Minimization of a class of matrix trace functions by means of refined majorization

A procedure is described for minimizing a class of matrix trace functions. The procedure is a refinement of an earlier procedure for minimizing the class of matrix trace functions using majorization. It contains a recently proposed algorithm by Koschat and Swayne for weighted Procrustes rotation as a special case. A number of trial analyses demonstrate that the refined majorization procedure is more efficient than the earlier majorization-based procedure. The research of H. A. L. Kiers has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences.     

MMPI中国常模中量表8及量表9K校正系数的探讨

本文针对MMPI中国常模中各临床量表一般不作K分校正的建议作了实证研究。其分析了912名被试的K、Sc及Ma量表数据资料,其中有345名精神分裂症患者,111名躁狂症患者以及456名正常被试。利用半和方区分比率法和频数分析法进行分析的结果表明,为了使MMPI在辅助诊断心理失常者时达到较好的效果,①量表8原始分数需加1K校正分数;②量表9原始分数以0.2K分数进行校正较不作校正为好。由此本研究建议,在应用MMPI时应采用加K分的中国常模。     

Majorization as a tool for optimizing a class of matrix functions

The problem of minimizing a general matrix, trace function, possibly subject to certain constraints, is approached by means of majorizing this function by one having a simple quadratic shape and whose minimum is easily found. It is shown that the parameter set that minimizes the majorizing function also decreases the matrix trace function, which in turn provides a monotonically convergent algorithm for minimizing the matrix trace function iteratively. Three algorithms based on majorization for solving certain least squares problems are shown to be special cases. In addition, by means of several examples, it is noted how algorithms may be provided for a wide class of statistical optimization tasks for which no satisfactory algorithms seem available.The Netherlands organization for scientific research(NWO) is gratefully acknowledged for funding this project. This research was conducted while the author was supported by a PSYCHON-grant (560-267-011) from this organization. The author is obliged to Jos ten Berge, Willem Heiser, and Wim Krijnen for helpful comments on an earlier version of this paper.     

The nonconvergence of factals: A nonmetric common factor analysis

FACTALS is a nonmetric common factor analysis model for multivariate data whose variables may be nominal, ordinal or interval. In FACTALS an Alternating Least Squares algorithm is utilized which is claimed to be monotonically convergent.In this paper it is shown that this algorithm is based upon an erroneous assumption, namely that the least squares loss function (which is in this case a nonscale free loss function) can be transformed into a scalefree loss function. A consequence of this is that monotonical convergence of the algorithm can not be guaranteed.     

The analysis of multitrait-multimethod matrices via constrained components analysis

Multitrait-Multimethod (MTMM) matrices are often analyzed by means of confirmatory factor analysis (CFA). However, fitting MTMM models often leads to improper solutions, or non-convergence. In an attempt to overcome these problems, various alternative CFA models have been proposed, but with none of these the problem of finding improper solutions was solved completely. In the present paper, an approach is proposed where improper solutions are ruled out altogether and convergence is guaranteed. The approach is based on constrained variants of components analysis (CA). Besides the fact that these methods do not give improper solutions, they have the advantage that they provide component scores which can later on be used to relate the components to external variables. The new methods are illustrated by means of simulated data, as well as empirical data sets.This research has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences to the first author. The authors are obliged to three anonymous reviewers and an associate editor for constructive suggestions on the first version of this paper.     

A study of algorithms for covariance structure analysis with specific comparisons using factor analysis

Several algorithms for covariance structure analysis are considered in addition to the Fletcher-Powell algorithm. These include the Gauss-Newton, Newton-Raphson, Fisher Scoring, and Fletcher-Reeves algorithms. Two methods of estimation are considered, maximum likelihood and weighted least squares. It is shown that the Gauss-Newton algorithm which in standard form produces weighted least squares estimates can, in iteratively reweighted form, produce maximum likelihood estimates as well. Previously unavailable standard error estimates to be used in conjunction with the Fletcher-Reeves algorithm are derived. Finally all the algorithms are applied to a number of maximum likelihood and weighted least squares factor analysis problems to compare the estimates and the standard errors produced. The algorithms appear to give satisfactory estimates but there are serious discrepancies in the standard errors. Because it is robust to poor starting values, converges rapidly and conveniently produces consistent standard errors for both maximum likelihood and weighted least squares problems, the Gauss-Newton algorithm represents an attractive alternative for at least some covariance structure analyses.Work by the first author has been supported in part by Grant No. Da01070 from the U. S. Public Health Service. Work by the second author has been supported in part by Grant No. MCS 77-02121 from the National Science Foundation.     

Evaluation of multiple‐alternative prompts during tact training

This study compared 2 methods of fading prompts while teaching tacts to 3 individuals who had been diagnosed with autism spectrum disorder (ASD). The 1st method involved use of an echoic prompt and prompt fading. The 2nd method involved providing multiple‐alternative answers and fading by increasing the difficulty of the discrimination. An adapted alternating‐treatments design showed that both procedures were more effective than a no‐intervention control condition. Providing multiple alternatives did not increase error rates or teaching time, and better maintenance was shown for tacts taught with the multiple‐alternative prompt.     

重复启动条件下语义启动效应减少的机制

采用重复启动范式探讨语义启动效应减少的机制。结果发现,在SOA为短的167ms条件下,重复/相同和重复/相关的启动效应无差异,且重复/相关启动并未导致语义启动效应的减少;而在SOA为长的1200ms条件下,重复/相同和重复/相关的启动效应差异显著,且重复/相关启动导致了语义启动效应的减少,而重复/相同未导致语义启动效应的减少。在抵消了期望和消退的作用后,在SOA为1200ms条件下所得到的重复/相同和重复/相关启动效应的差异,是由于扩散受到了阻碍而导致的。这一结论支持了激活扩散是有一定条件的观点。     

Tests of homogeneity of means and covariance matrices for multivariate incomplete data

Existing test statistics for assessing whether incomplete data represent a missing completely at random sample from a single population are based on a normal likelihood rationale and effectively test for homogeneity of means and covariances across missing data patterns. The likelihood approach cannot be implemented adequately if a pattern of missing data contains very few subjects. A generalized least squares rationale is used to develop parallel tests that are expected to be more stable in small samples. Three factors were varied for a simulation: number of variables, percent missing completely at random, and sample size. One thousand data sets were simulated for each condition. The generalized least squares test of homogeneity of means performed close to an ideal Type I error rate for most of the conditions. The generalized least squares test of homogeneity of covariance matrices and a combined test performed quite well also.Preliminary results on this research were presented at the 1999 Western Psychological Association convention, Irvine, CA, and in the UCLA Statistics Preprint No. 265 (http://www.stat.ucla.edu). The assistance of Ke-Hai Yuan and several anonymous reviewers is gratefully acknowledged.     

The rigid orthogonal Procrustes rotation problem

The problem of rotating a matrix orthogonally to a best least squares fit with another matrix of the same order has a closed-form solution based on a singular value decomposition. The optimal rotation matrix is not necessarily rigid, but may also involve a reflection. In some applications, only rigid rotations are permitted. Gower (1976) has proposed a method for suppressing reflections in cases where that is necessary. This paper proves that Gower’s solution does indeed give the best least squares fit over rigid rotation when the unconstrained solution is not rigid. Also, special cases that have multiple solutions are discussed. The author is obliged to Henk Kiers and Alwin Stegeman for helpful comments on a previous draft.