The Gaussian Graphical Model in Cross-Sectional and Time-Series Data

We discuss the Gaussian graphical model (GGM; an undirected network of partial correlation coefficients) and detail its utility as an exploratory data analysis tool. The GGM shows which variables predict one-another, allows for sparse modeling of covariance structures, and may highlight potential causal relationships between observed variables. We describe the utility in three kinds of psychological data sets: data sets in which consecutive cases are assumed independent (e.g., cross-sectional data), temporally ordered data sets (e.g., n = 1 time series), and a mixture of the 2 (e.g., n > 1 time series). In time-series analysis, the GGM can be used to model the residual structure of a vector-autoregression analysis (VAR), also termed graphical VAR. Two network models can then be obtained: a temporal network and a contemporaneous network. When analyzing data from multiple subjects, a GGM can also be formed on the covariance structure of stationary means—the between-subjects network. We discuss the interpretation of these models and propose estimation methods to obtain these networks, which we implement in the R packages graphicalVAR and mlVAR. The methods are showcased in two empirical examples, and simulation studies on these methods are included in the supplementary materials.     

Toward a unified theory of decision criterion learning in perceptual categorization

Optimal decision criterion placement maximizes expected reward and requires sensitivity to the category base rates (prior probabilities) and payoffs (costs and benefits of incorrect and correct responding). When base rates are unequal, human decision criterion is nearly optimal, but when payoffs are unequal, suboptimal decision criterion placement is observed, even when the optimal decision criterion is identical in both cases. A series of studies are reviewed that examine the generality of this finding, and a unified theory of decision criterion learning is described (Maddox & Dodd, 2001). The theory assumes that two critical mechanisms operate in decision criterion learning. One mechanism involves competition between reward and accuracy maximization: The observer attempts to maximize reward, as instructed, but also places some importance on accuracy maximization. The second mechanism involves a flat-maxima hypothesis that assumes that the observer's estimate of the reward-maximizing decision criterion is determined from the steepness of the objective reward function that relates expected reward to decision criterion placement. Experiments used to develop and test the theory require each observer to complete a large number of trials and to participate in all conditions of the experiment. This provides maximal control over the reinforcement history of the observer and allows a focus on individual behavioral profiles. The theory is applied to decision criterion learning problems that examine category discriminability, payoff matrix multiplication and addition effects, the optimal classifier's independence assumption, and different types of trial-by-trial feedback. In every case the theory provides a good account of the data, and, most important, provides useful insights into the psychological processes involved in decision criterion learning.     

Power and sample size calculations for multivariate linear models with random explanatory variables

This article considers the problem of power and sample size calculations for normal outcomes within the framework of multivariate linear models. The emphasis is placed on the practical situation that not only the values of response variables for each subject are just available after the observations are made, but also the levels of explanatory variables cannot be predetermined before data collection. Using analytic justification, it is shown that the proposed methods extend the existing approaches to accommodate the extra variability and arbitrary configurations of the explanatory variables. The major modification involves the noncentrality parameters associated with the F approximations to the transformations of Wilks likelihood ratio, Pillai trace and Hotelling-Lawley trace statistics. A treatment of multivariate analysis of covariance models is employed to demonstrate the distinct features of the proposed extension. Monte Carlo simulation studies are conducted to assess the accuracy using a child’s intellectual development model. The results update and expand upon current work in the literature.The author wishes to thank the associate editor and the referees for comments which improve the paper considerably. This research was partially supported by a grant from the Natural Science Council of Taiwan.     

Global optimality of the successive Maxbet algorithm

The Maxbet method is an alternative to the method of generalized canonical correlation analysis and of Procrustes analysis. Contrary to these methods, it does not maximize the inner products (covariances) between linear composites, but also takes their sums of squares (variances) into account. It is well-known that the Maxbet algorithm, which has been proven to converge monotonically, may converge to local maxima. The present paper discusses an eigenvalue criterion which is sufficient, but not necessary for global optimality. However, in two special cases, the eigenvalue criterion is shown to be necessary and sufficient for global optimality. The first case is when there are only two data sets involved; the second case is when the inner products between all variables involved are positive, regardless of the number of data sets.The authors are obliged to Henk Kiers for critical comments on a previous draft.     

Factor analysis and Psychometrika: Major developments

Papers on factor analysis appearing inPsychometrika reflect the initial efforts of the Thurstonians to reformulate psychology as a quantitative science. The Thurstonians' emphasis on the development of factor analysis as an exploratory methodology was not new with them but was taken from British statisticians and psychologists who preceded them, whose literature the Thurstonians otherwise tended to ignore. The Thurstonians' rejection of general factors and focus on rotation to simple structure reflected an attempt to avoid statistical artifact and to identify factors with psychological substance. Much of the literature on factor analysis inPsychometrika concerned solving technical problems in the exploratory factor analysis method. Factor analysis took a major shift in direction in the 1970's with the development of confirmatory methodologies, many of which now receive greater attention than the method of exploratory factor analysis, most of the problems of which are now resolved.     

A note on Roy's largest root

The name Roy's largest root and similar names are used in practice to label two different but functionally related statistics—one proportional to anF, and the other, a squared canonical correlation. This note presents the logic that leads to the two formulations, states which statistic some popular statistical packages use, and shows the possible source of this inconsistency in the original work of Roy (1953) and Heck (1960).     

The mixed model for multivariate repeated measures: validity conditions and an approximate test

Repeated measures on multivariate responses can be analyzed according to either of two models: a doubly multivariate model (DMM) or a multivariate mixed model (MMM). This paper reviews both models and gives three new results concerning the MMM. The first result is, primarily, of theoretical interest; the second and third have implications for practice. First, it is shown that, given multivariate normality, a condition called multivariate sphericity of the covariance matrix is both necessary and sufficient for the validity of the MMM analysis. To test for departure from multivariate sphericity, the likelihood ratio test can be employed. The second result is an approximation to the null distribution of the likelihood ratio test statistic, useful for moderate sample sizes. Third, for situations satisfying multivariate normality, but not multivariate sphericity, a multivariate correction factor is derived. The correction factor generalizes Box's and can be used to construct an adjusted MMM test.I am grateful to an anonymous referee for carefully attending to the mathematical details of this paper.     

Stability of the internal structure of the millon clinical multiaxial inventory

The Millon Clinical Multiaxial Inventory (MCMI) has become increasingly popular in clinical use. Along with this, there has been more interest in the internal structure of the 20 scales of the original 175-item MCMI-I. The literature reports some agreement on four components, although both three- and five-component solutions have been reported. The degree of similarity of these components across populations remains arguable, as none of the previous studies have used quantitative measures of component similarity. The present study reports on two new samples of psychiatric patients, one of 82 cases from a general hospital and the other of 145 inpatients from a psychiatric hospital. It also reanalyzes the data from nine samples from the literature, using Tucker's coefficient of congruence and ten Berge's analysis of principal component weights (PCW). The congruence analyses showed good agreement of the first three components across samples and notably lower agreement for the fourth. The PCW analyses showed two major types of structure matrices. In the first, there was a large and dominant first component, with three smaller ones. In the second, the variance was distributed more evenly across the four components. The results are discussed in terms of the overlapping scales of the MCMI-I.This work was supported by funds from the Departments of Psychiatry and Psychology, The University of Western Ontario.     

A general model for preferential and triadic choice in terms of centralF distribution functions

A model for preferential and triadic choice is derived in terms of weighted sums of centralF distribution functions. This model is a probabilistic generalization of Coombs' (1964) unfolding model and special cases, such as the model of Zinnes and Griggs (1974), can be derived easily from it. This new form extends previous work by Mullen and Ennis (1991) and provides more insight into the same problem that they discussed.     

A distance-based variety of nonlinear multivariate data analysis, including weights for objects and variables

In the distance approach to nonlinear multivariate data analysis the focus is on the optimal representation of the relationships between the objects in the analysis. In this paper two methods are presented for including weights in distance-based nonlinear multivariate data analysis. In the first method, weights are assigned to the objects while the second method is concerned with differential weighting of groups of variables. When each analysis variable defines a group the latter method becomes a variable weighting method. For objects the weights are assumed to be given; for groups of variables they may be given, or estimated. These weighting schemes can also be combined and have several important applications. For example, they make it possible to perform efficient analyses of large data sets, to use the distance-based variety of nonlinear multivariate data analysis as an addition to loglinear analysis of multiway contingency tables, and to do stability studies of the solutions by applying the bootstrap on the objects or the variables in the analysis. These and other applications are discussed, and an efficient algorithm is proposed to minimize the corresponding loss function.This study is funded by The Netherlands Organization for Scientific Research (NWO) by grant nr. 030-56403 for the PIONEER project Subject Oriented Multivariate Analysis to the third author.