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.

Keywords:

• constraint interaction
• equality constraints
• interpretation of parameter estimates

Acknowledgments

The authors are grateful for valuable remarks from Sandra Baar, Martin Becker, and Mireille Soliman.

Notes

¹ Constraint interaction may also appear in more general structural equation models. Studying these, however, is beyond the scope of this paper.

² Notice that the hypotheses to be considered are concerned with unstandardized quantities. If one was interested in standardized quantities, one would be investigating equality of $\left(A\to X_2\right)\frac{\sqrt{\mathrm{V}\mathrm{a}\mathrm{r}(A)}}{\sqrt{\mathrm{V}\mathrm{a}\mathrm{r}(X_2)}}$ and $\left(B\to X_4\right)\frac{\sqrt{\mathrm{V}\mathrm{a}\mathrm{r}(B)}}{\sqrt{\mathrm{V}\mathrm{a}\mathrm{r}(X_4)}}$. Constraint interaction typically does only appear when investigating unstandardized quantities.

³ In all estimations below, $S$ is interpreted as a maximum likelihood estimate of the covariance matrix, i.e. as having been calculated using the number of data points in the denominator of the corresponding formula.

⁴ Estimations were carried out using the freely available statistical software R, version 3.3.2, in combination with package lavaan, version 0.5–22, see R Core Team (2016) and Rosseel (2012).

⁵ Notice that this is a specific feature of the example at hand: in other examples, it might well be the case that the fixed factor method produces the best model fit, while the other scaling methods deliver bad model fits.

⁶ For ease of exposition, we mainly focus on the ${\chi }^2$-difference-statistics and the corresponding $p$-values. This does not cause any loss of generality, because constraint interaction emerges in exactly the same way when changes of fit indices are used for testing equality constraints.

⁷ Correspondingly, in this example, the loss in log-likelihood, which always equals half of the ${\chi }^2$-difference, is large under fixed factor scaling.

⁸ Of course, one may also use for instance the first indicator, $X_{11}$, for scaling the first factor, $A_1$, while using the third indicator, $X_{32}$, for scaling the second factor, $A_2$. Such a ’mixed’ choice of marker variables, however, is rarely used in practice. We thus refrain from including these variants, although it would be perfectly possible to do so.

⁹ We will elaborate on the reasons for the likely occurrence of constraint interaction in practice in section 4 below.

¹⁰ We postpone explaining the reasons for the residual variances’ invariance to the scaling method to the end of this subsection.

¹¹ See the Appendix. This equivalence of the results under the fixed marker method and effects coding is due to the fact that the corresponding factors have only two indicators. For the more complex example with four indicators per factor, such an equivalence does not hold, see below.

¹² Completely analogous reasoning shows why it is impossible to empirically test whether $A_1\to X_{21}$ and $A_2\to X_{22}$ coincide in the second example.