A method of optimal scaling for multivariate ordinal data and its extensions

This paper develops a method of optimal scaling for multivariate ordinal data, in the framework of a generalized principal component analysis. This method yields a multidimensional configuration of items, a unidimensional scale of category weights for each item and, optionally, a multidimensional configuration of subjects. The computation is performed by alternately solving an eigenvalue problem and executing a quasi-Newton projection method. The algorithm is extended for analysis of data with mixed measurement levels or for analysis with a combined weighting of items. Numerical examples and simulations are provided. The algorithm is discussed and compared with some related methods.Earlier results of this research appeared in Saito and Otsu (1983). The authors would like to acknowledge the helpful comments and encouragement of the editor.     

Principals versus osmod: A comment on saito and otsu

Saito and Otsu (1988) compared their OSMOD method of nonmetric principal-component analysis to an early and incorrect implementation of the PRINCIPALS algorithm of Young, Takane, and de Leeuw (1978). In this comment we present results from the current, correct implementations of the algorithm.     

The optimality of the centroid method

The aim of this note is to show that the centroid method has two optimality properties. It yields loadings with the highest sum of absolute values, even in absence of the constraint that the squared component weights be equal. In addition, it yields scores with maximum variance, subject to the constraint that none of the squared component weights be larger than 1.This research is financed by NSERC of Canada. The author is grateful to Michel Tenenhaus for pointing the similarity of the procedures in the centroid method and Q-mode PCA in L₁. The author also thanks the editor and associate editor for providing shorter proofs of the theorems, along with the referees for their helpful comments.     

Penalized optimal scaling for ordinal variables with an application to international classification of functioning core sets

Ordinal data occur frequently in the social sciences. When applying principal component analysis (PCA), however, those data are often treated as numeric, implying linear relationships between the variables at hand; alternatively, non-linear PCA is applied where the obtained quantifications are sometimes hard to interpret. Non-linear PCA for categorical data, also called optimal scoring/scaling, constructs new variables by assigning numerical values to categories such that the proportion of variance in those new variables that is explained by a predefined number of principal components (PCs) is maximized. We propose a penalized version of non-linear PCA for ordinal variables that is a smoothed intermediate between standard PCA on category labels and non-linear PCA as used so far. The new approach is by no means limited to monotonic effects and offers both better interpretability of the non-linear transformation of the category labels and better performance on validation data than unpenalized non-linear PCA and/or standard linear PCA. In particular, an application of penalized optimal scaling to ordinal data as given with the International Classification of Functioning, Disability and Health (ICF) is provided.     

Social networks with two sets of actors

Traditional network research analyzes relational ties within a single group of actors: the models presented in this paper involve relational ties exist beteen two distinct sets of actors. Statistical models for traditional networks in which relations are measured within a group simplify when modeling unidirectional relations measured between groups. The traditional paradigm results in a one-mode socionatrix; the network paradigm considered in this paper results in a two-mode socionatrix; A statistical model is presented, illustrated on a sample data set, and compared to its traditional counterpart. Extensions are discussed, including those that model multivariate relations simultaneously, and those that allow for the inclustion of attributes of the individuals in the group.We thank the Editor and two anonymous reviewers for their helpful comments. We are also grateful to George Barnett for allowing us to analyze his data.     

On the optimality of the simultaneous redundancy transformations

The objective of this paper is to introduce and motivate additional properties and interpretations for the redundancy variables. It is shown that these variables can be derived by application of certain invariance arguments and without reference to the index of redundancy. In addition, an optimality property for the variables is presented which is important whenever one restricts attention in a study to a subset of the redundancy variables. This optimality property pertains to the subset rather than to the individual variables.This paper is based in part on the author's doctoral dissertation, Department of Statistics, Princeton, University. Research was conducted under the supervision of Lawrence S. Mayer.     

Longitudinal Dynamics of Substance Use and Psychiatric Symptoms in Count Data with Zero Inflation

We examine the dynamics of substance use and psychiatric symptoms from childhood to adolescence using a longitudinal study of Mexican-origin families (N = 674). We present a longitudinal model capturing the trajectories of substance use and psychiatric symptoms (depression, conduct disorder), as well as the interrelations between these trajectories over time. Such a model is an extension of latent change score models designed to account for the occurrence of psychiatric symptoms while also accommodating a large amount of zeros for nonoccurrence and characterizing the changes over time in the count data. We compare this model with a more traditional approach based on a log transformation of the data. We describe differences between these approaches and highlight the benefits of using the two-part model when the data include a large amount of zeros for nonoccurrence of the behavior.     

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.     

A general approach to categorical data analysis with missing data,using generalized linear models with composite links

A general approach for analyzing categorical data when there are missing data is described and illustrated. The method is based on generalized linear models with composite links. The approach can be used (among other applications) to fill in contingency tables with supplementary margins, fit loglinear models when data are missing, fit latent class models (without or with missing data on observed variables), fit models with fused cells (including many models from genetics), and to fill in tables or fit models to data when variables are more finely categorized for some cases than others. Both Newton-like and EM methods are easy to implement for parameter estimation.The author thanks the editor, the reviewers, Laurie Hopp Rindskopf, and Clifford Clogg for comments and suggestions that substantially improved the paper.     

Logit models and logistic regressions for social networks: I. An introduction to Markov graphs andp

Spanning nearly sixty years of research, statistical network analysis has passed through (at least) two generations of researchers and models. Beginning in the late 1930's, the first generation of research dealt with the distribution of various network statistics, under a variety of null models. The second generation, beginning in the 1970's and continuing into the 1980's, concerned models, usually for probabilities of relational ties among very small subsets of actors, in which various simple substantive tendencies were parameterized. Much of this research, most of which utilized log linear models, first appeared in applied statistics publications.But recent developments in social network analysis promise to bring us into a third generation. The Markov random graphs of Frank and Strauss (1986) and especially the estimation strategy for these models developed by Strauss and Ikeda (1990; described in brief in Strauss, 1992), are very recent and promising contributions to this field. Here we describe a large class of models that can be used to investigate structure in social networks. These models include several generalizations of stochastic blockmodels, as well as models parameterizing global tendencies towards clustering and centralization, and individual differences in such tendencies. Approximate model fits are obtained using Strauss and Ikeda's (1990) estimation strategy.In this paper we describe and extend these models and demonstrate how they can be used to address a variety of substantive questions about structure in social networks.This research was supported by grants from the Australian Research Council and the National Science Foundation (#SBR93-10184). This paper was presented at the 1994 Annual Meeting of the Psychometric Society, Champaign, Illinois, June 1994. Special thanks go to Sarah Ardu for programming assistance, Laura Koehly and Garry Robins for help with this research, and to Shizuhiko Nishisato and three reviewers for their comments. INTERNETemail addresses:pattison@psych.unimelb.edu.au (PP);stanwass@uiuc.edu (SW). Affiliations: Department of Psychology, University of Melbourne (PP); Department of Psychology, Department of Statistics, and The Beckman Institute for Advanced Science and Technology, University of Illinois (SW).     

Kernel principal component analysis combining rotation forest method for linearly inseparable data

Rotation forest (RoF) is an ensemble classifier combining linear analysis theories and decision tree algorithms. In recent existing works, RoF was widely applied to various fields with outstanding performance compared to traditional machine learning techniques, given that a reasonable number of base classifiers is provided. However, the conventional RoF algorithm suffers from classifying linearly inseparable datasets. In this study, a hybrid algorithm integrating kernel principal component analysis (KPCA) and the conventional RoF algorithm is proposed to overcome the classification difficulty for linearly inseparable datasets. The radial basis function (RBF) is selected as the kernel for the KPCA method to establish the nonlinear mapping for linearly inseparable data. Moreover, we evaluate various kernel parameters for better performance. Experimental results show that our algorithm improves the performance of RoF with linearly inseparable datasets, and therefore provides higher classification accuracy rates compared with other ensemble machine learning methods.     

A note on the investigation of local and global identifiability

In general, nonlinear models such as those commonly employed for the analysis of covariance structures, are not globally identifiable. Any investigation of local identifiability must either yield a mapping of identifiability onto the entire parameter space, which will rarely be feasible in any applications of interest, or confine itself to the neighbourhood of such points of special interest as the maximum likelihood point.The author would like to thank J. Jack McArdle and Colin Fraser for their comments on this paper.     

Adaptation to Reversal of Retinal Feedback of Eyemovements

Both optical and hybrid-computer methods were devised to reverse the visual feedback which the eye gets of its own movements. Optical reversal by a contact lens device caused persisting, unstable, skittered, blurred vision, to which the subject made no effective adaptation. Using a hybrid analog-digital-analog computer, eyemovement-retinal feedback was reversed by yoking oscilloscope targets to the transduced electrical signals of ocular movements and reversing the sign of input-output control of the computer system. Vision with such computer-reversed feedback from ocular movements duplicated the experiences with the contact-lens method. Practice over five sessions of 50 trials indicated that three Ss showed no significant improvement in eye-tracking accuracy with the reversed vision. Perception of direction in vision and the basic signal processes of vision were thus found to be dependent on built-in direction specificity of eyemovement-retinal feedback interactions which also govern dynamic specialization and control of pursuit, saccadic, and fixation movements.     

Retrieving the correlation matrix from a truncated PCA solution: The inverse principal component problem

Whenr Principal Components are available fork variables, the correlation matrix is approximated in the least squares sense by the loading matrix times its transpose. The approximation is generally not perfect unlessr =k. In the present paper it is shown that, whenr is at or above the Ledermann bound,r principal components are enough to perfectly reconstruct the correlation matrix, albeit in a way more involved than taking the loading matrix times its transpose. In certain cases just below the Ledermann bound, recovery of the correlation matrix is still possible when the set of all eigenvalues of the correlation matrix is available as additional information.     

The indirect effect of cognitive reserve on the relationship between age and cognition in pathological ageing: A cross-sectional retrospective study in an unselected and consecutively enrolled sample

Cognitive reserve (CR) allows individuals to maintain cognitive functionality even in the presence of pathologies. The compensation hypothesis suggests that CR plays an indirect role between age and cognitive decline, contrasting the negative effect of ageing on cognition. We test this hypothesis in an unselected and consecutively enrolled sample of memory clinic attendees (n = 134) who completed the CR Index questionnaire and three neuropsychological tests assessing global cognition (MMSE, FAB, CDT). Participants were divided into two groups based on standard diagnostic criteria (DSM-5): those who were cognitively impaired (n = 92) and those who were preserved (n = 42). A principal component analysis was used to extract a composite measure of global cognitive functioning from the three neuropsychological tests, and mediation analysis was used to examine the relationship between CR, age and global cognitive functioning in the two groups. Results revealed that: (i) age had a significant direct negative effect on the global cognitive score in both groups; (ii) the three socio-behavioural proxies of CR together suppress the direct negative relationship between age and global cognitive score in cognitively impaired patients but not in cognitively preserved participants. This study confirms the association between CR, age and cognition and allows us to validate its role in a population with cognitive impairment and extend findings to a low-to-middle educated population. These results hold important implications for public health and wellness promotion, emphasising the beneficial role of maintaining healthy and active physical, cognitive and social lifestyles.     

Component Models for Fuzzy Data

The fuzzy perspective in statistical analysis is first illustrated with reference to the “Informational Paradigm” allowing us to deal with different types of uncertainties related to the various informational ingredients (data, model, assumptions). The fuzzy empirical data are then introduced, referring to J LR fuzzy variables as observed on I observation units. Each observation is characterized by its center and its left and right spreads (LR₁ fuzzy number) or by its left and right “centers” and its left and right spreads (LR₂ fuzzy number). Two types of component models for LR₁ and LR₂ fuzzy data are proposed. The estimation of the parameters of these models is based on a Least Squares approach, exploiting an appropriately introduced distance measure for fuzzy data. A simulation study is carried out in order to assess the efficacy of the suggested models as compared with traditional Principal Component Analysis on the centers and with existing methods for fuzzy and interval valued data. An application to real fuzzy data is finally performed. We would like to express our gratitude to the Editor, the Associate Editor, and the Referees whose comments and suggestions improved significantly the quality of the paper.     

Interactions of persistent slip bands with a grain boundary on the common primary slip plane in a copper bicrystal

In this letter, the dislocation patterns on the common primary slip plane in a fatigued \[1-34]-\[182-7] copper bicrystal with a Sigma=19b grain boundary (GB) have been investigated using the electron channelling contrast technique in a scanning electron microscope. The results show that the two-phase dislocation structure, such as veins and persistent slip band (PSB) walls, embedded within veins, can be clearly seen on the common primary slip plane. In particular, the interactions of PSBs with the GB are clearly revealed. It is found that there are three kinds of interaction mode between the GB and the dislocations during cyclic deformation, and those are discussed. It is suggested that the dislocations carried by PSBs cannot transfer through the GB continuously even though the bicrystal has a common primary slip plane and its surface slip bands are continuous across the GB.     

The Short Mood and Feelings Questionnaire (SMFQ): A Unidimensional Item Response Theory and Categorical Data Factor Analysis of Self-Report Ratings from a Community Sample of 7-through 11-Year-Old Children

Item response theory (IRT) and categorical data factor analysis (CDFA) are complementary methods for the analysis of the psychometric properties of psychiatric measures that purport to measure latent constructs. These methods have been applied to relatively few child and adolescent measures. We provide the first combined IRT and CDFA analysis of a clinical measure (the Short Mood and Feelings Questionnaire—SMFQ) in a community sample of 7-through 11-year-old children. Both latent variable models supported the internal construct validity of a single underlying continuum of severity of depressive symptoms. SMFQ items discriminated well at the more severe end of the depressive latent trait. Item performance was not affected by age, although age correlated significantly with latent SMFQ scores suggesting that symptom severity increased within the age period of 7–11. These results extend existing psychometric studies of the SMFQ and confirm its scaling properties as a potential dimensional measure of symptom severity of childhood depression in community samples.