Redundancy analysis for qualitative variables

Redundancy analysis (also called principal components analysis of instrumental variables) is a technique for two sets of variables, one set being dependent of the other. Its aim is maximization of the explained variance of the dependent variables by a linear combination of the explanatory variables. The technique is generalized to qualitative variables; it then gives implicitly a simultaneous optimal scaling of the dependent, qualitative variables. Examples are taken from the Dutch Life Situation Survey 1977, using Satisfaction with Life and Happiness as dependent variables. The analysis leads to one well-being scale, defined by the explanatory variables Marital status, Schooling, Income and Activity.The views expressed in this paper are those of the author and do not necessarily reflect the policies of the Netherlands Central Bureau of Statistics.     

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.     

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.     

Fostering learning from instructional video in a second language

This study investigated the most effective way to present an instructional video that contains words in the students' second language. Korean‐speaking university students received a 16‐min video lesson on Antarctica that included English narration (video + narration group), English text subtitles (video + text group), or English narration with simultaneous text subtitles (video + narration + text group). On a comprehension test, the video + text group scored higher than each of the other two groups, in contrast to the modality effect; and the video + narration + text group outscored the video + narration group, in contrast to the redundancy effect. Each of the lessons that included text was rated as less difficult than the lesson with narration only. The narration + text group reported lower effort than each of the other groups. Results highlight boundary conditions for two principles of multimedia instructional design that apply for college students who are learning in a second language. Theoretical implications are discussed.     

Aurelius Augustine on Imagination

The evaluation of a person seeking or volunteering information in cooperatively or competitively oriented problem solving groups was investigated in 63 white and black, male and female, high school and college students. It was hypothesized that under cooperative conditions the seeking of information would be evaluated more positively than would the volunteering of information, while under competitive conditions the volunteering of information would be evaluated more positively than would the seeking of information. A highly trained confederate (C) engaged in either the seeking of information or the volunteering of information in four-person problem solving groups in which Ss were either cooperatively or competitively oriented. Ss rated Cs on eight bipolar scales. The results confirmed the hypothesis.     

Model-based recursive partitioning of extended redundancy analysis with an application to nicotine dependence among US adults

Extended redundancy analysis (ERA) is used to reduce multiple sets of predictors to a smaller number of components and examine the effects of these components on a response variable. In various social and behavioural studies, auxiliary covariates (e.g., gender, ethnicity) can often lead to heterogeneous subgroups of observations, each of which involves distinctive relationships between predictor and response variables. ERA is currently unable to consider such covariate-dependent heterogeneity to examine whether the model parameters vary across subgroups differentiated by covariates. To address this issue, we combine ERA with model-based recursive partitioning in a single framework. This combined method, MOB-ERA, aims to partition observations into heterogeneous subgroups recursively based on a set of covariates while fitting a specified ERA model to data. Upon the completion of the partitioning procedure, one can easily examine the difference in the estimated ERA parameters across covariate-dependent subgroups. Moreover, it produces a tree diagram that aids in visualizing a hierarchy of partitioning covariates, as well as interpreting their interactions. In the analysis of public data concerning nicotine dependence among US adults, the method uncovered heterogeneous subgroups characterized by several sociodemographic covariates, each of which yielded different directional relationships between three predictor sets and nicotine dependence.     

Some properties of two measures of multivariate association

Two kinds of measures of multivariate association, based on Wilks' and the Bartlett-Nanda-Pillai trace criterionV, respectively, are compared in terms of properties of the univariateR ² which they generalize. A unified set of derivations of the properties is provided which are self-contained and not restricted to decompositions in canonical variates. One conclusion is that asymmetric index based on allows generalization of the multiplicative decomposition ofR ² in terms of squared partial correlations, but not the additive decomposition in terms of squared semipartial correlations, while the reverse is true for anasymmetric index based onV.We are indebted to Jos M. F. ten Berge for some fruitful discussions.     

Linear relations amongk sets of variables

A family of solutions for linear relations amongk sets of variables is proposed. It is shown how these solutions apply fork=2, and how they can be generalized from there tok3.The family of solutions depends on three independent choices: (i) to what extent a solution may be influenced by differences in variances of components within each set; (ii) to what extent the sets may be differentially weighted with respect to their contribution to the solution—including orthogonality constraints; (iii) whether or not individual sets of variables may be replaced by an orthogonal and unit normalized basis.Solutions are compared with respect to their optimality properties. For each solution the appropriate stationary equations are given. For one example it is shown how the determinantal equation of the stationary equations can be interpreted.     

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.     

Bayesian Extended Redundancy Analysis: A Bayesian Approach to Component-based Regression with Dimension Reduction

Abstract

Extended redundancy analysis (ERA) combines linear regression with dimension reduction to explore the directional relationships between multiple sets of predictors and outcome variables in a parsimonious manner. It aims to extract a component from each set of predictors in such a way that it accounts for the maximum variance of outcome variables. In this article, we extend ERA into the Bayesian framework, called Bayesian ERA (BERA). The advantages of BERA are threefold. First, BERA enables to make statistical inferences based on samples drawn from the joint posterior distribution of parameters obtained from a Markov chain Monte Carlo algorithm. As such, it does not necessitate any resampling method, which is on the other hand required for (frequentist’s) ordinary ERA to test the statistical significance of parameter estimates. Second, it formally incorporates relevant information obtained from previous research into analyses by specifying informative power prior distributions. Third, BERA handles missing data by implementing multiple imputation using a Markov Chain Monte Carlo algorithm, avoiding the potential bias of parameter estimates due to missing data. We assess the performance of BERA through simulation studies and apply BERA to real data regarding academic achievement.