Comparing Latent Means Without Mean Structure Models: A Projection-Based Approach

Abstract

The conventional setup for multi-group structural equation modeling requires a stringent condition of cross-group equality of intercepts before mean comparison with latent variables can be conducted. This article proposes a new setup that allows mean comparison without the need to estimate any mean structural model. By projecting the observed sample means onto the space of the common scores and the space orthogonal to that of the common scores, the new setup allows identifying and estimating the means of the common and specific factors, although, without replicate measures, variances of specific factors cannot be distinguished from those of measurement errors. Under the new setup, testing cross-group mean differences of the common scores is done independently from that of the specific factors. Such independent testing eliminates the requirement for cross-group equality of intercepts by the conventional setup in order to test cross-group equality of means of latent variables using chi-square-difference statistics. The most appealing piece of the new setup is a validity index for mean differences, defined as the percentage of the sum of the squared observed mean differences that is due to that of the mean differences of the common scores. By analyzing real data with two groups, the new setup is shown to offer more information than what is obtained under the conventional setup.

Appendix

In this appendix, we will first obtain the parameters under the conventional setup (i.e., ${\varvec{\kappa }}^{(1)}$, ${\varvec{\gamma }}^{(1)}$, ${\varvec{\kappa }}^{(2)}$, ${\varvec{\gamma }}^{(2)}$) as functions of those under the new setup (i.e., ${\varvec{\tau }}^{(1)}$, ${\varvec{\nu }}^{(1)}$, ${\varvec{\tau }}^{(2)}$, ${\varvec{\nu }}^{(2)}$), and then obtain the reversed functional relationship, as given in Equations (12) and (13).

It follows from Equations (2) and (3) under the conventional setup that

$$\begin{aligned} {\varvec{\kappa }}^{(1)}=\mathbf{0},\;\; {\varvec{\gamma }}^{(1)} = {\varvec{\mu }}^{(1)} = {\varvec{\Lambda }}{\varvec{\tau }}^{(1)}+{\varvec{\nu }}^{(1)}, \end{aligned}$$

and

$$\begin{aligned} {\varvec{\gamma }}^{(2)}={\varvec{\gamma }}^{(1)}={\varvec{\mu }}^{(1)}={\varvec{\Lambda }}{\varvec{\tau }}^{(1)}+{\varvec{\nu }}^{(1)}. \end{aligned}$$

By equating at ${\varvec{\mu }}^{(2)}$ and via Equation (3), we also have

$$\begin{aligned} {\varvec{\mu }}^{(2)}={\varvec{\gamma }}^{(2)}+{\varvec{\Lambda }}{\varvec{\kappa }}^{(2)} = {\varvec{\gamma }}^{(1)}+{\varvec{\Lambda }}{\varvec{\kappa }}^{(2)} ={\varvec{\Lambda }}{\varvec{\tau }}^{(2)}+{\varvec{\nu }}^{(2)}. \end{aligned}$$

Thus,

$$\begin{aligned} {\varvec{\Lambda }}({\varvec{\kappa }}^{(2)}-{\varvec{\tau }}^{(2)}) = {\varvec{\nu }}^{(2)}-{\varvec{\gamma }}^{(1)} = {\varvec{\nu }}^{(2)} - {\varvec{\mu }}^{(1)}. \end{aligned}$$

(23)

Notice that the vector ${\varvec{\nu }}^{(2)}$ is in the orthogonal space of that spanned by the columns of ${\varvec{\Lambda }}$, multiplying the left and right sides of (23) by ${\varvec{\Lambda }}'$ yields

$$\begin{aligned} {\varvec{\Lambda }}'{\varvec{\Lambda }}({\varvec{\kappa }}^{(2)}-{\varvec{\tau }}^{(2)}) = {\varvec{\Lambda }}'({\varvec{\nu }}^{(2)}-{\varvec{\mu }}^{(1)}) =-{\varvec{\Lambda }}'{\varvec{\mu }}^{(1)}. \end{aligned}$$

(24)

Since ${\varvec{\Lambda }}'{\varvec{\Lambda }}$ is non-singular, multiplying the left and right sides of (24) by $({\varvec{\Lambda }}'{\varvec{\Lambda }})^{-1}$ and solving for ${\varvec{\kappa }}^{(2)}$ yield

$$\begin{aligned} {\varvec{\kappa }}^{(2)} = {\varvec{\tau }}^{(2)}-({\varvec{\Lambda }}'{\varvec{\Lambda }})^{-1} {\varvec{\Lambda }}'{\varvec{\mu }}^{(1)} = {\varvec{\tau }}^{(2)}-{\varvec{\tau }}^{(1)}. \end{aligned}$$

For the parameterization under the new setup, with $\mathbf{Q}_{\lambda }$ being defined following Equation (13), we have

$$\begin{aligned} {\varvec{\tau }}^{(1)}= & {} ({\varvec{\Lambda }}'{\varvec{\Lambda }})^{-1}{\varvec{\Lambda }}'{\varvec{\mu }}^{(1)} = ({\varvec{\Lambda }}'{\varvec{\Lambda }})^{-1}{\varvec{\Lambda }}'{\varvec{\gamma }}^{(1)},\;\; {\varvec{\nu }}^{(1)}=\mathbf{Q}_{\lambda }{\varvec{\mu }}^{(1)}=\mathbf{Q}_{\lambda }{\varvec{\gamma }}^{(1)}, \\ {\varvec{\tau }}^{(2)}= & {} ({\varvec{\Lambda }}'{\varvec{\Lambda }})^{-1}{\varvec{\Lambda }}'{\varvec{\mu }}^{(2)} = ({\varvec{\Lambda }}'{\varvec{\Lambda }})^{-1}{\varvec{\Lambda }}' ({\varvec{\gamma }}^{(1)} + {\varvec{\Lambda }}{\varvec{\kappa }}^{(2)}) = {\varvec{\kappa }}^{(2)} + ({\varvec{\Lambda }}'{\varvec{\Lambda }})^{-1}{\varvec{\Lambda }}' {\varvec{\gamma }}^{(1)}, \end{aligned}$$

and

$$\begin{aligned} {\varvec{\nu }}^{(2)} = \mathbf{Q}_{\lambda }{\varvec{\mu }}^{(2)} = \mathbf{Q}_{\lambda } ({\varvec{\gamma }}^{(1)} + {\varvec{\Lambda }}{\varvec{\tau }}^{(2)})= \mathbf{Q}_{\lambda }{\varvec{\gamma }}^{(1)}. \end{aligned}$$