Robert Cudeck

Alternative Ways of Assessing Model Fit

This article is concerned with measures of fit of a model. Two types of error involved in fitting a model are considered. The first is error of approximation which involves the fit of the model, with optimally chosen but unknown parameter values, to the population covariance matrix. The second is overall error which involves the fit of the model, with parameter values estimated from the sample, to the population covariance matrix. Measures of the two types of error are proposed and point and interval estimates of the measures are suggested. These measures take the number of parameters in the model into account in order to avoid penalizing parsimonious models. Practical difficulties associated with the usual tests of exact fit or a model are discussed and a test of “close fit” of a model is suggested.

Single Sample Cross-Validation Indices for Covariance Structures.

This article considers single sample approximations for the cross-validation coefficient in the analysis of covariance structures. An adjustment for predictive validity which may be employed in conjunction with any correctly specified discrepancy function is suggested. In the case of maximum likelihood estimation under normality assumptions the coefficient obtained is a simple linear function of the Akaike Information Criterion. Results of a random sampling experiment are reported.

Cross-Validation of Covariance Structures.

This paper examines methods for comparing the suitability of alternative models for covariance matrices. A cross-validation procedure is suggested and its properties are examined. To motivate the discussion, a series of examples is presented using longitudinal data.

Analysis of correlation matrices using covariance structure models.

It is often assumed that covariance structure models can be arbitrarily applied to sample correlation matrices as readily as to sample covariance matrices. Although this is true in many cases and leads to an analysis that is mostly correct, it is not permissible for all structures. This article reviews three interrelated problems associated with the analysis of structural models using a matrix of sample correlations. Depending upon the model, applying a covariance structure to a matrix of correlations may (a) modify the model being studied, (b) produce incorrect values of the omnibus test statistic, or (c) yield incorrect standard errors. An important class of models are those that are scale invariant (Browne, 1982), for then Errors a and b cannot occur when a correlation matrix is analyzed. A number of examples based on restricted factor analysis are presented to illustrate the concepts described in the article.

Personality and behavioral vulnerabilities associated with risk status for eating disorders in adolescent girls.

This article presents first-year cross-sectional findings from a study of the development of eating disorders. Adolescent female (N = 937) 7th through 10th graders completed measures that included information on personality, self-concept, eating patterns, and attitudes. A risk status score was calculated on the basis of comprehensive information regarding DSM-III-R eating disorders criteria and other weight and attitudinal data. All personality measures showed significant differences according to risk, based on subject classification into high, moderate, and mild risk status and comparison groups. Early puberty was not associated with increased risk. The strongest predictor variables for risk were body dissatisfaction, negative emotionality, and lack of interoceptive awareness. The possible diathesis of personality including temperamental factors in the later development of an eating disorder is discussed.

A confirmatory test of a model of performance determinants

The total variance in any observed measure of performance can be attributed to 3 sources: (a) the correlation of the measure with the latent variable of interest (i.e., true score variance), (b) reliable but irrelevant variance due to contamination, and (c) error. A model is proposed that specifies 3, and only 3, determinants of the relevant variance: declarative knowledge, procedural knowledge and skill, and volitional choice (motivation). The 3 determinants are defined, and their implications for performance measurement are discussed. Using data from the U.S. Army Selection and Classification Project (Project A), the authors found that the model fits a simplex pattern to the criterion data matrix. The predictor-determinant correlations are also estimated

MODEL SELECTION IN COVARIANCE STRUCTURES ANALYSIS AND THE PROBLEM OF SAMPLE SIZE : A CLARIFICATION

Complex models for covariance matrices are structures that specify many parameters, whereas simple models require only a few. When a set of models of differing complexity is evaluated by means of some goodness of fit indices, structures with many parameters are more likely to be selected when the number of observations is large, regardless of other utility considerations. This is known as the sample size problem in model selection decisions. This article argues that this influence of sample size is not necessarily undesirable. The rationale behind this point of view is described in terms of the relationships among the population covariance matrix and 2 model-based estimates of it. The implications of these relationships for practical use are discussed.

Applications of standard error estimates in unrestricted factor analysis: Significance tests for factor loadings and correlations.

Estimates of standard errors of factor loadings and factor correlations in the unrestricted factor analysis model can be computed for oblique or orthogonal solutions under maximum likelihood. This information can be used to test individual coefficients for significance, to evaluate whether an orthogonal or oblique structure is most consistent with sample data, or to compute confidence intervals for single parameters or confidence regions for arbitrary groups of coefficients. Because the number of parameters estimated in factor analysis is approximately the product of number of variables multiplied by number of factors, a Bonferroni correction for the critical point of the individual test statistics is recommended to control the probability of a Type I error. Several examples are presented.

Exploratory Factor Analysis

Publisher Summary Factor analysis is a collection of methods for explaining the correlations among variables in terms of more fundamental entities called factors. It grew out of the observation that variables from a carefully formulated domain, such as tests of human ability or measures of interpersonal functioning, are often correlated with each other. According to the factor analytic perspective, variables correlate because they are determined in part by common, but unobserved influences. These influences must be superordinate to the variables that are actually measured because they account for the individual differences in the tests. The goals of factor analysis are to determine the number of fundamental influences underlying a domain of variables, to quantify the extent to which each variable is associated with the factors, and to obtain information about their nature from observing which factors contribute to performance on which variables. The chapter also illustrates how factor analysis resembles the partial correlation problem in several ways.

Structural Equations Modeling

I would like to hear comments from more experienced experimental researchers about standard practices for recruiting and compensating participants in consumer and marketing experiments. What are the pros and cons of using student participants? (I know there was a debate about this in the literature a few years ago, but what is the current prevailing opinion?) Is there a difference between using undergraduate students (business majors or nonbusiness majors) and graduate students? When using student participants, is it better to compensate them with extra course credit or to pay them? And, is one time of the semester or quarter (i.e., beginning, middle, or end) preferable for using student participants? I am especially interested to know if anyone has conducted a systematic study of these last two issues. I have recently run experiments using student samples from the same population, but paying one sample and giving extra credit to the other, which definitely affected the rate at which students showed up for their assigned sessions. It may also have affected the variance in the quality of students that chose to participate. Also, in an experiment that I recently ran at the end of a semester (during the last week and a half of class meetings before the final exam week), I collected informal statements from participants in debriefing sessions that indicated that they were no busier or more distracted than they would have been in the middle of the semester. Also, what are the standard practices for recruiting and compensating nonstudent participants (e.g., ordinary folks off the street)? And, for experimental marketing and organizational research (on which I am presently embarking), what are the equivalent standards for industry-based samples (i.e., executives, managers, executive MBA students)? (This information is critical for budgeting grant proposals. I recently called the National Science Foundation and they could not offer much help on this point.) Also, does anyone have any great suggestions for increasing our success rate for getting such populations to participate in experimental research? I was discouraged by a recent conversation with George Day and David Montgomery, who said that even they are finding it increasingly difficult to recruit managerial research participants in the executive courses at the Wharton School and Stanford University. (So, where does that leave the rest of us?) Professor Prashant Malaviya University of Illinois at ChicagoIn the spring of 1990, the Hubble Space Telescope was put into orbit, the culmination of work by a multitude of astronomers, engineers, technicians, and researchers over a period of many years. Its proponents hail it as a key tool to understanding the universe, while its critics write it off as a monumental waste of resources that will never fulfill the expectations of those who designed it. Almost immediately after it went on-line, concern arose about the robustness of its inner workings, yet the demand for access to this device is immense.This special issue poses anonymous questions, then provides the answers and a discussion of the issues by the expert who responded. The answers are not anonymous--partly to give credit to the experts and partly to encourage future communication and debate on whatever lingering controversies may arise. After a number of questions, the special issue concludes with a discussion by the guest editor that summarizes the answers and provides straightforward answers to questions that were not addressed by the experts.