Canonical/redundancy factoring analysis

The interrelationships between two sets of measurements made on the same subjects can be studied by canonical correlation. Originally developed by Hotelling [1936], the canonical correlation is the maximum correlation betweenlinear functions (canonical factors) of the two sets of variables. An alternative statistic to investigate the interrelationships between two sets of variables is the redundancy measure, developed by Stewart and Love [1968]. Van Den Wollenberg [1977] has developed a method of extracting factors which maximize redundancy, as opposed to canonical correlation.A component method is presented which maximizes user specified convex combinations of canonical correlation and the two nonsymmetric redundancy measures presented by Stewart and Love. Monte Carlo work comparing canonical correlation analysis, redundancy analysis, and various canonical/redundancy factoring analyses on the Van Den Wollenberg data is presented. An empirical example is also provided.Wayne S. DeSarbo is a Member of Technical Staff at Bell Laboratories in the Mathematics and Statistics Research Group at Murray Hill, N.J. I wish to express my appreciation to J. Kettenring, J. Kruskal, C. Mallows, and R. Gnanadesikan for their valuable technical assistance and/or for comments on an earlier draft of this paper. I also wish to thank the editor and reviewers of this paper for their insightful remarks.     

A distance-based variety of nonlinear multivariate data analysis, including weights for objects and variables

In the distance approach to nonlinear multivariate data analysis the focus is on the optimal representation of the relationships between the objects in the analysis. In this paper two methods are presented for including weights in distance-based nonlinear multivariate data analysis. In the first method, weights are assigned to the objects while the second method is concerned with differential weighting of groups of variables. When each analysis variable defines a group the latter method becomes a variable weighting method. For objects the weights are assumed to be given; for groups of variables they may be given, or estimated. These weighting schemes can also be combined and have several important applications. For example, they make it possible to perform efficient analyses of large data sets, to use the distance-based variety of nonlinear multivariate data analysis as an addition to loglinear analysis of multiway contingency tables, and to do stability studies of the solutions by applying the bootstrap on the objects or the variables in the analysis. These and other applications are discussed, and an efficient algorithm is proposed to minimize the corresponding loss function.This study is funded by The Netherlands Organization for Scientific Research (NWO) by grant nr. 030-56403 for the PIONEER project Subject Oriented Multivariate Analysis to the third author.     

Alternating least squares algorithms for simultaneous components analysis with equal component weight matrices in two or more populations

Millsap and Meredith (1988) have developed a generalization of principal components analysis for the simultaneous analysis of a number of variables observed in several populations or on several occasions. The algorithm they provide has some disadvantages. The present paper offers two alternating least squares algorithms for their method, suitable for small and large data sets, respectively. Lower and upper bounds are given for the loss function to be minimized in the Millsap and Meredith method. These can serve to indicate whether or not a global optimum for the simultaneous components analysis problem has been attained.Financial support by the Netherlands organization for scientific research (NWO) is gratefully acknowledged.     

密集追踪数据分析：模型及其应用

在心理学、教育学和临床医学等领域, 越来越多的研究者开始关注个体内部的行为、心理、临床效果等随时间而产生的动态变化, 重视针对个体的差异化建模。密集追踪是一种在短时间内对个体进行多个时间节点密集追踪测量的方法, 更适合用于研究个体内部心理过程等的动态变化及其作用机制。近年来, 密集追踪成为心理学研究的一大热点, 但许多密集追踪的研究分析仍停留在较为传统的方法。方法学领域已涌现出较多用于密集追踪数据分析的模型方法, 较为主流的模型包括以动态结构方程模型(Dynamic Structural Equation Model, DSEM)为代表的自上而下的建模方法, 以及以组迭代多模型估计(Group Iterative Multiple Model Estimation, GIMME)为代表的自下而上的建模方法。二者均可以方便地对密集追踪数据中的自回归及交叉滞后效应进行建模。     

Canonical Correlation and Chi-Square: Relationships and Interpretation

A 2 × 2 chi-square can be computed from a phi coefficient, which is the Pearson correlation between two binomial variables. Similarly, chi-square for larger contingency tables can be computed from canonical correlation coefficients. The authors address the following series of issues involving this relationship: (a) how to represent a contingency table in terms of a correlation matrix involving r - 1 row and c - 1 column dummy predictors; (b) how to compute chi-square from canonical correlations solved from this matrix; (c) how to compute loadings for the omitted row and column variables; and (d) the possible interpretive advantage of describing canonical relationships that comprise chi-square, together with some examples. The proposed procedures integrate chi-square analysis of contingency tables with general correlational theory and serve as an introduction to some recent methods of analysis more widely known by sociologists.     

Interpreting canonical correlation analysis through biplots of structure correlations and weights

This paper extends the biplot technique to canonical correlation analysis and redundancy analysis. The plot of structure correlations is shown to the optimal for displaying the pairwise correlations between the variables of the one set and those of the second. The link between multivariate regression and canonical correlation analysis/redundancy analysis is exploited for producing an optimal biplot that displays a matrix of regression coefficients. This plot can be made from the canonical weights of the predictors and the structure correlations of the criterion variables. An example is used to show how the proposed biplots may be interpreted.     

Constrained k‐means on cluster proportion and distances among clusters for longitudinal data analysis

Clustering individuals by measures of similarity or dissimilarity at trajectories of changes in longitudinal data enables determination of typical patterns of development and growth. The present research proposes a new constrained k‐means method with lower bound constraints on cluster proportions and distances among clusters at focused variables and time points to fulfill various needs in clustering longitudinal data. The method assumes a large number of clusters at the onset and iteratively deletes and combines clusters according to these constraints. An additional property of the proposed constrained k‐means includes direct estimation of the unknown number of clusters. Simulation results clearly show the usefulness of the method for extracting clusters in plausible, real‐life analysis including non‐normality within clusters, and the proposed algorithm works well and convergence of the estimates is satisfactory. An actual example using Japanese longitudinal data regarding sleep habits and mental health is presented to verify the utility of the proposed constrained k‐means.     

纵向数据的调节效应分析

目前调节效应检验主要是基于截面数据, 本文讨论纵向(追踪)数据的调节效应分析。如果自变量X和因变量Y有纵向数据, 调节效应可分为三类：调节变量Z不随时间变化、Z随时间变化、调节变量从自变量或因变量中产生。评介了基于多层模型、多层结构方程模型、交叉滞后模型和潜变量增长模型的纵向数据的多种调节效应分析方法。调节效应的分解和潜调节结构方程法的使用是纵向数据的调节效应分析的两大特点。对基于四类模型的调节效应分析方法进行综合比较后, 总结出一个纵向数据的调节效应分析流程。随后用实际例子演示如何进行纵向数据的调节效应分析, 并给出相应的Mplus程序。随后展望了纵向数据的调节效应分析的拓展方向, 例如基于动态结构方程模型的密集追踪数据的调节效应分析。     

Generalizations of the partial,part and bipartial canonical correlation analysis

Theg ₁- andg ₂-bipartial canonical correlation analyses are developed as generalizations of the partial, part, and bipartial canonical correlation analysis. Illustrative examples are provided.     

用典型相关进一步研究大学生亲社会倾向和亲社会推理的关系

本研究使用修订后的亲社会倾向量表(PTM)和亲社会客观推理量表(PROM),对大学生群体进行了测查,并采用简单相关和典型相关探讨亲社会倾向和亲社会推理的关系。结果表明:大学生的亲社会倾向由高到低依次为:紧急的、利他的、情绪的、依从的、匿名的和公开的;他们的亲社会推理由高到低依次是:需要定向、刻板定向、内化价值定向、享乐主义定向和赞扬定向。6种亲社会倾向都与刻板定向推理、内化价值定向推理正相关;紧急的、依从的亲社会倾向与需要定向推理正相关;匿名的亲社会倾向与享乐主义定向推理负相关。从典型相关分析结果来看,两者间的整体关联程度没有理论预期的高,典型变量所代表的亲社会倾向主要是紧急的、匿名的、依从的亲社会倾向,而这些亲社会倾向主要由内化价值定向推理、刻板定向推理、需要定向推理来解释。     

Some symmetric,invariant measures of multivariate association

A distinction is drawn between redundancy measurement and the measurement of multivariate association for two sets of variables. Several measures of multivariate association between two sets of variables are examined. It is shown that all of these measures are generalizations of the (univariate) squared-multiple correlation; all are functions of the canonical correlations, and all are invariant under linear transformations of the original sets of variables. It is further shown that the measures can be considered to be symmetric and are strictly ordered for any two sets of observed variables. It is suggested that measures of multivariate relationship may be used to generalize the concept of test reliability to the case of vector random variables.     

A one-step method for modelling longitudinal data with differential equations

Differential equation models are frequently used to describe non-linear trajectories of longitudinal data. This study proposes a new approach to estimate the parameters in differential equation models. Instead of estimating derivatives from the observed data first and then fitting a differential equation to the derivatives, our new approach directly fits the analytic solution of a differential equation to the observed data, and therefore simplifies the procedure and avoids bias from derivative estimations. A simulation study indicates that the analytic solutions of differential equations (ASDE) approach obtains unbiased estimates of parameters and their standard errors. Compared with other approaches that estimate derivatives first, ASDE has smaller standard error, larger statistical power and accurate Type I error. Although ASDE obtains biased estimation when the system has sudden phase change, the bias is not serious and a solution is also provided to solve the phase problem. The ASDE method is illustrated and applied to a two-week study on consumers’ shopping behaviour after a sale promotion, and to a set of public data tracking participants’ grammatical facial expression in sign language. R codes for ASDE, recommendations for sample size and starting values are provided. Limitations and several possible expansions of ASDE are also discussed.     

Estimating latent correlations between repeated testings

A model for longitudinal latent structure analysis is proposed. We assume that test scores for a given mental or attitudinal test are observed for the same individuals at two different points in time. The purpose of the analysis is to fit a model that combines the values of the latent variable at the two time points in a two-dimensional latent density. The correlation coefficient between the two values of the latent variable can then be estimated. The theory and methods are illustrated by a Danish dataset concerning psychic vulnerability.     

A Latent Gaussian process model for analysing intensive longitudinal data

Intensive longitudinal studies are becoming progressively more prevalent across many social science areas, and especially in psychology. New technologies such as smart-phones, fitness trackers, and the Internet of Things make it much easier than in the past to collect data for intensive longitudinal studies, providing an opportunity to look deep into the underlying characteristics of individuals under a high temporal resolution. In this paper we introduce a new modelling framework for latent curve analysis that is more suitable for the analysis of intensive longitudinal data than existing latent curve models. Specifically, through the modelling of an individual-specific continuous-time latent process, some unique features of intensive longitudinal data are better captured, including intensive measurements in time and unequally spaced time points of observations. Technically, the continuous-time latent process is modelled by a Gaussian process model. This model can be regarded as a semi-parametric extension of the classical latent curve models and falls under the framework of structural equation modelling. Procedures for parameter estimation and statistical inference are provided under an empirical Bayes framework and evaluated by simulation studies. We illustrate the use of the proposed model though the analysis of an ecological momentary assessment data set.     

Exploratory longitudinal factor analysis in multiple populations

For multiple populatios, a longtidinal factor analytic model which is entirely exploratory, that is, no explicit identification constraints, is proposed. Factorial collapse and period/practice effects are allowed. An invariant and/or stationary factor pattern is permitted. This model is formulated stochastically. To implement this model a stagewise EM algorithm is developed. Finally a numerical illustration utilizing Nesselroade and Baltes' data is presented.The authors wish to thank Barbara Mellers and Henry Kaiser for their helpful comments and John Nesselroade for providing us the data for our illustration. This research wwa supported in part by a grant (No. AG03164) from the National Institute on Aging to William Meredith. Details of the derivations and a copy of the PROC MATRIX program are available upon request from the first author.     

Canonical Extensions and Relational Representations of Lattices with Negation

This work is part of a wider investigation into lattice-structured algebras and associated dual representations obtained via the methodology of canonical extensions. To this end, here we study lattices, not necessarily distributive, with negation operations.We consider equational classes of lattices equipped with a negation operation ¬ which is dually self-adjoint (the pair (¬,¬) is a Galois connection) and other axioms are added so as to give classes of lattices in which the negation is De Morgan, orthonegation, antilogism, pseudocomplementation or weak pseudocomplementation. These classes are shown to be canonical and dual relational structures are given in a generalized Kripke-style. The fact that the negation is dually self-adjoint plays an important role here, as it implies that it sends arbitrary joins to meets and that will allow us to define the dual structures in a uniform way.Among these classes, all but one—that of lattices with a negation which is an antilogism—were previously studied by W. Dzik, E. Or?owska and C. van Alten using Urquhart duality.In some cases in which a given axiom does not imply that negation is dually self-adjoint, canonicity is proven with the weaker assumption of antitonicity of the negation.     

Component analysis in cross-sectional and longitudinal data

An extension of component analysis to longitudinal or cross-sectional data is presented. In this method, components are derived under the restriction of invariant and/or stationary compositing weights. Optimal compositing weights are found numerically. The method can be generalized to allow differential weighting of the observed variables in deriving the component solution. Some choices of weightings are discussed. An illustration of the method using real data is presented.Preparation of this article was supported in part by PSC-CUNY Grant #665365 to Roger E. Millsap and by National Institute of Aging Grant NIA-AG03164-03 to William Meredith. The authors thank John Nesselroade for permitting the use of the data presented in the article.     

Influence in canonical correlation analysis

The perturbation theory of the generalized eigenproblem is used to derive influence functions of each squared canonical correlation coefficient and the corresponding canonical vector pair. Three sample versions of these functions are described and some properties are noted. As particular applications, the influence function of the squared multiple correlation coefficient and influence functions of eigenvalues and eigenvectors in correspondence analysis are obtained. Three numerical examples are briefly discussed.We thank the Editor and the anonymous reviewers for their helpful comments. This research was carried out with the financial support of the Italian Ministry of the University and the National Research Council.     

The second order approximation to sample influence curve in canonical correlation analysis

A second order approximation to the sample influence curve (SIC) in canonical correlation analysis has been derived in the literature. However, it does not seem satisfactory for some cases. In this paper, we present a more accurate second order approximation. As a particular case, the proposed method is exact for the SIC of the squared multiple correlation coefficient. An example is given. The authors are most grateful to the associate editor and three reviewers for valuable comments and suggestions which improved the presentation of the paper considerably. The first author was partly supported by a RGC earmarked research grant of Hong Kong.