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.     

Remarks on the method of paired comparisons: I. The least squares solution assuming equal standard deviations and equal correlations

Thurstone's Case V of the method of paired comparisons assumes equal standard deviations of sensations corresponding to stimuli and zero correlations between pairs of stimuli sensations. It is shown that the assumption of zero correlations can be relaxed to an assumption of equal correlations between pairs with no change in method. Further the usual approach to the method of paired comparisons Case V is shown to lead to a least squares estimate of the stimulus positions on the sensation scale.This research was performed in the Laboratory of Social Relations under a grant made available to Harvard University by the RAND Corporation under the Department of the Air Force, Project RAND.     

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.     

A least squares solution for paired comparisons with incomplete data

A precise and rapid procedure has been devised for dealing with a matrix of incomplete data in paired comparisons. This method should increase the general applicability of paired comparisons since experiments involving large numbers of stimuli may now be shortened to feasible experimental proportions. Also, we may now use sets of stimuli which cover a wide range, resulting in a considerable number of 100 per cent vs. 0 per cent judgments, and still give a precise solution depending equally on each of the observations.This study was supported in part by the Office of Naval Research Contract N6onr-270-20 with Princeton University and by the National Science Foundation Grant G-642. The opinions expressed are, of course, those of the author and do not represent attitudes or policies of the Office of Naval Research or of the National Science Foundation.The author wishes to acknowledge helpful suggestions and comments received from Frederic M. Lord, Warren S. Torgerson, and Ledyard R Tucker. Thanks are also due to Mrs. Gertrude Diederich for some of the tabulating and computing for the illustrative problem.     

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 least squares algorithm for fitting additive trees to proximity data

A least squares algorithm for fitting additive trees to proximity data is described. The algorithm uses a penalty function to enforce the four point condition on the estimated path length distances. The algorithm is evaluated in a small Monte Carlo study. Finally, an illustrative application is presented.The author is Aspirant of the Belgian Nationaal Fonds voor Wetenschappelijk Onderzoek. The author is indebted to Professor J. Hoste for providing computer facilities at the Institute of Nuclear Sciences at Ghent.     

A bayesian alternative to least squares and equal weighting coefficients in regression

This paper details a Bayesian alternative to the use of least squares and equal weighting coefficients in regression. An equal weight prior distribution for the linear regression parameters is described with regard to the conditional normal regression model, and resulting posterior distributions for these parameters are detailed. Some interesting connections between this Bayesian procedure and several other methods for estimating optimal weighting coefficients are discussed. In addition, results are presented of a Monte Carlo investigation which compared the effectiveness of the Bayesian procedure relative to least squares, equal weight, ridge, and Bayesian exchangeability estimations.     

A generalized least squares regression approach for computing effect sizes in single-case research: application examples

A new method for deriving effect sizes from single-case designs is proposed. The strategy is applicable to small-sample time-series data with autoregressive errors. The method uses Generalized Least Squares (GLS) to model the autocorrelation of the data and estimate regression parameters to produce an effect size that represents the magnitude of treatment effect from baseline to treatment phases in standard deviation units. In this paper, the method is applied to two published examples using common single case designs (i.e., withdrawal and multiple-baseline). The results from these studies are described, and the method is compared to ten desirable criteria for single-case effect sizes. Based on the results of this application, we conclude with observations about the use of GLS as a support to visual analysis, provide recommendations for future research, and describe implications for practice.     

Beyond principal component analysis: A trilinear decomposition model and least squares estimation

The paper derives sufficient conditions for the consistency and asymptotic normality of the least squares estimator of a trilinear decomposition model for multiway data analysis.     

Problematic standard errors and confidence intervals for skewness and kurtosis

Many statistics packages print skewness and kurtosis statistics with estimates of their standard errors. The function most often used for the standard errors (e.g., in SPSS) assumes that the data are drawn from a normal distribution, an unlikely situation. Some textbooks suggest that if the statistic is more than about 2 standard errors from the hypothesized value (i.e., an approximate value for the critical value from the t distribution for moderate or large sample sizes when α = 5%), the hypothesized value can be rejected. This is an inappropriate practice unless the standard error estimate is accurate and the sampling distribution is approximately normal. We show distributions where the traditional standard errors provided by the function underestimate the actual values, often being 5 times too small, and distributions where the function overestimates the true values. Bootstrap standard errors and confidence intervals are more accurate than the traditional approach, although still imperfect. The reasons for this are discussed. We recommend that if you are using skewness and kurtosis statistics based on the 3^rd and 4^th moments, bootstrapping should be used to calculate standard errors and confidence intervals, rather than using the traditional standard. Software in the freeware R for this article provides these estimates.     

An alternating least squares method for the weighted approximation of a symmetric matrix

Bailey and Gower examined the least squares approximationC to a symmetric matrixB, when the squared discrepancies for diagonal elements receive specific nonunit weights. They focussed on mathematical properties of the optimalC, in constrained and unconstrained cases, rather than on how to obtainC for any givenB. In the present paper a computational solution is given for the case whereC is constrained to be positive semidefinite and of a fixed rankr or less. The solution is based on weakly constrained linear regression analysis.The authors are obliged to John C. Gower for stimulating this research.     

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.     

An alternative two stage least squares (2SLS) estimator for latent variable equations

The Maximum-likelihood estimator dominates the estimation of general structural equation models. Noniterative, equation-by-equation estimators for factor analysis have received some attention, but little has been done on such estimators for latent variable equations. I propose an alternative 2SLS estimator of the parameters in LISREL type models and contrast it with the existing ones. The new 2SLS estimator allows observed and latent variables to originate from nonnormal distributions, is consistent, has a known asymptotic covariance matrix, and is estimable with standard statistical software. Diagnostics for evaluating instrumental variables are described. An empirical example illustrates the estimator. I gratefully acknowledge support for this research from the Sociology Program of the National Science Foundation (SES-9121564) and the Center for Advanced Study in the Behavioral Sciences, Stanford, California. This paper was presented at the Interdisciplinary Consortium for Statistical Applications at Indiana University at Bloomington (March 2, 1994) and at the RMD Conference on Causal Modeling at Purdue University, West Lafayette, Indiana (March 3-5, 1994).