Diagnosis for Covariance Structure Models by Analyzing the Path

Abstract

When a covariance structure model is misspecified, parameter estimates will be affected. It is important to know which estimates are systematically affected and which are not. The approach of analyzing the path is both intuitive and informative for such a purpose. Different from path analysis, analyzing the path uses path tracing and elementary numerical analysis to identify affected parameters when a 1-way or 2-way arrow in a path diagram is omitted. It not only characterizes how a misspecification affects model parameters but also facilitates a good understanding of the relation among different parts of the model. This article introduces and studies this technique and, for commonly used models, provides detailed analysis to identify the directions of change for various model parameters. Examples based on real data show that the technique of analyzing the path can reliably predict the direction of change in parameter estimates even when the true model is unknown. Conditions that interfere with the results are also discussed and advice is provided for its proper application.

ACKNOWLEDGMENT

The research was supported by NSF Grant DMS04-37167 and by the James McKeen Cattell Fund. We are thankful to two referees for their constructive comments that have led to a significant improvement of this article over the previous version.

Notes

¹We call any one-way or two-way arrow in a path diagram a path. A path represents either a fixed or a free parameter in the model. A nonzero path corresponds to a nonzero value of the parameter.

² Saris, Satorra, and Sörbom (1987), Luijben and Boomsma (1988), and Kaplan (1990) suggested that the model modification index or LM test should be used in conjunction with expected parameter change (EPC) or standardized EPC. Bentler (1990) and Bollen (1990) provided alternative views about EPC/standardized EPC.

³Paths between indicators and errors may also exist (see Bentler, 1995, p. 103). These are seldom employed in practice and we do not consider them here.

⁴For commonly used models, the diagonal elements of the residual covariance matrix are all zero (Shapiro & Browne, 1990). When the diagonal elements of the residual covariance matrix are not zero, a greater factor loading may not imply a smaller error variance.

⁵When λ has double nonzero subscripts, the first indicates the order of the variable and the second indicates the order of the factor.