Improved Standard Errors of Standardized Parameters in Covariance Structure Models: Implications for Construct Explication

Abstract

Because measurement scales of observed variables in social and behavioral sciences are often arbitrary, and because the sample correlation matrix, unlike the sample covariance matrix, is scale independent, analyses of correlation structures based on sample correlation matrices are desirable, and often done (e.g., Velicer & Jackson, 1990; Goffin & Jackson, 1992). The problem is that most popular covariance structure software is built around the sampling theory designed to analyze covariance structures. This theory is based on the multivariate sampling distribution of a covariance matrix, which is not the same as that of a correlation matrix. A number of papers have addressed problems relevant to this issue in the contexts of exploratory factor analysis (EFA), and confirmatory factor analysis (CFA). These address problems of parameter estimation, tests of hypotheses, and standard error estimation. Our main focus here will be on the problem of standard error estimation. This is critical to accurate decision-making regarding the interpretation of constructs and the necessity of parameters such as factor loadings as indicators of such constructs. Since typical practice in factor analysis is to interpret a factor by its highly loading variables in a standardized solution, it is also important to verify that such loadings are statistically significantly different from zero. For example, if one uses confirmatory factor analysis in a sequential system for personality scale construction (e.g., Jackson, 1970), it would be essential to be able to determine whether or not the loading of a particular item on the trait factor that it is hypothesized to reflect is significantly different from zero. Furthermore, confirmatory factor analyses of the higher-order structure of personality scales (e.g., Jackson, Paunonen, Fraboni, & Goffin, 1996) could not be meaningfully interpreted in the absence of accurate estimates of the standard errors of the parameters. The previous examples are only two of a multitude of possible examples where the estimation of standard errors assumes a prominent role in the evaluation of construct validity. In this paper, we provide some new methods to achieve this purpose.

Douglas Jackson was Bentier’s dissertation chair on a problem in personality assessment at Stanford University 35 years ago, and ever since has provided inspiration and encouragement for work on improved quantitative methods in psychology.