A major problem encountered in covariance structure analyses involves decisions concerning whether or not a given theoretical model adequately represents the data used for its assessment. Given that X2 goodness-of-fit tests are joint functions of the difference between theoretical and empirical covariance structures and sample size, gauging the impact of sample size on such tests is essential. In this paper, we propose a simple index (critical N) and tentative acceptance criterion, which, by focusing on sample size, provide an improved method for assessing goodness-of-fit. Both small- and large-sample examples are presented, illustrating the utility of the proposed method for assessing theoretical models.

The Analysis of Covariance Structures Goodness-of-Fit Indices

Various methods for determining unidimensionality are reviewed and the rationale of these methods is as sessed. Indices based on answer patterns, reliability, components and factor analysis, and latent traits are reviewed. It is shown that many of the indices lack a rationale, and that many are adjustments of a previous index to take into account some criticisms of it. After reviewing many indices, it is suggested that those based on the size of residuals after fitting a two- or three-parameter latent trait model may be the most useful to detect unidimensionality. An attempt is made to clarify the term unidimensional, and it is shown how it differs from other terms often used inter changeably such as reliability, internal consistency, and homogeneity. Reliability is defined as the ratio of true score variance to observed score variance. Inter nal consistency denotes a group of methods that are intended to estimate reliability, are based on the vari ances and the covariances of test items, and depend on on...

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Methodology review: Assessing unidimensionality of tests and items

A new computational method for the maximum likelihood solution in factor analysis is presented. This method takes into account the fact that the likelihood function may not have a maximum in a point of the parameter space where all unique variances are positive. Instead, the maximum may be attained on the boundary of the parameter space where one or more of the unique variances are zero. It is demonstrated that suchimproper (Heywood) solutions occur more often than is usually expected. A general procedure to deal with such improper solutions is proposed. The proposed methods are illustrated using two small sets of empirical data, and results obtained from the analyses of many other sets of data are reported. These analyses verify that the new computational method converges rapidly and that the maximum likelihood solution can be determined very accurately. A by-product obtained by the method is a large sample estimate of the variance-covariance matrix of the estimated unique variances. This can be used to set up approximate confidence intervals for communalities and unique variances.

https://www.onlinelibrary.wiley.com/doi/pdf/10.1002/j.2333-8504.1966.tb00539.x

Some contributions to maximum likelihood factor analysis

Several issues related to the reliability and validity of self-report delinquency measures are raised and discussed. These include problems associated with the use of internal consistency as the measure of reliability, the level of reliability or precision required for different types of analyses, problems with the content validity of self-report measures, problems of overreporting and underreporting, problems with the use of official records as a validity check on self-reports, and the lack of any good criterion as a major obstacle in assessing the empirical validity of self-report measures. In the light of these problems, some cautions about the use of self-report measures are made.

Reassessing the reliability and validity of self-report delinquency measures

On exploratory factor analysis: A review of recent evidence, an assessment of current practice, and recommendations for future use

In order to determine the robustness of the maximum-likelihood estimation procedure, random variates from six distributions were generated independently on a high speed computer and then used to represent the common and specific factors in a factor analysis model in which the coefficients of these factors had been specified. Using Lawley's approximatex
2 statistic in evaluating the estimates obtained, the estimation procedure is found to be insensitive to changes in the distributions considered.

Robustness of the maximum-likelihood estimation procedure in factor analysis.

This paper considers the problem of computing estimates of factor loadings, specific variances, and communalities for a factor analytic model. The equations for maximum-likelihood estimators are discussed. Iterative formulas are developed to solve the maximum-likelihood equations and a simple and efficient method of implementation on a digital computer is described. Use of the iterative formulas and computing techniques for other estimators of factor loadings and communalities is also considered to provide a very general approach for this aspect of factor analysis.

W. J. Hemmerle

Papers

Robustness of the maximum-likelihood estimation procedure in factor analysis.

Obtaining maximum-likelihood estimates of factor loadings and communalities using an easily implemented iterative computer procedure.