A case of extreme simplicity of the core matrix in three-mode principal components analysis

In three-mode Principal Components Analysis, theP ×Q ×R core matrixG can be transformed to simple structure before it is interpreted. It is well-known that, whenP=QR,G can be transformed to the identity matrix, which implies that all elements become equal to values specified a priori. In the present paper it is shown that, whenP=QR − 1,G can be transformed to have nearly all elements equal to values spectified a priori. A cllsed-form solution for this transformation is offered. Theoretical and practical implications of this simple structure transformation ofG are discussed. Constructive comments from anonymous reviewers are gratefully acknowledged.     

A comment on ‘some standard errors in item response theory’

In maximum likelihood estimation the standard error of the location parameter of the three parameter logistic model can be large, due to inaccurate estimation of the lower asymptote. Thissen and Wainer who demonstrated this effect, suggested that the introduction of a prior distribution for the lower asymptote might alleviate the problems. Here it is demonstrated in some detail that the standard error of the location parameter can be made acceptably small in this way.The author thanks Pieter Vijn for his helpful comments.     

Structural analysis of covariance and correlation matrices

A general approach to the analysis of covariance structures is considered, in which the variances and covariances or correlations of the observed variables are directly expressed in terms of the parameters of interest. The statistical problems of identification, estimation and testing of such covariance or correlation structures are discussed.Several different types of covariance structures are considered as special cases of the general model. These include models for sets of congeneric tests, models for confirmatory and exploratory factor analysis, models for estimation of variance and covariance components, regression models with measurement errors, path analysis models, simplex and circumplex models. Many of the different types of covariance structures are illustrated by means of real data.1978 Psychometric Society Presidential Address.This research has been supported by the Bank of Sweden Tercentenary Foundation under the project entitledStructural Equation Models in the Social Sciences, Karl G. Jöreskog, project director.     

MSA for a special spearman matrix

Kaiser's measure of sampling adequacy is applied to a special Spearman matrix and a specialq-cluster generalization. The result supports the contention that the measure should be no less than .5 for data to be appropriate for factor analysis.The research reported in this note was supported in part by the Program in Applied Mathematics and Statistics, National Science Foundation, and by the National Institute of Education, Department of Health, Education, and Welfare.     

认知诊断模型Q矩阵修正：完整信息矩阵的作用

Q矩阵是CDM的核心元素之一,反映了测验的内部结构和内容设计,通常由领域专家根据经验进行主观界定,因此需要对可能存在的错误进行修正。本研究提出了一种新的Q矩阵修正方法——基于完整经验交叉相乘信息矩阵的Wald-XPD方法。采用Monte Carlo模拟检验了新方法的表现,并与同类方法进行了比较。研究表明：新开发的Wald-XPD方法在Q矩阵恢复率、保留正确标定属性的比例以及修正错误标定属性的比例这3个主要指标上均有较好的表现,且整体上优于其他方法,尤其是在修正错误标定的属性方面。通过实证数据展示了Wald-XPD方法在Q矩阵修正中的良好表现。总之,本研究为Q矩阵修正提供了有效的方法。     

Sentence comprehension and production in children with cochlear implants: Errorless procedures and equivalence-based instruction

Research supports equivalence-based instruction (EBI) and matrix training for increased listening and speaking skills in children with cochlear implants (CI). We incorporated errorless procedures to optimize the EBI and evaluated the effects on the auditory comprehension and verbal responding at-sentences level in six CI children who were readers and showed inaccurate tacting. Subject-verb-object sentences were arranged in two matrices; diagonal combinations were trained and evaluated non-diagonal combinations. EBI included stimulus fading and exclusion to directly teach the dictated sentence-picture matching; written-sentences construction upon dictation also was taught. Probes assessed the derived stimulus-stimulus and stimulus-response relations. All participants learned the auditory-visual discriminations and the exclusion resulted in fewer errors than on a stimulus fading procedure. Five participants formed equivalence classes and increased verbal responding to equivalent stimuli, especially pictures tacting. They also showed recombinative generalization for both matrices. Professionals can incorporate errorless procedures to EBI for improving sentence comprehension, tacting, and productivity in CI children.     

A new quantile estimator with weights based on a subsampling approach

Quantiles are widely used in both theoretical and applied statistics, and it is important to be able to deploy appropriate quantile estimators. To improve performance in the lower and upper quantiles, especially with small sample sizes, a new quantile estimator is introduced which is a weighted average of all order statistics. The new estimator, denoted NO, has desirable asymptotic properties. Moreover, it offers practical advantages over four estimators in terms of efficiency in most experimental settings. The Harrell–Davis quantile estimator, the default quantile estimator of the R programming language, the Sfakianakis–Verginis SV2 quantile estimator and a kernel quantile estimator. The NO quantile estimator is also utilized in comparing two independent groups with a percentile bootstrap method and, as expected, it is more successful than other estimators in controlling Type I error rates.     

Compact integer-programming models for extracting subsets of stimuli from confusion matrices

This paper presents an integer linear programming formulation for the problem of extracting a subset of stimuli from a confusion matrix. The objective is to select stimuli such that total confusion among the stimuli is minimized for a particular subset size. This formulation provides a drastic reduction in the number of variables and constraints relative to a previously proposed formulation for the same problem. An extension of the formulation is provided for a biobjective problem that considers both confusion and recognition in the objective function. Demonstrations using an empirical interletter confusion matrix from the psychological literature revealed that a commercial branch-and-bound integer programming code was always able to identify optimal solutions for both the single-objective and biobjective formulations within a matter of seconds. A further extension and demonstration of the model is provided for the extraction of multiple subsets of stimuli, wherein the objectives are to maximize similarity within subsets and minimize similarity between subsets.     

A Unified Approach to Power Calculation and Sample Size Determination for Random Regression Models

The underlying statistical models for multiple regression analysis are typically attributed to two types of modeling: fixed and random. The procedures for calculating power and sample size under the fixed regression models are well known. However, the literature on random regression models is limited and has been confined to the case of all variables having a joint multivariate normal distribution. This paper presents a unified approach to determining power and sample size for random regression models with arbitrary distribution configurations for explanatory variables. Numerical examples are provided to illustrate the usefulness of the proposed method and Monte Carlo simulation studies are also conducted to assess the accuracy. The results show that the proposed method performs well for various model specifications and explanatory variable distributions. The author would like to thank the editor, the associate editor, and the referees for drawing attention to pertinent references that led to improved presentation. This research was partially supported by National Science Council grant NSC-94-2118-M-009-004.