Picture of all Solutions of Successive 2-Block Maxbet Problems

The Maxbet method is a generalized principal components analysis of a data set, where the group structure of the variables is taken into account. Similarly, 3-block[12,13] partial Maxdiff method is a generalization of covariance analysis, where only the covariances between blocks (1, 2) and (1, 3) are taken into account. The aim of this paper is to give the global maximum for the 2-block Maxbet and 3-block[12,13] partial Maxdiff problems by picking the best solution from the complete solution set for the multivariate eigenvalue problem involved. To do this, we generalize the characteristic polynomial of a matrix to a system of two characteristic polynomials, and provide the complete solution set of the latter via Sylvester resultants. Examples are provided.     

Orthogonal rotations to maximal agreement for two or more matrices of different column orders

Methods for orthogonal Procrustes rotation and orthogonal rotation to a maximal sum of inner products are examined for the case when the matrices involved have different numbers of columns. An inner product solution offered by Cliff is generalized to the case of more than two matrices. A nonrandom start for a Procrustes solution suggested by Green and Gower is shown to give better results than a random start. The Green-Gower Procrustes solution (with nonrandom start) is generalized to the case of more than two matrices. Simulation studies indicate that both the generalized inner product solution and the generalized Procrustes solution tend to attain their global optima within acceptable computation times. A simple procedure is offered for approximating simple structure for the rotated matrices without affecting either the Procrustes or the inner product criterion.The authors are obliged to Charles Lewis for helpful comments on a previous draft of this paper and to Frank Brokken for preparing a computer program that was used in this study.     

Generalized approaches to the maxbet problem and the maxdiff problem,with applications to canonical correlations

Van de Geer has reviewed various criteria for transforming two or more matrices to maximal agreement, subject to orthogonality constraints. The criteria have applications in the context of matching factor or configuration matrices and in the context of canonical correlation analysis for two or more matrices. The present paper summarizes and gives a unified treatment of fully general computational solutions for two of these criteria, Maxbet and Maxdiff. These solutions will be shown to encompass various well-known methods as special cases. It will be argued that the Maxdiff solution should be preferred to the Maxbet solution whenever the two criteria coincide. Horst's Maxcor method will be shown to lack the property of monotone convergence. Finally, simultaneous and successive versions of the Maxbet and Maxdiff solutions will be treated as special cases of a fully flexible approach where the columns of the rotation matrices are obtained in successive blocks.The author is obliged to Henk Kiers for computational assistance and helpful comments.     

Bootstrapping the estimated latent distribution of the two‐parameter latent trait model

This paper focuses on the two‐parameter latent trait model for binary data. Although the prior distribution of the latent variable is usually assumed to be a standard normal distribution, that prior distribution can be estimated from the data as a discrete distribution using a combination of EM algorithms and other optimization methods. We assess with what precision we can estimate the prior from the data, using simulations and bootstrapping. A novel calibration method is given to check that near optimality is achieved for the bootstrap estimates. We find that there is sufficient information on the prior distribution to be informative, and that the bootstrap method is reliable. We illustrate the bootstrap method for two sets of real data.     

Evidence for an outflow theory of skill.

We argue that central monitoring of efferent signals (CME) is a necessary and perhaps sufficient condition for the most accurate reproduction of a linear movement. In a movement duplication task where the subject sets or is given a criterion movement which he then attempts to reproduce, we argue that voluntary criterion and reproduction movements may be based upon some ‘pre-set’ instruction whereas passive movement duplication must be based upon proprioceptive information. We show that subjects can perform a visual signal detection task during the voluntary criterion movement which does not affect accuracy of reproduction. The same task carried out during the passive criterion movement does affect reproduction. These findings suggest that pick-up of proprioceptive information is not involved in voluntary movement.     

Optimality And Concurrent Variable-interval Variable-ratio Schedules

Despite claims to the contrary, all leading theories about operant choice may be seen as models of optimality. Although melioration is often contrasted with global maximization, both make the same core assumptions as other versions of optimality theory, including momentary maximizing, hill climbing, and the various versions of optimal foraging theory. The present experiment aimed to test melioration against more global optimality and to apply the visit-by-visit analysis suggested by foraging theory. Rats were exposed to concurrent schedules in which one alternative was always variable-ratio 10 and the other alternative was a variable-interval schedule. Although choice relations varied from rat to rat, the overall results roughly confirmed the matching law, a result often taken to support melioration. Pooling the data across sessions and across rats, however, resulted in no increment in unsystematic variance, lending support to the contention by Ziriax and Silberberg (1984) that the choice relation is partly constrained. When the data were analyzed at the level of visits, the results either disconfirmed predictions of melioration or showed regularities about which melioration is silent. Instead, performance tended toward a rough optimization, in which responding favored the variable ratio, but with relatively brief visits to the variable interval. There were no asymmetries in travel or variability that would indicate that different processes were involved in generating visits at the two different schedules. The findings point toward a more global optimality model than melioration and demonstrate the value of per-visit analysis in the study of concurrent performances.     

Functional Multiple-Set Canonical Correlation Analysis

We propose functional multiple-set canonical correlation analysis for exploring associations among multiple sets of functions. The proposed method includes functional canonical correlation analysis as a special case when only two sets of functions are considered. As in classical multiple-set canonical correlation analysis, computationally, the method solves a matrix eigen-analysis problem through the adoption of a basis expansion approach to approximating data and weight functions. We apply the proposed method to functional magnetic resonance imaging (fMRI) data to identify networks of neural activity that are commonly activated across subjects while carrying out a working memory task.     

Convergence in mean square of factor predictors

Sufficient conditions for mean square convergence of factor predictors in common factor analysis are given by Guttman, by Williams, and by Schneeweiss and Mathes. These conditions do not hold for confirmatory factor analysis or when an error variance equals zero (Heywood cases). Two sufficient conditions are given for the three basic factor predictors and a predictor from rotated principal components analysis to converge to the factors of the model for confirmatory factor analysis, including Heywood cases. For certain model specifications the conditions are necessary. The conditions are sufficient for the existence of a unique true factor. A geometric interpretation is given for factor indeterminacy and mean square convergence of best linear factor prediction.     

Sufficient conditions for the uniqueness of parameter estimates from binary-response data

The procedure of fitting parameterized models to experimental data is that of extremalizing a statistically meaningful scalar-valued vector function. The existence of multiple local extrema can greatly complicate the search for the global solution. Sufficient conditions for uniqueness of the parameter estimates are usually determined from the convexity of the criterion surface: the convexity properties are determined by the statistical criterion, the structure of the model, the underlying distribution, and the observations (data). In this paper, we seek the combinations of criteria, models, and distributions which yield sufficient conditions for unique parameter estimates regardless of the observed binary-response data values.Under mild sufficient conditions usually satisfied in practice, the Maximum Likelihood, Minimum Chi Square, and Minimum Transform Chi Square criteria are convex functions when the parameters appear linearly. These results are applied to equalvariance models of signal detection/recognition, sequential response, and additive learning models with implications on the experimental design. Unequal-variance models and models of discrete-sensory processing (rectilinear ROC curves) lead to nonconvex criteria for some observations (saddlepoints are demonstrated). Although convexity cannot be assured for these cases, the results suggest an efficient search procedure in a lower dimensional subspace to find global extrema. The extension of these results to more than two response levels is discussed.     

The Problem of Necessary and Sufficient Conditions and Conceptual Analysis

In this article the standard philosophical method involving intuition‐driven conceptual analysis is challenged in a new way. This orthodox approach to philosophy takes analysanda to be the specifications of the content of concepts in the form of sets of necessary and sufficient conditions. Here it is argued that there is no adequate account of what necessary and sufficient conditions are. So, the targets of applications of the standard philosophical method so understood are not sufficiently well understood for this method to be dependable.     

A general solution of the weighted orthonormal procrustes problem

A general solution for weighted orthonormal Procrustes problem is offered in terms of the least squares criterion. For the two-demensional case. this solution always gives the global minimum; for the general case, an algorithm is proposed that must converge, although not necessarily to the global minimum. In general, the algorithm yields a solution for the problem of how to fit one matrix to another under the condition that the dimensions of the latter matrix first are allowed to be transformed orthonormally and then weighted differentially, which is the task encountered in fitting analogues of the IDIOSCAL and INDSCAL models to a set of configurations.The authors are grateful to the Editor and the anonymous reviewers for their helpful comments on an earlier draft of this paper.     

Semi-automated Rasch analysis using in-plus-out-of-questionnaire log likelihood

Rasch analysis is a popular statistical tool for developing and validating instruments that aim to measure human performance, attitudes and perceptions. Despite the availability of various software packages, constructing a good instrument based on Rasch analysis is still considered to be a complex, labour-intensive task, requiring human expertise and rather subjective judgements along the way. In this paper we propose a semi-automated method for Rasch analysis based on first principles that reduces the need for human input. To this end, we introduce a novel criterion, called in-plus-out-of-questionnaire log likelihood (IPOQ-LL). On artificial data sets, we confirm that optimization of IPOQ-LL leads to the desired behaviour in the case of multi-dimensional and inhomogeneous surveys. On three publicly available real-world data sets, our method leads to instruments that are, for all practical purposes, indistinguishable from those obtained by Rasch analysis experts through a manual procedure.     

Orthogonal procrustes rotation for two or more matrices

Necessary and sufficient conditions for rotating matrices to maximal agreement in the least-squares sense are discussed. A theorem by Fischer and Roppert, which solves the case of two matrices, is given a more straightforward proof. A sufficient condition for a best least-squares fit for more than two matrices is formulated and shown to be not necessary. In addition, necessary conditions suggested by Kristof and Wingersky are shown to be not sufficient. A rotation procedure that is an alternative to the one by Kristof and Wingersky is presented. Upper bounds are derived for determining the extent to which the procedure falls short of attaining the best least-squares fit. The problem of scaling matrices to maximal agreement is discussed. Modifications of Gower's method of generalized Procrustes analysis are suggested.     

Detecting latent taxa: Monte Carlo comparison of taxometric, mixture model, and clustering procedures

A Monte Carlo evaluation of four procedures for detecting taxonicity was conducted using artificial data sets that were either taxonic or nontaxonic. The data sets were analyzed using two of Meehl's taxometric procedures, MAXCOV and MAMBAC, Ward's method for cluster analysis in concert with the cubic clustering criterion and a latent variable mixture modeling technique. Performance of the taxometric procedures and latent variable mixture modeling were clearly superior to that of cluster analysis in detecting taxonicity. Applied researchers are urged to select from the better procedures and to perform consistency tests.     

A closer look at the probabilities of the notorious three prisoners.

The "problem of three prisoners", a counterintuitive teaser, is analyzed. It is representative of a class of probability puzzles where the correct solution depends on explication of underlying assumptions. Spontaneous beliefs concerning the problem and intuitive heuristics are reviewed. The psychological background of these beliefs is explored. Several attempts to find a simple criterion to predict whether and how the probability of the target event will change as a result of obtaining evidence are examined. However, despite the psychological appeal of these attempts, none proves to be valid in general. A necessary and sufficient condition for change in the probability of the target event, following observation of new data, is proposed. That criterion is an extension of the likelihood-ratio principle (which holds in the case of only two complementary alternatives) to any number of alternatives. Some didactic implications concerning the significance of the chance set-up and reliance on analogies are discussed.     

Global perceptual processing in problem solving: The case of the traveling salesperson

The traveling salesperson problem (TSP) consists of finding the shortest tour around a set of locations and is an important task in computer science and operations research. In four experiments, the relationship between processes implicated in the recognition of good figures and the identification of TSP solutions was investigated. In Experiment 1, a linear relationship was found between participants’ judgments of good figure and the optimality of solutions to TSPs. In Experiment 2, identification performance was shown to be a function of solution optimality and problem orientation. Experiment 3 replicated these findings with a forced-pace method, suggesting that global processing, rather than a local processing strategy involving point-by-point analysis of TSP solutions, is the primary process involved in the derivation of best figures for the presented TSPs. In Experiment 4, the role of global precedence was confirmed using a priming method, in which it was found that short (100 msec) primes facilitated solution identification, relative to no prime or longer primes. Effects of problem type were found in all the experiments, suggesting that local features of some problems may disrupt global processing. The results are discussed in terms of Sanocki’s (1993) global-to-local contingency model. We argue that global perceptual processing may contribute more generally to problem solving and that human performance can complement computational TSP methods.     

Global perceptual processing in problem solving: the case of the traveling salesperson.

The traveling salesperson problem (TSP) consists of finding the shortest tour around a set of locations and is an important task in computer science and operations research. In four experiments, the relationship between processes implicated in the recognition of good figures and the identification of TSP solutions was investigated. In Experiment 1, a linear relationship was found between participants' judgments of good figure and the optimality of solutions to TSPs. In Experiment 2, identification performance was shown to be a function of solution optimality and problem orientation. Experiment 3 replicated these findings with a forced-pace method, suggesting that global processing, rather than a local processing strategy involving point-by-point analysis of TSP solutions, is the primary process involved in the derivation of best figures for the presented TSPs. In Experiment 4, the role of global precedence was confirmed using a priming method, in which it was found that short (100 msec) primes facilitated solution identification, relative to no prime or longer primes. Effects of problem type were found in all the experiments, suggesting that local features of some problems may disrupt global processing. The results are discussed in terms of Sanocki's (1993) global-to-local contingency model. We argue that global perceptual processing may contribute more generally to problem solving and that human performance can complement computational TSP methods.     

Riemannian Newton and trust-region algorithms for analytic rotation in exploratory factor analysis

In exploratory factor analysis, latent factors and factor loadings are seldom interpretable until analytic rotation is performed. Typically, the rotation problem is solved by numerically searching for an element in the manifold of orthogonal or oblique rotation matrices such that the rotated factor loadings minimize a pre-specified complexity function. The widely used gradient projection (GP) algorithm, although simple to program and able to deal with both orthogonal and oblique rotation, is found to suffer from slow convergence when the number of manifest variables and/or the number of latent factors is large. The present work examines the effectiveness of two Riemannian second-order algorithms, which respectively generalize the well-established truncated Newton and trust-region strategies for unconstrained optimization in Euclidean spaces, in solving the rotation problem. When approaching a local minimum, the second-order algorithms usually converge superlinearly or even quadratically, better than first-order algorithms that only converge linearly. It is further observed in Monte Carlo studies that, compared to the GP algorithm, the Riemannian truncated Newton and trust-region algorithms require not only much fewer iterations but also much less processing time to meet the same convergence criterion, especially in the case of oblique rotation.     

Multidimensional rotation and scaling of configurations to optimal agreement

An integrated method for rotating and rescaling a set of configurations to optimal agreement in subspaces of varying dimensionalities is developed. The approach relates existing orthogonal rotation techniques as special cases within a general framework based on a partition of variation which provides convenient measures of agreement. In addition to the well-known Procrustes and inner product optimality criteria, a criterion which maximizes the consensus among subspaces of the configurations is suggested. Since agreement of subspaces of the configurations can be examined and compared, rotation and rescaling is extended from a data transformation technique to an analytical method.     

Specious causal attributions in the social sciences: the reformulated stepping-stone theory of heroin use as exemplar

The claims based on causal models employing either statistical or experimental controls are examined and found to be excessive when applied to social or behavioral science data. An exemplary case, in which strong causal claims are made on the basis of a weak version of the regularity model of cause, is critiqued. O'Donnell and Clayton claim that in order to establish that marijuana use is a cause of heroin use (their "reformulated stepping-stone" hypothesis), it is necessary and sufficient to demonstrate that marijuana use precedes heroin use and that the statistically significant association between the two does not vanish when the effects of other variables deemed to be prior to both of them are removed. I argue that O'Donnell and Clayton's version of the regularity model is not sufficient to establish cause and that the planning of social interventions both presumes and requires a generative rather than a regularity causal model. Causal modeling using statistical controls is of value when it compels the investigator to make explicit and to justify a causal explanation but not when it is offered as a substitute for a generative analysis of causal connection.