Concise formulas for the standard errors of component loading estimates

Concise formulas for the asymptotic standard errors of component loading estimates were derived. The formulas cover the cases of principal component analysis for unstandardized and standardized variables with orthogonal and oblique rotations. The formulas can be used under any distributions for observed variables as long as the asymptotic covariance matrix for sample covariances/correlations is available. The estimated standard errors in numerical examples were shown to be equivalent to those by the methods using information matrices.The author is indebted to anonymous reviewers for the corrections and suggestions on this study, which have led to improvements of earlier versions of this article.     

Sensitivity analysis in factor analysis: Difference between using covariance and correlation matrices

Influence curves of some parameters under various methods of factor analysis have been given in the literature. These influence curves depend on the influence curves for either the covariance or the correlation matrix used in the analysis. The differences between the influence curves based on the covariance and the correlation matrices are derived in this paper. Simple formulas for the differences of the influence curves, based on the two matrices, for the unique variance matrix, factor loadings and some other parameter are obtained under scale-invariant estimation methods, though the influence curves themselves are in complex forms.The authors are most grateful to the referees, the Associate Editor, the Editor and Raymond Lam for helpful suggestions for improving the clarity of the paper.     

The evaluation of multiple and partial correlation coefficients from the factorial matrix

This paper shows how to compute multiple correlation coefficients, partial correlation coefficients, and regression coefficients from the factorial matrix. Special emphasis is given to computation technique and to approximation formulas. The method is extremely flexible in application since it may be applied to any subset of the original set of observed variables. It is also extremely useful when many of these coefficients are desired.     

Factor Analysis with EM Algorithm Never Gives Improper Solutions when Sample Covariance and Initial Parameter Matrices Are Proper

Rubin and Thayer (Psychometrika, 47:69–76, 1982) proposed the EM algorithm for exploratory and confirmatory maximum likelihood factor analysis. In this paper, we prove the following fact: the EM algorithm always gives a proper solution with positive unique variances and factor correlations with absolute values that do not exceed one, when the covariance matrix to be analyzed and the initial matrices including unique variances and inter-factor correlations are positive definite. We further numerically demonstrate that the EM algorithm yields proper solutions for the data which lead the prevailing gradient algorithms for factor analysis to produce improper solutions. The numerical studies also show that, in real computations with limited numerical precision, Rubin and Thayer’s (Psychometrika, 47:69–76, 1982) original formulas for confirmatory factor analysis can make factor correlation matrices asymmetric, so that the EM algorithm fails to converge. However, this problem can be overcome by using an EM algorithm in which the original formulas are replaced by those guaranteeing the symmetry of factor correlation matrices, or by formulas used to prove the above fact.     

Probabilistic contingency theories of animal conditioning: A critical analysis

Seven contingency theories of classical and instrumental conditioning were defined in relation to the contingency matrix and to six separate probabilities which can be derived from this matrix. These theories were compared on the basis of formal similarities and differences, and were judged against three separate empirical variables which have been discussed in the contingency literature: the duration of the intertrial interval, partial reinforcement, and negative contingencies between the two events of conditioning. All previous theories had some difficulty predicting the effects of one or more of these variables upon animal conditioning. Also, some formulas make the unusual prediction that there will be less conditioning when there is more than one conditioning trial. The paper concluded with an extensive discussion of the problems that are created by conditioning events with temporal extension; events which are then categorized by a contingency matrix which has no temporal character. Some possible solutions to the problems were discussed.     

考试评分缺失数据的概化理论分析

考试评分缺失数据较为常见,如何有效利用现有数据进行统计分析是个关键性问题。在考试评分中,题目与评分者对试卷得分的影响不容忽视。根据概化理论原理,按考试评分规则推导出含有缺失数据双侧面交叉设计（p×i×r）方差分量估计公式,用Matlab7.0软件模拟多组缺失数据,验证此公式的有效性。结果发现：（1）推导出的公式较为可靠,估计缺失数据的方差分量偏差相对较小,即便数据缺失率达到50％以上,公式仍能对方差分量进行较为准确地估计;（2）题目数量对概化理论缺失数据方差分量的估计影响最大,评分者次之,当题目和评价者数量分别为6和5时,公式能够趋于稳定地估计;（3）学生数量对各方差分量的估计影响较小,无论是小规模考试还是大规模考试,概化理论估计缺失数据的多个方差分量结果相差不大。     

Reliability formulas for noncompleted or speeded tests

New formulas are developed to give lower bounds to the reliability of a test, whether or not all respondents attempt all items. The formulas apply in particular, then, to completed tests, pure speed tests, pure power tests, and any mixture of speed and power. For the case of completed tests, the formulas give the same answer as certain standard ones; for noncompleted tests the formulas give a correct answer where previous standard formulas are inappropriate. The formulas hold both in the sense of retest reliability and of parallel tests.This research was facilitated by an uncommitted grant-in-aid to the writer from the Behavioral Sciences Division of the Ford Foundation.     

On the determination of redundancies in sociometric chains

The use of a matrix to represent a relationship between the members of a group is well known in sociometry. If this matrix is raised to a certain power, the elements appearing give the total number of connecting paths between each pair of members. In general, some of these paths will be redundant. Methods of finding the number of such redundant paths have been developed for three- and four-step chains by Luce and Perry (3) and Katz (2), respectively. We have derived formulas for the number of redundant paths of five and six steps; and in addition, an algorithm for determining the number of redundant paths of any given length.The research leading to this paper was supported by a grant from the Rockefeller Foundation.     

Translation of first order formulas into ground formulas via a completion theory

A translation technique is presented which transforms a class of First Order Logic formulas, called Restricted formulas, into ground formulas. For the formulas in this class the range of quantified variables is restricted by Domain formulas.If we have a complete knowledge of the predicates involved in the Domain formulas their extensions can be evaluated with the Relational Algebra and these extensions are used to transform universal (respectively existential) quantifiers into finite conjunctions (respectively disjunctions).It is assumed that the complete knowledge is represented by Completion Axioms and Unique Name Axioms à la Reiter. These axioms involve the equality predicate. However, the translation allows to remove the equality in the ground formulas and for a large class of formulas their consequences are the same as the initial First Order formulas. This result open the door for the design of efficient deduction techniques.     

Approximate sample size formulas for the two‐sample trimmed mean test with unequal variances

Yuen's two‐sample trimmed mean test statistic is one of the most robust methods to apply when variances are heterogeneous. The present study develops formulas for the sample size required for the test. The formulas are applicable for the cases of unequal variances, non‐normality and unequal sample sizes. Given the specified α and the power (1?β), the minimum sample size needed by the proposed formulas under various conditions is less than is given by the conventional formulas. Moreover, given a specified size of sample calculated by the proposed formulas, simulation results show that Yuen's test can achieve statistical power which is generally superior to that of the approximate t test. A numerical example is provided.     

Frame Based Formulas for Intermediate Logics

In this paper we define the notion of frame based formulas. We show that the well-known examples of formulas arising from a finite frame, such as the Jankov-de Jongh formulas, subframe formulas and cofinal subframe formulas, are all particular cases of the frame based formulas. We give a criterion for an intermediate logic to be axiomatizable by frame based formulas and use this criterion to obtain a simple proof that every locally tabular intermediate logic is axiomatizable by Jankov-de Jongh formulas. We also show that not every intermediate logic is axiomatizable by frame based formulas. Presented by Johan van Benthem     

A Bayesian hierarchical model for the measurement of working memory capacity

Working memory is the memory system that allows for conscious storage and manipulation of information. The capacity of working memory is extremely limited. Measurements of this limit, and what affects it, are critical to understanding working memory. Cowan (2001) and Pashler (1988) suggested applying multinomial tree models to data from change detection paradigms in order to estimate working memory capacity. Both Pashler and Cowan suggested simple formulas for estimating capacity with these models. However, in many cases, these simple formulas are inadequate, and may lead to inefficient or biased estimation of working memory capacity. I propose a Bayesian hierarchical alternative to the Pashler and Cowan formulas, and show that the hierarchical model outperforms the traditional formulas. The models are easy to use and appropriate for a wide range of experimental designs. An easy-to-use graphical user interface for fitting the hierarchical model to data is available.     

Measures of attitudinal polarization

Abstract.— Criteria for measures of attitudinal polarization, i. e., degree of opposition among people on a specific issue, are proposed and some formulas, including the standard deviation, are evaluated in relation to the criteria. The formulas were also tested on empirical data with respect to level, dispersion and agreement of received values. The measures on the whole showed a high degree of agreement. There clearly exist instances where the standard deviation is not an adequate measure of attitudinal polarization. Some guidelines are given for the choice of constant values in one of the formulas.     

Note on the sampling error of the difference between correlated proportions or percentages

Two formulas are presented for judging the significance of the difference between correlated proportions. The chi square equivalent of one of the developed formulas is pointed out.     

Measuring coherence using LP-models

This paper introduces a technique for measuring the degree of (in)coherence of inconsistent sets of propositional formulas. The coherence of these sets of formulas is calculated using the minimal models of those sets in G. Priest's Logic of Paradox. The compatibility of the information expressed by a set of formulas with the background or domain knowledge can also be measured with this technique. In this way, Hunter's objections to many-valued paraconsistent logics as instruments for measuring (in)coherence are addressed.     

Communality of a variable: Formulation by cluster analysis

The communality of a variable represents the degree of its generality acrossn – 1 behaviors. Domain-sampling principles provide a fundamental conception and definition of the communality. This definition may be alternatively stated in eight different ways. Three definitions lead to precise formulas that determine thetrue value of the communality: (i) from thek necessary and sufficient dimensions derived by iterated factoring, (ii) from then – 1 remaining variable-domains, and (iii) fromk' multiple clusters of then variables. Seven definitions provide approximation formulas: (i) one from thek dimensions as initially factored, (ii) one from then – 1 remaining variables, and (iii) five from a single cluster. Rank of the matrix is not a desiratum in some definitions. Using an example designed by Guilford to illustrate multiple-factor analysis, applications of the formulas based on the three precise definitions recover the true communalities, and five approximation formulas each gives values closer than thead hoc estimates usually employed in factor analysis.The writer wishes to express his indebtedness to C. F. Wrigley and H. Kaiser for their many helpful constructive criticisms.     

The reliability of speeded tests

Some methods are presented for estimating the reliability of a partially speeded test without the use of a parallel form. The effect of these formulas on some test data is illustrated. Whenever an odd-even reliability is computed it is probably desirable to use one of the formulas noted in Section 2 of this paper in addition to the usual Spearman-Brown correction. Since the formulas given here involve the mean and the standard deviation of the “number unattempted score,” a method is given in Section 4 for computing this mean and standard deviation from item analysis data. If the item analysis data are available, this method will save considerable time as compared with rescoring answer sheets.     

Determining the degree of inconsistency in a set of paired comparisons

Consistency in paired comparison data is defined. Two types of inconsistency which may arise are defined. Computational formulas for these types of inconsistency are derived, and examples illustrating the use of these formulas are presented.These ideas were developed while the first author was on the staff of the Research Center for Human Relations. The work was made possible by the ONR contract NONR 285(10). The authors are indebted to Jack Moshman for his helpful critical suggestions. The United States Government is authorized to reprint this article in whole or in part.     

层次图(混合Horn)公式的部分极大可满足性

本文为解决一类混合Horn公式([13,14]),又称为层次图公式([15])的MAXSAT问题进行了基于随机局部搜索过程的经验研究。具体地,我们首先在随机CNF公式的MAX2SAT及MAX3SAT问题上进行WalkSAT和Tabu-Sat(及其变种)的比较,其次,我们在层次图公式上比较了上述过程的改进版本,这些公式编码了随机生成层次图的最小化跨边问题。本文所引入的Tabu-Sat过程,当在搜索空间中检测到一个圈的时候,动态地改变Tabu长度参数。另一个被称为Vector-Tabu-Sat的过程,对所有的布尔变元进行独立的Tabu长度参数管理。一些数值实验的结果显示,我们改进的Tabu-Sat变种在子句个数增长的时候优于Walksat.