APL functions for interactive data analysis: Preparation of data files

APL functions to support preparation of data files are presented: a function to manage the entry of raw data, functions to display the entered data in formats convenient for checking, a function to support correction of errors, a function to organize the data into tables and file them, and a user interface function that provides menu selection of the data preparation functions. General-purpose support functions to assist in file use and in menu selection are also provided.     

Principal components analysis of sampled functions

This paper describes a technique for principal components analysis of data consisting ofn functions each observed atp argument values. This problem arises particularly in the analysis of longitudinal data in which some behavior of a number of subjects is measured at a number of points in time. In such cases information about the behavior of one or more derivatives of the function being sampled can often be very useful, as for example in the analysis of growth or learning curves. It is shown that the use of derivative information is equivalent to a change of metric for the row space in classical principal components analysis. The reproducing kernel for the Hilbert space of functions plays a central role, and defines the best interpolating functions, which are generalized spline functions. An example is offered of how sensitivity to derivative information can reveal interesting aspects of the data.This research was supported by Grant PA 0320 to the second author by the Natural Sciences and Research Council of Canada. We are grateful to the reviewers of an earlier version and to J. B. Kruskal and S. Winsberg for many helpful comments concerning exposition.     

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.     

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.     

Common functional principal components analysis: a new approach to analyzing human movement data

In many human movement studies angle-time series data on several groups of individuals are measured. Current methods to compare groups include comparisons of the mean value in each group or use multivariate techniques such as principal components analysis and perform tests on the principal component scores. Such methods have been useful, though discard a large amount of information. Functional data analysis (FDA) is an emerging statistical analysis technique in human movement research which treats the angle-time series data as a function rather than a series of discrete measurements. This approach retains all of the information in the data. Functional principal components analysis (FPCA) is an extension of multivariate principal components analysis which examines the variability of a sample of curves and has been used to examine differences in movement patterns of several groups of individuals. Currently the functional principal components (FPCs) for each group are either determined separately (yielding components that are group-specific), or by combining the data for all groups and determining the FPCs of the combined data (yielding components that summarize the entire data set). The group-specific FPCs contain both within and between group variation and issues arise when comparing FPCs across groups when the order of the FPCs alter in each group. The FPCs of the combined data may not adequately describe all groups of individuals and comparisons between groups typically use t-tests of the mean FPC scores in each group. When these differences are statistically non-significant it can be difficult to determine how a particular intervention is affecting movement patterns or how injured subjects differ from controls. In this paper we aim to perform FPCA in a manner allowing sensible comparisons between groups of curves. A statistical technique called common functional principal components analysis (CFPCA) is implemented. CFPCA identifies the common sources of variation evident across groups but allows the order of each component to change for a particular group. This allows for the direct comparison of components across groups. We use our method to analyze a biomechanical data set examining the mechanisms of chronic Achilles tendon injury and the functional effects of orthoses.     

Assessing the stability of principal components using regression

This paper presents an analysis, based on simulation, of the stability of principal components. Stability is measured by the expectation of the absolute inner product of the sample principal component with the corresponding population component. A multiple regression model to predict stability is devised, calibrated, and tested using simulated Normal data. Results show that the model can provide useful predictions of individual principal component stability when working with correlation matrices. Further, the predictive validity of the model is tested against data simulated from three non-Normal distributions. The model predicted very well even when the data departed from normality, thus giving robustness to the proposed measure. Used in conjunction with other existing rules this measure will help the user in determining interpretability of principal components.The authors would like to thank the four anonymous reviewers and the two editors for their valuable comments. Atanu R. Sinha gratefully acknowledges the research support received from the Marketing Studies Center, AGSM, UCLA. Send requests for reprints to Atanu R. Sinha, B418 Gold Hall, 110 Westwood Plaza, Los Angeles, CA 90095.     

Principal component analysis with external information on both subjects and variables

A method for structural analysis of multivariate data is proposed that combines features of regression analysis and principal component analysis. In this method, the original data are first decomposed into several components according to external information. The components are then subjected to principal component analysis to explore structures within the components. It is shown that this requires the generalized singular value decomposition of a matrix with certain metric matrices. The numerical method based on the QR decomposition is described, which simplifies the computation considerably. The proposed method includes a number of interesting special cases, whose relations to existing methods are discussed. Examples are given to demonstrate practical uses of the method.The work reported in this paper was supported by grant A6394 from the Natural Sciences and Engineering Research Council of Canada to the first author. Thanks are due to Jim Ramsay, Haruo Yanai, Henk Kiers, and Shizuhiko Nishisato for their insightful comments on earlier versions of this paper. Jim Ramsay, in particular, suggested the use of the QR decomposition, which simplified the presentation of the paper considerably.     

Some mathematical notes on three-mode factor analysis

The model for three-mode factor analysis is discussed in terms of newer applications of mathematical processes including a type of matrix process termed the Kronecker product and the definition of combination variables. Three methods of analysis to a type of extension of principal components analysis are discussed. Methods II and III are applicable to analysis of data collected for a large sample of individuals. An extension of the model is described in which allowance is made for unique variance for each combination variable when the data are collected for a large sample of individuals.     

Functional principal components analysis of workload capacity functions

Workload capacity, an important concept in many areas of psychology, describes processing efficiency across changes in workload. The capacity coefficient is a function across time that provides a useful measure of this construct. Until now, most analyses of the capacity coefficient have focused on the magnitude of this function, and often only in terms of a qualitative comparison (greater than or less than one). This work explains how a functional extension of principal components analysis can capture the time-extended information of these functional data, using a small number of scalar values chosen to emphasize the variance between participants and conditions. This approach provides many possibilities for a more fine-grained study of differences in workload capacity across tasks and individuals.     

Nonlinear principal components analysis: introduction and application

The authors provide a didactic treatment of nonlinear (categorical) principal components analysis (PCA). This method is the nonlinear equivalent of standard PCA and reduces the observed variables to a number of uncorrelated principal components. The most important advantages of nonlinear over linear PCA are that it incorporates nominal and ordinal variables and that it can handle and discover nonlinear relationships between variables. Also, nonlinear PCA can deal with variables at their appropriate measurement level; for example, it can treat Likert-type scales ordinally instead of numerically. Every observed value of a variable can be referred to as a category. While performing PCA, nonlinear PCA converts every category to a numeric value, in accordance with the variable's analysis level, using optimal quantification. The authors discuss how optimal quantification is carried out, what analysis levels are, which decisions have to be made when applying nonlinear PCA, and how the results can be interpreted. The strengths and limitations of the method are discussed. An example applying nonlinear PCA to empirical data using the program CATPCA (J. J. Meulman, W. J. Heiser, & SPSS, 2004) is provided.     

Evoked potential and visual half-field evidence for task-dependent cerebral asymmetries in congenitally deaf adults

Evoked potentials to laterally presented stimuli were collected from left and right tempero-parietal sites during performance of two visual half-field tasks, lexical decision, and line orientation discrimination. Reaction time and accuracy data were simultaneously collected. The behavioral data indicated the development of a right field advantage for the lexical decision task as a function of practice. A principal components analysis revealed three independent evoked potential components which displayed task-dependent hemispheric asymmetries. Multiple regression analyses revealed that visual half-field asymmetries in response accuracy were closely related to hemispheric asymmetries on several independent evoked response components. Subject's scores on independent tests of verbal reasoning and spatial relations were also found to be closely related to hemispheric asymmetry on several independent evoked response components. These data support a multidimensional concept of cerebral specialization. They also suggest that visual field asymmetries reflect the confluence of several underlying processes which have independent lateralization distributions across the population. In general, the results underscore the need for further research on the nature of the relationship between cerebral and perceptual asymmetries.     

InterFace: A software package for face image warping,averaging, and principal components analysis

We describe InterFace, a software package for research in face recognition. The package supports image warping, reshaping, averaging of multiple face images, and morphing between faces. It also supports principal components analysis (PCA) of face images, along with tools for exploring the “face space” produced by PCA. The package uses a simple graphical user interface, allowing users to perform these sophisticated image manipulations without any need for programming knowledge. The program is available for download in the form of an app, which requires that users also have access to the (freely available) MATLAB Runtime environment.     

Coh-metrix: analysis of text on cohesion and language.

Advances in computational linguistics and discourse processing have made it possible to automate many language- and text-processing mechanisms. We have developed a computer tool called Coh-Metrix, which analyzes texts on over 200 measures of cohesion, language, and readability. Its modules use lexicons, part-of-speech classifiers, syntactic parsers, templates, corpora, latent semantic analysis, and other components that are widely used in computational linguistics. After the user enters an English text, CohMetrix returns measures requested by the user. In addition, a facility allows the user to store the results of these analyses in data files (such as Text, Excel, and SPSS). Standard text readability formulas scale texts on difficulty by relying on word length and sentence length, whereas Coh-Metrix is sensitive to cohesion relations, world knowledge, and language and discourse characteristics.     

The Hopelessness Scale: a factor analysis

A confirmatory factor analysis of the Beck Hopelessness Scale in a sample of 340 Italian students did not support the 3-factor model reported for previous samples of psychiatric patients. A follow-up principal axis factor analysis yielded two interpretable correlated factors, suggesting that the structure of the scale may differ across clinical and nonclinical groups and as a function of nationality.     

Computer-augmented decision making

The principal judgmental components of multiattribute decision making are examined here with specific reference to how these components can be captured electronically. Once captured, a function, rule, or algorithm may be executed for the integration of this information and the selection of the optimal alternative(s). Two kinds of algorithms are discussed: one based on linear models, the other on fuzzy-set theory and ratio scaling. With on-line support and certain assumptions about human biases (which lead to nonoptimal decisions), the quality of decisions can be enhanced considerably. The principal concerns are with end-user acceptance of computer augmented decisions.     

Charting the process of change: A primer on survival analysis

Survival analysis is a powerful and useful technique for understanding qualitative change. This article provides a practical, nontechnical introduction to the use of survival analysis for social scientists. Important issues in using survival analysis are discussed, including research design, data preparation and management, and data analysis. Attendance data from a self-helf organization are used to illustrate common survival analysis tasks such as describing the overall survival and hazard functions, examining covariate effects, and modeling the form of the hazard function over time. An appendix that discusses the strengths and weaknesses of current survival analysis computer programs is included. Editor's note: Edward Seidman served as action editor for this article while serving as Associate Editor for Methodology.I thank William Davidson, Susan Englund, Bruce Rapkin, Kurt Ribisl, and three anonymous reviewers for their helpful comments on earlier drafts. The example data presented here were collected with the support of an NIMH grant (MH37390) awarded to Julian Rappaport and Ed Seidman.     

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.