Standard errors for the class of orthomax-rotated factor loadings: Some matrix results

The partial derivative matrices of the class of orthomax-rotated factor loadings with respect to the unrotated maximum likelihood factor loadings are derived. The reported results are useful for obtaining standard errors of the orthomax-rotated factor loadings, with or without row normalization (standardization) of the initial factor loading matrix for rotation. Using a numerical example, we verify our analytic formulas by comparing the obtained standard error estimates with that from some existing methods. Some advantages of the current approach are discussed.Authorship is determined by alphabetical order. The authors contributed equally to the research. Kentaro Hayashi is now at the Department of Mathematics, Bucknell University, Lewisburg, PA 17837 (email: khayashi@Bucknell.edu). Yiu-Fai Yung is now at the SAS Institute, Inc., SAS Campus Drive, Cary, NC 27513 (email: yiyung@wnt.sas.com).Part of the research was completed while Yiu-Fai Yung was a visiting scholar at the Department of Psychology, the Ohio State University. The visit was supported in part by grant N4856118101 from the NIMH and the Mason and Linda Stephenson Travel Award from the Department of Psychology, University of North Carolina at Chapel Hill. The authors are grateful to Michael Browne who suggested some relevant references and provided valuable comments on the research, and to Robert Cudeck who provided the FAS program for the numerical comparison. The expert comments by the reviewers are deeply appreciated.     

掩蔽刺激的空间参考框架对视觉表象旋转的影响

本实验通过将具有参考框架信息的前置刺激改为掩蔽刺激,进一步探讨参考框架的变化和表象旋转的过程。结果表明：１、因掩蔽刺激而引起的主观参考框架的产生是瞬间完成的,而不是由原先直立的参考框架经过连续旋转后形成的。２、显示了倾斜字母的加工不完全是整体表象的线性旋转过程,而可能是整字一成分的混合加工过程。     

Orthogonal procrustes rotation for two or more matrices

Necessary and sufficient conditions for rotating matrices to maximal agreement in the least-squares sense are discussed. A theorem by Fischer and Roppert, which solves the case of two matrices, is given a more straightforward proof. A sufficient condition for a best least-squares fit for more than two matrices is formulated and shown to be not necessary. In addition, necessary conditions suggested by Kristof and Wingersky are shown to be not sufficient. A rotation procedure that is an alternative to the one by Kristof and Wingersky is presented. Upper bounds are derived for determining the extent to which the procedure falls short of attaining the best least-squares fit. The problem of scaling matrices to maximal agreement is discussed. Modifications of Gower's method of generalized Procrustes analysis are suggested.     

身体转向模式对方位效应和相对位置效应的影响

在阅读故事而产生的想象场景中,观察者身体的物理转向（ｐｈｙｓｉｃａｌｒｏｔａｔｉｏｎ）模式或想象身体转向（ｉｍａｇｉｎｅｄｒｏｔａｔｉｏｎ）模式对物体搜索方位效应和相对位置效应产生影响。结果发现：方位效应独立于身体转向模式,说明对空间关系的理解是以想象自我（ｒｅｐｒｅｓｅｎｔｅｄｓｅｌｆ）为参照系的。相对位置效应独立于身体转向模式,说明物体间的拓扑结构导致了相对位置效应。     

The weighted oblimin rotation

Cureton & Mulaik (1975) proposed the Weighted Varimax rotation so that Varimax (Kaiser, 1958) could reach simple solutions when the complexities of the variables in the solution are larger than one. In the present paper the weighting procedure proposed by Cureton & Mulaik (1975) is applied to Direct Oblimin (Clarkson & Jennrich, 1988), and the rotation method obtained is called Weighted Oblimin. It has been tested on artificial complex data and real data, and the results seem to indicate that, even though Direct Oblimin rotation fails when applied to complex data, Weighted Oblimin gives good results if a variable with complexity one can be found for each factor in the pattern. Although the weighting procedure proposed by Cureton & Mulaik is based on Landahl's (1938) expression for orthogonal factors, Weighted Oblimin seems to be adequate even with highly oblique factors. The new rotation method was compared to other rotation methods based on the same weighting procedure and, whenever a variable with complexity one could be found for each factor in the pattern, Weighted Oblimin gave the best results. When rotating a simple empirical loading matrix, Weighted Oblimin seemed to slightly increase the performance of Direct Oblimin.The author is obliged to Henk A. L. Kiers and three anonymous reviewers for helpful comments on an earlier version of this paper.     

心理旋转训练对5~6岁儿童空间能力和数学能力的影响

采用2（时间：前测/后测）×2（分组：实验组/控制组）的两因素混合实验设计,考察心理旋转游戏训练对5~6岁儿童空间及数学能力的影响。基于心理旋转的可塑性以及空间能力与数学能力之间的密切关系设计心理旋转干预方案,将36名被试分为实验组和控制组进行为期8周共24次的干预研究。研究结果表明：（1）空间能力与数学能力之间显著正相关;（2）心理旋转游戏有效地提高了5~6岁儿童的空间能力和数学能力,该训练方案具有良好的即时效果。     

Rotation in the dynamic factor modeling of multivariate stationary time series

A special rotation procedure is proposed for the exploratory dynamic factor model for stationary multivariate time series. The rotation procedure applies separately to each univariate component series of aq-variate latent factor series and transforms such a component, initially represented as white noise, into a univariate moving-average. This is accomplished by minimizing a so-called state-space criterion that penalizes deviations of the rotated solution from a generalized state-space model with only instantaneous factor loadings. Alternative criteria are discussed in the closing section. The results of an empirical application are presented in some detail.This research was supported by the Institute for Developmental and Health Research Methodology, University of Virginia.     

自我和物体为参照系的心理旋转分离：内旋效应

采用虚拟的旋转不同角度左、右手模型,构建“左右手判断(Left and right hand judgment: LR)”任务和“相同－不同判断(same and different judgment: SD)”任务,考察这两种实验任务是否都存在内旋效应和角度效应,以此推论被试采用何种旋转策略。结果发现：（1） 两种实验任务结果均表现出显著的角度效应。（2）在LR任务条件下,存在显著的内旋效应,而在SD任务中不存在内旋效应。从而表明当人手图片作为心理旋转材料时,它具有双重角色。被试心理旋转加工时究竟选用何种参照系的旋转策略,与实验材料和实验任务两者都密不可分     

弱智儿童心理旋转的研究

该研究探讨了一组9~12岁弱智儿童的“心理旋转”能力,该组儿童平均年龄10.1岁,平均智商（IQ）53.8（中度智力残疾）。研究结果与正常儿童（均龄8岁）作了比较。结果显示,弱智儿童的心理旋转能力明显较之正常儿童为差,智力缺陷对视觉图形空间定向能力（心理旋转）有着明显的影响。但无论弱智儿童或是正常儿童的结果均未发现性别差异。