Generalized Structured Component Analysis with Latent Interactions

Generalized structured component analysis (GSCA) is a component-based approach to structural equation modeling. In practice, researchers may often be interested in examining the interaction effects of latent variables. However, GSCA has been geared only for the specification and testing of the main effects of variables. Thus, an extension of GSCA is proposed to effectively deal with various types of interactions among latent variables. In the proposed method, a latent interaction is defined as a product of interacting latent variables. As a result, this method does not require the construction of additional indicators for latent interactions. Moreover, it can easily accommodate both exogenous and endogenous latent interactions. An alternating least-squares algorithm is developed to minimize a single optimization criterion for parameter estimation. A Monte Carlo simulation study is conducted to investigate the parameter recovery capability of the proposed method. An application is also presented to demonstrate the empirical usefulness of the proposed method.     

Structural Equation Models in a Redundancy Analysis Framework With Covariates

A recent method to specify and fit structural equation modeling in the Redundancy Analysis framework based on so-called Extended Redundancy Analysis (ERA) has been proposed in the literature. In this approach, the relationships between the observed exogenous variables and the observed endogenous variables are moderated by the presence of unobservable composites, estimated as linear combinations of exogenous variables. However, in the presence of direct effects linking exogenous and endogenous variables, or concomitant indicators, the composite scores are estimated by ignoring the presence of the specified direct effects.

To fit structural equation models, we propose a new specification and estimation method, called Generalized Redundancy Analysis (GRA), allowing us to specify and fit a variety of relationships among composites, endogenous variables, and external covariates. The proposed methodology extends the ERA method, using a more suitable specification and estimation algorithm, by allowing for covariates that affect endogenous indicators indirectly through the composites and/or directly. To illustrate the advantages of GRA over ERA we propose a simulation study of small samples. Moreover, we propose an application aimed at estimating the impact of formal human capital on the initial earnings of graduates of an Italian university, utilizing a structural model consistent with well-established economic theory.     

Relationships between redundancy analysis,canonical correlation,and multivariate regression

This paper attempts to clarify the nature of redundancy analysis and its relationships to canonical correlation and multivariate multiple linear regression. Stewart and Love introduced redundancy analysis to provide non-symmetric measures of the dependence of one set of variables on the other, as channeled through the canonical variates. Van den Wollenberg derived sets of variates which directly maximize the between set redundancy. Multivariate multiple linear regression on component scores (such as principal components) is considered. The problem is extended to include an orthogonal rotation of the components. The solution is shown to be identical to van den Wollenberg's maximum redundancy solution.This research was supported in part by U.S. Environmental Protection Agency contract 68-02-3402. The author gratefully acknowledges the stimulation of Maurice Tatsuoka and Beth Dawson-Saunders in first interesting him in redundancy analysis, as well as a useful change suggested by Warren Sarle.     

A unique role of endogenous visual-spatial attention in rapid processing of multiple targets

Visual spatial attention can be exogenously captured by a salient stimulus or can be endogenously allocated by voluntary effort. Whether these two attention modes serve distinctive functions is debated, but for processing of single targets the literature suggests superiority of exogenous attention (it is faster acting and serves more functions). We report that endogenous attention uniquely contributes to processing of multiple targets. For speeded visual discrimination, response times are faster for multiple redundant targets than for single targets because of probability summation and/or signal integration. This redundancy gain was unaffected when attention was exogenously diverted from the targets but was completely eliminated when attention was endogenously diverted. This was not a result of weaker manipulation of exogenous attention because our exogenous and endogenous cues similarly affected overall response times. Thus, whereas exogenous attention is superior for processing single targets, endogenous attention plays a unique role in allocating resources crucial for rapid concurrent processing of multiple targets.     

The effect of affine transformation on redundancy analysis

Canonical redundancy analysis provides an estimate of the amount of shared variance between two sets of variables and provides an alternative to canonical correlation. The proof that the total redundancy is equal to the average squared multiple correlation coefficient obtained by regressing each variable in the criterion set on all variables in the predictor set is generalized to the case in which there are a larger number of criterion than predictor variables. It is then shown that the redundancy for the criterion set of variables is invariant under affine transformation of the predictor variables, but not invariant under transformation of the criterion variables.     

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.     

A unified approach to multiple‐set canonical correlation analysis and principal components analysis

Multiple‐set canonical correlation analysis and principal components analysis are popular data reduction techniques in various fields, including psychology. Both techniques aim to extract a series of weighted composites or components of observed variables for the purpose of data reduction. However, their objectives of performing data reduction are different. Multiple‐set canonical correlation analysis focuses on describing the association among several sets of variables through data reduction, whereas principal components analysis concentrates on explaining the maximum variance of a single set of variables. In this paper, we provide a unified framework that combines these seemingly incompatible techniques. The proposed approach embraces the two techniques as special cases. More importantly, it permits a compromise between the techniques in yielding solutions. For instance, we may obtain components in such a way that they maximize the association among multiple data sets, while also accounting for the variance of each data set. We develop a single optimization function for parameter estimation, which is a weighted sum of two criteria for multiple‐set canonical correlation analysis and principal components analysis. We minimize this function analytically. We conduct simulation studies to investigate the performance of the proposed approach based on synthetic data. We also apply the approach for the analysis of functional neuroimaging data to illustrate its empirical usefulness.     

Regularized Partial and/or Constrained Redundancy Analysis

Methods of incorporating a ridge type of regularization into partial redundancy analysis (PRA), constrained redundancy analysis (CRA), and partial and constrained redundancy analysis (PCRA) were discussed. The usefulness of ridge estimation in reducing mean square error (MSE) has been recognized in multiple regression analysis for some time, especially when predictor variables are nearly collinear, and the ordinary least squares estimator is poorly determined. The ridge estimation method was extended to PRA, CRA, and PCRA, where the reduced rank ridge estimates of regression coefficients were obtained by minimizing the ridge least squares criterion. It was shown that in all cases they could be obtained in closed form for a fixed value of ridge parameter. An optimal value of the ridge parameter is found by G-fold cross validation. Illustrative examples were given to demonstrate the usefulness of the method in practical data analysis situations. We thank Jim Ramsay for his insightful comments on an earlier draft of this paper. The work reported in this paper is supported by Grants 10630 from the Natural Sciences and Engineering Research Council of Canada to the first author.     

Endogenous versus exogenous change: Change detection,self and agency

The goal of this study is to characterize observers’ abilities to discriminate between endogenous (i.e., self-produced) and exogenous changes. To do so, we developed a new experimental paradigm. On each trial, participants were shown a dot pattern on the screen. Next, the pattern disappeared and participants were to reproduce it. Changes were surreptuously introduced in the stimulus, either by presenting participants anew with the dot pattern they had themselves produced on the previous trial (endogenous change) or by presenting participants with a slightly different dot pattern (exogenous changes). We analyzed awareness of the changes and behavioral adaptation to them in a dynamical manner. We observe (1) signal attenuation in the presence of endogenous change, (2) dissociation between self-attribution reports and behavioral effect of agency. We discuss the source of this sensitive attenuation as well as the relation between a minimal or core self and an extended, narrative or autobiographical self.     

Fuzzy partition models for fitting a set of partitions

Methodology is described for fitting a fuzzy consensus partition to a set of partitions of the same set of objects. Three models defining median partitions are described: two of them are obtained from a least-squares fit of a set of membership functions, and the third (proposed by Pittau and Vichi) is acquired from a least-squares fit of a set of joint membership functions. The models are illustrated by application to both a set of hard partitions and a set of fuzzy partitions and comparisons are made between them and an alternative approach to obtaining a consensus fuzzy partition proposed by Sato and Sato; a discussion is given of some interesting differences in the results.We are grateful to Dr. M.G. Pittau for carrying out the analyses of the macroeconomic data using the method of Sato and Sato (1994).     

Masked stimuli modulate endogenous shifts of spatial attention

Unconscious stimuli can influence participants’ motor behavior but also more complex mental processes. Recent research has gradually extended the limits of effects of unconscious stimuli. One field of research where such limits have been proposed is spatial cueing, where exogenous automatic shifts of attention have been distinguished from endogenous controlled processes which govern voluntary shifts of attention. Previous evidence suggests unconscious effects on mechanisms of exogenous shifts of attention. Here, we applied a cue-priming paradigm to a spatial cueing task with arbitrary cues by centrally presenting a masked symmetrical prime before every cue stimulus. We found priming effects on response times in target discrimination tasks with the typical dynamic of cue-priming effects (Experiments 1 and 2) indicating that central symmetrical stimuli which have been associated with endogenous orienting can modulate shifts of spatial attention even when they are masked. Prime–Cue Congruency effects of perceptual dissimilar prime and cue stimuli (Experiment 3) suggest that these effects cannot be entirely reduced to perceptual repetition priming of cue processing. In addition, priming effects did not differ between participants with good and poor prime recognition performance consistent with the view that unconscious stimulus features have access to processes of endogenous shifts of attention.     

Contributions to factor analysis of dichotomous variables

A new method is proposed for the factor analysis of dichotomous variables. Similar to the method of Christoffersson this uses information from the first and second order proportions to fit a multiple factor model. Through a transformation into a new set of sample characteristics, the estimation is considerably simplified. A generalized least-squares estimator is proposed, which asymptotically is as efficient as the corresponding estimator of Christoffersson, but which demands less computing time.This research was supported by the Bank of Sweden Tercentenary Foundation under project Structural Equation Models in the Social Sciences, project director Karl G. Jöreskog.     

CPCA: A program for principal component analysis with external information on subjects and variables

A program is described for principal component analysis with external information on subjects and variables. This method is calledconstrained principal component analysis (CPCA), in which regression analysis and principal component analysis are combined into a unified framework that allows a full exploration of data structures both within and outside known information on subjects and variables. Many existing methods are special cases of CPCA, and the program can be used for multivariate multiple regression, redundancy analysis, double redundancy analysis, dual scaling with external criteria, vector preference models, and GMANOVA (growth curve models).     

FUNCTIONAL ANALYSIS OF SEPARATE TOPOGRAPHIES OF ABERRANT BEHAVIOR

We conducted a functional analysis of distinct topographies of aberrant behavior displayed by 4 clients. We first analyzed the behaviors in an aggregate fashion and then separated the behaviors to formulate hypotheses about the maintaining variables for each behavior. The procedures were used in a two-phase experiment. During Phase 1, two extended functional analyses were completed, one in an inpatient unit and one in a special education classroom. During Phase 2, two brief functional analyses were completed in an outpatient clinic. Results indicated that hypotheses of separate functions for distinct behaviors can be generated using both extended and brief functional analyses when the results are graphed in the aggregate and are separated by response topography. The results also suggest that these methods can improve the accuracy of data interpretation and treatment selection.     

Generalized full-information item bifactor analysis

Full-information item bifactor analysis is an important statistical method in psychological and educational measurement. Current methods are limited to single-group analysis and inflexible in the types of item response models supported. We propose a flexible multiple-group item bifactor analysis framework that supports a variety of multidimensional item response theory models for an arbitrary mixing of dichotomous, ordinal, and nominal items. The extended item bifactor model also enables the estimation of latent variable means and variances when data from more than 1 group are present. Generalized user-defined parameter restrictions are permitted within or across groups. We derive an efficient full-information maximum marginal likelihood estimator. Our estimation method achieves substantial computational savings by extending Gibbons and Hedeker's (1992) bifactor dimension reduction method so that the optimization of the marginal log-likelihood requires only 2-dimensional integration regardless of the dimensionality of the latent variables. We use simulation studies to demonstrate the flexibility and accuracy of the proposed methods. We apply the model to study cross-country differences, including differential item functioning, using data from a large international education survey on mathematics literacy.     

一个多值逻辑的一阶谓词系统

鞠实儿曾提出一个开放类三值命题逻辑系统,这一逻辑也可以推广到任意m值逻辑情形,成为一个联结词函数完全的逻辑。本文将对推广的命题逻辑系统L^＊建立一种一阶谓词系统,并证明其可靠性、完全性。     

State-trace analysis can be an appropriate tool for assessing the number of cognitive systems: A reply to Ashby (2014)

Ashby (2014) has argued that state-trace analysis (STA) is not an appropriate tool for assessing the number of cognitive systems, because it fails in its primary goal of distinguishing single-parameter and multiple-parameter models. We show that this is based on a misunderstanding of the logic of STA, which depends solely on nearly universal assumptions about psychological measurement and clearly supersedes inferences based on functional dissociation and the analysis of interactions in analyses of variance. We demonstrate that STA can be used to draw inferences concerning the number of latent variables mediating the effects of a set of independent variables on a set of dependent variables. We suggest that STA is an appropriate tool to use when making arguments about the number of cognitive systems that must be posited to explain behavior. However, no statistical or inferential procedure is able to provide definitive answers to questions about the number of cognitive systems, simply because the concept of a “system” is not defined in an appropriate way.     

Exogenous and endogenous spatial attention effects on visuospatial working memory

In this study, we investigate how exogenous and endogenous orienting of spatial attention affect visuospatial working memory (VSWM). Specifically, we focused on two attentional effects and their consequences on storage in VSWM, when exogenous (Experiment 1) or endogenous (Experiment 2) orienting cues were used. The first effect, known as the meridian effect, is given by a decrement in behavioural performance when spatial cues and targets are presented in locations separated by vertical and/or horizontal meridians. The second effect, known as the distance effect, is given by a decrement in the orienting effects as a function of the spatial distance between cues and targets. Our results revealed a dissociation between exogenous and endogenous orienting mechanisms in terms of both meridian and distance effects. We found that meridian crossing affects performance only when endogenous cues were used. Specifically, VSWM performance with endogenous cueing depended more on the number of meridian crossings than on distance between cue and target. By contrast, a U-shaped distance dependency was observed using exogenous cues. Our findings therefore suggest that exogenous and endogenous orienting mechanisms lead to different forms of attentional bias for storage in VSWM.     

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.     

Optimal scaling by alternating length-constrained nonnegative least squares, with application to distance-based analysis

An important feature of distance-based principal components analysis, is that the variables can be optimally transformed. For monotone spline transformation, a nonnegative least-squares problem with a length constraint has to be solved in each iteration. As an alternative algorithm to Lawson and Hanson (1974), we propose the Alternating Length-Constrained Non-Negative Least-Squares (ALC-NNLS) algorithm, which minimizes the nonnegative least-squares loss function over the parameters under a length constraint, by alternatingly minimizing over one parameter while keeping the others fixed. Several properties of the new algorithm are discussed. A Monte Carlo study is presented which shows that for most cases in distance-based principal components analysis, ALC-NNLS performs as good as the method of Lawson and Hanson or sometimes even better in terms of the quality of the solution. Supported 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. We would like to thank the anonymous referees for their valuable remarks that have improved the quality of this paper.