Michael W. Browne

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.

Power analysis and determination of sample size for covariance structure modeling.

A framework for hypothesis testing and power analysis in the assessment of fit of covariance structure models is presented. We emphasize the value of confidence intervals for fit indices, and we stress the relationship of confidence intervals to a framework for hypothesis testing. The approach allows for testing null hypotheses of not-good fit, reversing the role of the null hypothesis in conventional tests of model fit, so that a significant result provides strong support for good fit. The approach also allows for direct estimation of power, where effect size is defined in terms of a null and alternative value of the root-mean-square error of approximation fit index proposed by J. H. Steiger and J. M. Lind (1980). It is also feasible to determine minimum sample size required to achieve a given level of power for any test of fit in this framework. Computer programs and examples are provided for power analyses and calculation of minimum sample sizes.

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.

The use of causal indicators in covariance structure models: some practical issues.

In conventional representations of covariance structure models, indicators are defined as linear functions of latent variables, plus error. In an alternative representation, constructs can be defined as linear functions of their indicators, called causal indicators, plus an error term. Such constructs are not latent variables but composite variables, and they have no indicators in the conventional sense. The presence of composite variables in a model can, in some situations, result in problems with identification of model parameters. Also, the use of causal indicators can produce models that imply zero correlation among many measured variables, a problem resolved only by the inclusion of a potentially large number of additional parameters. These phenomena are demonstrated with an example, and general principles underlying them are discussed. Remedies are described so as to allow for the evaluation of models that contain causal indicators.

An Overview of Analytic Rotation in Exploratory Factor Analysis

The use of analytic rotation in exploratory factor analysis will be examined. Particular attention will be given to situations where there is a complex factor pattern and standard methods yield poor solutions. Some little known but interesting rotation criteria will be discussed and methods for weighting variables will be examined. Illustrations will be provided using Thurstones 26 variable box data and other examples.

ON THE MULTIVARIATE ASYMPTOTIC DISTRIBUTION OF SEQUENTIAL CHI-SQUARE STATISTICS

The multivariate asymptotic distribution of sequential Chi-square test statistics is investigated. It is shown that: (a) when sequential Chi-square statistics are calculated for nested models on the same data, the statistics have an asymptotic intercorrelation which may be expressed in closed form, and which is, in many cases, quite high; and (b) sequential Chi-squaredifference tests are asymptotically independent. Some Monte Carlo evidence on the applicability of the theory is provided.

Testing Differences between Nested Covariance Structure Models: Power Analysis and Null Hypotheses.

For comparing nested covariance structure models, the standard procedure is the likelihood ratio test of the difference in fit, where the null hypothesis is that the models fit identically in the population. A procedure for determining statistical power of this test is presented where effect size is based on a specified difference in overall fit of the models. A modification of the standard null hypothesis of zero difference in fit is proposed allowing for testing an interval hypothesis that the difference in fit between models is small, rather than zero. These developments are combined yielding a procedure for estimating power of a test of a null hypothesis of small difference in fit versus an alternative hypothesis of larger difference.

Circumplex Models for Correlation Matrices.

Structural models that yield circumplex inequality patterns for the elements of correlation matrices are reviewed. Particular attention is given to a stochastic process defined on the circle proposed by T. W. Anderson. It is shown that the Anderson circumplex contains the Markov Process model for a simplex as a limiting case when a parameter tends to infinity.Andersons model is intended for correlation matrices with positive elements. A replacement for Andersons correlation function that permits negative correlations is suggested. It is shown that the resulting model may be reparametrzed as a factor analysis model with nonlinear constraints on the factor loadings. An unrestricted factor analysis, followed by an appropriate rotation, is employed to obtain parameter estimates. These estimates may be used as initial approximations in an iterative procedure to obtain minimum discrepancy estimates.Practical applications are reported.

Specification and Estimation of Mean- and Covariance-Structure Models

The analysis of moment structures originated with the factor analysis model and with some simple pattern hypotheses concerning equality of elements of mean vectors and covariance matrices. They have more recently received considerable attention and been expanded to incorporate a variety of additional models. Covariance structures, some with associated mean structures, occur in psychology, economics, education, marketing, sociology, biometrics, and other disciplines. Most models involving covariance structures that are in current use are related to the factor analysis model in some way, either by being special cases with restrictions on parameters or, more commonly, extensions incorporating additional assumptions.