Constructing Approximate Confidence Intervals for Parameters With Structural Equation Models

Abstract

Confidence intervals (CIs) for parameters are usually constructed based on the estimated standard errors. These are known as Wald CIs. This article argues that likelihood-based CIs (CIs based on likelihood ratio statistics) are often preferred to Wald CIs. It shows how the likelihood-based CIs and the Wald CIs for many statistics and psychometric indexes can be constructed with the use of phantom variables (Rindskopf, 1984) in some of the current structural equation modeling (SEM) packages. The procedures to form CIs for the differences in correlation coefficients, squared multiple correlations, indirect effects, coefficient alphas, and reliability estimates are illustrated. A simulation study on the Pearson correlation is used to demonstrate the advantages of the likelihood-based CI over the Wald CI. Issues arising from this SEM approach and extensions of this approach are discussed.

ACKNOWLEDGMENTS

Preparation of this work was supported by the Academic Research Fund Tier 1 (R-581-000-064-112) from the Ministry of Education, Singapore. Portions of this article were presented at the International Meeting of the Psychometric Society, July 4–9, 2005, in Tilburg, The Netherlands. I would like to thank Michael Neale and Gerhard Mels for providing advice on the use of Mx and LISREL, and Tenko Raykov for providing useful comments on the article. I would also like to express my sincere gratitude to several reviewers for their valuable suggestions on improving the article.

Notes

¹The expected Fisher information, which is defined as −E(d ²(log L(>))/d > ²), can also be used to estimate standard errors. However, the observed Fisher information is usually preferred (see Azzalini, 1996; Pawitan, 2001).

²It should be noted that when the parameter is tested at a boundary, say zero for variance and ±1 for correlation, the LR statistic is not distributed as χ²(1). It is distributed as a mixture of χ²(0) and χ²(1) (see Stoel, Galindo-Garre, Dolan, & van den Wittenboer, 2006). Thus, both Wald and likelihood CIs might be incorrect.

³Equivalently, it is also possible to set the phantom variable so that it is uncorrelated with other variables. The variance of the phantom variable can then be used to “store” the parameter of interest by imposing suitable constraints.

⁴LISREL (Jöreskog & Sörbom, 1996, pp. 345–347) uses the delta method to calculate the of the parameters with constraints, and Mplus also employs the delta method to calculate the of the parameters with constraints.

⁵To implement all of the examples discussed in this article, the SEM packages should be able to implement linear and nonlinear constraints on the parameters. The current versions of AMOS (6.0) and EQS (6.1) do not allow nonlinear constraints.

⁶The complete codes and output in Mplus, LISREL, and Mx are available at http://courses.nus.edu.sg/course/psycwlm/internet/.

⁷Dr. Phillip Wood posted some Mplus code to calculate the same reliability estimate at SEMNET on November 17, 2004.