Explaining Constraint Interaction: How to Interpret Estimated Model Parameters Under Alternative Scaling Methods

Abstract

In this paper, we explain the reasons behind constraint interaction, which is the phenomenon that the results of testing equality constraints may depend heavily on the scaling method used. We find that the scaling methods interfere with the testing procedures because scaling methods determine which transformations of population quantities model parameters actually estimate. We therefore also develop rules on how to correctly interpret estimates of model parameters under alternative scaling methods.

The Impact of Scaling Methods on the Properties and Interpretation of Parameter Estimates in Structural Equation Models with Latent Variables

“…Consequently, on the one hand, the example shows that, given the knowledge of the population regression coefficient in this hypothetical example, the estimated regression cannot be interpreted in the same way as in the regression model with manifest variables only. On the other hand, we show that the problem mentioned by Bielby (1986) also occurs in the context of EC and FF scaling, and, drawing on results provided in Klößner and Klopp (2019), the question arises which algebraic transformation of population parameters the estimated regression coefficient represents under the different scaling methods.…”

“…6 The considerations presented so far represent a systematization and extension of ideas that have been discussed in the SEM literature, e.g., Jöreskog (1978); Klößner and Klopp (2019); Mulaik (2010, p. 443); SanMartin, Gonzalez, and Tuerlinckx (2015); Schweizer (2011); Schweizer, Troche, and DiStefano (2019); ; von Oertzen (2010); Yoon and Millsap (2007).…”

“…In subsection 3.1 we demonstrate how to convert the estimated parameters obtained under one scaling method to their equivalent obtained under another scaling method. In subsection 3.2, we exemplify the interpretation of the estimated parameters for all three scaling methods, extending the interpretation of estimated parameters described in Klößner and Klopp (2019). 17 Finally, in subsection 3.3, we deal with the issue of the interpretation of estimated parameters in the context of misspecified models.…”

“…Consequently, the interpretation of estimated parameters also depends on the choice of the scaling method. The issue of the interpretation of parameter estimates in SEMs involving latent variables has been recently discussed by Klößner and Klopp (2019) in an explanation of the phenomenon of constraint interaction (Steiger, 2002). Klößner and Klopp demonstrated the dependency of the estimated parameters on the scaling method.…”

A Tutorial on Testing the Equality of Standardized Regression Coefficients in Structural Equation Models using Wald Tests with lavaan

“…Both scaling issues cause the estimated regression coefficients to depend on the given metric of the manifest variables and the scaling method used to scale the latent variable. Fortunately, the standardized regression coefficients are also invariant towards the particular applied scaling method for the latent variables (see Klößner & Klopp, 2019;Klopp & Klößner, 2020).…”