Combinatorial data analysis: Association and partial association

A combinatorial data analysis strategy is reviewed that is designed to compare two arbitrary measures of proximity defined between the objects from some set. Based on a particular cross-product definition of correspondence between these two numerically specified notions of proximity (typically represented in the form of matrices), extensions are then pursued to indices of partial association that relate the observed pattern of correspondence between the first two proximity measures to a third. The attendant issues of index normalization and significance testing are discussed; the latter is approached through a simple randomization model implemented either through a Monte Carlo procedure or distributional approximations based on the first three moments. Applications of the original comparison strategy and its extensions to partial association may be developed for a variety of methodological and substantive tasks. Besides rank correlation, we emphasize the topics of spatial autocorrelation for one variable and spatial association between two and mention the connection to the usual randomization approach for one-way analysis-of-variance.     

Matrices with a given number of violations of Regular Minimality

A row (or column) of an n×n matrix complies with Regular Minimality (RM) if it has a unique minimum entry which is also a unique minimum entry in its column (respectively, row). The number of violations of RM in a matrix is defined as the number of rows (equivalently, columns) that do not comply with RM. We derive a formula for the proportion of n×n matrices with a given number of violations of RM among all n×n matrices with no tied entries. The proportion of matrices with no more than a given number of violations can be treated as the p-value of a permutation test whose null hypothesis states that all permutations of the entries of a matrix without ties are equiprobable, and the alternative hypothesis states that RM violations occur with lower probability than predicted by the null hypothesis. A matrix with ties is treated as being represented by all matrices without ties that have the same set of strict inequalities among their entries.     

Identifying a Reordering of Rows and Columns for Multiple Proximity Matrices Using Multiobjective Programming

This paper is concerned with a problem where K (n×n) proximity matrices are available for a set of n objects. The goal is to identify a single permutation of the n objects that provides an adequate structural fit, as measured by an appropriate index, for each of the K matrices. A multiobjective programming approach for this problem, which seeks to optimize a weighted function of the K indices, is proposed, and illustrative examples are provided using a set of proximity matrices from the psychological literature. These examples show that, by solving the multiobjective programming model under different weighting schemes, the quantitative analyst can uncover information about the relationships among the matrices and often identify one or more permutations that provide good to excellent index values for all matrices.     

The approximation of two-mode proximity matrices by sums of order-constrained matrices

A least-squares strategy is proposed for representing a two-mode proximity matrix as an approximate sum of a small number of matrices that satisfy certain simple order constraints on their entries. The primary class of constraints considered define Q-forms (or anti-Q-forms) for a two-mode matrix, where after suitable and separate row and column reorderings, the entries within each row and within each column are nondecreasing (or nonincreasing) to a maximum (or minimum) and thereafter nonincreasing (or nondecreasing). Several other types of order constraints are also mentioned to show how alternative structures can be considered using the same computational strategy.     

Graph-theoretic representations for proximity matrices through strongly-anti-Robinson or circular strongly-anti-Robinson matrices

There are various optimization strategies for approximating, through the minimization of a least-squares loss function, a given symmetric proximity matrix by a sum of matrices each subject to some collection of order constraints on its entries. We extend these approaches to include components in the approximating sum that satisfy what are called the strongly-anti-Robinson (SAR) or circular strongly-anti-Robinson (CSAR) restrictions. A matrix that is SAR or CSAR is representable by a particular graph-theoretic structure, where each matrix entry is reproducible from certain minimum path lengths in the graph. One published proximity matrix is used extensively to illustrate the types of approximation that ensue when the SAR or CSAR constraints are imposed.The authors are indebted to Boris Mirkin who first noted in a personal communication to one of us (LH, April 22, 1996) that the optimization method for fitting anti-Robinson matrices in Hubert and Arabie (1994) should be extendable to the fitting of strongly anti-Robinson matrices as well.     

Generalized concordance

Based on a simple nonparametric procedure for comparing two proximity matrices, a measure of concordance is introduced that is appropriate whenK independent proximity matrices are available. In addition to the development of a general concept of concordance and specific techniques for its evaluation within and between the subsets of a partition of theK matrices, several methods are also suggested for comparing and/or for fitting a particular structure to the given data. Finally, brief indications are provided as to how the well-known notion of concordance forK rank orders can be included within the more general framework.Partial support for this research was supplied by the National Science Foundation through SOC-77-28227.     

A relationship between harris factors and guttman's sixth lower bound to reliability

Using an approach nearly identical to one adopted by Guttman, it is established that within the framework of classical test theory the squared multiple correlation for predicting an element of a composite measure from then — 1 remaining elements is a lower-bound to the reliability of the element. The relationship existing between the reliabilities of the elements of a composite measure and their squared-multiple correlations with remaining elements is used to derive Guttman's sixth lower bound (λ ₆) to the reliability of a composite measure. It is shown that Harris factors of a correlation matrixR are associated with a set of (observable) uncorrelated latent variables having maximum coefficientsλ ₆.     

ConPar: a method for identifying groups of concordant subject proximity matrices for subsequent multidimensional scaling analyses

A common representation of data within the context of multidimensional scaling (MDS) is a collection of symmetric proximity (similarity or dissimilarity) matrices for each of M subjects. There are a number of possible alternatives for analyzing these data, which include: (a) conducting an MDS analysis on a single matrix obtained by pooling (averaging) the M subject matrices, (b) fitting a separate MDS structure for each of the M matrices, or (c) employing an individual differences MDS model. We discuss each of these approaches, and subsequently propose a straightforward new method (CONcordance PARtitioning—ConPar), which can be used to identify groups of individual-subject matrices with concordant proximity structures. This method collapses the three-way data into a subject×subject dissimilarity matrix, which is subsequently clustered using a branch-and-bound algorithm that minimizes partition diameter. Extensive Monte Carlo testing revealed that, when compared to K-means clustering of the proximity data, ConPar generally provided better recovery of the true subject cluster memberships. A demonstration using empirical three-way data is also provided to illustrate the efficacy of the proposed method.     

Nearest neighbor analysis of point processes: Simulations and evaluations

Tversky, Rinott, and Newman (Journal of Mathematical Psychology, 1983, 27, 000) examine the asymptotic behavior of a measure of the centrality of the nearest neighbor relation. The applicability of their conclusions when the number of dimensions (d) and the number of points (n) take on the small-to-moderate values commonly encountered in the analysis of proximity data is investigated. The results suggest that convergence is fast when n is large relative to d and slow when d is large relative to n.     

The dynamics of choice among multiple alternatives

We consider neurally based models for decision-making in the presence of noisy incoming data. The two-alternative forced-choice task has been extensively studied, and in that case it is known that mutually inhibited leaky integrators in which leakage and inhibition balance can closely approximate a drift-diffusion process that is the continuum limit of the optimal sequential probability ratio test (SPRT). Here we study the performance of neural integrators in n?2 alternative choice tasks and relate them to a multihypothesis sequential probability ratio test (MSPRT) that is asymptotically optimal in the limit of vanishing error rates. While a simple race model can implement this ‘max-vs-next’ MSPRT, it requires an additional computational layer, while absolute threshold crossing tests do not require such a layer. Race models with absolute thresholds perform relatively poorly, but we show that a balanced leaky accumulator model with an absolute crossing criterion can approximate a ‘max-vs-ave’ test that is intermediate in performance between the absolute and max-vs-next tests. We consider free and fixed time response protocols, and show that the resulting mean reaction times under the former and decision times for fixed accuracy under the latter obey versions of Hick's law in the low error rate range, and we interpret this in terms of information gained. Specifically, we derive relationships of the forms log(n-1), log(n), or log(n+1) depending on error rates, signal-to-noise ratio, and the test itself. We focus on linearized models, but also consider nonlinear effects of neural activities (firing rates) that are bounded below and show how they modify Hick's law.     

Expanding on cross-price elasticity: Understanding tobacco product demand and substitution from the cross-price purchase task

We examine whether cigarettes serve as substitutes for electronic nicotine delivery systems (ENDS) among ENDS users and demonstrate methodological extensions of data from a cross-price purchase task to inform policies and interventions. During a clinical laboratory study, n = 19 exclusive ENDS users and n = 17 dual cigarette/ENDS users completed a cross-price purchase task with cigarettes available at a fixed price while prices of own-brand ENDS increased. We estimated cross-price elasticity using linear models to examine substitutability. We defined five additional outcomes: nonzero cross-price intensity (purchasing cigarettes if ENDS were free), constant null demand (not purchasing cigarettes at any ENDS price), cross-product crossover point (first price where participants purchased more cigarettes than ENDS), dual-demand score (percentage of prices where both products were purchased), and dual-use break point (minimum relative price to force complete substitution). The cross-price elasticity results indicated that cigarettes could serve as substitutes for ENDS among ENDS users on average, but this average effect masked substantial heterogeneity in profiles of demand (here, a measure of the drug's reinforcement potential). Policies and regulations that increase ENDS prices appear unlikely to steer most exclusive ENDS users toward cigarette use, as most would not purchase cigarettes at any ENDS price, but they could prompt some dual users to substitute cigarettes completely while others remain dual users. This heterogeneity in consumer responses suggests additional indices of cross-product demand are useful to characterize the anticipated and unanticipated effects of tobacco price policies more fully.     

Two-generator semigroups of binary relations

We study upper bounds on the size of the semigroups generated two randomly chosen n × n Boolean matrices having exactly N one entries. In general, as $\text{N}\text{n}$ increases the semigroups tend to be smaller. If N is unrestricted, or is a function which is at least [$\text{((2 + ?)n}{}^3\text{log}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$] the probability tends to one that the size of the semigroup is exactly 3. If N is a function which is at least [(r + 1 + ?)n log n] the probability tends to one that the size of the semigroup will be no more than $\text{2}{}^{\mathrm{<mtext>(</mtext><mtext>(n?1)</mtext><mtext>r</mtext><mtext>)+1</mtext>}}\text{? 1}$. However, if N is any function w(n) such that $\text{w(n)}\text{n}{}^2\text{→ 0}$ and w(n) > n, the average size of the resulting semigroups will be at least . This phenomenon is caused by some semigroups of extremely large size. It is thought that w(n) > n and $\text{w(n)}\text{n}{}^2\text{→ 0}$ hold for those matrices usually encountered in sociology and psychology.     

Fitting the off-diagonal dedicom model in the least-squares sense by a generalization of the harman and jones minres procedure of factor analysis

Harshman's DEDICOM model providesa framework for analyzing square but asymmetric materices of directional relationships amongn objects or persons in terms of a small number of components. One version of DEDICOM ignores the diagonal entries of the matrices. A straight-forward computational solution for this model is offered in the present paper. The solution can be interpreted as a generalized Minres procedure suitable for handing asymmetric matrices.     

Complex span and n-back measures of working memory: A meta-analysis

Working memory is a construct of primary relevance to many areas of psychology. Two types of tasks have been used to measure working memory, primarily in different research areas: Complex span tasks are commonly used in behavioral studies in the cognitive and individual-differences literature, whereas n-back tasks have been used more frequently in cognitive neuroscience studies investigating the neural underpinnings of working memory. Despite both categories of tasks being labeled as “working memory” measures, previous empirical studies have provided mixed evidence regarding the shared amount of overlapping processes between complex span and n-back tasks. The present meta-analysis showed that the complex span and n-back tasks are weakly correlated, although significant heterogeneity across studies was observed. A follow-up analysis of unpublished data indicated that the sample composition affects the relationship between the complex span and n-back tasks, following the law of diminishing returns. Finally, a separate meta-analysis indicated that the simple span and n-back tasks are correlated to the same extent as are the complex span and n-back tasks. The present findings indicate that the complex span and n-back tasks cannot be used interchangeably as working memory measures in research applications.     

A new method for analyzing aesthetic preferences: Some theoretical considerations

The aesthetic preferences of a group of persons are obtained from their orders of sets of pictures and patterns according to “liking.” The same pictures are ordered independently by a team of experts, according to certain artistic criteria such as naturalism, composition, color, rhythm, etc. The orders of preference and orders according to the criteria are compared by correlation and matrices of correlation formed from (1) correlations between the persons' orders of preference; (2) correlations between the orders of preference and orders according to artistic criteria; and (3) correlations between the criterion orders. These matrices are symbolised byR _p ,R ₀, andR _c , respectively, and combined to form a single matrix

$$\left[ \begin{gathered} R_p R_o \hfill \\ R'_o R_c \hfill \\ \end{gathered} \right]$$     

The role of implicit and explicit components of exerciser self-schema in the prediction of exercise behaviour

ObjectivesThis study investigated implicit and explicit components of exerciser self-schema and their role in the prediction of exercise behaviour. In addition, the effect of implicit and explicit exerciser self-schema on intention to exercise was examined. Moderation and mediation effects involving exerciser self-schema, in both its implicit and explicit forms, were investigated.MethodMembers of fitness centres and other regular exercisers (N = 98, n_males = 37, n_females = 65) completed a written questionnaire to measure explicit exerciser self-schema and exercise intentions. An exercise Implicit Association Test (IAT) was also administered to measure implicit exerciser self-schema. Actual exercise occurrence was measured one week and two weeks following the initial testing phase.ResultsCorrelations indicated that implicit and explicit components of exerciser self-schema were related but distinct constructs. Hierarchical regression analysis revealed that high levels of both implicit and explicit exerciser self-schema improved absolute levels of exercise behaviour. Regression analyses examining the possibility that self-schema moderated the intention–behaviour relationship were not significant for implicit or explicit forms of self-schema. Forced entry regression analyses provided evidence for explicit exerciser self-schema being mediated by intention in its relationship with behaviour.ConclusionsIt is beneficial to consider both implicit and explicit exerciser self-schema when attempting to understand and predict exercise behaviour. Both implicit and explicit exercise self-schema have a direct effect on behaviour, although explicit self-schema also indirectly influences behaviour via intention to exercise.     

Complexity Results for Modal Dependence Logic

Modal dependence logic was introduced recently by Väänänen. It enhances the basic modal language by an operator = (). For propositional variables p ₁, . . . , p _n , = (p ₁, . . . , p _n-1, p _n ) intuitively states that the value of p _n is determined by those of p ₁, . . . , p _n-1. Sevenster (J. Logic and Computation, 2009) showed that satisfiability for modal dependence logic is complete for nondeterministic exponential time. In this paper we consider fragments of modal dependence logic obtained by restricting the set of allowed propositional connectives. We show that satisfiability for poor man’s dependence logic, the language consisting of formulas built from literals and dependence atoms using ${\wedge, \square, \lozenge}$ (i. e., disallowing disjunction), remains NEXPTIME-complete. If we only allow monotone formulas (without negation, but with disjunction), the complexity drops to PSPACE-completeness. We also extend Väänänen’s language by allowing classical disjunction besides dependence disjunction and show that the satisfiability problem remains NEXPTIME-complete. If we then disallow both negation and dependence disjunction, satisfiability is complete for the second level of the polynomial hierarchy. Additionally we consider the restriction of modal dependence logic where the length of each single dependence atom is bounded by a number that is fixed for the whole logic. We show that the satisfiability problem for this bounded arity dependence logic is PSPACE-complete and that the complexity drops to the third level of the polynomial hierarchy if we then disallow disjunction. In this way we completely classify the computational complexity of the satisfiability problem for all restrictions of propositional and dependence operators considered by Väänänen and Sevenster.     

Measuring and testing the agreement of matrices

The problem of comparing the agreement of two n × n matrices has a variety of applications in experimental psychology. A well-known index of agreement is based on the sum of the element-wise products of the matrices. Although less familiar to many researchers, measures of agreement based on within-row and/or within-column gradients can also be useful. We provide a suite of MATLAB programs for computing agreement indices and performing matrix permutation tests of those indices. Programs for computing exact p-values are available for small matrices, whereas resampling programs for approximate p-values are provided for larger matrices.