When can information from ordinal scale variables be integrated?

Many theoretical constructs of interest to psychologists are multidimensional and derive from the integration of several input variables. We show that input variables that are measured on ordinal scales cannot be combined to produce a stable weakly ordered output variable that allows trading off the input variables. Instead a partial order is obtained in which the amount of ordering depends on the number and nature of the input variables and the relationship between them. However, if trade-offs are excluded, it is still possible to obtain a weak order using lexicographic ordering of the input variables. An implication is that psychological processes that integrate information from different input variables and that produce consistent output require that the input variables be measured on more than ordinal scales. A further implication is that the level of measurement of the input variables affects the kind of psychological model that can be applied to the process.     

The effect of affine transformation on redundancy analysis

Canonical redundancy analysis provides an estimate of the amount of shared variance between two sets of variables and provides an alternative to canonical correlation. The proof that the total redundancy is equal to the average squared multiple correlation coefficient obtained by regressing each variable in the criterion set on all variables in the predictor set is generalized to the case in which there are a larger number of criterion than predictor variables. It is then shown that the redundancy for the criterion set of variables is invariant under affine transformation of the predictor variables, but not invariant under transformation of the criterion variables.     

How to Generate Non-normal Data for Simulation of Structural Equation Models

A procedure for generating non-normal data for simulation of structural equation models is proposed. A simple transformation of univariate random variables is used for the generation of data on latent and error variables under some restrictions for the elements of the covariance matrices for these variables. Data on the observed variables is then computed from latent and error variables according to the model. It is shown that by controlling univariate skewness and kurtosis on pre-specified random latent and error variables, observed variables can be made to have a relatively wide range of univariate skewness and kurtosis characteristics according to the pre-specified model. Univariate distributions are used for the generation of data which enables a user to choose from a large number of different distributions. The use of the proposed procedure is illustrated for two different structural equation models and it is shown how PRELIS can be used to generate the data.     

A note on McDonald's generalization of principal components analysis

It is shown that McDonald's generalization of classical Principal Components Analysis to groups of variables maximally channels the total variance of the original variables through the groups of variables acting as groups. A useful equation is obtained for determining the vectors of correlations of theL2 components with the original variables. A calculation example is given.     

Observed and latent variables: Discussion of a basic LISREL paradigm

The aim of the present paper is to discuss a general paradigm of LISREL: that observed psychologocal variables are causally determined by non-observed, latent variables. An example from the LISREL manual is used to show that this basic assumption is not generally meaningful. The possibility of understanding the theoretical variables of LISREL in an instrumentalist way is also considered. But instrumentalism, it is concluded, does not provide a tenable way of defending the LISREL paradigm. For LISREL conceives latent variables as causes of observed variables, and this is not compatible with an instrumentalist understanding of theoretical terms.     

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.     

A MULTIDIMENSIONAL APPROACH FOR EVALUATING VARIABLES IN ORGANIZATIONAL RESEARCH AND PRACTICE

One of the most difficult tasks facing industrial-organizational psychologists is evaluating the importance of variables, especially new variables, to be included in the prediction of some outcome. When multiple regression is used, common practices suggest evaluating the usefulness of new variables by showing incremental validity beyond the set of existing variables. This approach assures that the new variables are not statistically redundant with this existing set, but this approach attributes any shared criterion-related validity to the existing set of variables and none to the new variables. More importantly, incremental validity alone fails to answer the question directly about the importance of variables included in a regression model—arguably the more important statistical concern for practitioners. To that end, the current article reviews 2 indices of relative importance, general dominance weights and relative weights, which may be used to complement incremental validity evidence and permit organizational decision makers to make more precise and informed decisions concerning the usefulness of predictor variables. We illustrate our approach by reanalyzing the correlation matrices from 2 published studies.     

Indeterminacy Problems in the Lisrel Model

The latent variables and errors of the Lisrel model are indeterminate even when the parameters of the model are perfectly identified. The reason for the indeterminacy is that the Lisrel model gives a solution in terms of estimation of latent variables by means of observed variables. The indeterminacy is relevant also in practice; the minimum correlation between equivalent latent variables, is often negative in empirical examples. The degree of indeterminacy of the latent variables depends on the data. The average minimum correlation is a linear combination of the eigenvalues of the correlation matrix of solutions and it is always included in weak bounds which depend on the same eigenvalues.     

Bayesian modeling of measurement error in predictor variables using item response theory

It is shown that measurement error in predictor variables can be modeled using item response theory (IRT). The predictor variables, that may be defined at any level of an hierarchical regression model, are treated as latent variables. The normal ogive model is used to describe the relation between the latent variables and dichotomous observed variables, which may be responses to tests or questionnaires. It will be shown that the multilevel model with measurement error in the observed predictor variables can be estimated in a Bayesian framework using Gibbs sampling. In this article, handling measurement error via the normal ogive model is compared with alternative approaches using the classical true score model. Examples using real data are given.This paper is part of the dissertation by Fox (2001) that won the 2002 Psychometric Society Dissertation Award.     

The temporal stability of some Rorschach features

One hundred nonpatient adults, screened for evidence of personality disorganization, were retested after a three-year interval to study the temporal consistency of the Rorschach. In general, the correlational analysis for 19 basic variables studied, and a directionality analysis for five ratios, illustrate a considerable sturdiness over time. It is postulated that these variables can be separated into situational related indices (state variables), and more stable scores indicative of durable response styles (trait variables), based on the assumption that variables with lower correlations would identify state variables, while the higher correlations would signify the trait features. Nine of the 19 variables yield retest correlations in excess of.     

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.     

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.     

Forgotten variables in memory theory and research

Abstract

Memory research, like other scientific research, disregards many variables in order to bring the full force of the scientific method to bear on clearly important variables. The reasons why memory research attends to certain variables and disregards others emanate largely from theoretical assumptions that distinguish memory systems from other psychological systems, and that distinguish variables intrinsic to memory from those extrinsic to memory. Nevertheless, a number of these ‘forgotten’ variables affect memory performance. Regardless of past practice, it is a mistake for memory research to continue to ignore relevant variables. Doing so introduces measurement error that contaminates memory performance measures, and classification error that precludes the discovery of legitimate memory variables. It is proposed here that if forgotten memory variables are controlled, manipulated, and measured more extensively, then future memory research will have greater power and memory theories will have greater validity.     

The wellness behavior interaction model

A cognitive-behavioral model for conceptualizing wellness is proposed consisting of situational/antecedent variables, person variables, and consequence variables which, taken individually or together, may serve to describe and explain wellness behavior. The model also is interactional in the sense that different categories of variables or factors within a category may impact on each other in a multidirectional fashion. A selective review of research relating to each variable is provided to demonstrate the need for such a multidimensional conceptual scheme for wellness.     

Note on the squared multiple correlation as a lower bound to communality

It is demonstrated that the squared multiple correlation of a variable with the remaining variables in a set of variables is a function of the communalities and the squared canonical correlations between the observed variables and common factors. This equation is shown to imply a strict inequality between the squared multiple correlation and communality.     

Using Principal Components as Auxiliary Variables in Missing Data Estimation

To deal with missing data that arise due to participant nonresponse or attrition, methodologists have recommended an “inclusive” strategy where a large set of auxiliary variables are used to inform the missing data process. In practice, the set of possible auxiliary variables is often too large. We propose using principal components analysis (PCA) to reduce the number of possible auxiliary variables to a manageable number. A series of Monte Carlo simulations compared the performance of the inclusive strategy with eight auxiliary variables (inclusive approach) to the PCA strategy using just one principal component derived from the eight original variables (PCA approach). We examined the influence of four independent variables: magnitude of correlations, rate of missing data, missing data mechanism, and sample size on parameter bias, root mean squared error, and confidence interval coverage. Results indicate that the PCA approach results in unbiased parameter estimates and potentially more accuracy than the inclusive approach. We conclude that using the PCA strategy to reduce the number of auxiliary variables is an effective and practical way to reap the benefits of the inclusive strategy in the presence of many possible auxiliary variables.     

Demonstration of properties of a suppressor variable

A general logical model of properties of suppressor variables is proposed. Consistent exploration of possible manifestations of suppressor variables within this theoretical framework accounts for extant classifications of suppressor variables into the classical, net, and cooperative categories and suggests existence of new subcategories, not detected previously. The discussed model leads to consistent identification and classification of suppressor variables and facilitates computer simulation.     

Learning to visually perceive the relative mass of colliding balls in globally and locally constrained task ecologies

Novice observers differ from each other in the kinematic variables they use for the perception of kinetic properties, but they converge on more useful variables after practice with feedback. The colliding-balls paradigm was used to investigate how the convergence depends on the relations between the candidate variables and the to-be-perceived property, relative mass. Experiment 1 showed that observers do not change in the variables they use if the variables with which they start allow accurate performance. Experiment 2 showed that, at least for some observers, convergence can be facilitated by reducing the correlations between commonly used nonspecifying variables and relative mass but not by keeping those variables constant. Experiments 3a and 3b further demonstrated that observers learn not to rely on a particular nonspecifying variable if the correlation between that variable and relative mass is reduced.     

Detection of one-, two-, and three-dimensional Markov constraints in visual displays II: Multistate displays.

The study of the depth of visual information processing is here extended to multistate displays. Two classes of variables are distinguished: display variables which may either be fixed at a single level or varied over its possible levels, e.g., a numeric character, its brightness, and its orientation; and spatial-temporal variables which assume all possible states within each display, e.g., the x- and y-coordinates of the display, and the time-coordinate, t, representing successive frames of the display. Information was encoded in terms of constraints upon combinations of variables. Excellent discrimination is achieved for detecting constraints among two, but not three, display variables; or for detecting constraints among one, but not two, display variables and up to three spatial-temporal variables. Comparisons are made with previous tests of the depth of visual information processing with binary-coded materials within the spatial-temporal microstructure of the display.     

Latent variables should remain as such: Evidence from a Monte Carlo study

Use of subject scores as manifest variables to assess the relationship between latent variables produces attenuated estimates. This has been demonstrated for raw scores from classical test theory (CTT) and factor scores derived from factor analysis. Conclusions on scores have not been sufficiently extended to item response theory (IRT) theta estimates, which are still recommended for estimation of relationships between latent variables. This is because IRT estimates appear to have preferable properties compared to CTT, while structural equation modeling (SEM) is often advised as an alternative to scores for estimation of the relationship between latent variables. The present research evaluates the consequences of using subject scores as manifest variables in regression models to test the relationship between latent variables. Raw scores and three methods for obtaining theta estimates were used and compared to latent variable SEM modeling. A Monte Carlo study was designed by manipulating sample size, number of items, type of test, and magnitude of the correlation between latent variables. Results show that, despite the advantage of IRT models in other areas, estimates of the relationship between latent variables are always more accurate when SEM models are used. Recommendations are offered for applied researchers.