means of likelihood ratio tests is discussed. Finally, the algorithm is applied to some existing data sets for illustrative purposes.     

Properties of the maximum likelihood solution in factor analysis regression

Algebraic properties of the normal theory maximum likelihood solution in factor analysis regression are investigated. Two commonly employed measures of the within sample predictive accuracy of the factor analysis regression function are considered: the variance of the regression residuals and the squared correlation coefficient between the criterion variable and the regression function. It is shown that this within sample residual variance and within sample squared correlation may be obtained directly from the factor loading and unique variance estimates, without use of the original observations or the sample covariance matrix.     

The validity of the successive intervals method of psychometric scaling

The degree to which scale values computed by the method of successive intervals diverge from theoretically true values is seen to be due to three types of error: error due to inequalities in variances of the distributions from which the scale values are computed, error due to non-normality of the distributions, and sampling error. The contribution of each type of error to the total error is evaluated; the latter is seen to be surprisingly small under appropriate conditions. Certain aspects of the formal methodology underlying scaling procedures are also briefly considered.This paper reports research undertaken in cooperation with the Quartermaster Food and Container Institute for the Armed Forces, and has been assigned number 475 in the series of papers approved for publication. The views or conclusions contained in this report are those of the authors. They are not to be construed as necessarily reflecting the views or indorsement of the Department of Defense.     

A feasible method for standard errors of estimate in maximum likelihood factor analysis

A jackknife-like procedure is developed for producing standard errors of estimate in maximum likelihood factor analysis. Unlike earlier methods based on information theory, the procedure developed is computationally feasible on larger problems. Unlike earlier methods based on the jackknife, the present procedure is not plagued by the factor alignment problem, the Heywood case problem, or the necessity to jackknife by groups. Standard errors may be produced for rotated and unrotated loading estimates using either orthogonal or oblique rotation as well as for estimates of unique factor variances and common factor correlations. The total cost for larger problems is a small multiple of the square of the number of variables times the number of observations used in the analysis. Examples are given to demonstrate the feasibility of the method.The research done by R. I. Jennrich was supported in part by NSF Grant MCS 77-02121. The research done by D. B. Clarkson was supported in part by NSERC Grant A3109.     

A maximum likelihood method for latent class regression involving a censored dependent variable

The standard tobit or censored regression model is typically utilized for regression analysis when the dependent variable is censored. This model is generalized by developing a conditional mixture, maximum likelihood method for latent class censored regression. The proposed method simultaneously estimates separate regression functions and subject membership in K latent classes or groups given a censored dependent variable for a cross-section of subjects. Maximum likelihood estimates are obtained using an EM algorithm. The proposed method is illustrated via a consumer psychology application.     

The estimation of the discriminal dispersion in the method of successive intervals

A new algebraic formula is derived for estimation of the discriminal dispersion in the method of successive intervals. The legitimate use of the formula requires that as many normal deviates as possible be present in the matrix. For this reason, it is recommended that deviates corresponding to the interval (0.01, 0.99) of the cumulative proportions be used, instead of those corresponding to (0.05, 0.95), the interval used by Edwards and Thurstone. Computations on data published by Edwards and Thurstone showed that when adjustment was made for variability in dispersions calculated by the formula of this paper, a reduction of fifty per cent in mean absolute discrepancy was produced. Since the formula is easy to use and avoids the disadvantages of its predecessors, it should have fairly wide applicability in psychological research.This research was supported in part by the United States Air Force under Contract No. AF 33(038)-25726 monitored by Air Force Personnel and Training Research Center. Permission is granted for reproduction, translation, publication, use and disposal in whole and in part by or for the United States Government. The writer is grateful to Dr. A. L. Edwards for a critical reading of an earlier version of this paper, and to Dr. L. H. Lanier and Dr. L. M. Stolurow for editorial advice on the present version, which was written at the University of Illinois. The editors ofPsychometrika have informed the writer that H. J. A. Rimoldi and M. Hormaeche (7) have independently derived the same formula for the discriminal dispersion from a different set of postulates using the law of comparative judgment.     

A marginal maximum likelihood method for the vector threshold model to analyze dichotomous choice data

Abstract: At least two types of models, the vector model and the unfolding model can be used for the analysis of dichotomous choice data taken from, for example, the pick any/ n method. The previous vector threshold models have a difficulty with estimation of the nuisance parameters such as the individual vectors and thresholds. This paper proposes a new probabilistic vector threshold model, where, unlike the former vector models, the angle that defines an individual vector is a random variable, and where the marginal maximum likelihood estimation method using the expectation-maximization algorithm is adopted to avoid incidental parameters. The paper also attempts to discuss which of the two models is more appropriate to account for dichotomous choice data. Two sets of dichotomous choice data are analyzed by the model.     

An internal consistency check for scale values determined by the method of successive intervals

The method of successive intervals is a psychological scaling procedure in which stimuli are classified into successive intervals according to the degree of some defined attribute which they are judged to possess. A psychological continuum is defined and the scale values are then taken as the medians of the distributions of judgments on the psychological continuum. It is assumed that the distributions of judgments for each stimulus are normal on the psychological continuum as defined.An internal consistency check indicates that the cumulative distributions of empirical judgments for the various stimuli can be reproduced by means of a limited number of parameters with an average error that compares favorably with that usually reported for paired comparison data. Furthermore, the scale values obtained by successive interval scaling, for the data reported, are shown to be linearly related to those obtained by the method of paired comparisons.This paper was written while the first author was a post-doctoral Research Training Fellow of the Social Science Research Council studying at the University of Chicago. It reports research undertaken in cooperation with the Quartermaster Food and Container Institute for the Armed Forces. The views or conclusions contained in this report are those of the authors. They are not to be construed as necessarily reflecting the views or endorsement of the Department of Defense.     

A reliability coefficient for maximum likelihood factor analysis

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. This research was supported by the Personnel and Training Research Programs Office of the Office of Naval Research under contract US NAVY/00014-67-A-0305-0003. Critical review of the development and suggestions by Richard Montanelli were most helpful.     

A Newton-Raphson algorithm for maximum likelihood factor analysis

This paper demonstrates the feasibility of using a Newton-Raphson algorithm to solve the likelihood equations which arise in maximum likelihood factor analysis. The algorithm leads to clean easily identifiable convergence and provides a means of verifying that the solution obtained is at least a local maximum of the likelihood function. It is shown that a popular iteration algorithm is numerically unstable under conditions which are encountered in practice and that, as a result, inaccurate solutions have been presented in the literature. The key result is a computationally feasible formula for the second differential of a partially maximized form of the likelihood function. In addition to implementing the Newton-Raphson algorithm, this formula provides a means for estimating the asymptotic variances and covariances of the maximum likelihood estimators. This research was supported by the Air Force Office of Scientific Research, Grant No. AF-AFOSR-4.59-66 and by National Institutes of Health, Grant No. FR-3.     

Combining standardized mean differences using the method of maximum likelihood

A maximum likelihood procedure for combining standardized mean differences based on a noncentratt-distribution is proposed. With a proper data augmentation technique, an EM-algorithm is developed. Information and likelihood ratio statistics are discussed in detail for reliable inference. Simulation results favor the proposed procedure over both the existing normal theory maximum likelihood procedure and the commonly used generalized least squares procedure.     

Note on the least squares solution for the method of successive categories

The problem of estimation in the method of successive categories is reconsidered and a new least squares solution is obtained. An empirical comparison of this solution with Gulliksen's solution is presented.Preparation of this paper has been supported by the Quartermaster Food and Container Institute for the Armed Forces. Views or conclusions expressed herein are those of the author and do not necessarily reflect the views or indorsement of the Department of Defense.     

An application of confidence intervals and of maximum likelihood to the estimation of an examinee's ability

A mathematical definition of the theoretical relation between the examinee's actual responses to the test items and his “true ability” is selected. A maximum-likelihood solution is obtained for estimating the examinee's “true ability” from his responses to the items. The standard error of the maximum-likelihood estimate is obtained, its relation to the discriminating power of the test is pointed out, and some generalizations are drawn as to the optimum level of item difficulty. The Neyman-Pearson power function is applied to determine which of two psychological tests is the most powerful for the selection of “successful” examinees.     

A method of estimating item characteristic functions using the maximum likelihood estimate of ability

A method of estimating item characteristic functions is proposed, in which a set of test items, whose operating characteristics are known and which give a constant test information function for a substantially wide range of ability, are used. The method is based on the maximum likelihood estimates of ability for a group of several hundred examinees. Throughout the present study the Monte Carlo method is used.